Chemical Equilibria in Analytical Chemistry: The Theory of Acid–Base, Complex, Precipitation and Redox Equilibria [1st ed. 2019] 978-3-030-17179-7, 978-3-030-17180-3

Table of contents :
Front Matter ....Pages i-vii
Introduction (Fritz Scholz, Heike Kahlert)....Pages 1-2
Chemical Equilibrium (Fritz Scholz, Heike Kahlert)....Pages 3-15
Acid–Base Equilibria (Fritz Scholz, Heike Kahlert)....Pages 17-91
Complex Formation Equilibria (Fritz Scholz, Heike Kahlert)....Pages 93-105
Solubility Equilibria (Fritz Scholz, Heike Kahlert)....Pages 107-134
Redox Equilibria (Fritz Scholz, Heike Kahlert)....Pages 135-168
Titrations (Fritz Scholz, Heike Kahlert)....Pages 169-242
Back Matter ....Pages 243-251

Citation preview

Fritz Scholz · Heike Kahlert

Chemical Equilibria in Analytical Chemistry The Theory of Acid–Base, Complex, Precipitation and Redox Equilibria

Chemical Equilibria in Analytical Chemistry

Fritz Scholz • Heike Kahlert

Chemical Equilibria in Analytical Chemistry The Theory of Acid–Base, Complex, Precipitation and Redox Equilibria

123

Fritz Scholz Institut für Biochemie Universität Greifswald Greifswald, Germany

Heike Kahlert Institut für Biochemie Universität Greifswald Greifswald, Germany

ISBN 978-3-030-17179-7 ISBN 978-3-030-17180-3 https://doi.org/10.1007/978-3-030-17180-3

(eBook)

Translation from the German language edition: Chemische Gleichgewichte in der Analytischen Chemie by Fritz Scholz and Heike Kahlert, © Springer-Verlag GmbH Germany, part of Springer Nature 2018. All Rights Reserved. © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 Chemical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Genesis of the Theory of Chemical Equilibrium . . . . . . . . . . 2.2 Why Are Chemical Equilibria so Important for Chemical Analysis? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Can the Rate of a Chemical Reaction in Approaching Equilibrium Be Used for Analysis? . . . . . . . . . . . . . . . . . . . 2.4 The Thermodynamics of Chemical Equilibrium . . . . . . . . . . 2.5 The Effect of Entropy on Equilibrium . . . . . . . . . . . . . . . . . 2.6 The Kinetics of Chemical Equilibrium . . . . . . . . . . . . . . . . . 2.7 Reversibility and Irreversibility of Chemical Reactions (Equilibria) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Catalysis—The “Magic” Way to Enhance the Establishment of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Guidelines for Calculations of Equilibria . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.... ....

3 4

....

7

3 Acid–Base Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Glimpses of the History of Acid–Base Theories . . . . . . . . . 3.2 Quantification of Acid and Base Strength According to the Brønsted–Lowry Theory . . . . . . . . . . . . . . . . . . . . . 3.2.1 Acidity and Basicity Constants . . . . . . . . . . . . . . . . 3.2.2 The pH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 The Strength of Acids and Bases . . . . . . . . . . . . . . 3.2.4 Principles Defining the Strength of Inorganic Acids . 3.2.5 Principles Governing the Strength of Organic Acids and Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Non-aqueous Solvents . . . . . . . . . . . . . . . . . . . . . . 3.3 The Mathematical and Graphical Description of Acid–Base Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Monobasic and Polybasic Acids . . . . . . . . . . . . . . . 3.3.2 pH-logci Diagrams . . . . . . . . . . . . . . . . . . . . . . . . .

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7 8 10 12

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13 14 14

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17 17

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28 28 33 35 38

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50 52

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56 57 59

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v

vi

Contents

3.3.3 pH Calculations Using Approximation Equations 3.3.4 Calculating the pH of Salt Solutions . . . . . . . . . . 3.4 The Degree of Protolysis and Ostwald’s Law of Dilution 3.5 Acid–Base Equilibria of Amino Acids . . . . . . . . . . . . . . 3.6 Acid–Base Equilibria at the Surface of Solids . . . . . . . . 3.7 Buffer Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Complex Formation Equilibria . . . . . . . . . . . . . . 4.1 Monodentate and Multidentate Ligands . . . . . 4.2 Side Reaction Coefficients and Conditional Stability Constants . . . . . . . . . . . . . . . . . . . . 4.2.1 Side Reactions of Ligands . . . . . . . . . 4.2.2 Side Reactions of the Metal Ions . . . . 4.2.3 Conditional Stability Constants . . . . . . 4.2.4 The pH Dependence of Side Reaction Coefficients and Conditional Constants 4.3 The Chelate Effect . . . . . . . . . . . . . . . . . . . . 4.4 Applications of Complex Equilibria . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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64 69 72 77 82 86 90

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93 93

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96 96 98 99

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5 Solubility Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Saturation Concentration . . . . . . . . . . . . . . . . . . . . . . . 5.2 The pH Dependence of Saturation Concentration . . . . . . . . . 5.2.1 pH Dependence of the Saturation Concentration of Metal Hydroxides, Oxide Hydroxides, and Oxides 5.2.2 Graphical Presentation of the Solubilities of Metal Hydroxides as a Function of pH . . . . . . . . . . . . . . . . 5.2.3 Solubility Equilibria of Metal Hydroxides in the Presence of Complexing Agents . . . . . . . . . . . 5.2.4 pH Dependence of the Solubility of Metal Sulfide . . . 5.3 Coprecipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Redox Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Quantitative Treatment of Redox Equilibria . . . . . . . . . . . . 6.2 Calculating the Equilibrium Constants of Redox Equilibria . 6.3 Formal Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Formal Potentials That Include Activity Coefficients 6.3.2 Formal Potentials That Include Activity Coefficients and Side Reaction Coefficients . . . . . . . . . . . . . . . .

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120 125 131 133

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135 137 142 145 146

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Contents

vii

6.4 pH-Dependent Redox Potentials . . . . . . . . . . . . . . . . . . . . . 6.4.1 Redox Equilibria with Coupled Acid–Base Equilibria That Can Be Experimentally Separated . . . . . . . . . . . 6.4.2 Redox Equilibria with Inherently Coupled Acid–Base Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Relations Between the Standard Potentials of an Element Having Various Oxidation States: Luther’s Rule . . . . . . . . . . 6.6 Biochemical Standard Potentials . . . . . . . . . . . . . . . . . . . . . 6.7 Redox Potentials in Non-aqueous Solvents . . . . . . . . . . . . . . 6.8 Graphical Presentation of Redox Equilibria . . . . . . . . . . . . . 6.9 Kinetic Aspects of Redox Equilibria . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Titrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The History of Titrations . . . . . . . . . . . . . . . . . . . . . . . . 7.2 General Theory of Titrations . . . . . . . . . . . . . . . . . . . . . 7.3 Titration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Direct Titration . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Inverse Titration . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Back Titration . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Substitution Titration . . . . . . . . . . . . . . . . . . . . . 7.3.5 Indirect Titration . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Theoretical Considerations and Graphical Representation of Titration Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Acid–Base Titrations . . . . . . . . . . . . . . . . . . . . . 7.4.2 Complexometric Titrations . . . . . . . . . . . . . . . . . 7.4.3 Precipitation Titrations . . . . . . . . . . . . . . . . . . . . 7.4.4 Redox Titrations . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Indication Methods for Titrations . . . . . . . . . . . . . . . . . . 7.5.1 Classical Methods . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Titration Errors in Classical Titrations . . . . . . . . . 7.5.3 Instrumental Methods of Indication for Titrations References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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158 160 161 163 166 167

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169 169 173 175 175 175 176 179 180

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180 182 189 196 201 208 208 221 229 241

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

1

Introduction

Chemical equilibria are based on the laws of chemical thermodynamics and so they are part of physical chemistry. They need to be calculated in analytical, environmental, industrial, and pharmaceutical chemistry, as well as in biochemistry and in all sciences where chemical reactions play a role. In university curricula, they are usually taught in analytical chemistry and inorganic chemistry. Understanding chemical equilibria is fundamental to all wet chemical analyses, e.g., titrations and gravimetry—often where students hear about this subject for the first time. Later, students appreciate that understanding and calculating chemical equilibria are both necessary for environmental chemistry, chemical synthesis, biochemistry, and whatever else they study in chemistry-related fields. In this textbook calculations are presented that are necessary to treat acid–base, complex, precipitation, and redox equilibria in solutions. The main reference point is chemical analysis, as performed in undergraduate laboratory courses. However, wherever appropriate, topics from environmental chemistry and biochemistry are also considered, so that students can see the role chemical equilibria play in advanced topics in the curriculum. In each chapter and section, the most important parts are highlighted, and examples given, as boxed material. This textbook also includes literature references, so that students can easily expand their understanding. We are in debt to PD Dr. Richard Thede, University of Greifswald, Prof. Dr. Ingo Krossing, University of Freiburg, and Dr. Gisela Boeck, University of Rostock, for their most valuable comments on parts of this textbook. We acknowledge the extensive help provided by Anja Albrecht in preparing figures, literature acquisition, and index preparation.

© Springer Nature Switzerland AG 2019 F. Scholz and H. Kahlert, Chemical Equilibria in Analytical Chemistry, https://doi.org/10.1007/978-3-030-17180-3_1

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2

1

Introduction

This textbook is dedicated to Prof. Dr. Günter Henrion on the occasion of his 85th birthday on May 3, 2018. Both authors remember, with highest esteem, the lectures on chemical equilibria he delivered at Humboldt University, Berlin.

2

Chemical Equilibrium

In the different fields of natural, social, and applied sciences, the term EQUILIBRIUM is used with different meanings. However, in all cases it describes a state which does not change, under certain external conditions. In economics, the equilibrium of markets expresses that demand equals supply. Naturally, a market equilibrium includes all the time flows of goods, services, and money between producers and consumers, and so it is not static but dynamic, i.e., it is a steady state equilibrium. In politics, the term balance of power means that two opposite parties (states, confederations, etc.) possess equal military, economic, or political power, so that no active conflicts result. In science, systems are distinguished by their ability to exchange matter and energy with the environment: a system is called isolated when it neither exchanges energy nor matter with its environment. Closed systems can exchange energy, but not matter, with the environment; open systems can exchange both energy and matter with their environment. In a closed system a static equilibrium is, for example, realized for the two sides of a mechanical balance—when equal masses on the two weighing pans and equal arm lengths of the pivoted lever establish equal forces. In a closed system at constant system parameters (volume v, temperature T, and pressure p) the chemical equilibrium between the starting compounds and the products of a chemical reaction is characterized: (1) macroscopically by a complete constancy of concentrations of all participating species, and (2) microscopically (on an atomic or molecular scale) by equal rates of forward and backward reactions. The equal reaction rates define a dynamic equilibrium. Hence, in thermodynamics, the criterion for equilibrium is the constancy of an extensive state function, e.g., the Gibbs free energy (also called Gibbs energy and Gibbs function), and in kinetics the criterion is the equality of forward and backward reaction rates. In biological systems, chemical equilibria are omnipresent; however, due to a higher level of organization that is always found in open systems, dynamic (steady state) equilibria dominate. A dynamic equilibrium is established in the part of a system when certain parameters, e.g., temperature, pressure, or the concentration of © Springer Nature Switzerland AG 2019 F. Scholz and H. Kahlert, Chemical Equilibria in Analytical Chemistry, https://doi.org/10.1007/978-3-030-17180-3_2

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4

2 Chemical Equilibrium

a compound, are kept constant as a result of a flow of matter across the border of that part of the system or by a chemical conversion. The active maintenance of a dynamic equilibrium is also called autoregulation or homeostasis in living systems.

2.1

Genesis of the Theory of Chemical Equilibrium

Since the most ancient of times, the differences between the chemical reactions of different compounds have been the subject of curiosity of learned people [1]. In the Middle Ages, the term affinity (later also called elective affinity) was used to explain differences in reactivities by differences of kinship. Obviously, these ideas were developed in tenebrous times. Modern science defines reactivity by: (1) the thermodynamic state of equilibrium (the equilibrium constant), or (2) kinetically by the rate of establishing equilibrium (the rate constant). This book is not the place to plot the long historical pathway leading to a clear-cut definition of affinity in terms of thermodynamics and kinetics. On this route, scientists have realized that for any chemical reactions it is essential to distinguish between the state of equilibrium (the equilibrium constant) and the rate of establishing equilibrium (rate constant), i.e., to distinguish between thermodynamics and kinetics. For the thermodynamic treatment of chemical equilibria, the quantity affinity is not necessary. It suffices to use the Gibbs free energy and the equilibrium constants. Affinity Modern chemical thermodynamics defines affinity A as the partial derivative of the molar Gibbs free energy G with respect to the extent of reaction n (Greek letter xi): A ¼ ð@G=@nÞT;p

ð2:1Þ

According to de Donder (Théophile Ernest de Donder, 1872–1957, Belgian physicist and physical chemist) the product of affinity and change in extent of reaction is the change of uncompensated heat Q0 : dQ0 ¼ Adn

ð2:2Þ

The uncompensated heat is related to the entropy production di S of a chemical reaction: di S ¼ dQ0 =T

ð2:3Þ

This gives the temporal change of entropy production: di S 1 dQ0 A dn ¼ ¼ dt T dt T dt

ð2:4Þ

2.1 Genesis of the Theory of Chemical Equilibrium

5

Rearrangement gives: dQ0 A¼ dt




dn dt

ð2:5Þ

The affinity is a state function. For a reaction A + B  C + D Eq. 2.1 is equivalent with AðnÞ ¼ lA ðnÞ þ lB ðnÞ  lC ðnÞ  lD ðnÞ. Here, li ðnÞ is the chemical potential of the chemical species i at a certain extent of reaction (l: Greek letter mu). The affinity is zero at equilibrium, it is positive when the reaction proceeds from left to right, and negative when it proceeds from right to left when approaching equilibrium. The affinity is an expression of the driving force of a reaction. Some milestones on the way to our modern understanding of chemical equilibrium [2, 3] include: 1704: Isaac Newton (1643–1726) speculated that the reactivities of chemical substances are rooted in forces, similar to the gravitational force [4]. 1718: Étienne François Geoffroy (1672–1731) published tables of chemical affinities, where he ordered the compounds according to their reactivities [5]. 1777: Carl Friedrich Wenzel (1740–1793) used the rates of chemical reactions to judge affinities and realized that rates depended on the quantity(!) of reactants as well as their nature (quality) [6]. ~1800: Claude Louis Berthollet (1748–1822) described invertible (reversible) reactions and realized that it was not only affinities, but also the amounts of reacting compounds [7], that play a role. 1862: To quantify affinities Marcelin Pierre Eugène Berthelot (1827–1907) and Léon Péan de Saint-Gilles (1832–1863) published studies on the formation and decomposition of ethers [8]. 1867: Cato Maximilian Guldberg (1836–1902) and Peter Waage (1833–1900) derived the law of mass action (LMA) [9]. 1869–1881: August Friedrich Horstmann (1842–1929) applied the 2nd law of thermodynamics to chemical reactions [10]. 1874–1878: Josiah Willard Gibbs (1839–1903) formulated the basics of thermodynamic equilibrium [11]; however, his publications did not receive much attention because they appeared in a journal with a small circulation. Gibbs introduced the state function “chemical potential” li of component i. It is related to the Gibbs free energy G as follows:   @GT;P;nj li ¼ @ni

ð2:6Þ

6

2 Chemical Equilibrium

1877: Hermann von Helmholtz (1821–1894) published the equation: DE ¼ k log

ca cc

ð2:7Þ

which gives the potential difference DE of a galvanic concentration cell, where ca and cc are the concentration of metal ions in the anode and cathode compartments, respectively; however, at that time, he did not yet understand the meaning of the constant k [12]. 1882: Hermann von Helmholtz (1821–1894) derived an equation, presented in different forms, e.g.: ð@F=@T Þv ¼ S; ð@G=@T Þp ¼ S

ð2:8Þ

DH ¼ DG þ TDS

ð2:9Þ

and

where F is the molar Helmholtz free energy; H is the molar enthalpy; G is the molar Gibbs free energy; S is molar entropy; and T is absolute temperature. This equation is known as the Gibbs–Helmholtz equation, although, until 1882, only Helmholtz had derived it [13]. Later it was shown that the equation is implicitly present in Gibb’s earlier publication [11, 14, 15]. 1884: Jacobus Henricus van’t Hoff (1852–1911) came to the conclusion that the maximum amount of useful work in a chemical reaction is a measure of the affinity of the reacting compounds toward one another. He derived an equation for the temperature dependence of equilibrium constants [16]. 1889: Walther Nernst (1864–1941) derived an equation, which describes the potential of a metal/metal ion electrode [17, 18]. In modern terms this Nernst equation reads: —  EMen þ =Me ¼ EMe þ nþ =Me

RT ln aMen þ nF

ð2:10Þ

—  is the where EMen þ =Me is the potential versus standard hydrogen electrode; EMe nþ =Me standard potential; R is the molar gas constant; n is the number of transferred electrons; F is the Faraday constant; and aMen þ is the activity of metal ions in solution. Later it was understood that the equation could be generalized for any charge transfer equilibrium at an interface. 1892: Walther Nernst showed [19] that the potential difference at the interface of a solid salt and its solution can be calculated with a similar equation, and in 1902 Nernst’s student Hermann Riesenfeld (1877–1957) demonstrated in his Ph.D. thesis that the potential difference at the interface of immiscible electrolyte solutions can also be calculated with the Nernst equation [20].

2.1 Genesis of the Theory of Chemical Equilibrium

7

1893: Rudolf Peters (1869–1937) applied the Nernst equation to solutions containing a dissolved redox pair and described the potential of an inert electrode [21] with the following version of the Nernst equation: —  EMem þ =MeðmnÞ þ ¼ EMe þ mþ =MeðmnÞ þ

RT a mþ ln Me nF aMeðmnÞ þ

ð2:11Þ

(here formulated for metal ions in two oxidation states). 1901, 1907, 1908: Gilbert Newton Lewis (1875–1946) [22–24] found that the partial pressure of a real gas, or the concentration of a dissolved species, are only approximations for calculating the “tendency of a species to leave the phase” in a distribution equilibrium. He suggested the term fugacity, which he later restricted to gaseous species [23]—for dissolved species he suggested the term activity. The fugacity is the partial pressure multiplied by the fugacity coefficient, and the activity is the concentration multiplied by the activity coefficient. Using fugacities and activities allowed using the LMA and other thermodynamic relations for an exact description of the behavior of species. Initially, the fugacity and activity coefficients were empirical data; however, later a complete framework of theories, e.g., the Debye–Hückel theory and its advanced versions, were developed for their calculation [25].

2.2

Why Are Chemical Equilibria so Important for Chemical Analysis?

The aim of chemical analysis is the determination of concentrations or amounts of a substance with as low an error as possible. Hence, these concentrations or amounts should be well defined and constant under certain conditions (temperature, pressure, other chemical species, etc.). This can only be guaranteed when chemical equilibrium has been established. For systems that are not in chemical equilibrium, the most minute changes in conditions, e.g., temperature and pressure, or the presence of catalysts or inhibitors, may result in appreciable changes in concentrations or amounts of a species targeted in an analysis. This is the reason why analytical methods relying on an established chemical equilibrium form the most reliable basis for chemical analysis.

2.3

Can the Rate of a Chemical Reaction in Approaching Equilibrium Be Used for Analysis?

Yes, this is possible, and it is used in all kinetic methods of analysis. Such methods, applied to inorganic catalysts, are often summarized as catalymetry, when applied to biochemical catalysts or substrates as enzymatic analysis, or generally as kinetic methods of analysis [26, 27]. These methods afford a very precise control of the reaction conditions. They are not covered in this textbook.

8

2.4

2 Chemical Equilibrium

The Thermodynamics of Chemical Equilibrium

For a chemical reaction proceeding at constant pressure and temperature, the state function Gibbs free energy G is defined as follows: G ¼ U þ pV  TS

ð2:12Þ

Here and henceforth, capital letters indicate molar quantities. The sum of the inner energy U and volume work (pressure p times molar volume V) is the enthalpy H. Equation 2.12 can thus also be written as follows: G ¼ H  TS

ð2:13Þ

For a chemical reaction the changes of the molar quantities are: DR G ¼ DR H  TDR S

ð2:14Þ

It is impossible to give the absolute value of the Gibbs free energy, since that would require knowledge of its value at zero temperature on the thermodynamic scale. However, the relative changes DG can be accessed accurately. Figure 2.1 depicts the changes of the Gibbs free energy with the degree of reaction of a chemical equilibrium reaction: A + B  C + D

ðEquilibrium 2:1Þ

Fig. 2.1 Gibbs free energy of a chemical equilibrium A + B  C + D as function of the n dni , where n is the extent of reaction n ¼ and nmax the maximum degree of reaction a ¼ nmax mi extent of the reaction (ni is the amount of substance i and aeq is the degree of reaction at equilibrium)

2.4 The Thermodynamics of Chemical Equilibrium

9

Here it is possible to separate DR G into two parts: a standard quantity DR G— and a term depending on the activity ai of the chemical species i: DR G ¼ DR G— þ RT ln

Y

ami i

ð2:15Þ

i

where m (Greek letter nu) are the stoichiometric coefficients in the equilibrium reaction. Using the equilibrium activities ai;eq in Eq. 2.15 it is possible to calculate the equilibrium constant K: K¼

Y i

i ami;eq

ð2:16Þ

This is the general notation of the LMA. Since DR G is zero, when the equilibrium is established, application of Eq. 2.16 to Equilibrium 2.1 yields: K¼

DR G aC;eq aD;eq ¼ e RT aA;eq aB;eq

 —

ð2:17Þ

with stoichiometric coefficients: mA ¼ mB ¼ 1 and mC ¼ mD ¼ þ 1. Figure 2.1 depicts the change of Gibbs free energy regardless of the reaction path, i.e., without considering the Gibbs free energy of activation. For most of the reactions used in analytical chemistry the Gibbs free energies of activation are so small at room temperature that equilibrium is established very quickly, in many cases practically instantaneously, because the thermal energy is sufficient to overcome the activation barrier. Exceptions will be discussed in Sect. 2.7. Please note that in Fig. 2.1 the back reaction starts with a mixture of C and D only! What Quantities Are Equal in a Chemical Equilibrium? Considering the reaction mA A + mB B  mC C + mD D, the term “equilibrium” might be easily misunderstood as indicating, that in equilibrium, the concentrations or amounts of educts and products are equal. This, of course, is completely wrong! The answer is given by chemical thermodynamics, which defines the chemical potential of a component (chemical species) i as follows:  li ¼

@g @ni

 T;p;nj

ð2:18Þ

The chemical potential is an expression of the differential change of Gibbs free energy g that is caused by a differential change of the number of particles

10

2 Chemical Equilibrium

Fig. 2.2 The state of equilibrium is defined by the equality of the sums of chemical potentials multiplied by the stoichiometric coefficients of the educts on one side and products on the other

ni of component i, at constant temperature, pressure, and constancy of the number nj of all other components. Since the state of equilibrium is characterized by the condition ðdgÞT;p ¼ 0, it follows that the chemical potentials must fulfil the following equation: jmA jlA  jmB jlB þ jmC jlC þ jmD jlD ¼ 0

ð2:19Þ

where the first two terms possess a negative sign because the educts are consumed, when the products are formed. When Eq. 2.19 is written in the form: jmA jlA þ jmB jlB ¼jmC jlC þ jmD jlD

ð2:20Þ

it becomes clear that in the state of equilibrium, the sums of chemical potentials multiplied by the stoichiometric coefficients of the educts are equal to respective data of the products (Fig. 2.2).

2.5

The Effect of Entropy on Equilibrium

Rudolf Clausius (1822–1888) introduced the term entropy for the ratio dq=T of the amount of heat and absolute temperature. He showed that entropy is a state function, which for isolated systems distinguishes spontaneous from non-spontaneous (enforced) processes. Spontaneous processes always lead to an increase of entropy in isolated systems. In closed systems, there can occur spontaneous processes

2.5 The Effect of Entropy on Equilibrium

11

accompanied by a decrease of entropy, if this is overcompensated by a very negative enthalpy. For many reactions, the entropy is of immense importance to the position of the equilibrium. This is obvious when the following equations are considered: DR G— ¼ DR H —  TDR S— K ¼ e K¼e

ð2:14Þ

— DR G  RT

ð2:17Þ

— — DR H  þ TDR S RT

ð2:21Þ

When a reaction has a positive DR S— , the term TDR S— may be much more significant than DR H — , and the exponent in Eq. 2.21 will be positive, i.e., the equilibrium constant will be large. It generally holds that for small values of DR H — , the degree of order of the educts and products can give a clear indication of the position of the equilibrium (cf. Fig. 2.3). Good examples of the importance of entropy are: (1) the condensation reactions of metal hydroxo complexes (Chap. 3.2.4), and (2) the chelate effect (Chap. 4.3). The entropy increase is also important for the so-called hydrophobic effect: hydrophobic molecules (or hydrophobic parts of molecules) only weakly interact with the surrounding water molecules. These water molecules assume a structure which has similarities to the structure of water molecules in ice. They possess fewer orientations compared with water molecules in liquid water. This “water shell” is called iceberg water. The iceberg water has a lower entropy than “liquid water,” because of the higher degree of order. When hydrophobic molecules (or parts of molecules) attach to one another, this iceberg water is released forming “liquid

Relation of entropies of Changes of the order in the course of a chemical equilibrium reaction

educts and products, and resulting equilibrium constant K

educts

decrease of entropy ⎯⎯⎯⎯⎯⎯ → ←⎯⎯⎯⎯⎯ ⎯ increase of entropy

products

O

ΔS reaction = O

low degree of order O

Seducts

educts

high degree of order >

⎯⎯⎯⎯⎯⎯ → ←⎯⎯⎯⎯⎯ ⎯ decrease of entropy increase of entropy

O

S products

products

K is small O

ΔS reaction = O

high degree of order O

Seducts

low degree of order
0 K is large

Fig. 2.3 Effect of the degree of order of educts and products on the position of the equilibrium, when the reaction enthalpy is negligible compared with that of the entropy

12

2 Chemical Equilibrium

water,” and the entropy of the system increases. Clearly, this is the driving force for amphiphilic molecules to form vesicles, like micelles or liposomes, and membranes. Completely hydrophobic molecules, e.g., hexane, cannot assemble in layers, but simply form a second liquid phase. The folding of proteins is also partially driven by the hydrophobic effect.

2.6

The Kinetics of Chemical Equilibrium

When the forward and backward reaction of the equilibrium: A + BC + D

ðEquilibrium 2:1Þ

proceeds unconstrained, in the equilibrium state (the temporary constancy of activities ai;eq of all components) the rates rf and rb of the forward and backward reactions, respectively, must be the same. In kinetics, the concentrations, e.g., molar concentrations in mol L–1, are used, and the following relation holds: rf ¼ kf cA;eq cB;eq ¼ rb ¼ kb cC;eq cD;eq

ð2:22Þ

where kf and kb are the rate constants and ci;eq the equilibrium concentrations of the components. This allows calculation of the equilibrium constant according to: K¼

cC;eq cD;eq kf ¼ cA;eq cB;eq kb

ð2:23Þ

The kinetic interpretation of equilibrium constants dates back to the Austrian physicist Leopold Pfaundler von Hadermur (1839–1920), the French chemist Marcelin Pierre Eugène Berthelot (1827–1907), the Norwegian Chemist Peter Waage (1833–1900), the Norwegian mathematician Cato Maximilian Guldberg (1836–1902), and the Dutch chemist Jacobus Henricus van’t Hoff (1852–1911) [28]. Understanding chemical equilibria in terms of kinetics was very important for the development of chemistry, and it is still of great value. However, for the quantitative description of equilibria, thermodynamics are of greater importance and value, because the equilibrium constants can be calculated from the experimentally accessible thermodynamic state functions (cf. Eqs. 2.14 and 2.17). It is also very helpful that equilibrium constants can be understood as a result of the reaction enthalpy and entropy (Chap. 4).

2.7 Reversibility and Irreversibility of Chemical Reactions (Equilibria)

2.7

13

Reversibility and Irreversibility of Chemical Reactions (Equilibria)

Frequently, chemical reactions are classified as reversible systems, where the forward and backward reactions proceed, or as irreversible systems, where only one reaction, e.g., the forward reaction, proceeds. This is in apparent contradiction to Eq. 2.23, which ascribes to a chemical equilibrium a certain ratio of forward and backward reaction rates, i.e., finite reaction rates in both directions. Thus, even for the oxidation of glucose with oxygen, forming carbon dioxide and water, a finite back reaction rate is supposed to exist. Of course, in reality the back-reaction rate is absolutely zero. Equation 2.23 makes sense only for reversible equilibria. The crucial point is the timescale for which the equilibrium is established: if this happens within the time of measurement, the equilibrium appears reversible; if it does not happen during this time of observation, it is not reversible. On this point, electrochemistry is unique, because it is possible to vary the timescale of measurement, and with this the rate of mass transport to the electrode relative to the charge transfer at the electrode. In this way an electrochemical reaction can be shifted from reversibility (relatively slow mass transport compared with a fast charge transfer) to irreversibility (relatively fast mass transport compared with a slow charge transfer). The establishment of electrochemical equilibrium can also be varied by catalysis, i.e., by enhancing the rate of charge transfer. Affecting the Mass Transport Rate in Electrochemistry by Varying the Rate of Measurement When the potential of the electrode is varied very quickly, diffusion is given only a short time to increase diffusion layer thickness. This means that a very steep concentration gradient in the diffusion layer prompts a very fast mass transport toward the electrode surface. A slow potential change, however, allows the diffusion layer thickness to grow much more, and then the concentration gradient is more gently inclined and the mass transport rate lower. The rate of charge transfer does not depend on the mass transport rate but only on the system (the nature of the redox system, solvent, and electrode material). (Both the mass transport rate and charge transfer rate of course depend on temperature and pressure.) This shows that in electrochemistry a system may appear irreversible under one condition and reversible under another.

2.8

Catalysis—The “Magic” Way to Enhance the Establishment of Equilibrium

Students are certainly already acquainted with catalysis from lectures on inorganic chemistry, where the Haber–Bosch synthesis of ammonia will have been discussed. In this technical process a very active form of metallic iron is used to increase the

14

2 Chemical Equilibrium

reaction rate, i.e., to shorten the time for establishing equilibrium. Nowadays ruthenium–carbon catalysts are used for this. Among all the equilibria presented in this textbook, there are very few that normally show such a slow reaction rate that they need to be catalyzed. Some hydration reactions, e.g., the hydration of carbon dioxide to carbonic acid (and its back-reaction, i.e., dehydration) are indeed slow (Chap. 3). Another example is the hydrolysis of the condensation products of chromium(III)-aqua complexes. Some complex formation and redox equilibria are also slow. The kinetics of such slow reactions play an important role in some analytical applications, in biochemistry (e.g., carbon dioxide hydration and its back-reaction), and in environmental chemistry. Finally, it needs to be stressed that catalysis lowers the energetic barrier (the activation free energy) because the catalyst offers reaction pathways that are closed in the absence of a catalyst.

2.9

Guidelines for Calculations of Equilibria

For the successful calculation of equilibria, the following procedure is recommended and followed in this book: 1. Formulate all chemical equilibria with correct stoichiometry. Take the simplest equilibrium reactions, for which equilibrium constants are tabulated. 2. Write down the LMA for each equilibrium. 3. Write down balance equations, which relate the concentrations of all the involved species to one another. For example, find the equation which relates the overall concentration of an acid to all its equilibrium species. 4. In the case that ions are involved in the equilibrium reactions, write down the equations for charge balance, because all macro systems have no net charge. 5. Combine the LMA equations with the balance equations and solve for the concentration of the species of interest. With these equations, graphical presentations can be constructed.

References 1. 2. 3. 4.

Quílez J (2017) ChemTexts 3:3 Lindauer MW (1962) J Chem Educ 39:384–390 Quílez J (2006) Bull Hist Chem 31:45–57 Newton I (1998) Opticks: or, a treatise of the reflections, refractions, inflexions and colours of light. Also two treatises of the species and magnitude of curvilinear figures. Commentary by Nicholas Humez (Octavo ed.). Palo Alto (Opticks was originally published in 1704) 5. Geoffroy EF (1718) Table des Differents Rapports Observés en Chimie entre Differentes Substances. Memoires de l’Académie Royale des Sciences 202–212 6. Wenzel CF (1777) Lehre von der Verwandtschaft der Körper. Gerlach, Dresden 7. Berthollet CL (1803) Essai de Statique Chimique. Demonvill et Soeurs, Paris

References

15

8. Ostwalds Klassiker der Naturwissenschaften, No. 173: Untersuchungen über die Affinitäten. Über Bildung und Zersetzung der Äther. M. Berthellot und L. Péan de Saint-Gilles. Annales de Chimie et Physique. 3e série, Tome 65, p 385; 66, p 5 et 68, p 225. Leipzig, 1910 9. A translation into English has been published here: Waage P, Guldberg CM (1986) J Chem Educ 63:1044 10. (a) Horstmann A (1903) Abhandlungen zur Thermodynamik chemischer Vorgänge. In: van’t Hoff JH (ed). Engelmann, Leipzig. (b) Jensen WB (2009) Bull Hist Chem 34:83–91 11. (a) Gibbs JW (1874–1878) Trans Connecticut Acad Arts Sci 3:108–248, 343–524. (b) The Scientific Papers of Willard Gibbs J (1993) Vol 1, Thermodynamics. Woodbridge, Ox Bow Press. (c) German translation (1892) Thermodynamische Studien. Unter Mitwirkung des Verfassers übersetzt von W. Ostwald. Engelmann, Leipzig 12. Helmholtz H (1877) Monatsber kgl preuss Akad Wiss 713–726 (also included in: Planck M (ed) (1902) Abhandlungen zur Thermodynamik von H. Helmholtz. Leipzig, Engelmann (reprinted: 2013. Servus Verlag, Hamburg) 13. Helmholtz H (1882) Monatsber kgl preuss Akad Wiss 22–39 (also included in: Abhandlungen zur Thermodynamik von H. Helmholtz (1902) Planck M (ed) Leipzig, Engelmann (reprinted: 2013. Servus Verlag, Hamburg) 14. Jensen WB (2016) ChemTexts 2:1 15. Keszei E (2016) ChemTexts 2:15 16. (a) Hoff JH van’t (1884) Études de dynamique chimique. Muller & Co, Amsterdam. German translation: (b) Hoff JH van’t (1896) Studies in chemical dynamics. Revised and enlarged by E. Cohen and Th. Ewan. F. Muller, Amsterdam 17. Nernst W (1889) Z Phys Chem (Leipzig) 4:129–189 18. Scholz F (2017) J Solid State Electrochem 21:1847–1859 19. Nernst W (1892) Z Phys Chem (Leipzig) 9:137–142 20. Riesenfeld EH (1902) Ann Phys 313:616–624 21. Peters R (1898) Z Phys Chem (Leipzig) 26:193–236 22. Lewis GN (1901) Proc Am Acad Arts Sci 37:49–69 23. Lewis GN (1907) Proc Am Acad Arts Sci 43:259–293 24. Lewis GN (1908) J Am Chem Soc 30:668–683 25. Burgot J-L (2017) The notion of activity in chemistry. Springer, Berlin 26. Müller H, Otto M, Werner G (1980) Katalytische Methoden in der Spurenanalyse. Akadem Verlagsgesell, Leipzig 27. Bergmeyer HU (ed) (1984) Methods of enzymatic analysis. Verlag Chemie, Weinheim, Deerfield Beach, Basel 28. Califano S (2012) Pathways to modern chemical physics. Springer, Berlin

3

Acid–Base Equilibria

3.1

Glimpses of the History of Acid–Base Theories

The adjective “sour” (“acidic”) is most probably one of the oldest words in all languages, as it was a very early sensation noted when eating fruits and other foods. In modern scientific English, the adjective “acidic” and the noun “acid” are used. These terms entered English via the French acide (16th century) and were derived from the Latin acidus for “sour, sharp.” It was a very long way to go from the perception of a sour taste to a theory of acids [1]. Here we can give only a very brief sketch of this history. The German chemist/pharmacist Johann Rudolph Glauber (1604–1670) (Fig. 3.1) realized that acids and bases represent “opposite principles,” and stated that salts are the product of a reaction between these “opposite principles.” The term “base” originated from the French chemist Guillaume-François Rouelle (1703–1770), with the meaning “basic” being a term going back to the Latin basis (foundation), which itself is derived from the Greak bahlό1 (“bathmos”) meaning “step, pedestal.” The other term for base is alkali/alkaline. This term is a creation from the late 14th century, meaning soda ash. The medieval Latin term alkali was formed from the Arabic al-qaliy “the ashes, burnt ashes” of saltwort, a plant growing in alkaline soils. In Arabic qala means “to roast in a pan.” The Anglo-Irish philosopher and scientist Robert Boyle (1626–1691) (Fig. 3.2) characterized acids as exhibiting the following reactions: (1) they turn blue plant dyes red; (2) they precipitate sulfur from solutions of “liver of sulfur” (a mixture of polysulfides, sulfides, and thiosulfate, obtained by melting sulfur with potassium carbonate); (3) they neutralize alkaline solutions; and (4) they react with chalk (calcium carbonate) by liberating a gas [2]. The French chemist Antoine-Laurent Lavoisier (1743–1794) (Fig. 3.3) supposed that all acids contain an element which he named oxygène, derived from the Greak word ὀnύ1 (oxys) for sharp, sour, and cemή1 (genês) for “born,” “derived from.” Lavoisier’s oxygen theory of acids had a strong impact on the chemical nomenclature of many languages: whereas in English the name oxygen was introduced, © Springer Nature Switzerland AG 2019 F. Scholz and H. Kahlert, Chemical Equilibria in Analytical Chemistry, https://doi.org/10.1007/978-3-030-17180-3_3

17

18

3

Acid–Base Equilibria

Fig. 3.1 Johann Rudolph Glauber. © Austrian National Library/INTERFOTO

other languages translated it as “Sauerstoff” (German meaning “sour matter”) and “zuurstoff” (Dutch, same meaning). Here it is interesting to note that it is only in Polish that a completely different term exists: “tlen,” a term introduced by the alchemist Michał Sędziwój (1566–1636, latinized name Michael Sendivogius). In 1604 Sędziwój discovered that there exists a special gas (oxygen) that supports glowing and fire and life(!), and he derived the name from the Polish word “tleć” for glowing [3]. Although he published his discovery in Latin, the language of science of the Middle Ages, it remained unnoticed for a very long time—Priestley (1733–1804) and Scheele (1742–1786) are still most frequently referred to as the discoverers of oxygen. In 1812, the Swedish chemist Jöns Jacob Berzelius (1779–1848) (Fig. 3.4) published his so-called electrochemical theory of chemical affinity [4]. From his electrolysis experiments he concluded that “oppsite electricities” (charges) are responsible for keeping chemical compounds together. He assumed that the properties of being acidic and alkaline were somehow related to “electricities,” because the electrolysis of a neutral salt solution in water produces an acid at the positive pole and an alkaline solution at the negative pole. The assumption of a relation between “electricities” and acid–base properties was a genius idea, but it remained unexplained at his time, because even the very nature of charges was unknown.

3.1 Glimpses of the History of Acid–Base Theories Fig. 3.2 Robert Boyle. © Georgios Kollidas/Fotolia

Fig. 3.3 Antoine-Laurent Lavoisier and his wife Marie-Anne Pierrette Paulze Lavoisier. © Erich Lessing/akg-images/picture alliance

19

20

3

Acid–Base Equilibria

Fig. 3.4 Jöns Jacob Berzelius. © Georgios Kollidas/stock.adobe.com

Berzelius wrote with respect to electrolysis “We do not yet know how electricities act and we have to content ourself with conjectures” (free translation). In the first two decades of the 19th century, the Cornish chemist Humphry Davy (1778–1829) (Fig. 3.5) shattered Lavoisier’s view—that acids always contain oxygen. Davy showed that HF, HCl, and HI (in modern writing) do not contain oxygen [5] and made hydrogen responsible for the acidity of acids. Clearly, many chemists objected that not all compounds containing hydrogen are acids. The German chemist Justus von Liebig (1803–1873) (Fig. 3.6) also refuted Lavoisier’s theory and rendered Davy's theory more precisely by defining an acid as a compound containing hydrogen, that can be substituted by metals, e.g., by liberating hydrogen gas by the action of a metal on a solution of an acid. The next player in this process was the Swede Svante August Arrhenius (1859–1927) (Fig. 3.7), the founder of the dissociation theory of electrolytes [6]. Now it was possible to define acids as compounds producing hydrogen ions (H+) by dissociation, and bases as producing hydroxide ions (OH−). In 1923, the Danish physical chemist Johannes Nicolaus Brønsted (1879–1947) (Fig. 3.8) and the English physical chemist Thomas Martin Lowry (1874–1936) (Fig. 3.9), independent of one another, made the most profound contribution to the theory of acids and bases [7, 8]. According to their theory, acids were compounds which could donate protons, and bases were those which could accept protons. Because of the great importance of the Brønsted–Lowry theory it is given detailed consideration in Sect. 3.2.

3.1 Glimpses of the History of Acid–Base Theories

21

Fig. 3.5 Humphry Davy. © Archivist/stock.adobe.com

Fig. 3.6 Justus von Liebig. © Sammlung Rauch/INTERFOTO

In the same year, 1923, the US physical chemist Gilbert Newton Lewis (1875– 1946) (Fig. 3.10) published his book Valence and the Structure of Atoms and Molecules [9], in which he also expressed his views on the nature of acids and

22 Fig. 3.7 Svante August Arrhenius. © Science and Society/INTERFOTO

Fig. 3.8 Johannes Nicolaus Brønsted. © Royal Society of Chemistry (Source Bell R. P. (1950) J Chem Soc, 409– 419)

3

Acid–Base Equilibria

3.1 Glimpses of the History of Acid–Base Theories

23

Fig. 3.9 Thomas Martin Lowry (Printed with permission. Copyright: The Royal Society)

Fig. 3.10 Gilbert Newton Lewis. © Welch Foundation (Source Calvin M. (1976) Proc The Robert A Welch Fond Conf on Chem Res, XXth American Chemistry-Bicentennial (Nov 8–10, 1976, Houston Texas) 116–150)

bases. Not aware of the papers by Brønsted and Lowry, he also claimed that acids donate protons and bases accept them, but he went further: “It seems to me that with complete generality we may say that a basic substance is one which has a lone pair of electrons which may be used to complete the stable group of another atom, and

24

3

Acid–Base Equilibria

Fig. 3.11 Edward Curtis Franklin (Courtesy of the Department of Special Collections, Stanford Libraries)

that an acid substance is one which can employ a lone pair from another molecule in completing the stable group of one of its own atoms. In other words, the basic substance furnishes a pair of electrons for a chemical bond, the acid substance accepts such a pair” [9]. Lewis further extended the understanding of acids and bases by including the donor–acceptor reactions of anions and cations: “Another definition of acid and base in any given solvent would be the following: An acid is a substance which gives off the cation or combines with the anion of the solvent; a base is a substance which gives off the anion or combines with the cation of the solvent” [9]. US-born Edward Curtis Franklin (1862–1937) (Fig. 3.11), Hamilton Perkins Cady (1874–1943) (Fig. 3.12), and Howard McKee Elsey (1891–1982) (Fig. 3.13) along with the German Gerhart Jander (1892–1961) (Fig. 3.14) defined acids and bases for non-aqueous solvents. This theory is now referred to as the solvo acid–solvo base theory: a solvo acid is the cation formed in the autosolvolysis of a compound while a solvo base is the anion [10, 11]. In the case of the solvent bromtrifluoride BrF3 , the cation BrF2þ is the acid and the anion BrF 4 is the base. The equilibrium equation for the autosolvolysis reaction is: BrF3 þ BrF3  BrF2þ þ BrF 4

ðEquilibrium 3:1Þ

3.1 Glimpses of the History of Acid–Base Theories Fig. 3.12 Hamilton Perkins Cady (Courtesy of the University of Kansas, Kenneth Spencer Research Library)

Fig. 3.13 Howard McKee Elsey (Smithsonian Institution Archives. Acc. 18-094, Science Service, Records, E&MP 4.001)

25

26

3

Acid–Base Equilibria

Fig. 3.14 Gerhart Jander (Universitätsarchiv Greifswald, Fotosammlung)

The German chemist Hermann Lux (1904–1999) (Fig. 3.15) and the Norwegian chemist Håkon Flood (1905–2001) (Fig. 3.16) defined acids and bases for oxidic melts. According to them an acid is an oxide ion acceptor and a base is an oxide ion donator [12, 13]. This allows for the understanding of the dissolution of alumina in a carbonate melt as being an acid–base reaction: Al2 O3 acid1

þ þ

CO2 3 base2



2AlO 2 base1

þ þ

CO2 acid2

ðEquilibrium 3:2Þ

The Russian chemist Mikhail Il’ich Usanovich (1894–1981) (Fig. 3.17) generalized all these acid–base concepts using the following definition [14]: Acids are either cation donators, or anion or electron acceptors. Bases are either cation acceptors, or anion or electron donators. A comparison of this definition with what Lewis stated in his book in 1923 [9], identifies significant agreement. The only difference is that Lewis only considered electron pairs and did not include redox reactions with a single electron (or multi-electron) transfer.

3.1 Glimpses of the History of Acid–Base Theories Fig. 3.15 Hermann Lux (Historisches Archiv der Technischen Universität München)

Fig. 3.16 Håkon Flood (Photo by: Schrøder/NTNU University Library)

27

28

3

Acid–Base Equilibria

Fig. 3.17 Mikhail Il’ich Usanovich (Archive of Al-Farabi University, Almaty, Kazakhstan)

3.2

Quantification of Acid and Base Strength According to the Brønsted–Lowry Theory

3.2.1 Acidity and Basicity Constants The Brønsted–Lowry theory is well suited for tackling acid–base equilibria in protic solvents, e.g., water. According to Brønsted and Lowry, an acid–base equilibrium is a proton transfer reaction: HB a1

þ

H2 O b2



B b1

þ

H3 O þ a2

ðEquilibrium 3:3Þ

The letters a and b below the chemical formulae stand for acid and base, respectively, and the numbers 1 and 2 indicate the chemical compounds that form the acid/base pairs. The pairs a1/b1 and a2/b2 are called corresponding acid–base pairs, as the members can be interconverted by proton transfer. Table 3.1 provides some examples. Equilibrium 3.3 is called a protolysis equilibrium because in the forward reaction a proton is separated from HB (the Greek kύri1 means separating, dissolving). In the backward reaction, a proton is separated from H3 O þ and transferred to B . H3 O þ is called a hydronium ion. The water molecule in Equilibrium 3.3 acts as a base since it accepts a proton. In other reactions, it can act as an acid and form a

3.2 Quantification of Acid and Base Strength … Table 3.1 Examples of corresponding acid–base pairs

29

Corresponding acid

Corresponding base

HCl H2 SO4 HSO 4

Cl− HSO 4 SO2 4 NH3
þ FeðH2 OÞ5 OH

NH4þ
2 þ FeðH2 OÞ6

NH 2 HCOO (formiate) C2 O4 H (hydrogen oxalate) H2 O

NH3 HCOOH (formic acid) C2 O4 H2 (oxalic acid) H3 O þ

hydroxide ion HO by losing a proton. In the solvent water, both hydronium and hydroxide ions are formed by proton transfer between the water molecules: H2 O a1

þ

H2 O b2



HO b1

þ

H3 O þ a2

ðEquilibrium 3:4Þ

Solvents that are able to donate and accept protons are called protic solvents. This distinguishes them from solvents which have covalently bonded hydrogen, e.g., benzene, rendering them unable to transfer protons. The acid a1 does not need to be a neutral molecule, but can be a cation: H2 B þ a1

þ

H2 O b2



HB b1

þ

H3 O þ a2

ðEquilibrium 3:5Þ

H3 O þ a2

ðEquilibrium 3:6Þ

An example for this is the ammonium cation: H4 N þ a1

þ

H2 O b2



H3 N b1

þ

It is also possible that acid a1 is an anion: HB a1

þ

H2 O b2



B2 b1

þ

H3 O þ a2

ðEquilibrium 3:7Þ

Hydrogencarbonate (bicarbonate) is an example of an anion acid: HCO 3 a1

þ

H2 O b2



CO2 3 b1

þ

H3 O þ a2

ðEquilibrium 3:8Þ

30

3

Acid–Base Equilibria

Here, water is the proton acceptor. Many solvents undergo similar reactions to those of water, e.g., ethanol ðCH3 CH2 OHÞ, methanol ðCH3 OHÞ, liquid hydrogen sulfide ðH2 SÞ, and liquid ammonia ðNH3 Þ, etc. For these solvents, the following equations can be written: In ethanol HB a1

þ



B b1

þ

CH3 CH2 OH2þ a2



B b1

þ

CH3 OH2þ a2

ðEquilibrium 3:10Þ

H2 S b2



B b1

þ

H3 S þ a2

ðEquilibrium 3:11Þ

H3 N b2



B b1

þ

H4 N þ a2

ðEquilibrium 3:12Þ

CH3 CH2 OH b2

ðEquilibrium 3:9Þ

In methanol HB a1

þ

CH3 OH b2

In liquid hydrogen sulfide HB a1

þ

In liquid ammonia HB a1

þ

Protonated solvent molecules possess lyonium ions, e.g., H4 N þ ammonium, H3 S þ sulfonium, CH3 OH2þ methoxonium, CH3 CH2 OH2þ ethoxonium, and H4 P þ phosphonium. The general name given to deprotonated solvent molecules is lyate ions. In many cases they have specific names, like OH hydroxide, SH bisulfide, CH3 O methanolate, CH3 CH2 O ethanolate, and NH 2 amide. The reactions of the bases B− with the solvent can be written as the back reactions of Equilibrium 3.3 through Equilibrium 3.12, e.g.: B b1

þ

H3 O þ a2



HB a1

þ

H2 O b2

ðEquilibrium 3:13Þ

However, since the concentration of H3 O þ is very small in pure water, it makes sense to consider the reaction of the base with the main constituent of the solvent, i.e., here H2 O: B b1

þ

H2 O a2



HB a1

þ

OH b2

ðEquilibrium 3:14Þ

3.2 Quantification of Acid and Base Strength …

31

All the proton transfer reactions obey the law of mass action (LMA) and the proton transfer belongs to the fastest chemical reactions [15]. The LMA of Equilibrium 3.3 reads as follows: Ka ¼

a B a H 3 O þ aHB aH2 O

ð3:1Þ

where Ka is the equilibrium constant of the protolysis of HB in water. The activities a of the chemical species appearing in Eq. 3.1 are defined as follows: ci ac;i ¼ fi — c

ð3:2Þ

The index c of a indicates that these activities are based on molar concentrations. Therefore, they are also called concentration activities. ci is the molar concentration (also called amount concentration) in mol L1 and fi is the concentration-dependent dimensionless activity coefficient. c— is by definition 1 mol L1 , which is the concentration of a compound in the thermodynamic reference state (although, theoretically it is supposed to have the properties of an infinitely diluted solution). The activity coefficient approaches unity at infinite dilution: lim fi ¼ 1

ð3:3Þ

ci !0

The concentration activity ac;i is dimensionless (cf. Eq. 3.2). The concentration activity of the solvent cannot be defined on a strict thermodynamic basis [16]. Therefore, the molar ratio activity ax;H2 O has to be used for water in Eq. 3.1. The n molar ratio activity is based on the molar ratio xH2 O ¼ nH OHþ2 OnHB . Since the molar 2

ratio of pure water is xH2 O ¼ 1, the molar ratio activity of pure water is unity ax;H2 O ¼ 1.1 This allows writing Eq. 3.1 as follows: Ka ¼

ac;B ac;H3 O þ ac;HB ax;H2 O

ð3:4Þ

This equation defines the dimensionless thermodynamic equilibrium constant Ka , called the acidity constant, of HB (here in water). Since the molar ratio activity of water approaches unity when the concentration of HB is very small, it is possible to write 1 instead of ax;H2 O , or simply:

1

Generally, the molar ratio activity of any pure phase, be it liquid, solid, or gas, is unity. This is also important in the solubility equilibria of solid phases.

32

3

Ka ¼

ac;B ac;H3 O þ ac;HB

Acid–Base Equilibria

ð3:5Þ

At very low concentrations of HB the activity coefficients of all species will approach unity, so that the molar concentrations can be used instead of molar activities: Ka ¼

c B c H 3 O þ cHB

ð3:6Þ

The so defined acidity constant is strictly taken as not constant anymore, since the term on the right-hand side of the equation depends on the concentration of HB. However, for diluted solutions (roughly below 0:01 mol L1 ), the deviations between concentrations and activities are negligible, if there is not a very high demand for accuracy. When activity coefficients need to be considered, they should be calculated, otherwise experimental data will need to be used (for this see one of the many physical chemistry textbooks). In strict analogy to Eq. 3.6 the basicity constant is defined for Equilibrium 3.14: Kb ¼

cHB cOH c B

ð3:7Þ

Ka and Kb are related to one another as follows: Ka Kb ¼ cH3 O þ cOH

ð3:8Þ

This relation follows from a multiplication of the terms on the right-hand side of Eqs. 3.6 and 3.7. The exact relation is: Ka Kb ¼ ac;H3 O þ ac;OH

ð3:9Þ

Equation 3.9 is clearly the LMA for Equilibrium 3.4. The equilibrium constant is called the autoprotolyis constant (here of water): Kw ¼ ac;H3 O þ ac;OH ¼ Ka Kb

ð3:10Þ

Another name for this constant is the ion product (here of water). Because the acidity and basicity constants vary over many orders of magnitude, the negative decadic logarithms, are used very frequently: pKa ¼  log Ka

ð3:11Þ

3.2 Quantification of Acid and Base Strength … Table 3.2 Temperature dependence of pKw at a pressure of 0.1 MPa [17]

Table 3.3 Pressure dependence of pKw at 25 °C [18]

33

T (°C)

pKw

0 10 15 18 20 25 30 50 75 100

14.938 14.528 14.340 14.233 14.163 13.995 13.836 13.275 12.711 12.264

p (MPa)

pKw

0, 1 25 50 100

13.995 13.908 13.824 13.668

pKb ¼  log Kb

ð3:12Þ

pKw ¼  log Kw

ð3:13Þ

pKw ¼ pKa þ pKb

ð3:14Þ

With this, Eq. 3.10 yields:

Since Kw at 25 °C is almost precisely 1  1014 , i.e., pKw ¼ 14, the acidity and basicity constants are related at this temperature by the following equation: pKa þ pKb ¼ 14

ð3:15Þ

The autoprotolysis constant depends on temperature and pressure. Table 3.2 gives data at different temperatures (at a constant pressure) and Table 3.3 gives data for different pressures (at a constant temperature).

3.2.2 The pH In 1909 the Danish chemist Søren Peter Lauritz Sørensen (Fig. 3.18) introduced the symbol p for “–log” [19] to present concentrations of hydrogen ions on a simple

34

3

Acid–Base Equilibria

Fig. 3.18 Søren Peter Lauritz Sørensen (1868– 1939) (Royal Library, Copenhagen, Denmark)

scale. Initially he wrote pHþ ¼  log cH þ ; however, the symbol pHþ was later replaced by pH: ð3:16Þ

pH ¼  log cH þ

For a long period of time it was believed that Sørensen used p as an abbreviation of the term “potential” (in the sense of the “potential to do something”), but this is wrong, as an historical study proved [20]. Sørensen chose p as the variable giving þ proton concentrations as 10pH ; or in modern notation 10pH . The modern definition of pH, accepted by IUPAC [21], is based on the molal activity am;H þ of hydrogen ions (protons): pH ¼  log am;H þ ¼  log

mH þ fH þ m—

ð3:17Þ

where mH þ is the molality of hydrogen ions in mol kg1 , fH þ is the activity coefficient of hydrogen ions, and m— is the standard molality (1 mol kg1 ). This definition has the weakness that activity coefficients and activities of single sorts of ions are, for reasons of principle, not accessible; however, they can be estimated, both experimentally and theoretically, with a sufficient degree of correctness. Note, that the symbol pH is always written upright. In what follows, pH is taken according the relation:

3.2 Quantification of Acid and Base Strength …

pH   log ac;H þ ¼  log ac;H3 O þ   log cH3 O þ

35

ð3:18Þ

(For small concentrations, when solutions almost have the density of pure water, molar and molal concentrations do not significantly deviate.) Furthermore, the nature of the hydrated proton is described by the formula H3 O þ . IUPAC uses H þ because several hydrates can be formulated, e.g., H3 O þ , H5 O2þ , and H7 O3þ . In the chemical literature it became customery to use H3 O þ . The most recent studies have revealed that the best description of a hydrated proton in water is the so-called Zundel complex H5 O2þ [22]. The German physicist (as well as being an entrepreneur and philanthropist) Georg Zundel (1931–2007) postulated this formula on the basis of infrared spectra. By analogy to pH, pOH is defined as follows: pOH   log ac;OH   log cOH

ð3:19Þ

This definition leads Eq. 3.10 to assume the form: pKW ¼ pH þ pOH

ð3:20Þ

The data provided in Table 3.2 yields the following pH = pOH values for pure water at different temperatures: 10 °C, pH ¼ 7:26; 20 °C, pH ¼ 7:08; 25 °C, pH ¼ 7:00; 100 °C, pH ¼ 6:13 At a temperature of 25 °C, solutions having a pH below 7.00 are called acidic, those with a higher pH are called basic, and solutions where pH = 7.00 are neutral. In reality, tap water or even deionized water normally has a pH below 7.00 because of the dissolved carbon dioxide it contains. From the temperature dependence of the autoprotolysis constant of water, the neutrality point is 7.47 at 0 °C and 6.13 at 100 °C.

3.2.3 The Strength of Acids and Bases When looking at Equilibrium 3.3 it is important to understand what quantitative relations can be deduced from this chemical equation: 1. HB constantly reacts with H2 O forming B and H3 O þ . Simultaneously, B reacts with H3 O þ forming HB and H2 O. However, this does not mean that B necessarily collides with the H3 O þ ions. There is, however, a higher probability that B collides with H2 O molecules, forming HB and HO ions, which react with H3 O þ to form H2 O. This is a mechanistic discussion, which does not need consideration here, where only the thermodynamics of the reactions are attended to.

36

3

Acid–Base Equilibria

2. The following interpretation is very important. When one HB molecule reacts with one H2 O molecule, exactly one B ion and one H3 O þ ion are produced. If only this reaction is considered, the ratio c cBþ is unity and H3 O

hence cB ¼ cH3 O þ . However, the chemical equation does not state how many HB molecules undergo this reaction. Naturally, the chemical reaction is only written with resepct to the reacting particles, since an equation for the non-reacting particles, i.e., HB þ H2 O  HB þ H2 O, is not helpful to any calculations. 3. From point 2 above it follows that it is impossible to make any quantic tative conclusion about the ratios ccBHB , c cHBþ , and c H2 Oþ . For this it is H3 O

H3 O

mandatory to know the equilibrium constant (Eq. 3.6): Ka ¼

cB  cH3 O cHB

Because this equilibrium constant expresses the ratio of concentrations of products to educts, it is a measure of the strength of the acid HB. When the acid produces a lot of H3 O þ ions, i.e., when the solution is very acidic, the acid is regarded as stronger than another acid which produces at the same overall concentration fewer H3 O þ ions. 4. Since the quantification of acid strength is based on Equilibrium 3.3, this quantification is valid only when water is the solvent. In other solvents, the acid has a different strength.

Table 3.4 gives a compilation of the pKa values for some acids in water. The pKa data that falls in the range of 0–14 are known with better precision than those smaller than 0 and larger than 14. The reason for this is simple: pKa  0 means Ka  1, and pKa  14 means Ka  1014 , i.e., the Ka values are either very large or very small, and this poses problems for a precise determination, especially when Ka is, for example, 1010 (as for perchloric acid) or 1023 (as for ammonia). When the pKa of an acid HB is below zero, the strength of this acid is leveled of (this is called the leveling effect of water) because all acids with pKa values smaller than zero possess practically the same strength, as they are practically completely protolyzed and converted to B and H3 O þ . This means that different acids with pKa values smaller than zero will produce the same concentration of H3 O þ ions, when their overall concentrations are identical. Similarily, acids with pKa values above 14 are such weak acids in water, that it is reasonable to say that they are not acids at all. Some textbooks give strict definitions to classify acids as: (1) very strong, (2) strong, (3) medium strong, (4) weak, and (5) very weak. Setting the boarders for such

3.2 Quantification of Acid and Base Strength …

37

Table 3.4 pKa values for some acids in water. Data are mainly from [23] Corresponding acid–base pair

pKa (ionic strength)a (25 °C)

HClO4 =ClO 4 

ca. –10 ca. –10 ca. –9 ca. –7 –3 [0] –1.34 [0] 1.271

2 HSO 4 =SO4 H3 PO4 =H2 PO 4 2 þ
3 þ
FeðH2 OÞ6 = FeðH2 OÞ5 OH

1.99 [0]; 1.55 [0.1]

HI/I HBr/Br HCl/Cl H2 SO4 =HSO 4 HNO3 =NO 3 C2 O4 H2 =C2 O4 H (oxalic acid/hydrogenoxalate)

2.148 [0]; 2.0 [0.1] 2.19 [0]; 2.83 [0.1]

HF=F HCOOH=HCOO (formic acid/formiate)

3.17 [0]; 2.92 [0.1]; 2.96 [1.0] 3.752

C2 O4 H =C2 O2 2 (hydrogen oxalate/oxalate) H3 CCOOH=H3 CCOO (acetic acid/acetate) 2 þ
3 þ
AlðH2 OÞ6 = AlðH2 OÞ5 OH

4.266

H2 CO3 =HCO 3 H2 S=HS

2 H2 PO 4 =HPO4

4.756 4.99 [0]; 5.69 [0.1] 6.35 [0]; 6.16 [0.1]; 6.02 [1.0] 7.02 [0]; 6.83 [0.1]; 6.61 [1.0] 7.199[0]; 6.72 [0.1]; 6.46 [1.0]

NH4þ =NH3 HCN=CN þ
2 þ
FeðH2 OÞ6 = FeðH2 OÞ5 OH

9.244 [0]; 9.29 [0.1]; 9.40 [1.0]

2 HCO 3 =CO3

10.33 [0]; 10.0 [0.1]; 9.57 [1.0]

3 HPO2 4 =PO4  2

12.35 [0]; 11.74 [0.1]; 10.79 [3.0]

9.21 [0]; 9.01 [0.1]; 8.95 [1.0] 9.5 [0]

13.9 [0]; 13.8 [1.0]b HS =S NH3 =NH ca. 23 2 a The ionic strength can perceptibly affect the pKa values. Often, the reported data have been extrapolated for an ionic strength of zero b In the literature values of even 16 and 18 are reported as the pKa of the second protolysis of hydrogen sulfide in water

categories is rather arbitrary and is avoided here. Figure 3.19 gives approximate ranges of acids according to their strength. The most expedient differences exist between acids for which their strength is leveled off by water and for which their strength is differentiated. In water, the cation H3 O þ (hydronium ion) is the strongest acid and the anion OH (hydroxide ion) is the strongest base that can exist in analytically relevant concentrations. Generally, in protic solvents, these are the lyonium and lyate ions.

38

3

Acid–Base Equilibria

Fig. 3.19 Classification of acids according to their strength

3.2.4 Principles Defining the Strength of Inorganic Acids There are some helpful principles to assess the relative strength of inorganic acids. The point is always to consider the bond strength of the proton. In the case of monoprotic acids not containing oxygen, e.g., HCl and HBr, two parameters are important: electronegativity and the size of the atom to which the proton is bonded. If an atom is strongly electronegative, it will strongly attract bonding electrons and facilitate the donation of a proton to another molecule, like water. The effect of atomic size is even more important: in a large atom the electrons are spread over a larger space and thus the bond between the atom and the proton is much weaker than in the case of a smaller atom. When hydrogen is bonded to oxygen, the electronegativity of the nearest neighbor is decisive. Formally, the arrangement HOZ can be considered: when Z is a metal atom, the electron pair between O and Z is completely in possession of O, and Z is a cation, like in Na+OH−. In an aqueous solution a dissociation takes place because the ions are hydrated (generally solvated), forming hydrated metal aqua ions and hydrated hydroxide ions. The hydrated ions are formed because water molecules are dipoles interacting with ions —the process of solvation decreases the Gibbs free energy. When Z is a non-metal ion with high electronegativity, the bond between O and Z is dominantly covalent and cannot be easily broken. Due to the electron attraction of O and Z, the bond strength toward hydrogen is decreased—facilitating the loss of a proton. This effect gets stronger with the increasing electronegativity of Z. When additional oxygens are bonded to Z, the acid strength increases further, as in the following series: hypochlorous acid HClO (pKa = 7.54); chlorous acid HClO2 (pKa = 1.97); chloric acid HClO3 (pKa = −2.7); and perchloric acid HClO4 (pKa = −10). Further details of the relation between structure and the strength of inorganic acids can be found in inorganic chemistry textbooks. Polyprotic (also called polybasic) acids and polyacidic bases are compounds which can donate and accept more than one proton. Thus, sulfuric acid H2 SO4 and phosphoric acid H3 PO4 are polyprotic acids and sulfate ions SO2 4 and phosphate

3.2 Quantification of Acid and Base Strength …

39

ions PO3 4 are polyacidic bases. Each deprotonation and protonation step has its respective acidity and basicity constant. The sequence is shown here for phosphoric acid: First protolysis of H3 PO4 þ H3 PO4 þ H2 O  H2 PO 4 + H3 O

Ka1 ¼

aH2 PO4 aH3 O þ ; pKa1 ¼  log Ka1 aH3 PO4

ðEquilibrium 3:15Þ ð3:21Þ

Second protolysis of H3 PO4 2 þ H2 PO 4 þ H2 O  HPO4 þ H3 O

Ka2 ¼

aHPO2 aH 3 O þ 4 aH2 PO4

; pKa2 ¼  log Ka2

ðEquilibrium 3:16Þ ð3:22Þ

Third protolysis of H3 PO4 3 þ HPO2 4 þ H2 O  PO4 þ H3 O

Ka3 ¼

aPO3 aH 3 O þ 4 aHPO2 4

; pKa3 ¼  log Ka3

ðEquilibrium 3:17Þ ð3:23Þ

To be completely unambiguous, the pKa data can be designated as follows: pKa1;H3 PO4 , pKa2;H3 PO4 , and pKa3;H3 PO4 . For the phosphate ion, the following sequence applies for protonation steps: First protonation of PO3 4 2  PO3 4 þ H2 O  HPO4 þ OH

Kb1 ¼

aHPO2 aOH 4 aPO3 4

; pKb1 ¼  log Kb1

ðEquilibrium 3:18Þ ð3:24Þ

Second protonation of PO3 4   HPO2 4 þ H2 O  H2 PO4 þ OH

Kb2 ¼

aH2 PO4 aOH ; pKb2 ¼  log Kb2 aHPO2 4

ðEquilibrium 3:19Þ ð3:25Þ

40

3

Acid–Base Equilibria

Third protonation of PO3 4  H2 PO 4 þ H2 O  H3 PO4 þ OH

Kb3 ¼

aH3 PO4 aOH ; pKb3 ¼  log Kb3 aH2 PO2 4

ðEquilibrium 3:20Þ ð3:26Þ

, pKb2;PO3 , and The following symbols provide even more precision: pKb1;PO3 4 4 pKb3;PO3 . 4 When protons are bonded to one atom (e.g., to sulfur in hydrogen sulfide), or to oxygen atoms which are bonded to one atom (like in phosphoric acid), the Gibbs free energy of losing a proton increases which each step because the negative charge of the remaining bases is increasing. The acid strength decreases roughly by a factor of 105 with each protolysis step (e.g., consider the pKa data of the protolysis steps of phosphoric acid in Table 3.4). The base strength decreases in the same way with increasing protonation. Hydrofluoric acid Table 3.4 shows some peculiarities: it is striking that hydrofluoric acid has in comparison to HCl, HBr, and HI a surprisingly large pKa value of 3.7 (for an extrapolated ionic strength of 0). Looking at the pKa data of HCl, HBr, and HI in water, one might expect a value around –4. This discrepancy can be explained by considering the following equilibria [24, 25]: HF þ H2 O  F þ H3 O þ

ðEquilibrium 3:21Þ

with Ka;HF ¼

c F c H 3 O þ cHF

F þ H3 O þ  ½F H3 O þ 

ð3:27Þ ðEquilibrium 3:22Þ

with Kip ¼

c½F H3 O þ  c H 3 O þ c F

ð3:28Þ

Here, ½F H3 O þ  is a very stable ion pair with a stability constant Kip (also called the association constant). Equation 3.27 only contains concentrations of fluoride ions, hydronium ions, and neutral HF. Experimental techniques as applied for the determination of acidity constants, e.g., potentiometry and conductometry, cannot distinguish HF from the ion pair ½F H3 O þ , and thus Eq. 3.27 yields only an

3.2 Quantification of Acid and Base Strength …

41

apparent acidity constant Ka;HF;apparent , that does not correctly express the tendency of HF to lose a proton: Ka;HF;apparent ¼

cF cH3 O þ cF cH3 O þ Ka;HF ¼ ¼ ¼ 103:45 mol L1 cHF;apparent cHF þ c½F H3 O þ  1 þ Ka;HF Kip ð3:29Þ

Instead of using the true concentration of HF ðcHF Þ, it uses the sum cHF þ c½F H3 O þ  , which can be called the apparent concentration cHF;apparent . Therefore, the then resulting acidity constant is the apparent acitidy constant Ka;HF;apparent . The formation of ion pairs ½F H3 O þ  is also favored because the hydration of fluoride ions has a very negative entropy. At a high HF concentration, another equilibrium plays an important role: ½F H3 O þ  þ HF  H3 O þ þ HF 2, increasing the concentration of H3 O þ and providing a higher apparent acidity.2 Carbonic acid In Table 3.4, there is another acid with an apparent acidity constant: carbonic acid. In a solution of carbon dioxide in water, most of the latter is present simply as dissolved gas, without reacting with the solvent—the water. The equilibrium constant of hydration of CO2, according to the reaction: !

CO2 þ H2 O  H2 CO3

ðEquilibrium 3:23Þ

is K ¼ 1:7  103 , clearly indicating that most of the CO2 is present just as a dissolved gas. The true acidity constant of H2 CO3 , calculated for the following equilibrium: !

þ H2 CO3 þ H2 O  HCO 3 þ H3 O

ðEquilibrium 3:24Þ

is Ka1;H2 CO3 ;real ¼

cHCO3 cH3 O þ ¼ 2:5  104 mol L1 cH2 CO3

ð3:30Þ

That is, pKa1;H2 CO3 ;real ¼ 3:6. Here, and in the following equations, equilibrium constants are given with units, because concentrations are used instead of activities —because the numeric values based on concentrations approach the values based on activities at very low concentrations. When the equilibrium concentrations of dissolved CO2 ðcCO2 Þ and of carbonic acid ðcH2 CO3 Þ are not distinguished, and the 2

We owe most of our current understanding of the acid–base equilibria of hydrofluoric acid to the Canadian chemist Paul-Antoin Giguère (1910‒1987).

42

3

Acid–Base Equilibria

sum of both is used as an apparent concentration cH2 CO3 ;apparent , a much smaller acidity constant Ka1;H2 CO3 ;apparent results: Ka1;H2 CO3 ;apparent ¼

cHCO3 cH3 O þ cHCO3 cH3 O þ ¼ ¼ 4:6  107 mol L1 cH2 CO3 ;apparent cH2 CO3 þ cCO2

ð3:31Þ

and the resulting pKa1;H2 CO3 ;apparent  6:37. The use of this apparent acidity constant is in practice well reasoned, as this value relates to the overall CO2 concentration, which is more easily accessible than the two concentrations cCO2 and cH2 CO3 . The kinetics of equilibrium establishment in the case of carbonic acid Establishment of the equilibrium: !

þ H2 CO3 þ H2 O  HCO 3 þ H3 O

ðEquilibrium 3:24Þ

is normally very slow. For life, this is an enormous problem, because respiration needs the fast removal of formed CO2 from the blood for the homeostasis of blood pH. For the non-catalytic hydration of CO2 and the dehydration of H2CO3 the following rate constants have been determined: CO2 þ H2 O ! H2 CO3 : kforward = 0.039 s−1 H2 CO3 ! CO2 þ H2 O: kbackward = 23 s−1 Since these rates are completely insufficient to transport carbon dioxide and achieve homeostasis of pH, organisms have specific enzymes, the carboanhydrases, to accelerate the two reactions: CO2 þ H2 O ! H2 CO3 : kforward,cat = 1,000,000 s−1 H2 CO3 ! CO2 þ H2 O: kbackward,cat = 400,000 s−1 The mechanism of this catalytic cycle is depicted in Fig. 3.20. The acidity of metal aqua ions Table 3.4 also lists metal aqua ions, showing that they can react as Brønsted acids according to the following examples:
n þ
ðn1Þ þ MeðOH2 Þx þ H2 O  MeðOH2 Þx1 ðOHÞ þ H3 O þ ðEquilibrium 3:25Þ
ðn2Þ þ
ðn1Þ þ þ H2 O  MeðOH2 Þx2 ðOHÞ2 þ H3 O þ MeðOH2 Þx1 ðOHÞ ðEquilibrium 3:26Þ Of course, the deprotonations do not stop at this stage, but can continue, although at this point condensation reactions (see below) become more and more important. The reason for the acidity of metal aqua ions is the polarizing action of the positive metal ion. Figure 3.21 depicts schematically how the positive charge of the

3.2 Quantification of Acid and Base Strength …

43

Fig. 3.20 Catalytic cycle of the hydration of carbon dioxide and the dehydration of carbonic acid by carboanhydrases. The water molecule coordinated to the zinc ion is more acidic than free water molecules

metal ion Men þ weakens the bond of the protons to oxygen in the water ligand. This means that the protons of the coordinated water are more easily transferred to a solvent water molecule, than the proton of the solvent water molecule to another solvent water molecule. Looking at Fig. 3.21 it is not surprising that the Ka values are proportional to the charge and inversely proportional to the radius of the metal ions: Ka

charge of metal ion qMen þ radius of metal ion rMen þ

ð3:32Þ

q

The ratio r Menn þþ is frequently called the ion potential of the metal ion. This Me quantity appears in many thermodynamic equations for ions and ionic solids. Figure 3.22 demonstrates, for a large number of metal ions, the linear relation q between the pKa1 data and the ratio r Menn þþ . The fact that a few metal ions do not Me follow the linear relation is due to the more or less pronounced covalent character of their bonds to water. There also exists an interesting correlation between the pKa1 data and the n-th ionization potentials of the metal ions (Fig. 3.23). These ionization potentials relate to the metal ions in vacuum and give the energy necessary to q remove the n-th electron. This energy also depends on the ratio r Menn þþ . Me

44

3

Acid–Base Equilibria

Fig. 3.21 The strong polarizing power of the metal ion Men þ weakens the bond between oxygen and hydrogen in the coordinated water molecule by shifting electron density toward the metal ion

Fig. 3.22 Correlation between pKa1 data and the ratio

qMen þ rMen þ

for different metal ions

Figure 3.23 shows that a number of metal aqua ions are more acidic than acetic acid, which has a pKa of 4.75. The acidity of these metal aqua ions has far reaching implications for the chemistry of aqueous solutions: the hydroxo complexes formed as a result of the reactions in Equilibria 3.25 and 3.26 are able to undergo condensation reactions:
ðn1Þ þ
ð2n2Þ þ 2 MeðOH2 Þx1 ðOHÞ  ðH2 OÞx1 MeOMeðOH2 Þx1 þ H2 O ðEquilibrium 3:27Þ h ið2n4Þ þ
ðn2Þ þ  ðH2 OÞx2 MehO iMe ð OH Þ þ 2H2 O 2 MeðOH2 Þx2 ðOHÞ2 2 O x2 ðEquilibrium 3:28Þ

3.2 Quantification of Acid and Base Strength …

45

Fig. 3.23 Correlation between pKa1 data and the n-th ionization potentials of metal ions (according to [26])

In the case of octahedral metal aqua complexes, the two octahedra of
ð2n2Þ þ ðH2 OÞ5 MeOMeðOH2 Þ5 are bridged across a corner by one oxygen h ið2n4Þ þ O the two octahedra are bridged atom, whereas in ðH2 OÞ4 MehO iMeðOH2 Þ4 by two oxygen atoms and share a common edge (Fig. 3.24). In principle, the con
ðn2Þ þ can also result in the formation of densation of the ions MeðOH2 Þx2 ðOHÞ2  
chains possessing the formula ðHO)ðH2 OÞx2 MeO MeðH2 OÞx2 m 

MeðOH2 Þx2 ðOH)ðm þ 2Þðn2Þ þ . In many cases, the bridging oxygen atoms can also be protonated.

Fig. 3.24 Schematic MeðOH2 Þ5 

ð2n2Þ þ

of the two oxygen-bridged complexes: h ið2n4Þ þ and b ðH2 OÞ4 MehO O iMeðOH2 Þ4

a




ðH2 OÞ5 MeO

46

3

Acid–Base Equilibria

If three or more hydroxide ligands are bonded to one metal ion, their condensation leads to three-dimensional products, as is the case for isopoly and heteropoly metal ions (Fig. 3.28). As the size of the condensation products grows, so the solubility decreases and finally solid metal oxide hydrates can form, which can even be completely dehydrated forming metal oxides. In this respect, iron(III) aqua ions are of exceptional environmental importance. The acid–base chemistry of iron(II) and iron(III) aqua ions  2 þ II The pKa1 of iron(II) hexaaqua ions FeðOH2 Þ6 is 9.5, i.e., they are slightly less acidic than ammonium ions, which have a pKa = 9.25. The pKa1 of iron(III) hex 3 þ III aaqua ions FeðOH2 Þ6 is 2.2, i.e., greater than 7 orders of magnitude more acidic than the iron(II) complex. In agreement with Eq. 3.32, this is a result of the 3 q ¼ 4:6  larger charge and the smaller ionic radius of iron(III): r Fe33 þþ ¼ Fe 65 pm 2 q 102 pm1 in comparison with r Fe22 þþ ¼ ¼ 2:6  102 pm1 . Under anaerobic Fe 78 pm conditions, e.g., in groundwater, iron is present in the oxidation state +2. In an aerobic environment iron(II) is oxidized to iron(III). This is precisely what happened about 3–4 billion years ago, when the first prokaryotic and later eukaryotic organisms started to produce oxygen by photosynthesis in the oceans. This oxygen was scavenged by the process of iron(II) oxidation: about 58% of all oxygen ever produced on Earth by photosynthesis, and about 38% was scavenged by the oxidation of sulfide and sulfur. Of course, other elements, like manganese, were also oxidized, but due to their lesser abundance, they did not play as important a role as iron. The early scavenging of oxygen in the oceans is the reason for banded iron formations ðFe2 O3 Þ in rocks—with an age of between 1 and 3.5 billion years. The initial concentration of iron(II) in the oceans, before the commencement of photosynthesis, was about 107 mol L1 Fe2 þ ; while the iron(III) concentration was practically zero. The present concentration of iron(III) in the oceans is roughly 1019 mol L1 Fe3 þ ; while iron(II) is now practically zero [27]! The process of iron (II) oxidation to iron(III), and the follow-up condensation reactions of the hydroxo complexes, can still be observed when iron-rich groundwater is pumped to the surface: the initially clear and colorless groundwater quickly (within hours) attains a yellowish-brown color and becomes increasingly turbid, finally exhibiting a yellow–brown precipitate that is iron(III) oxide hydrate. The oxidation of other iron(II) compounds can also lead to the formation of iron (III) oxide hydrates and finally to goethite and other iron(III) minerals. Figure 3.25 shows a goethite (a-FeOOH) nodule from the shore of the Baltic Sea at Rügen island, where it formed from a pyrite ðFeS2 Þ nodule. Pyrite nodules are present in the chalk cliffs on the island, and when they are released from the chalk, by weathering, they come into contact with the ambient air and the Fe2 þ and S2 2 ions are oxidized—the iron finally forming goethite and the disulfide producing sulfuric

3.2 Quantification of Acid and Base Strength …

47

Fig. 3.25 Goethite nodule formed by the oxidation and hydrolysis of a pyrite nodule at the shore of the Baltic Sea at Rügen island, where pyrite nodules are released by weathering from chalk cliffs

Fig. 3.26 Little goethite nodules (diameters ranging between 1 and 6 mm) formed by goethite precipitation around grains of sand in lake Schwielowsee near Potsdam (Germany), and collected at the beach

acid, which is washed away. Similar, but normally more compact and layered, goethite nodules frequently form when iron(II)-containing groundwater creeps up in the capillaries of the soil, especially in arid areas. The goethite then precipitates around pebbles (Fig. 3.26). Larger goethite nodules with such origins form the so-called bog iron ore or morass ore, which in Europe once served as an early source for iron production. The acid–base chemistry of higher charged metal cations When the charge of metal ions is 4+, 5+, 6+, 7+, or higher, the stability of water molecules in the coordination sphere decreases so much that finally they cannot keep their protons, with only O2 remaining as a ligand. Examples are

48

3

Acid–Base Equilibria

Fig. 3.27 Predominance ranges of the ligands H2 O, OH , and O2 of metal ions, depending on the charge qMen þ of the metal ion and pH. As examples, the complexes of chromium(III) and chromium(VI) are given (adapted from [28]). Cr(OH)3  aq represents sparingly soluble chromium (III)-oxide hydrate

VII

VI

VI

VI

2 2 2 permanganate Mn O 4 , chromate Cr O4 , molybdate Mo O4 , tungstate W O4 , in addition  to otheranions. For some of these ions, protonated forms are also known, VI

e.g., O3 Cr OH



(hydrogen chromate, or according to IUPAC: hydroxy-oxidoVI

dioxochromium); however, the acid H2 Cr O4 (chromic acid), better written as VI

O2 CrðOHÞ2 , in its first protolysis:   VI VI O2 CrðOHÞ2 þ H2 O  O3 Cr OH þ H3 O þ

ðEquilibrium 3:29Þ 

VI



is so strong an acid that in aqueous solutions only the ion O3 Cr OH can exist.   VI In acidic solutions the hydrogen chromate ions O3 Cr OH condense and form (similar to Equilibrium 3.27). Figure 3.27 depicts the dichromate ions Cr2 O2 7 predominance ranges of the ligands H2 O, OH , and O2 of metal ions, depending on the charge qMen þ of the metal ion and pH. Complexes of chromium(III) and chromium(VI) are given as examples. This, however, is only a rough description of predominance ranges—the exact dependence requires consideration of both the radii and the possible covalent contributions to the bonds. The protolysis and condensation reactions of many metal aqua complexes results in the formation of well-defined isopoly cations, e.g., the paratungstate ion ½H2 W12 O42 10 (Fig. 3.28). When anions like silicate, phosphate, or arsenate are simultaneously present during the protolysis and condensation reactions, heteropoly

3.2 Quantification of Acid and Base Strength …

49

Fig. 3.28 Structure of the paratungstate anion ½H2 W12 O42 10 . The protons are depicted as red balls. The blue octahedrals are WO6 units which are bridged via their edges and corners [30]

anions can be formed, e.g., dodeka molybdato phosphate ½PO4 Mo12 O36 3 . For details of isopoly and heteropoly compounds see [29] and inorganic chemistry textbooks. Kinetic features of the reactions of metal aqua complexes As for all proton-transfer reactions in aqueous solutions, those of metal aqua complexes are also very fast. However, the follow-up condensation reactions may be slow or even very slow, as to render them rather irreversible. The reactions of metal ions of oxidation state +3, especially the reactions of chromium(III) ions, exhibit a very slow exchange of coordinated water molecules, and once formed it is extremely difficult to hydrolyze the condensation products (isopoly ions) back to monomeric hexaaqua metal cations. For the preparation of very pure chromium(III) hexaaqua ion solutions, the following is recommended: reduce dichromate in perchlorate acid with the help of oxalic acid, always maintaining a strongly acidic medium—this is necessary, as the dissolution of any solid chromium(III) salts in water will inevitably lead to the formation of oligomeric complexes (condensation products). The disposition to quickly exchange ligands is called lability: lable complexes are those with a high rate of ligand exchange (the self-exchange of aqua complexes in water). Inert complexes are those with a very slow exchange rate. Chromium(II) and chromium(III) complexes are most different with respect to lability; a chromium(II) complex can be called labile, while a chromium(III) complexe is inert: the exchange rate of water as a ligand is 1014 times faster for chromium(II) than for chromium(III) [31]! Implications of the acid–base chemistry of metal ions for analytical and environmental chemistry and for toxicology The protolysis and condensation of metal aqua ions has far reaching implications for analysis: with exception of alkali and higher alkali earth metal solutions any dilution of metal ion solutions may lead to precipitations of metal oxide hydrates, which may

50

3

Acid–Base Equilibria

be practically invisible at very low concentrations. In very diluted solutions colloids (nanoparticles) may form, which can attach to the walls of vessels, or may not be available for reactions (in the case of irreversibly formed precipitates/colloids). In titrations of Sb3 þ and Bi3 þ solutions, it is highly important to keep solutions strongly acidic, otherwise colorless (white) precipitates of respective oxide hydrates form, which can no longer react with bromate ions for oxidation to metal(V) ions. In the case of bismuth(III) oxide hydrate, even attempts to dissolve the precipitate afterward in acid will fail because of the irreversibility of condensation reactions. Clearly, the formation of condensation products may strongly affect the uptake of metal ions by organisms—usually retarding the uptake.

3.2.5 Principles Governing the Strength of Organic Acids and Bases Many organic compounds possess acidic or basic groups. A comprehensive discussion of all functional groups is beyond the scope of this book and for this, textbooks on organic chemistry should be consulted. Here we only discuss some of the general principles governing the strength of some common organic acids and bases: for an assessment of acidity, the bond strength keeping the protons is key, and that depends to a great extend on inductive and resonance effects. Alcohols are derivatives of water, in which one proton is substituted by an alkyl group. If the organic remainder is an alkene or alkine, the resulting compounds are called alkenols and alkynols, respectively. When the OH group is directly bonded to an sp2-hybridized carbon atom, the compound is an enol, existing in a keto-enol tautomery equilibrium with the corresponding ketone. Generally, alcohols are weak acids in water (Table 3.5) and their strength decreases with growing alkyl chain length, which is a result of the inductive effect of the alkyl group. Branching of the alkyl chains further decreases acidity by steric hindrance. On the other hand, electron-withdrawing groups near to the OH functionality increase acidity. This inductive effect increases with the number of electron-withdrawing groups, and it decreases with the distance to the OH group. The electron pairs at the oxygen of the Table 3.5 pKa data for selected organic compounds

Compound

pKa

CH3OH CH3CH2OH (CH3)2CHOH CF3CH2OH HCOOH CH3COOH CH3CH2COOH Cl3CCOOH F3CCOOH

15.5 15.9 17.1 12.4 3.75 4.75 4.85 0.65 0.23

3.2 Quantification of Acid and Base Strength …

51

Scheme 3.1 Resonance stabilization of the carboxylate anion

OH group also make alcohols basic; however, the base strength is very low. The pKa values of corresponding oxonium ions are below zero. The rather acidic character of carboxylic acids is due to the inductive effect of the carbonyl moiety; however, the resonance stabilization of the carboxylate anion is even more important for acidity (Scheme 3.1). Also in the case of carboxylic acids the positive inductive effect of the alkyl chain leads to a decrease of acidity with growing chain length (see the data for formic, acetic, and propionic acids provided in Table 3.5), and strongly electronegative substituents increase their acidity. In Fig. 3.29 the pKa data of monocarboxylic and dicarboxylic acids (here only the pKa1 data) are plotted versus the molecular mass, in order to demonstrate the trend with growing chain length. When an organic compound has more than one carboxylic group, the difference between the pKa data depends on the distance between these groups in the molecule. The further they are separated, the closer the pKa data (cf. Fig. 3.30).

Fig. 3.29 pKa data of monocarboxylic (black dots) and dicarboxylic acids (here only the pKa1 data is provided) as function of molecular mass: C4 represents butyric acid (pKa = 4.82) and isobutyric acid (pKa = 4.86) and C5 represents isovaleric acid (pKa = 4.76), 2-methylbutyric acid (pKa = 4.79), valeric acid (pKa = 4.84), and pivalic acid (pKa = 5.02)

8 7 6 C2

C3 C4

C5 C6 C7

C8 C9

5 C1

pKa1 4

C4

C8 C9 C5 C6 C7

3 C3

2 1

C2

0 0

50

100

150

molecular weight [g mol−1]

200

52

3 6

Acid–Base Equilibria 6

pKa2 5

5

pKa1

4

pKa

4

3

3

2

2

ΔpKa

ΔpKa 1

1

0

2

4

nCH

6

8

2

Fig. 3.30 pKa1 and pKa2 data for dicarboxylic acids, and their differences ΔpKa as function of the number of CH2 groups between the carboxylic groups

3.2.6 Non-aqueous Solvents The acid–base theory of Brønsted and Lowry (acids = proton donators, bases = proton acceptors) is easily applicable to non-aqueous protic solvents, because acid–base reactions are simply proton-transfer reactions. The problem, however, is the quantification of acid and base strength, as that is always coupled with the reaction partner, i.e., in water it is water, but in other solvents the solvent needs to be given consideration (for water, see Eq. 3.1). By analogy to water, it is possible to write down the autoprotolysis of water-like solvents, e.g., of liquid ammonia and anhydrous acetic acid (glacial acetic acid): 2NH3  NH4þ þ NH 2

ðEquilibrium 3:30Þ

2CH3 COOH  CH3 COOH2þ þ CH3 COO

ðEquilibrium 3:31Þ

As mentioned earlier, the cations resulting from autoprotolysis are called lyonium ions, and the anions are called lyate ions. When an acid HB or a base B is dissolved in such an amphiprotic solvent LH, the following equilibria are established:

3.2 Quantification of Acid and Base Strength …

53

HB þ LH  B þ LH2þ

ðEquilibrium 3:32Þ

B þ LH  HB þ L

ðEquilibrium 3:33Þ

When these equilibria are far on the right-hand side, the main products are either lyonium ions (Equilibrium 3.32) or lyate ions (Equilibrium 3.33). As in the case of water, this is the result of the so-called leveling effect of the solvent. The lyonium and lyate ions are the strongest acid and base in the respective solvents. Although even stronger acids and bases may in principle exist in the solvent, they would only do so in much lower concentrations than lyonium and lyate ions, respectively. For quantifying the term acidity it is important to know that acidity can be defined in two ways: 1. The acidity of a dissolved compound (the acid) toward the solvent: in the case of Brønsted acids, this is the strength to donate protons to the solvent. In aqueous solutions this strength is quantified by the equilibrium constant Ka ðpKa Þ. 2. The acidity of the solvent (or a solution) toward the dissolved compound: this term is limited to cases of solutions containing a Brønsted acid, and it quantifies the tendency of the solution to protonate a reference base. Let us first discuss (2), i.e., the acidity of a solution. In aqueous solution the measure of acidity is the pH, since it is a measure of the activity of hydronium ions H3 O þ . To apply this approach to non-aqueous solutions Louis P. Hammett (1894– 1987, a US physical chemist) proposed a function, which now bears his name, the Hammett acidity function [32, 33]. To define this function, the degree of protonation of an indicator base, or the dissociation of the corresponding indicator acid, respectively, is determined. Hammett defined a so-called “simple” indicator base B as a non-ionized (neutral) compound, which can accept one proton per molecule without undergoing follow-up reactions. The extent of protonation can be based on the equilibrium: BH þ  B þ H þ

ðEquilibrium 3:34Þ

described as follows: Ka;BH þ ¼

aH þ aB cB fB ¼ aH þ þ aBH cBH þ fBH þ

ð3:33Þ

According to Hammett the acidity function H0 is then defined as:  H0  log aH þ

fB fBH þ

¼ pKa;BH þ þ log

cB cBH þ

ð3:34Þ

54

3

Acid–Base Equilibria

The ratio c cBþ can be experimentally determined with the help of spectrophoBH tometry, potentiometry, or conductometry. The assumption made is that the pKa;BH þ values (i.e., the pKa values for BH þ ) are the same in different solutions. This is not always the case, but for a series of similar solutions, e.g., solutions of sulfuric acid with different concentrations, this assumption is fulfilled to a reasonable extent. Another assumption is that various bases have, in the same solution, the same ratio of activity coefficients f fBþ . In dilute aqueous solutions H0 approaBH ches the pH. In 2010 Krossing et al. [34] proposed a generalized Brønsted acidity scale, based on the absolute chemical potential of protons in any medium. The absolute chemical standard potential of the reference state, a proton gas at 1 bar and 298.15 K, of protons for the absolute (maximal) acidity of protons in the gas phase (g) l—abs ðH þ ; gÞ was arbitrarily set to 0 kJ mol−1. Due to the interaction of protons with molecules of a certain phase, whether gaseous, liquid, or solid, the acidity and hence the chemical standard potential of the gaseous proton (H+, g) is lowered. This decrease in a solution (s: solvens) related to the gas phase is given by the standard solvation Gibbs energy Dsolv G— ðH þ ; sÞ (proton transfer from the ideal proton gas at 1 bar and 298.15 K to a infinitely diluted proton solution with the same properties, but having a molar ratio activity ax;H þ ¼ 1 , i.e., pH = 0). Using a model, these data have been calculated for various solvents. For water the standard solvation Gibbs energy is Dsolv G— ðH þ ; H2 OÞ ¼ ð1105  8Þ kJ mol1 [35]. A pH change of one unit equals a change of chemical potential of 5.71 kJ mol−1 (RTD ln aH þ ¼ 2:303 RT log 0:1 ¼ 5:71 kJ mol1 ). Now it is possible to give the chemical potential of solvated protons labs ðH þ ; solvÞ for any proton activity in any solvent:

 — labs ðH þ ; solvÞ ¼ Dsolv G ðH þ ; sÞ  pH  5:71 kJ mol1

ð3:35Þ

To do this it is only necessary to know the value of Dsolv G— ðH þ ; sÞ for respective solvent s (Table 3.6). In this way it is possible to compare the absolute acidities of solutions.

Table 3.6 Free standard solvatation Gibbs energy for different solvents, according to [32]

Solvent

Dsolv G— ðH þ ,sÞ/kJ mol−1

Benzene Dichloromethane Sulfur dioxide Hydrogen fluoride Sulfuric acid Acetonitrile Water DMSO

−816 −835 −898 −908 −966 −1056 −1105 −1120

3.2 Quantification of Acid and Base Strength …

55

Let us now discuss the acidity of a dissolved compound (acid) toward a solvent (case 1). For titrations it is important to know whether the relations of acid and base strength remain the same, and by what effects they are affected. For chemicaly related groups of compounds, it has been observed that the acid and base strength is shifted by roughly constant increments in relation to data where water is the solvent [36]. Deviations from this rule find their explanation in effects of dielectric constant, possible hydrogen bonds, and steric effects. The dielectric constant affects, more seriously, acids that are neutral or negatively charged, than those that are positively charged. The ions formed by the protolysis of a neutral acid (Equilibrium 3.36) are better solvated in a solvent with a higher dielectric constant than one with a lower dielectric constant. HB þ þ S  B þ SH þ

ðEquilibrium 3:35Þ

HB þ S  B þ SH þ

ðEquilibrium 3:36Þ

HB þ S  B2 þ SH þ

ðEquilibrium 3:37Þ

From this it follows that the pKa of a carboxylic acid is larger in an alcohol than in water. In case of a carboxylic acid like succinic acid, the two carboxyl groups can only be titrated in water together, because pKa1 ¼ 4:16 and pKa2 ¼ 5:61 are very close to one another. In a solvent possessing a lower dielectric constant, the acidity of negatively charged acid decreases more than that of neutral acid, giving rise to two well-developed and separated equivalence points in the titration. In the case of positively charged acids (Equilibrium 3.35), e.g., ammonium ions, the extent of protolysis depends mainly on the basicity of the solvent, and is almost independent of its dielectric constant. In weakly polar solvents, e.g., acetic acid, consideration should be given to the fact that ionization (proton transfer) happens, but that the ions form pairs which are poorly dissociated. An example is perchloric acid in glacial acetic acid: HClO4 þ CH3 COOH  CH3 COOH2þ þ ClO 4

ðEquilibrium 3:38Þ

The true acidity constant is: Ka;HClO4 ¼

aCH3 COOH2þ aClO4 aHClO4

ð3:36Þ

56

3

Acid–Base Equilibria

and
 þ  CH3 COOH2þ þ ClO 4  CH3 COOH2  ClO4

ðEquilibrium 3:39Þ

has the following ion pair formation constant: Kip ¼

a½CH3 COOH þ ClO 

ð3:37Þ

4

2

aCH3 COOH2þ aClO4

The apparent acidity constant of HClO4 follows, thus: Ka;HClO4 ;apparent ¼

aCH3 COOH2þ aClO4

aHClO4 þ a½CH3 COOH þ ClO  2

4

¼

Ka;HClO4 1 þ Ka;HClO4 Kip

ð3:38Þ

This case has similarities to HF solutions in water (Sect. 3.2.4). As in the case of HF, the experimentally accessible acidity constant is an apparent constant. When strong acids like HCl and HClO4 are dissolved in glacial acetic acid the protons are practically all transferred to the solvent, but the ion pairs
 ðH3 COOH2 Þ þ Cl are much more stable ðpKip;½ðH3 COOH2 Þ þ Cl  ¼ 8:55Þ than
 the ion pairs ðH3 COOH2 Þ þ ðClO4 Þ (with pKip;½ðH3 COOH2 Þ þ ðClO4 Þ  ¼ 4:87). This is also a kind of differentiating effect. In solvents having very low dielectric constant, associates may form, which can obscure the equivalence points of titrations. The specific effects of the solvents can be hydrogen bonding, electronic interactions (p-electron complexes), and steric effects.

3.3

The Mathematical and Graphical Description of Acid–Base Equilibria

For the mathematical and graphical description of acid–base equilibria, only the relevant laws of mass action and balance equations are needed (Chap. 2). The general first requirement is, however, the exact formulations of the chemical equilibria (reaction equations). An elegant way to describe acid–base equilibria is graphically using pH-logci coordinates. Such graphics are very handy for deriving mathematical approximations for pH calculations, for deriving titration curves, and for many other applications. Here, mathematical treatment and graphics are demonstrated for monobasic, dibasic, and tribasic acids.

3.3 The Mathematical and Graphical Description of Acid–Base Equilibria

57

3.3.1 Monobasic and Polybasic Acids Monobasic acids In the case of a monobasic acid, the following equilibrium must be formulated: HB a1

þ

H2 O b2



B b1

þ

H3 O þ a2

ðEquilibrium 3:3Þ

in addition to the autoprotolysis of water: HB a1

þ

H2 O b2



HO b1

þ

H3 O þ a2

ðEquilibrium 3:4Þ

The chemical equilibria have the following equilibrium constants: Ka ¼

ac;B ac;H3 O þ ac;HB ax;H2 O

ð3:4Þ

and Kw ¼ ac;H3 O þ ac;OH ¼ Ka Kb

ð3:10Þ

From Equilibrium 3.3 follows the balance equation of the overall concentration  CHB of the acid. This overall concentration is frequently called the “analytical concentration” because it is determined by titration (classical analysis):  ¼ cHB þ cB CHB

ð3:39Þ

To derive this equation it is sufficient to understand that the acid can be present only as HB or as B , i.e., as the sum of cHB and cB . Sometimes Equilibrium 3.3 is incorrectly understood. The only correct interpretation is as follows: if one molecule HB reacts with one molecule of H2 O, exactly one ion B and one ion H3 O þ are formed; however, how many molecules HB are reacting with, can only be calculated by using the LMA. If the pH of a solution of acid in water needs to be calculated, it is necessary to realize that both Equilibria 3.3 and 3.4 yield H3 O þ ions. In one equilibrium the number of H3 O þ ions equals the number of B ions, and in the other the number of

58

3

Acid–Base Equilibria

formed H3 O þ ions equals the number of OH ions. From this follows the balance equation: cH3 O þ ¼ cOH þ cB

ð3:40Þ

The last four equations can be combined and result in a cubic equation for cH3 O þ (we shall continue with using concentrations instead of activities): Kw ¼ cH3 O þ cOH ! cOH ¼

Kw cH3 O þ

Insertion into Eq. 3.40 gives: cH3 O þ ¼ Ka ¼

cB cH3 O þ c B c H 3 O þ ! cHB ¼ cHB Ka c B  ¼ cH

 CHB 3

Oþ

Ka

Kw þ c B cH3 O þ insertion in Eq: 3:39

insertion in Eq: 3:41

!

þ1

c2H3 O þ ¼Kw þ c2H3 O þ

ð3:41Þ

!

cH3 O þ ¼

 CHB ¼

Kw cH 3 O þ

þ

cB cH3 O þ þ c B Ka  CHB cH O þ 3 Ka þ 1

 CHB cH3 O þ cH

3O

þ

þ Ka

Ka

  cH 3 O þ þ K a cH 3 O þ þ K a  cH3 O þ ¼Kw þ CHB Ka Ka

c3H3 O þ þ Ka c2H3 O þ Ka

¼

Kw cH3 O þ þ Ka Kw  þ CHB cH3 O þ Ka

 c3H3 O þ þ Ka c2H3 O þ ¼ Kw cH3 O þ þ Ka Kw þ Ka CHB cH3 O þ

  c3H3 O þ þ Ka c2H3 O þ  Ka CHB þ Kw cH3 O þ  Ka Kw ¼ 0

ð3:42Þ

Polybasic acids In the case of polybasic acids, the number of variables (equilibrium concentrations) increases and thus so does the number of equations. Thus, di-, tri-, and tetrabasic acids result in quartic (fourth degree), fifth degree, and sixth degree equations for cH3 O þ . In Sect. 3.3.2 we demonstrate how approximate vales of pH can be derived from pH-logci diagrams for solutions of acids, bases, salts, and buffers, as well as

3.3 The Mathematical and Graphical Description of Acid–Base Equilibria

59

how approximation equations can be derived for these cases, in order to avoid solving cubic equations, Eq. 3.42 (or equations of an even higher degree).

3.3.2 pH-logci Diagrams A pH-logci diagram displays the concentrations of all chemical species of an acid– base system as a function of pH. In the case of a monobasic acid these are: HB, B , H2 O, H3 O þ , and OH . Because water is the main constituent of the solution, and its concentration is practically constant, there is no need to display its concentration in such diagrams (however, see the discussion of water further down). The activity of H3 O þ determines the pH of the solution (cf. Eq. 3.18). Equilibria 3.3 and 3.4 involve the species HB, B , H2 O, H3 O þ , and OH , i.e., it is not possible to change the activity (concentration) of one of them without also affecting the others. Therefore, it makes sense to display all species in one diagram. In aqueous solutions the pH range of 0–14 is of greatest importance because most solutions have a pH within that range. Only in very special cases does a pH exist below 0 or above 14. There is also a certain range of concentrations, which have relevance in most real cases: from 1 mol L−1 (upper limit) to 1.0  10−14 mol L−1 (lower limit). This translates on the logci axis to the range 0 to –14. This logci range and the pH range of 0–14 are situated in the 4th quadrant of the coordination system of Fig. 3.31—forming a regular square. Since all important cases can be treated in that part of the diagram, from here on only that square is displayed. From equation (definition) pH ¼  log cH3 O þ , a diagonal results with coordinates logci ¼ 0; pH ¼ 0 and logci ¼ 14; pH ¼ 14. For OH follows also a straight line (diagonal) when in Eq. 3.10 concentrations are used and Eq. 3.20 is rearranged: pOH   log cOH ¼ pKw  pH

ð3:43Þ

The results are shown in Fig. 3.32, provided they relate to pKw ¼ 14 for 25 °C. Figure 3.32 raises a question about the position of the line for H2 O, which is the corresponding base of H3 O þ and the corresponding acid of OH ! Without going into detail here (see the later discussion about HB), Fig. 3.33 provides an answer. In order to construct the diagram, the molar concentration of water in pure water is needed: from the density of water at 20 °C (0.998203 g cm−3) and the molar mass (18.0153 g mol−1), follows CH 2 O = 55.408 mol L−1 (i.e., log CH 2 O = 1.74). On a strictly thermodynamic basis it is not possible to calculate the pKa of H3 O þ and the pKb of OH [16]. However, using the molar concentration of water in water (log CH 2 O = 1.74), we arrive at Fig. 3.33. From this figure one can read the apparent pKa of H3 O þ (–1.74) and H2 O (15.74). Both data are based on the wrong assumption that the activity of water in water is equal to its molar concentration. Therefore, the figure is a rather formal presentation, just to show that the H3 O þ and OH lines correspond, of course, to a line for H2 O.

60

3

6

1st quadrant

2nd quadrant logci 4 2 -6

-4

-2

Acid–Base Equilibria

pH 0

2

4

6

8

10

12

14

-2 -4

3rd quadrant

-6

4th quadrant

-8 -10 -12 -14

Fig. 3.31 Coordination system of pH-logci diagrams: normally, only the 4th quadrant is displayed because the vast majority of real systems are situated within its range

pH 0

2

4

6

8

10

0 -2 -4

logci

-6 -8 -10 -12

OH−

H3O+

-14 Fig. 3.32 pH-logci diagram with the lines for H3 O þ and OH of water

12

14

3.3 The Mathematical and Graphical Description of Acid–Base Equilibria

H2O

2 -2

61

0

2

4

0

6 pH 8

10

12

14

16

-2 -4

logci

-6 -8 -10

OH -12

H3O+

-14 -16

Fig. 3.33 pH-logci of water using log CH 2 O ¼ 1:74 and the not completely accurate values pKa ðH3 O þ Þ ¼ 1:74 and pKa ðH2 OÞ ¼ 15:74

For displaying the lines of all other species, the following derivations must be made:

62

3

Acid–Base Equilibria

Equations 3.44 and 3.45 describe curves in the pH-logci diagrams which resemble hyperbolas. However, in the strict mathematical sense they are not hyperbolas. Regardless of this detail, they are here referred to as hyperbolas. They are normally called the HB and B lines, respectively. In the diagram the species HB and B should be written at the lines, and not cHB and cB , since in the diagram the logarithms of concentrations are plotted. Figure 3.34 depicts the HB and B lines for a special example. In Fig. 3.34 one can see that each hyperbola has two almost linear branches. Only around pH = pKa are the hyperbolas strongly bent. This allows plotting only the asymptotes of the hyperbolas. To derive the equations describing the asymptotes, it is necessary to understand what term in the sum ð1 þ 10pHpKa Þ of Eq. 3.44 and what term in the sum ð1 þ 10pH þ pKa Þ of Eq. 3.45 is dominating at pH values both smaller and larger than the pKa .

For the sake of simplicity, it is possible to plot only the asymptotes in the diagrams, described by Eqs. 3.46–3.49 in the case of monobasic acids (cf. Fig. 3.35). For constructing pH-logci diagrams for monobasic acids only two pieces of data  are needed: log CHB and pKa . At pH\pKa , the HB line is parallel to the pH axis  , and the B line is parallel to the OH line. At crossing the coordinate log CHB   , pH [ pKa , the B line is parallel to the pH axis crossing the coordinate log CHB þ and the HB line is parallel to the H3 O line crossing the coordinate pH ¼ pKa .

3.3 The Mathematical and Graphical Description of Acid–Base Equilibria HB

pH = pKa

0

2

4

63

pH 6

8

10

12

14

0 logc = logC i

O HB

-2 -4

B

-6

logci

HB

-8 -10 -12

H3O+

OH -14

Fig. 3.34 pH-logci diagram with the H3 O þ and OH lines of water and the hyperbolas of HB  = –1 and pKa = 4.75) and B , for the example of acetic acid (with log CHB HB

pH = pKa

0

2

4

pH 6

8

10

12

14

0 logc = logC i

O HB

-2 -4

logci

-6

B

HB

-8 -10 -12

OH -14

H3O+

Fig. 3.35 pH-logci diagram with the H3 O þ and OH lines of water and the asymptotes of HB  = –1 and pKa = 4.75) and B , for the example of acetic acid (with log CHB

64

3 H B

pH = pKa12

0

2

4

pH

Acid–Base Equilibria

H B

pH = pKa22

6

8

10

12

14

0 logc = logC i

O H 2B

-2 -4

logci

-6

HB B2

H2B

-8 -10

H3O+

-12

OH -14

Fig. 3.36 pH-logci diagram with the H3 O þ and OH lines of water, the asymptotes of all forms of H2 B, the case of log CH 2 B = –1, and pKa1 = 3 and pKa2 = 8

In pH-logci diagrams for polybasic acids, the slopes of the lines have some specificities, as one can see in Figs. 3.36, 3.37, and 3.38. The mathematics behind this is described in detail in [37], and not given here.

3.3.3 pH Calculations Using Approximation Equations Since the solution of the cubic equation:

  c3H3 O þ þ Ka c2H3 O þ  Ka CHB þ Kw cH3 O þ  Ka Kw ¼ 0

ð3:42Þ

is no longer a problem, one may argue that the derivation of approximation equations has little meaning. However, in many cases it is good to know such equations, and to understand the approximations behind them. Thus, the derivation of an approximation equation has a high didactic value. All necessary approximations relate to what term can be neglected in the two balance equations:  CHB ¼ cHB þ cB

ð3:39Þ

3.3 The Mathematical and Graphical Description of Acid–Base Equilibria H B

pH = pKa13 pH = pKH3B a2

0

2

4

6

pH

8

65 H B

pH = pKa33

10

12

14

0 logc = logC i

O H 3B

-2 -4 -6

logci

H2B

HB2

-8 -10

H3O+ OH

-12

B3

H3B

-14

Fig. 3.37 pH-logci diagram with the H3 O þ and OH lines of water, the asymptotes of all forms of H3 B, the case of log CH 3 B = –1, and pKa1 = 2, pKa2 = 5, and pKa3 = 12

H B

pH = pKa14

0

2

H B

H B pH = pKa24 pH = pKa34 4 6 8

pH 10

H B

pH = pKa44 12 14

0 logc = logC i

O H 4B

-2 -4

logci

H2B2

-6

H3B

-8

HB

3

-10

H3O+

-12 -14

OH

B4

H4B

Fig. 3.38 pH-logci diagram with the H3 O þ and OH lines of water, the asymptotes of all forms of H4 B, the case of log CH 4 B = –1, and pKa1 = 2, pKa2 = 4, pKa3 = 7, and pKa4 = 12

66

3

Acid–Base Equilibria

cH3 O þ ¼ cOH þ cB

ð3:40Þ

To resolve this, pH-logci diagrams are very useful. Here it is very important to understand that in pH-logci diagrams, the two acid–base systems are plotted independent of one another. Figure 3.39 shows the pH-logci diagram of a solution containing 0.1 mol L−1 HCl. The pKa of hydrochloric acid is −7. To illustrate what this value means, a pH range is illustrated to –10. Clearly, the concentration of HCl, i.e., of the acid, is extremely low in the range of pH 0–14. This allows neglecting the term cHB in Eq. 3.39. Thus, a decision is required for Eq. 3.40, regarding how the terms cOH and cB relate to one another. This question can also be answered using a pH-logci diagram: initially, looking at the contribution of hydrochloric acid, it is obvious that  cB is practically equal to CHB = 0.1 mol L−1. Next, it is necessary to check what additional contribution may come from the autoprotolysis of water, i.e., to check in Eq. 3.40, whether cOH may make a contribution comparable to that made by cB . Pure water has a hydroxide ion concentration of 10−7 mol L−1, and the hydrochloric acid alone provides a hydronium ion concentration of 10−1 mol L−1, i.e., the pH will be near to 1. Figure 3.39 shows that at pH = 1, cOH is 10−13 mol L−1— completely negligible compared with 10−1 mol L−1. This discussion makes it clear  that cH3 O þ ¼ CHB is an excellent approximation, i.e.:  pH ¼  log CHB

-10

-8

-6

-4

-2

0

2

4

ð3:50Þ

6

pH

8

10

12

14

P1

Cl

-2

HCl

-4 -6

logci

-8

P2

-10 -12 P 3 -14

OH

H3O

Fig. 3.39 pH-logci diagram of a solution containing 0.1 mol L−1 HCl (pKa = −7). Values of pH < 0 are only shown for didactic reasons—normally they are not displayed

3.3 The Mathematical and Graphical Description of Acid–Base Equilibria

-10

-8

-6

-4

-2

0

2

4

6

pH

8

67

10

12

14

-2

HCl

logci

-4 P1

-6 -8

Cl

P2

-10 -12

H3O+

OH

-14 Fig. 3.40 pH-logci diagram of a solution containing 10−6 mol L−1 HCl (pKa = −7)

However, if the concentration of hydrochloric acid is much lower, e.g., 10−6 mol L , it still applies that cHB is negligible compared with cB , but the contribution of water cOH is now comparable with the cH3 O þ contribution of hydrochloric acid (cf. Fig. 3.40). Rearranging the equation for the autoprotolysis of water as: −1

cOH ¼

Kw cH3 O þ

and inserting into Eq. 3.40 yields: cH3 O þ ¼

Kw cH3 O þ

þ c B

 Since cB is practically equal to CHB it follows that:

cH3 O þ ¼

Kw cH 3 O þ

 þ CHB

rearrangement gives:  cH3 O þ c2H3 O þ ¼ Kw þ CHB

68

3

Acid–Base Equilibria

 0 ¼ c2H3 O þ  CHB cH3 O þ  Kw

with the solution: cH 3 O þ

C ¼ HB  2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Þ2 ðCHB þ Kw 4

ð3:51Þ

The only real solution is the one with the positive square root term. Finally another approximation equation is derived: Fig. 3.41 shows the pH-logci diagram of a solution containing 10−1 mol L−1 ammonium chloride (pKa;NH4þ = 9.25). Since the chloride ions are an extremely weak base, they do not affect the pH, and only the protolysis of the ammonium ions needs to be considered. The ammonium ions produce as much H3 O þ as NH3 (P1 in Fig. 3.41), and water produces as much H3 O þ as OH (P2 in Fig. 3.41). Comparing cH3 O þ at P1 and P2 shows that the contribution of water to the overall concentration of cH3 O þ is negligible. A comparison of the concentration of NH4þ (P3) with that of NH3 (P1)

pH 0

2

4

6

8

10

12

14

0

P3

-2 -4

logci

-6

P1

NH+4

NH3

-8

P2

-10 -12

OH -14

H3O+

Fig. 3.41 pH-logci diagram of a solution containing 10−1 mol L−1 ammonium chloride (pKa;NH4þ = 9.25)

3.3 The Mathematical and Graphical Description of Acid–Base Equilibria

69

 proves that it is a good approximation to write cNH4þ  Cammonium chloride , or gen  þ erally, cHB ¼ CHB and cB ¼ cH3 O . As a consequence of this:

Ka ¼

c2H O þ cB cH3 O þ ¼ 3 cHB CHB

cH3 O þ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Ka CHB

ð3:52Þ

There are several other special cases, for which approximate equations make sense. Since these cases and equations have been presented in detail in [37] it is superfluous to repeat them here. In some cases it is not easy to decide what contribution to the overall concentration of the acid, i.e., cHB or cB , and what contribution to the overall concentration of H3 O þ , i.e., cB or cOH , can be neglected. For rough estimates, one may decide to neglect contributions below 1%; however, this is rather arbitrary. It is better to decide the acceptable error range for approximate solutions when compared with cubic equation solutions. This can be nicely  displayed in a diagram using a coordinate system of log CHB versus pKa . In such a diagram, areas can be designated to certain approximate equations, with the borders of the areas depending on accepted errors (cf. Fig. 3.42).

3.3.4 Calculating the pH of Salt Solutions Many salts dissolve in water and are completely dissociated in ions. The resulting solutions can be acidic, neutral, or basic, depending on the nature of the cations and anions. A solution of sodium chloride is neutral because the hydrated sodium ions are an extremely weak acid (practically no acid) and the chloride ions are an extremely weak base (practically no base). In contrast, a solution of ammonium chloride is acidic because the ammonium ions are acidic (Sect. 3.3.3). The pH of such a solution can be graphically assessed using the pH-logci diagram depicted in Fig. 3.41. In other cases, the cation may be an extremely weak acid (“no acid”), but the anion may be a strong enough base that it is to some extent protonated, with the solution becoming basic. An example is sodium acetate—the acetate ions react with water and form some acetic acid and OH ions. The pH coordinate of P1 in Fig. 3.43 gives the approximate pH of the solution. In addition, in the case that both ions undergo protolysis reactions, the pH of the salt solution can be derived from a pH-logci diagram, when both acid–base systems are included. Ammonium formiate can serve as an example: the ammonium ions are a weak acid, and the formiate ions are a weak base. In a solution containing 0.1 mol L−1 ammonium formiate, both acid–base systems, i.e., ammonium/ammonia (pKa;NH4þ = 9.25) and formic acid/formiate (pKa;formic acid = 3.75), are present in the same overall concentration. Figure 3.44 depicts the pH-logci diagram.

70

3

Fig. 3.42 Validity areas of approximate equations for pH calculations of solutions of monobasic acids. Within the fields, the error made using the specified approximate equation is smaller than 0.2 pH units

Acid–Base Equilibria

-10 °

°

-8 -6

°

logC oHB

°

-4 -2 o

0

0

2

4

6

8

10

12

8

10

12

14

pKa -10 -8

logCoHB

-6 o

-4 -2 0

0

2

4

6

14

pKa In aqueous solution, the following equilibria are operative: NH4þ þ H2 O  NH3 þ H3 O þ

ðEquilibrium 3:40Þ

HCOO þ H3 O þ  HCOOH þ H2 O

ðEquilibrium 3:41Þ

The overall equilibrium can be written as follows: NH4þ þ HCOO  NH3 þ HCOOH

ðEquilibrium 3:42Þ

3.3 The Mathematical and Graphical Description of Acid–Base Equilibria

71

pH 0

2

4

6

8

10

12

14

0

P3

-2 -4 -6

logci

P1

Ac

HAc

-8

P2 -10 -12

H3O+

OH -14

Fig. 3.43 pH-logci diagram of a 10−1 M solution of sodium acetate (pKa;acetic acid = 4.75). The pH coordinate of P1 gives the approximate pH of the solution

0 logc = logC i

O

0

2

HCOOH

4

NH4+

pH

HCOOH

pH = pKa

6

pH = pKa

8

HCOO

NH4HCOO

10

12

14

NH3

-2 -4

logci

HCOO

NH+4

P1

-6

HCOOH

-8 -10

NH3

-12

OH -14

H3O+

Fig. 3.44 pH-logci diagram of a solution containing 0.1 mol L−1 ammonium formiate. The pH coordinate of P1 is the approximate pH of the solution

72

3

Acid–Base Equilibria

1.0

0.8

α HB

0.6

0.4 HNO3

0.2

HClO2 CH3COOH

0.0

HClO HCN

0

2

4

6

8

10

12

14

pH Fig. 3.45 The pH dependence of the degeree of protolysis aHB according to Eq. 3.54 for different acids. The following pKa data have been used: pKa;HNO3 ¼ 1:37; pKa;HClO2 ¼ 1:97; pKa;CH3 COOH ¼ 4:75; pKa;HClO ¼ 7:54; and pKa;HCN ¼ 9:4

This means that in this solution the equilibrium concentrations of ammonia and formic acid are equal, and the pH is given by the point P1 in Fig. 3.44, i.e., pH = 6.5. Certainly, this is only an approximate value, as the autoprotolysis of water is not considered. At much smaller salt concentrations, the pH will be fixed almost by the water alone, and the pH will approach 7.0, provided that no other acid–base systems, e.g., carbon dioxide equilibria, are present.

3.4

The Degree of Protolysis and Ostwald’s Law of Dilution

The degree of protolysis aHB (Greek letter alpha) of an acid HB relates to Equilibrium 3.3: HB a1

þ

H2 O b2



B b1

þ

H3 O þ a2

3.4 The Degree of Protolysis …

73

and indicates the ratio of the equilibrium concentration of the base B to the overall  concentration of the acid CHB ¼ cHB þ cB : aHB ¼

c B  CHB

ð3:53Þ

The degree of protolysis is often also called the degree of dissociation, which is not completely accurate, as dissociation means splitting molecules (or ionic solids) into ions by the process of solvation. Protolysis, however, is a chemical reaction with a protic solvent. Of course, the difference is just gradual, as solvation is also a chemical reaction, but one that does not involve a proton transfer. Two dependences of the degeree of protolysis aHB on chemical concentrations are important: 1. The dependence of the degree of protolysis aHB on solution pH, because it indicates what fraction of the acid is protolysed at a certain pH. 2. The dependence of the degree of protolysis aHB on the overall concentration of the acid CHB , because it allows the experimental determination of the acidity constant Ka of the acid HB, when the solution contains only the acid and the pH is not affected by other solution constituents but is only the result of Equilibrium 3.3. It was Wilhelm Ostwald who understood that with the help of conductometry it is possible to determine the degree of protolysis and thus determine the acidity constants of acids. Details concerning (1). The dependence of the degree of protolysis aHB on pH can be derived as follows: Ka ¼

c B c H 3 O þ cHB

ð3:6Þ

Rearrangement gives:   cH3 O þ cHB CHB  cB CHB 1 ¼ ¼ ¼ 1¼ 1 aHB Ka cB c B c B

and then aHB ¼ cH

3O

Ka

1 þ

þ1

ð3:54Þ

74

3

Fig. 3.46 The dependence of the logarithm of the degeree of protolysis aHB according to Eq. 3.55 for different acids. The following pKa data have been used: pKa;HNO3 ¼ 1:37; pKa;HClO2 ¼ 1:97; pKa;CH3 COOH ¼ 4:75; pKa;HClO ¼ 7:54; and pKa;HCN ¼ 9:4

Acid–Base Equilibria

0

-2

logα HB -4

HNO3 HClO2

-6

CH3COOH HClO HCN

-8

-10 0

2

4

6

8

10

12

14

pH

log aHB

  pH

 cH 3 O þ 10 ¼  log þ 1 ¼  log þ 1 ¼  log 10pKa pH þ 1 Ka 10pKa ð3:55Þ c

þ

The case of 10pKa pH 1 (which is equal to the condition HK3 Oa 1 and pH pKa ) follows the approximation: log aHB ¼  log 1 ¼ 0, i.e., aHB ¼ 1. c þ Whereas 10pKa pH 1 (which is equal to the condition HK3 Oa 1 and pH pKa ) follows the approximation: log aHB ¼ pH  pKa , i.e., aHB ¼ c pKa pH

1

Ka

H3 O þ

. Figure 3.45

depicts the function aHB ¼ f ðpHÞ ¼ ð10 þ 1Þ and Fig. 3.46 the function pKa pH þ 1Þ, for different acids. log aHB ¼ f ðpHÞ ¼  logð10 From Eq. 3.54 it follows that for pH ¼ pKa , i.e., for cH3 O þ ¼ Ka , the degeree of protolysis is aHB ¼ 0:5, i.e., log aHB  0:3.  Details concerning (2). The dependence of aHB on the overall concentration CHB : Solving Eq. 3.54 for the concentration of protons and inserting it in the cubic equation for calculating pH (Eq. 3.42) gives a cubic equation for the degeree of protolysis as a function of the overall acid concentration:

   a3HB þ Ka2  Ka CHB  KW a2HB  2Ka2 aHB þ Ka2 0 ¼ Ka CHB

ð3:56Þ

3.4 The Degree of Protolysis …

75

When the pH of the solution is solely determined by the acid HB, i.e., no contribution from the autoprotolysis of water, it follows from Equilibrium 3.3 that: c H 3 O þ ¼ c B

ð3:57Þ

With the balance equation for the acid HB (Eq. 3.39) the LMA can be written as follows: Ka ¼

c2B c2  a2

B  ¼ HB  ¼ 1aHB  cB C 2 1  cB2 HB C C C

 CHB

HB

ð3:58Þ

HB

HB

a2HB C Ka ¼ ð1  aHB Þ HB

This relation is called Ostwald’s law of dilution [38, 39], as a consequence of Wilhelm Ostwald being the first to derive this relation and use it for the determination of acidity constants following conductometric measurements of the degeree of protolysis. According to Eq. 3.58, the degree of protolysis increases with increasing dilution (decreasing overall concentration). At very high overall concentration, the degree of protolysis approaches zero. However, this rule has limited validity: for very strong acids, which are almost completely protolysed (lim aHB ¼ 1), the rule is not applicable. In addition, for very weak acids the degree of protolysis approaches almost constant values at very low concentrations, as can  be seen in Fig. 3.47. This is because at overall concentrations CHB smaller than

Fig. 3.47 Dependence of the degree of protolysis of an acid on its overall concentration. The curves were calculated using the cubic equation 3.56 for different acids. The following pKa data were used: pKa;HNO3 ¼ 1:37; pKa;HClO2 ¼ 1:97; pKa;CH3 COOH ¼ 4:75; pKa;HClO ¼ 7:54; and pKa;HCN ¼ 9:4

1.0

0.8

α HB

HNO3 HClO2

0.6

CH3COOH HClO HCN

0.4

0.2

0.0 0

2

4

logC

6 O HB

8

76 Fig. 3.48 The pH of aqueous solutions of acids depending on the overall concentration of acid. The data were calculated with Eq. 3.42. The following pKa data were used: pKa;HNO3 ¼ 1:37; pKa;HClO2 ¼ 1:97; pKa;CH3 COOH ¼ 4:75; pKa;HClO ¼ 7:54; and pKa;HCN ¼ 9:4

3

Acid–Base Equilibria

8

6

NH+4 HClO CH3COOH

pH 4

HCOOH HNO2

2

HNO3 0

0

2

4

6

10

8

logCOHB

10−7 mol L−1, the pH is determined by the autoprotolysis of water and is practically constant (pH = 7), as depicted in Fig. 3.48. As can be seen in Fig. 3.49, acids possessing a pKa of around 7 (or larger), and having a concentration of 10−7 mol L−1 (or smaller) at pH 7, exist mostly in the form HB and never attain a degeree of protolysis of 1. Figure 3.49 is a good example of the usefulness of pH-logci diagrams: it is easy  to read off good approximations of the concentrations CHB and cB , in order to Fig. 3.49 pH-logci diagram of a solution containing 10−9 mol L−1 ammonium chloride. The pH is practically 7, and at this pH the concentration of ammonia is cNH3 ¼ 5  1012 mol L1 , giving a degeree of protolysis of ammonium ions aHB = 0.005

pH 0

2

4

6

8

10

12

0 -2

H3O

OH

-4

logci

-6 -8 -10 -12 -14

NH4

NH3

14

3.4 The Degree of Protolysis …

77

calculate aHB . At higher concentrations of acid, and/or for stronger acids, it is first  necessary to find the pH of the solution before reading off concentrations CHB and cB , in order to calculate aHB .

3.5

Acid–Base Equilibria of Amino Acids

Because of the immense importance of amino acids in biochemistry (proteins), they are discussed here in more detail. Amino acids posses a carboxylic group COOH that can transfer a proton to water, and an amino group NH2 that can accept a proton because it has a free electron pair on the nitrogen atom. An a-amino acid H2 NCHðRÞCOOH can therefore exist in the following forms: (1) In a neutral form (only the carboxylate group is protonated): H2 NCHðRÞCOOH, abbreviated as HA. (2) As a cation (the amino group and carboxylate group are protonated): þ

H3 N CH(R)COOH, abbreviated as H2 A þ . (3) As an anion (carboxylic group deprotonated; amino group not protonated): H2 NCH(R)COO , abbreviated as A . þ

(4) As an inner salt, or zwitterion: H3 N CH(R)COO , abbreviated as HA . It is a specific feature of amino acids that the basicity of the nitrogen atom is larger than the basicity of the carboxylate group. This has the consequence that an inner salt is formed—as indicated in point (4). Figure 3.50 depicts all equilibria. The equilibria are as follows: þ

Ka1;micro

H3 N CH(R)COOH

þ

H2 O

ðH2 A þ

þ

H2 O

þ



H3 N CH(R)COO

þ

H3 O þ



HA

þ

H3 O þ Þ

Ka1;micro

ðEquilibrium 3:43Þ þ

Ka2;micro

H3 N CH(R)COO

þ

H2 O

ðHA

þ

H2 O



H2 N CH(R)COO

þ

H3 O þ



A

þ

H3 O þ Þ

Ka2;micro

ðEquilibrium 3:44Þ þ

Ka3;micro

H3 N CH(R)  COOH

þ

H2 O

ðH2 A þ

þ

H2 O



H2 N  CH(R)  COOH

þ

H3 O þ



HA

þ

H3 O þ Þ

Ka3;micro

ðEquilibrium 3:45Þ

78

3

Acid–Base Equilibria

Fig. 3.50 The four forms in which amino acids are present in aqueous solutions, together with the microscopic acidity constants Ka1;micro to Ka4;micro and the equilibrium constant Kz of the zwitterion formation Ka4;micro

H2 NCH(R)COOH

þ

H2 O

ðHA

þ

H2 O



H2 NCH(R)COO



A

Ka4;micro

þ

H3 O þ

þ H3 O þ Þ ðEquilibrium 3:46Þ

Glycine has the following data: pKa1;micro = 2.31; pKa2;micro = 9.62; pKa3;micro = 7.62; pKa4;micro = 4.31 These are related to the concentrations of the species according to: Ka1;micro ¼

cHA cH3 O þ cH 2 A þ

ð3:59Þ

Ka2;micro ¼

cA cH3 O þ cHA

ð3:60Þ

Ka3;micro ¼

cHA cH3 O þ cH 2 A þ

ð3:61Þ

Ka4;micro ¼

cA cH3 O þ cHA

ð3:62Þ

3.5 Acid–Base Equilibria of Amino Acids

The

numerical

values

of

þ

79

pKa1;micro

and

pKa4;micro

are

such

that

H3 N C(H2 ÞCOOH is a stronger acid than H2 NCHðRÞCOOH. This is typical for amino acids and it prompts the equilibrium: Kz



H2 NCH(R)COOH

þ

H3 N CH(R)COO

Kz

ðHA



ðEquilibrium 3:47Þ

HA Þ

to lay far on the right-hand side, because the equilibrium constant of the zwitterion formation Kz is connected with the acidity constants as follows: c Kz ¼

þ

H3 N CHðRÞCOO

cH2 NCHðRÞCOOH

¼

Ka4;micro Ka1;micro ¼ Ka2;micro Ka3;micro

ð3:63Þ

It is important to note, and very interesting to consider, that this equilibrium constant Kz is pH independent! In the case of glycine, the constant is: Kz;glycine ¼

Ka1;micro;glycine 102:31 ¼ ¼ 105:31 Ka3;micro;glycine 107:62

ð3:64Þ

Since, in the case of amino acids, the concentration of the neutral form HA is normally very small in comparison with the concentration of the zwitterionic form HA , it is common practice to consider only the forms H2 A þ ; HA , and A . The acidity constants Ka1;micro to Ka4;micro are called microscopic acidity constants, because they relate to the deprotonation of specific acidic groups. From the titration curves of the acid H2 A þ with the final product A , it is also possible to derive acidity constants (Chap. 7): these constants are called macroscopic acidity constants Ka1;macro and Ka2;macro . In the titration, the zwitterionic form and the neutral form do not distinguish from one another, and the following two reactions proceed simultaneously at the first titration stage: H2 A þ þ OH  HA þ H2 O

ðEquilibrium 3:48Þ

H2 A þ þ OH  HA þ H2 O

ðEquilibrium 3:49Þ

At the second titration step the following two reactions also proceed simultaneously: HA þ OH  A þ H2 O

ðEquilibrium 3:50Þ

80

3

HA þ OH  A þ H2 O

Acid–Base Equilibria

ðEquilibrium 3:51Þ

The two titration steps are governed by the following two LMAs [40, 41] : Ka1;macro ¼

ðcHA þ cHA ÞcH3 O þ cH2 A þ

ð3:65Þ

cA cH3 O þ ðcHA þ cHA Þ

ð3:66Þ

Ka2;macro ¼

These two equilibrium constants relate to the following equilibria:

 H2 A þ þ H2 O  HA x þ ðHAÞ1x þ H3 O þ

HA

 x

ðEquilibrium 3:52Þ

þ ðHAÞ1x þ H2 O  A þ H3 O þ

ðEquilibrium 3:53Þ

When the equilibrium constant of the zwitterion formation is very large, i.e., c  K ¼ Ka1;micro is at least 100 or larger, which means that cHA cHA , the when Kz ¼ cHA HA a3;micro following approximation equations are valid: c c cA  c þ þ Ka1;macro  HAc Hþ3 O and Ka2;macro  c H3O , i.e., Ka1;macro  Ka1;micro and H2 A

HA

Ka2;macro  Ka2;micro . The following exact relationships hold for all macroscopic and microscopic constants: -For Ka1;macro : using the relation cHA ¼

Ka3;micro cH A þ 2 cH O þ

¼

3

Ka3;micro cHA cH O þ 3 cH O þ Ka1;micro

stitution of cHA in Eq. 3.65, yields:

Ka1;macro ¼

cHA þ

Ka3;micro cHA Ka1;micro



¼

3

cH3 O þ

cH A þ  2 Ka3;micro ¼ Ka1;micro 1 þ Ka1;micro

c  cH O þ ¼ HA 3 cH 2 A þ

Ka3;micro cHA Ka1;micro

for the sub-

 Ka3;micro 1þ Ka1;micro

and thus: Ka1;macro ¼ Ka1;micro þ Ka3;micro -For Ka2;macro : using the relation cHA ¼ cHA in Eq. 3.66, yields:

cA cH O þ 3 Ka4;micro

¼

Ka2;micro cH O þ cHA 3 cH O þ Ka4;micro 3

¼

ð3:67Þ

Ka2;micro cHA Ka4;micro

for substituting

3.5 Acid–Base Equilibria of Amino Acids Fig. 3.51 Logarithm of fraction ai ¼ Cci for all four

2

species (i = HA, H2 A þ , A , HA ) of glycine as a function  is the of solution pH. Cglycine overall concentration of glycine

0

81

glycine

-2 -4

log

-6

H2A+

-8

HA HA+/A-

-10 -12 -14 0

Ka2;macro ¼

cA  cH 3 O þ ¼ ðcHA þ cHA Þ c

2

4

6

pH

8

10

12

14

cA cH3 O þ 1

 ¼ Ka2;micro  Ka2;micro K 1 þ Ka2;micro HA 1 þ Ka4;micro a4;micro

Ka2;macro ¼

Ka2;micro Ka4;micro Ka2;micro þ Ka4;micro

ð3:68Þ

Microscopic acidity constants can be determined with the help of spectroscopic techniques (NMR, UV, IR) when the single species are detectable. In the case of amino acids it is customery to designate constants Ka1;macro and Ka2;macro simply as Ka1 and Ka2 . Figure 3.51 depicts the logarithm of fraction ai ¼ Cci for all four species glycine

(i = HA, H2 A þ , A , HA ) of glycine as a function of solution pH and Fig. 3.52 shows a titration curve. Figure 3.51 shows how small the contribution of HA really is to the overall concentration. The pH at which the concentrations of HA and HA have a maximum is called the isoelectric point: pHIE ¼

pKa1 þ pKa2 2

ð3:69Þ

The isoelectric point is of special importance in the case of proteins, which have many acidic and basic groups, as at this point the protein is overall uncharged (neutral) and does not move in an electric field. This means that in electrophoreses along a pH gradient, it will stop at pH ¼ pHIE (isoelectric focusing).

82

3

Fig. 3.52 Titration curve of glycine hydrochloride (H2 A þ  form, Cglycine ¼ 0:1 mol L1 )  with OH

Acid–Base Equilibria

14 12 10

pH

8 6 4 2 0 0.00

3.6

0.25

0.50

0.75

1.00

1.25

1.50

Acid–Base Equilibria at the Surface of Solids

All the acids discussed so far exist in homogeneous solutions. However, there are also many solid compounds and materials, which possess at their surface chemical groups that can transfer protons to water (or other solvents or dissolved species) or accept protons from water (or other solvents or dissolved species). These groups are surface-bound acids or bases according to the Brønsted–Lowry theory of acids and bases. Good examples are hydroxides, oxides, oxide hydrates, clay minerals, glasses, and zeolithes. The acid–base equilibria on their surfaces are of great importance as they can explain for example the function of glass electrodes, the electroosmotic flow, the buffering of soil pH, and the sorption of metal ions on mineral surfaces. The degree of surface protonation also decides the surface charge, and thus also determines the ability to form stable colloidal dispersions. In order to understand the binding of metal ions on surfaces, knowledge of the surface acid– base equilibria is as similarily important as knowledge of homogeneous acid–base equilibria for understanding homogeneous complex formation. Of course, surface acid–base equilibria can also be regarded as a kind of complex formation in which protons play the role of metal ions. In technical catalysis Brønsted–Lowry acids on solid surfaces are responsible for hydrations, dehydrations, and dehydrochlorations, to name but a few examples. By designating an acidic surface group of hydroxides or oxides of silicon, iron, aluminum, etc., as SOH, many surface acid–base reactions can be described by the following equations:

3.6 Acid–Base Equilibria at the Surface of Solids

83

Fig. 3.53 Possible acidic and basic surface groups. S is a solid-bound surface ion at the solid|solution interface (according to [42])

SOH2þ þ H2 O  SOH þ H3 O þ

ðEquilibrium 3:54Þ

SOH þ H2 O  SO þ H3 O þ

ðEquilibrium 3:55Þ

These equilibria can be characterized by acidity constants:  Ka1 ¼

c SOH cH3 O þ c SOH2þ

ð3:70Þ

 Ka2 ¼

c SO cH3 O þ c SOH

ð3:71Þ

Here, c SOH2þ , c SOH , and c SO are the concentrations of the surface groups given in moles per kilogram of solid material, and cH3 O þ is the concentration of hydronium ions in the surrounding solution. These Ka values lack a strict thermodynamic definition; however, they are useful for practical purposes. The model using the two equilibria (Equilibria 3.54 and 3.55) is commonly referred to as the “two pKa model.” This model is not operative for all surface OH groups. There are also systems with other configurations (e.g., those depicted in Fig. 3.53). In the case of glass, silicate, quartz, and silica gel, only SiOH and SiO groups are on the surface in contact with aqueous solutions. A protonation of

SiOH has not been observed. Equilibria 3.54 and 3.55 differ from equilibria in homogeneous solutions in one important way: the proton transfer leads to a charging of the surface and thus to a change of the electrical double layer (for further consideration of this see electrochemistry textbooks). Systems having two Ka values possess, similar to amino acids and proteins, an isoelectric point pHIE at which the excess surface charge is zero: pHIE ¼

  pKS1 þ pKS2 2

ð3:72Þ

When a powder of the solid material suspended in water is titrated with an acid or base, titration curves result, which exhibit the equilibrium solution pH as function of the amount of acid or base added (Fig. 3.54). For a concentration of suspended solid material cM , in kilograms per liter of suspension, the following relation holds between the added concentrations of acid

84

3

Acid–Base Equilibria

10 9 8

pH

7 6 5 4 3 1.0

cHClO

0.5 4

0.0

0.5

1.0

cNaOH

1.5

[mmol L 1]

Fig. 3.54 Titration of a goethite (a-FeOOH) suspension with an acid and base. The suspension contained 6 g L−1 goethite with an overall concentration of f SOHoverall g groups of 2  10−4 mol g−1. The titration was performed in the absence of specifically adsorbing ions and in the presence of the inert electrolyte NaClO4 at a concentration of 0.1 mol L−1 (according to [42])

and base, the   solution pH, and the concentrations of the surface groups SOH2þ and f SO g, in moles per kilogram of solid, and thus with the surface charge Q, in moles of unit charge per kilogram of solid:   cacid  cbase þ cOH  cH3 O þ ¼ SOH2þ  f SO g ¼ Q m

ð3:73Þ

In Fig. 3.55 the function of solution pH on surface charge Q is plotted. This figure shows that the surface of the solid has zero charge at a certain pH of the solution. This pH is called the pH of the point of zero charge (PZC) and has the symbol pHpzc. For each pH it is possible to calculate a microscopic acidity constant Ka1;micro according to: Ka1;micro ¼

ðf SOHoverall g  QÞcH3 O þ Q

ð3:74Þ

3.6 Acid–Base Equilibria at the Surface of Solids Fig. 3.55 Dependence between solution pH and surface charge of goethite as calculated from the titration curve shown in Fig. 3.54 (according to [42])

85

10 9 8 7

pH 6 5 4 3 10

5

0

Q ·10

-5

5

mol g

-10

-15

1

for pH values smaller than pHpzc, and Ka2;micro ¼

QcH3 O þ ðQ  f SOHoverall gÞ

ð3:75Þ

for pH values larger than pHpzc. The existence of microscopic acidity constants pKa;micro , which depend on the pH, is explainable by the effect of surface charge on the state of equilibria. Plotting the pKa;micro data as a function of surface charge Fig. 3.56 Plot of the pKa;micro data as a function of surface charge, calculated for the example shown in Fig. 3.54 using Eqs. 3.74 and 3.75 (according to [42])

11 10 9

pKa 8 7 6 5 4 3 10

5

0

Q ·10

-10

-5 5

mol g

1

-15

86

3

Acid–Base Equilibria

gives two straight lines (Fig. 3.56), which can be extrapolated to a zero surface charge, giving the two intrinsic quantities pKa1 and pKa2 . A deeper discussion of the experimental methods for studying the acid–base equilibria of surface groups and the underlying theory is beyond the scope of this textbook—readers wishing to consider this further should consult specific literature [42].

3.7

Buffer Solutions

In many chemical systems it is important to keep the pH rather constant in order to have optimal reaction rates (kinetics) or optimal states of equilibrium (thermodynamics). Enzymatic reactions often require a small range of pH to achieve the highest reaction rates. In analytical chemistry, protolysis equilibria may affect the state of equilibrium of other reactions, e.g., of complex formations, solubility, and redox equilibria (Chaps. 4, 5, and 6), so a rather constant pH is necessary to have well-defined reaction conditions. To realize this, such reactions need solutions with a buffered pH. In living organisms, all intracellular and extracellular liquids have pH buffers to regulate biochemical reactions. As an example, human blood has a pH of 7.4, and the main buffer system of blood is carbonic acid/hydrocarbonate. Stronger deviations from this pH can induce irreparable damage or even death. An acid–base buffer is a solution possessing the ability to keep its pH rather constant when an acid or base is added or when it is simply diluted. This can be accomplished when the solution contains a medium-strong or weak acid and the corresponding base, or a medium-strong base and its corresponding acid. Then, the pH is fixed by the ratio of concentrations of the constituents of this corresponding acid–base pair, as can be easily seen when the LMA (Eq. 3.6) is rearranged as follows: log Ka ¼ log cH3 O þ þ log log cH3 O þ ¼ log Ka  log

cB cHB

c B cHB ð3:76Þ

c B cHB pH ¼ pKa þ log ¼ pKa  log cHB c B The latter form of these equations is called the buffer equation or Henderson– Hasselbalch equation.3 For reasons of simplicity, concentrations are used here instead of activities.

3

Lawrence Joseph Henderson (1878‒1942, US chemist and biologist) and Karl Albert Hasselbalch (1874‒1962, Danish physicist and chemist) derived this equation and were the first to apply it to practical medical problems.

3.7 Buffer Solutions

87

Since buffer solutions usually contain rather high concentrations of buffer constituents, deviations of experimentally measured and concentration-based calculated pH values can be easily found, as the activity coeffiecients may deviate from unity. As discussed in Sect. 3.4, medium-strong and weak acids and bases at high concentrations are present almost always in the non-protolyzed form, so that the overall concentrations can be taken as the equilibrium concentrations. When, e.g., 0.1 mol acetic acid and 0.1 mol sodium acetate are dissolved in water to produce 1 L of buffer solution, the resulting concentrations cHAc and cAc are both 0.1 mol L−1. According to Eq. 3.76 the pH should stay constant upon dilution, as the ratio remains constant. This, however, is only approximately true—not for large dilutions. The buffer equation also shows that the pH of the buffer solution will be equal to the pKa when the concentration ratio is ccBHB ¼ 1. Any change of the ratio ccBHB by one order of magnitude changes the pH by one unit. Increasing the base concentration relative to that of the acid makes the solution more basic (increases the pH), whereas an increase of acid concentration relative to that of the base makes the solution more acidic (decreases the pH). Adding a strong acid to the buffer solution means an equal amount of the buffer base is transformed to the buffer acid: H3 O þ þ B  HB þ H2 O

ðEquilibrium 3:56Þ

Analogously, when a strong base is added to the buffer solution, an equal amount of buffer acid is transformed to buffer base. When the added amounts of acid or base are small, the pH changes only slightly. In contrast to this, the same amounts of acid or base added to pure water would affect the pH to a very large extend. So-called buffer curves (also named buffer titration curves) can be experimentally obtained when a weak acid is titrated with a strong base (or a weak base with a strong acid) and the pH recorded in the course of titration (Chap. 7). At pH = pKa the slope of the curve has a minimum (turning point). These curves illustrate the fact that a useful buffer capacity is confined to the range 0:1  ccBHB  10, i.e., to the pH range pH ¼ pKa  1. The dependence of buffer  solution pH on the molar ratio of the buffer base, i.e., pH ¼ f ðxB Þ ¼ f

nB nHB þ nB

, is displayed in Fig. 3.57.

The effect of a buffer is quantified by its buffer capacity b, which is defined as: 

b¼

dnH3 O þ dnOH ¼ dpH dpH

ð3:77Þ

Here, dnOH and dnH3 O þ are the amounts of strong base or acid which prompt, upon addition to the buffer, a pH shift of dpH.4 The quantities dnOH and dnH3 O þ have the units of moles per liter; however, they are not concentrations. They would The infinitesimal changes dnH3 O þ and dpH can be substituted by finite changes D nH3 O þ and D pH, provided the additons are really small.

4

88

3 14

Acid–Base Equilibria

HPO2− /PO3− 4 4

12

pH

10

NH+4/NH3

8

H2PO−4/HPO2− 4

6

H2CO3/HCO−3

4

CH3COOH/CH3COO−

2

H3PO4/H2PO−4

0 0.0

0.2

0.4

0.6

0.8

1.0

xB Fig. 3.57 Buffer curves of the buffer systems CH3 COOH=CH3 COO (HAc=Ac Þ with pKa;CH3  2 COOH ¼ 4:75; H3 PO4 =H2 PO 4 with pKa1;H3 PO4 ¼ 1:96; H2 PO4 =HPO4 with pKa2;H3 PO4 ¼ 7:12; 2 3 þ HPO4 =PO4 with pKa3;H3 PO4 ¼ 12:32; NH4 =NH3 with pKa;NH4þ ¼ 9:25; and H2 CO3 =HCO 3 with pKa1;H2 CO3 ¼ 6:52

be real concentrations if the strong acid or base would not react with the buffer constituents. Many textbooks use the symbol C for these quantities, which is unfortunate as it can lead to misunderstanding. The buffer capacity depends on the concentration ratio ccBHB and on the overall concentration C  ¼ cHB þ cB . The following equation allows calculation of the buffer capacity [43]:  cHB cB  þ b ¼ 2:303 cH3 O þ cOH þ cHB þ cB

ð3:78Þ

The strongly ascending parts of the curve at low and high pH values (Fig. 3.58) are due to the terms 2:303  cH3 O þ and 2:303  cOH . These parts of the curve demonstrate that solutions containing high concentrations of strong acids or bases, e.g., of hydrochloric acid and sodium hydroxide, possess strong buffer capacities. Such solutions are usually not referred to as buffer solutions, although they contain acid–base pairs in high concentrations, namely H3 O þ /H2 O and H2 O/OH , respectively. cHB cB In Eq. 3.78 the term 2:303 cHB þ cB is responsible for the bell-shaped part having a maximum at pH = pKa , where b ¼ 0:58  C  .

3.7 Buffer Solutions

89

Fig. 3.58 Buffer capacity of equimolar acetic acid/acetate solutions having different overall concentrations ðC  ¼ cHB þ cB Þ

0.30

0.25

CO = 0.5 mol L−1

0.20

CO = 0.4 mol L−1

0.15

CO = 0.2 mol L−1

0.10

0.05

CO = 0.1 mol L−1

0.00 0

2

4

6

pH = pKa

8

10

12

14

pH

In the range of the bell-shaped curve, the following equation can be used for the approximate calculation of the buffer capacity: b ¼ 2:303

cHB cB cHB þ cB

ð3:79Þ

For a buffer solution containing 0.1 mol acetate and 0.1 mol acetic acid per liter, the buffer capacity is: b ¼ 2:303

cHB cB 0:1  0:1 mol L1 ¼ 0:115 mol L1 ¼ 2:303 0:1 þ 0:1 cHB þ cB

Adding to 1 L of this solution 0.005 mol OH− ions, the pH will change by: dpH ¼

dnOH 0:005 ¼ 0:04 ¼ 0:115 b

By chosing different acid–base pairs with different pKa values, it is possible to prepare buffers with pH ranges between 2 and 12. The acid–base pair should to be chosen in such a way that the buffer solution has a maximum buffer capacity, i.e., a pH which is as near as possible to the pKa of the buffer system. Clearly, no buffer solution has the capacity to buffer the pH for unlimited additions of acid or base. In Table 3.7 examples of typical buffer systems are given with pH ranges which allow a good buffering effect.

90

3

Acid–Base Equilibria

Table 3.7 Examples of typical buffer systems Buffer system

pH range

CH3 COOH=CH3 COO

3.7–5.7 5.4–8.0

2 H2 PO 4 =HPO4 (PBS, phosphate buffered salt solution, also containing NaCl and KCl)

NH4þ =NH3 HEPES, 4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid/ 4(2-hydroxyethyl)-1-piperazineethanesulfonate TRIS, 2-amino-2-(hydroxymethyl)propane-1,3-diol hydrochloride/ 2-amino-2(hydroxymethyl)propane-1,3-diol (TRIS–glycine: contains additional glycine and sodium dodecylsulfate; TBS: TRIS-buffered salt solution containing additional NaCl for adjusting the ionic strength to the physiological value; TBS-T: TBS containing an additional tenside; TE: TRIS and EDTA)

8.2–10.2 6.8–8.2 7.2–9.0

Since the acidity constant of the buffer acid depends on temperature, the pH of buffers also depends on temperature, something which requires consideration when planning practical work.

References 1. Finston HL, Rychtman AC (1982) A new view of current acid-base theories. Wiley, New York 2. Boyle R (1661) The Sceptical Chymist. Cadwell, Crooke http://www.gutenberg.org/ebooks/ 22914?msg=welcome_stranger. The book can be downloaded here https://ia801908.us. archive.org/16/items/scepticalchymis00BoylA/scepticalchymis00BoylA.pdf 3. Sendivogii M (1604) Tractatus de lapide philosophorum sive novum lumen chymicum. Prague. English translation: Bujas J. http://www.levity.com/alchemy/newchem1.html 4. Berzelius J (1812) Schweiggers J 6:119–176. http://www.archive.org/stream/ journalfrchemie52unkngoog#page/n7/mode/2up 5. The Collected Works of Sir Humphry Davy (1840) John Davy (ed). Smith, Elder and Co, London 6. (a) “Recherches sur la conductibilité galvanique des électrolytes. Première partie: La conductibilité des solution aqueuses extrêmement diluées déterminée au moyen du dépolarisateur. Bihang till Kongl. Svenska vetenskaps-akademiens handlingar 8 (1984) No 13. Seconde partie: Théorie chimique des électrolytes. Bihang till Kongl. Svenska vetenskaps-akademiens handlingar 8 (1984) No. 14. Stockholm 1884. Kongl. Boktryckeriet. (b) Arrhenius SA (1887) Z physik Chem I:631–648 7. Brønsted JN (1923) Rec Trav Chim Pays-Bas 42:718–728 8. Lowry TM (1923) Ind Chem (London) 42:1048–1052 9. Lewis GN (1923) Valence and the structure of atoms and molecules. In: American chemical social monograph series. The Chemical Catalogue Company, New York, p 142 10. Franklin EC (1905) J Am Chem Soc 27:820–851; (1924) J Am Chem Soc 46:2137–2151 11. Jander G, Mesech H (1939) Z Physik Chem A 183:255 12. Lux H (1939) Z Elektrochem 45:303 13. Flood H, Förland T, Roald B (1947) Acta Chem Scand 1:790 14. Usanovich M (1938) Zh obshchey khim 9:182–192 15. Eigen M (1964) Angew Chem Int Ed 3:1–72

References 16. 17. 18. 19. 20. 21.

22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

91

Henrion G, Scholz F, Schmidt W, Zettler M (1982) Z Phys Chem (Leipzig) 263:634–636 Light TS, Licht SL (1987) Anal Chem 59:2327 Bandura AV, Lvov SN (2006) J Phys Chem Ref Data 35:15 Sørensen SPL (1909) Compt Rend Lab Carlsberg 8:1; and 8:396 Nørby JG (2000) Trends Biochem Sci 25:36–37 IUPAC. Compendium of Chemical Terminology, 2nd ed. (the “Gold Book”). Compiled by McNaught AD, Wilkinson A. Blackwell Scientific Publications, Oxford (1997). XML on-line corrected version: http://goldbook.iupac.org (2006–) created by Nic M, Jirat J, Kosata B; updates compiled by Jenkins A. ISBN 0-9678550-9-8. https://doi.org/10.1351/goldbook. Last update Feb 24 2014; version: 2.3.3. https://doi.org/10.1351/goldbook.p04524 Thämer M, De Marco L, Ramasesha K, Mandal A, Tokmakoff A (2015) Science 350:78–82 Kotrlý Šůcha L (1985) Handbook of chemical equilibria in analytical chemistry. E. Horwood Ltd., Chichester Giguère PA (1979) J Chem Educ 56:571–575 Giguère PA, Turrell S (1980) J Am Chem Soc 102:5473–5477 Tytko KH (1979) Chem unserer Zeit 13:184–194 Williams JP (2003) Fraústo da Silva JJR. J Theor Biol 220:323–343 Lunk HJ (2015) ChemTexts 1:6 Pope MT (1987) Isopolyanions and Heteropolyanions. Compr Coord Chem 3:S 1023–1058. Pergamon Press, Oxford Lunk HJ (2014) ChemTexts 1:3 Lincoln SF, Richens DT, Sykes AG (2004) Metal aqua ions. Compr Coordin Chem II 1 (515):01055 Hammett LP (1940) Physical organic chemistry. McGraw Hill, New York, p 1973 Cruse K (1955) Theoretische Grundlagen der pH-Messung. In: Methoden der Organischen Chemie (Houben-Weyl) Müler E (ed) Physikalische Methoden, Teil 2. Georg Thieme Verlag, Stuttgart, p 25–63 Himmel D, Goll SK, Leito I, Krossing I (2010) Angew Chem Int Ed 49:6885–6888 Kelly CP, Cramer CJ, Truhlar DG (2006) J Phys Chem B 110:16066 Streuli CA (1960) Anal Chem 32:407 Kahlert H, Scholz F (2013) Acid-base diagrams. Springer, Berlin Ostwald W (1888) Z Phys Chem 2:36–37 Stock JT (1997) J Chem Educ 74:865–867 Burgot JL (2012) Ionic equilibria in analytical chemistry. Springer, New York Scholz F, Kahlert H (2018) ChemTexts 4:6 Stumm W, Morgan JJ (1996) Aquatic chemistry. Chemical equilibria and rates in natural waters. 3rd edn, Wiley, New York, pp 533-538 Bliefert C (1978) pH-Wert-Berechnungen. Verlag Chemie, Weinheim, pp 163

4

Complex Formation Equilibria

This chapter focuses on the formation of complexes of metal ions with ligands. The mathematical description of the formation of complexes consisting of neutral molecules follows similar rules; however, it is not covered here because it is less important for analytical chemistry. This is also not the place to discuss the nature of chemical bonds in metal complexes, since it is well documented in standard textbooks on inorganic chemistry (e.g., [1]).

4.1

Monodentate and Multidentate Ligands

For the description of complex equilibria, the number of bonds a ligand is capable of forming with a metal ion is important. This property is called denticity, and leads to distinguishing between monodentate (or unidentate, i.e., one bond), bidentate (two bonds), tridentate (three bonds), tetradentate (4 ligands), pentadentate (five bonds), hexadentate (6 bonds) ligands, etc. (Table 4.1). Metal ligands can be neutral molecules or anions possessing free electron pairs, which can form bonds. In multidentate ligands several so-called donor atoms are present that have free electron pairs. Furthermore, these donor atoms have to be situated to one another in such way that a—more or less—strainless ring can be formed with the central metal ion. An ideal ring has five members (Fig. 4.1). To simplify the equations, the charges of the ligands and metal ions are omitted in the following text, i.e., Men þ  Me and Lm  L. When a metal ion is able to form complexes with a variable number of ligands, the following equations can be given:

© Springer Nature Switzerland AG 2019 F. Scholz and H. Kahlert, Chemical Equilibria in Analytical Chemistry, https://doi.org/10.1007/978-3-030-17180-3_4

93

94

4 Complex Formation Equilibria

Table 4.1 Examples of ligands of different denticity Monodentate ligands Bidentate ligands

Tridentate ligands

Tetradentate ligands

OH ; Cl ; Br ; I ; NH3 oxalate,

ethylene diamine

diethylene triamine,

terpyridine (2,2′;6′,2″-terpyridine)

triethylene tetramine

nitrilotriacetate

Pentadentate ligands

dibenzo-pentathia-tridecanoate

Hexadentate ligands

ethylenediaminetetraacetate

Octadentate ligands

diethylenetriaminepentaacetate

H2 N

NH2 Me

Fig. 4.1 Stressless five-membered ring of the ligand ethylene diamine and a metal ion. Charges are omitted for simplicity

4.1

Monodentate and Multidentate Ligands

95

Me þ L  MeL

ðEquilibrium 4:1Þ

MeL þ L  MeL2

ðEquilibrium 4:2Þ

... MeLn1 þ L  MeLn

ðEquilibrium 4:3Þ

The law of mass action (LMA) of each of these equilibria gives: aMeL cMeL  aMe aL cMe cL

ð4:1Þ

aMeL2 cMeL2  aMeL aL cMeL cL

ð4:2Þ

Kstab;1 ¼ Kstab;2 ¼

... Kstab;n ¼

aMeLn cMeLn  aMeLn1 aL cMeLn1 cL

ð4:3Þ

The equilibrium constants Kstab;n are called the stepwise stability constants or stepwise formation constants, or simply stability (or formation) constants. They are also called (stepwise) binding constants. In principle, Equilibria 4.1–4.3 can also be written in the reversed form: MeL  Me þ L

ðEquilibrium 4:4Þ

Then the LMA follows as: K1 ¼

aMe aL cMe cL  aMeL cMeL

ð4:4Þ

In this case the equilibrium constants are called instability constants and have the symbol Kn . They are related to the stability constants according to 1 . Instability constants are only mentioned because they are sometimes Kn ¼ Kstab;n used in the scientific literature (e.g., [2]). Another system to describe complex formation equilibria is based on the following overall reactions: Me þ L  MeL

ðEquilibrium 4:5Þ

Me þ 2L  MeL2

ðEquilibrium 4:6Þ

... Me þ nL  MeLn

ðEquilibrium 4:7Þ

96

4 Complex Formation Equilibria

with the respective equations for the LMA: b1 ¼

aMeL cMeL  aMe aL cMe cL

ð4:5Þ

b2 ¼

aMeL2 cMeL2  2 aMe aL cMe c2L

ð4:6Þ

... bn ¼

aMeLn cMeLn  aMe anL cMe cnL

ð4:7Þ

These equilibrium constants are called cumulative stability constants and are attributed the symbol bn (Greek letter beta). The relations between the stepwise and cumulative constants are b1 ¼ Kstab;1 , b2 ¼ Kstab;1 Kstab;2 , and generally: bn ¼

i¼n Y

Kstab;i

ð4:8Þ

i¼1

In principle, complex formation equilibria can be handled with only one of these systems, i.e., the stepwise stability constants or the cumulative stability constants. However, since both systems are used in chemistry, both have to be given here. As explained in Chap. 3, the thermodynamic constants are defined on the basis of activities, which have no units, and so these constants also have no units. Using molar concentrations instead of the activities leads to constants having units. For very low concentrations, their numerical values approach the thermodynamic equilibrium constants because the activity coefficients approach unity when the concentration approaches zero. For experimental determination of the stability constants of metal complexes, the book by Martell and Motekaitis should be consulted [3].

4.2

Side Reaction Coefficients and Conditional Stability Constants

Because complex formation equilibria are almost always coupled to other equilibria, it is important to treat all equilibria together.

4.2.1 Side Reactions of Ligands Most ligands are rather strong Brønsted bases, e.g., ammonia, oxalate, and ethylendiaminetetraacetate, and hence the acid–base equilibria of these ligands must

4.2 Side Reaction Coefficients and Conditional Stability Constants

97

be taken into consideration. Protonation of the ligand, i.e., the base, diminishes the concentration (activity) of the free ligand, which forms the metal complex. Therefore, it is necessary to calculate the concentration of the free ligand in relation to its overall concentration. This will be explained here taking ethylenediaminetetraacetic acid (EDTA) as an example, because this compound is of exceptional importance for complexometric titrations [4]:   C2 H4 N½CH2 COOH2 2  LH4   C2 H4 N½CH2 COOH2 ðN½CH2 COOH½CH2 COO Þ  LH 3     2 C2 H4 N½CH2 COOH2 N½CH2 COO  2  LH2   C2 H4 ðN½CH2 COOH½CH COO  Þ N½CH2 COO 2  LH3 2   4 C2 H4 N½CH2 COO 2 2  L In a solution containing the metal ions Me2 þ and EDTA, the following equilibrium is established: Me2 þ þ L4  ½MeL2

ðEquilibrium 4:8Þ

and the respective equilibrium constant is: Kstab;1 ¼

a½MeL2 aMe2 þ aL4



c½MeL2 cMe2 þ cL4

ð4:9Þ

 The overall concentration CEDTA of EDTA is the sum of all its forms:  CEDTA ¼ cLH4 þ cLH3 þ cLH2 þ cLH3 þ cL4 þ c½MeL2 2

ð4:10Þ

Since only cL4 is involved in the complex formation (Equilibrium 4.8), the ratio: aL ¼

cL4 c 4 ¼ L0 3 þ c 4 cLH4 þ cLH3 þ cLH2 þ c cL LH L 2

ð4:11Þ

has to be calculated. c0L is the sum of the concentrations of all EDTA forms, with exclusion of that of ½MeL2 . aL (Greek letter alpha) is called the side reaction coefficient of EDTA, and here one can also use the symbol aEDTA4 . The general equation for the side reaction coefficient of the ligand is: aL ¼

concentration of the free ligand sum of the concentrations of all forms of the ligand, excluding that of the metalligand complex

98

4 Complex Formation Equilibria

4.2.2 Side Reactions of the Metal Ions Metal ions can usually undergo a greater variety of side reactions. As presented in Chap. 3, they can form hydroxo complexes (the corresponding bases of metal aqua complexes), e.g., þ   2 þ  0  MeðOH2 Þ6 ; MeðOH2 Þ5 OH ; MeðOH2 Þ4 ðOHÞ2 , and MeðOH2 Þ3  0 ðOHÞ3  , etc. are simplified here as follows: ½Me2 þ ; ½MeOH þ ; MeðOHÞ2 ,   and MeðOHÞ3 . Note, that all these forms are dissolved species, even when they  0 are uncharged as MeðOHÞ2 . The latter can be regarded as an ion pair. The situation becomes more complicated by the ability of the hydroxo complexes to undergo condensation reactions, leading to dimers, oligomers, and eventually even to solid phases (precipitates). The condensation reactions can be very slow, so that equilibrium may not be achieved within the time frame of any experiments. The reaction rate depends mainly on the rate of ligand exchange, and thus one distinguishes labile complexes (fast ligand exchange) and inert complexes (slow ligand exchange). In the case of complexometric titrations, so-called auxiliary ligands (also called ancillary ligands) are used to prevent the formation of the condensation products of hydroxo complexes. When L0 denotes the auxiliary ligand the sum  CMe 2 þ of all forms of the metal ions is:  CMe 2 þ ¼ cMeOH þ cMeðOHÞ þ cMeðOHÞ þ    þ cMe2 O þ cMe3 O2 þ    þ cMeL0 þ cMeðL0 Þ þ    þ cMe2 þ þ c½MeL2 2 3 2

ð4:12Þ The sum cMeOH þ cMeðOHÞ2 þ cMeðOHÞ3 þ    relates to all hydroxo complexes, the sum cMe2 O þ cMe3 O2 þ    to all condensation products, and cMeL0 þ cMeðL0 Þ2 þ    to all complexes with the auxiliary ligand. cMe2 þ denotes the concentration of free metal ions, i.e., of the metal aqua complex, and c½MeL2 is the concentration of the complex of the metal ion with the ligand L. Now it is possible to calculate the side reaction coefficient aMe according to: aMe ¼

cMe2 þ c 2þ ¼ Me0 cMeOH þ cMeðOHÞ2 þ cMeðOHÞ3 þ    þ cMe2 O þ cMe3 O2 þ    þ cMeL0 þ cMeðL0 Þ2 þ    þ cMe2 þ cMe

ð4:13Þ where c0Me is the sum of concentrations of all metal complexes (remember that Me2 + is the metal aqua complex), excluding that of the metal–ligand complex ½MeL2 . It generally holds that: aMe ¼

concentration of free metal ions sum of concentrations of all metal complexes, excluding that of the metalligand complex

4.2 Side Reaction Coefficients and Conditional Stability Constants

99

For many ligands and metals, side reaction coefficients are available in reference books [5]. Definition of the side reaction coefficients of ligand and metal ions allows the definition of conditional equilibrium (stability) constants.

4.2.3 Conditional Stability Constants Equations 4.11 and 4.13 can be solved for cL4 and cMe2 þ : cL4 ¼ c0L aL

ð4:14Þ

cMe2 þ ¼ c0Me aMe

ð4:15Þ

Substitution of these terms in the LMA of complex formation (Eq. 4.9) leads to: Kstab;1 ¼ The ratio

c½MeL2 c0Me c0L

c½MeL2 0 cMe aMe c0L aL

¼

1 aMe aL



c½MeL2 c0Me c0L

ð4:16Þ

is the conditional (or effective) stability constant: Kcond ¼

c½MeL2 c0Me c0L

¼ Kstab;1 aMe aL

ð4:17Þ

The term conditional indicates that these constants are defined for defined conditions, especially for a defined solution composition. The term effective indicates that these are the really operative constants under defined solution conditions.

4.2.4 The pH Dependence of Side Reaction Coefficients and Conditional Constants How does solution pH affect aL and aMe ? Considering: aL ¼

cL4 c 4 ¼ L0 3 þ c 4 cLH4 þ cLH3 þ cLH2 þ c cL LH L 2

ð4:11Þ

it is clear from the acid–base equilibria of LH4 , that at a pH above pKa4 , the side reaction coefficient aL will approach unity. The lower the pH, the larger the percentage of protonated forms of the ligand. Below pKa1 the side reaction coefficient aL approaches zero. The exact dependence (Fig. 4.2) of the side reaction coefficient aEDTA4 , i.e., aL , of ethylenediaminetetraacetate on pH is determined by the pKa values of EDTA. In Eq. 4.13 the concentrations cMeOH þ cMeðOHÞ2 þ cMeðOHÞ3 þ    and cMe2 O þ cMe3 O2 þ    are pH dependent. This may be also the case for cMeL0 þ cMeðL0 Þ2 þ   ,

100

4 Complex Formation Equilibria 1.2

1.0

0.8

αEDTA

4−

0.6

α 0.4

0.2

0.0

-0.2 0

2 4 6 8 10 pH = pKa2 = 2.75 pH = pK = 6.24 a3

pH = pKa1 = 2.07

pH

12

14

pH = pKa4 = 10.34

Fig. 4.2 pH dependence of the aEDTA4 of ethylenediaminetetraacetate. The pKa values of EDTA are: pKa1 ¼ 2:07, pKa2 ¼ 2:75, pKa3 ¼ 6:24, and pKa4 ¼ 10:34

if the auxiliary ligands are Brønsted bases. From this it follows that in very acidic solutions, where the concentrations of hydroxo complexes and of their condensation products is negligible, the side reaction coefficient aMe of the metal ions will approach unity, provided the auxiliary ligand L0 is also a Brønsted base. If L0 is not a Brønsted base, the concentrations cMeL0 þ cMeðL0 Þ2 þ    are pH independent and the side reaction coefficient of the metal ions aMe will approach a constant value smaller than unity. With growing pH the side reaction coefficient aMe decreases and finally may approach zero. Figure 4.3 depicts the pH dependence of aEDTA4 and aAg þ and the resulting dependence of the conditional stability constant. Silver ions form two complexes   ½AgOH0 and AgðOHÞ2 with hydroxide ions that have stability constants b1 ¼ 102:00 L mol1 and b2 ¼ 103:99 L2 mol2 . The complex of silver ions with EDTA has a stability constant of Kstab;½AgEDTA3 ¼ 107:32 L mol1 . The conditional stability constant has a maximum at the optimal pH (here pHoptimum ¼ 11:12; cf. Fig. 4.5). Figure 4.4 clearly shows that the conditional stability constants exist for all pH values smaller than Kstab;½AgEDTE3 ¼ 107:32 L mol1 . At pHoptimum ¼ 11:12   the difference log Kstab;½AgEDTE3  log Kcond is only 0.13 (Fig. 4.5).

4.2 Side Reaction Coefficients and Conditional Stability Constants

101

1.2

1.0

0.8

αEDTA

4−

0.6

αAg

+

α 0.4

0.2

0.0

-0.2 0

2 4 6 8 10 12 14 pH = pKa2 = 2.75 pH = pKa3 = 6.24 pH = pKa4 = 10.34 pH = pKa1 = 2.07 pH

Fig. 4.3 pH dependence of the side reaction coefficients aEDTA4 and aAg þ for ethylenediaminetetraacetate and silver ions 10

1.2 log K[AgEDTA] = 7.32 3−

1.0 5 0.8

α

0.6

0

α EDTA

4−

logKcond

αAg

+

0.4

-5 0.2 -10

0.0 -0.2 0

2

4

6

8

10

12

14

pH = pKa2 = 2.75 pH = pKa3 = 6.24 pH = pKa4 = 10.34 pH = pKa1 = 2.07

pH

Fig. 4.4 pH dependence of the side reaction coefficients aEDTA4 and aAg þ for ethylendiaminetetraacetate and silver ions and the resulting pH dependence of the logarithm of the conditional stability constant Kcond ¼ Kstab;½AgEDTA3 aAg þ aEDTA4

102

4 Complex Formation Equilibria 8

1.2 log K[AgEDTA] = 7.32 3−

1.1

7

1.0

6

α

logKcond 5

0.9

αEDTA

4−

αAg

0.8

4

+

3

0.7 8

9

10

11

pH = pKa4 = 10.34

pH

12

13

pH = pHoptimum = 11.12

14

Fig. 4.5 As Fig. 4.4, however, depicted only for the pH range of 8–14

In a complexometric titration, in order to guarantee a maximum conditional stability constant, it is important to keep the pH at a value where this is given. In the course of many complexometric titrations the pH shifts if no buffers are present. To use buffers to keep the pH rather stable means that buffer capacity must be considered. This means that the buffer needs to be present in relatively high concentrations. Furthermore, it is necessary to consider the possible complex formations between metal ions and buffer constituents—in some cases these are desirable when they act as auxiliary ligands.

4.3

The Chelate Effect

The formation of metal complexes is always a substitution reaction. The ligand water ðOH2 Þ is substituted by the ligand L:  m þ  m þ MeðOH2 Þn + L  MeðOH2 Þn1 L + H2 O

ðEquilibrium 4:9Þ

When the ligand is hexadentate, like, e.g., EDTA4 , and the metal ion is present  2 þ as hexaaqua complex MeðOH2 Þ6 , the equilibrium reaction is:  2 þ + L4  ½MeL2 + 6H2 O MeðOH2 Þ6

ðEquilibrium 4:10Þ

4.3 The Chelate Effect

103

Here, two particles on the left-hand side are in equilibrium with seven particles on the right side. This means that when the reaction proceeds from the left-hand side to the right-hand side, the number of particles increases from two to seven. This means that disorder increases, especially because the six water molecules are fixed to the metal ion on the left-hand side (although they may be labile and quickly exchanged with the water molecules of the surrounding water). In the surrounding water, the water molecules are also fixed in the water network, but the number of possible arrangements is higher than in the complex. So the water molecules have more degrees of freedom in the water phase than in the metal aqua complex. This has far reaching consequences for thermodynamics: increasing disorder means increasing the standard entropy of the system by DR S . Considering the relation: DR G ¼ RTlnK

ð4:18Þ

and the Gibbs–Helmholtz equation: DR G ¼ DR H  TDR S

ð4:19Þ

it is clear that a large positive DR S can lead to negative, possibly even very negative, values of DR G ; and thus to very large equilibrium constants K. According to Eq. 4.19, the reaction enthalpy DR H is also important; however, in the case of complex formation, the reaction enthalpy is normally small compared with the term TDR S ; and so the increase in entropy is decisive. Since in all cases of polydentate ligands (chelate-forming ligands) an entropy increase is observed, this effect is called the chelate effect. According to Schwarzenbach [6, 7], the chelate effect can be quantified using the cumulative stability constant of the metal chelate complex and the metal complex with n monodentate ligands (with similar ligating groups) MeL0n : D log K ¼ log KMeL  log bMeL0n

ð4:20Þ

Table 4.2 shows a comparison of the standard reaction data and equilibrium constants for cadmium ions and ammonia and ethylenediamine (en), illustrating how sensitive the reaction entropy is to an increase in the number of particles in a chemical reaction. The anions of aminopolycarboxylic acids are particularly well suited to form chelates because they possess nitrogen and oxygen atoms with free electron pairs, capable of forming bonds with metal ions. At the same time, the ligating atoms are separated by carbon atoms, so that geometrically stable five-membered rings are formed. This is also relevant in biochemistry, where amino acids are frequently the ligating moieties for metal ions in metal–protein complexes (see photosynthesis, respiration, and nitrogen fixation in biochemistry textbooks [9]).

104

4 Complex Formation Equilibria

Table 4.2 Comparison of the standard reaction enthalpies, entropies, free energies, and logarithms of cumulative stability constants for the complex formation of cadmium ions with ammonia and ethylenediamine (en). The ratio x relates to the number of particles on the left-hand (reactants) nreactants and right-hand (product) side nproduct of the equilibrium (Data from [8]) Complex    

CdðOH2 Þ4 ðNH3 Þ2 CdðOH2 Þ4 ðenÞ

2 þ

CdðOH2 Þ2 ðNH3 Þ4 CdðOH2 Þ2 ðenÞ2

4.4

2 þ

2 þ

2 þ

Dr H (kJ mol−1)

Dr S (J K−1mol−1)

Dr G (kJ mol−1)

logb

x ¼ nnreactants products

4.95 ðlogb2 Þ 5.84 ðlogb1 Þ 7.44 ðlogb4 Þ 10.62 ðlogb2 Þ

3:3

−29.79

−5.19

−28.24

−29.41

+13.05

−33.30

−53.14

−35.5

−42.51

−56.48

+13.75

−60.67

2:3 5:5 3:5

Applications of Complex Equilibria

There are many different applications of complex equilibria. Some are listed here:

(1) The formation of a metal complex is used for titrimetric determinations (Chap. 7). (2) Complex formation is used to dissolve sparingly soluble precipitates, e.g., to dissolve boiler scale. Commercial powders for cleaning kettles usually contain chelators. Similarly, rust-removing chemicals contain chelators to bind iron(III) ions. (3) Laundry detergent powders contain chelators for calcium and magnesium ions to prevent the formation of salts of these metal ions with anionic tensides, as they deteriorate materials when dried. Nowadays, mainly zeolites are used (e.g., Na12((AlO2)12(SiO2)12)  27 H2O); however, in the past polyphosphates were used, which caused eutrophication of rivers and lakes. Later, aminopolycaboxylic acids were used, which also had a negative environmental impact: they remobilized metal ions from sediments. See also Chap. 7.5.2.1. (4) Chelating reagents can be used to mask metal ions to avoid undesirable reactions. An example is the prevention of sulfide precipitates of metal ions with H2S (Chap. 5). In electroanalytical chemistry, the masking of interfering metal ions is sometimes used in determinations. As an example, lead ions can be complexed by hydroxide ions to allow the voltammetric determination of thallium ions [10]. (5) Chelating reagents are used in chelation therapy for the removal of toxic metal ions from the human body [11].

4.4 Applications of Complex Equilibria

105

(6) In spectrophotometric analysis, chelating ligands are used to produce metal complexes that have very large absorption coefficients [12]. (7) In cell cultures, chelates can be used to make metal ions bioaccessible, especially when the metal ions can form polyoxometal cations or even sparingly soluble oxide hydrates (Chaps. 3 and 5). (8) In galvanic plating, chelators are important to adjust the reduction and plating potentials (Chap. 6.3.2).

References 1. Huheey JE, Keiter EA, Keiter RL, Medhi OK (2006) Inorganic chemistry: principles of structure and reactivity. Pearson Education, London 2. Yatsimirskii KB, Vasil’ev VP (1960) Instability constants of complex compounds. Pergamon Press Ltd, London 3. Martell AE, Motekaitis RJ (1992) Determination and use of stability constants, 2nd edn. Wiley-VCH, New York 4. West TS (1969) Complexometry with EDTA and related reagents, 3rd edn. BDH Chemical, Poole 5. Kragten J (1978) Atlas of metal-ligand equilibria in aqueous solution. Ellis Horwood, Chichester 6. Schwarzenbach G (1952) Helv Chim Acta 35:2344 7. Šůcha L, Kotrlý S (1972) Solution equilibria in analytical chemistry. Van Nostrand Reinhold, London 8. Spike CG, Parry RW (1953) J Amer Chem Soc 75:2726–2729 9. Berg JM, Stryer L, Tymoczko JL, Gatto GJ (2015) Biochemistry, 8th edn. Palgrave Macmillan 10. Kahlert H, Schröder U (2002) Practical voltammetry with the 757 VA computrace. Metrohm-Monograph, Herisau, Switzerland 11. Fauci AS, Braunwald E, Kasper DL, Hauser SL, Longo DL, Jameson JL, Loscalzo J (eds) (2008) Harrison principles of internal medicine, 17th edn. Mc Graw Hill Medical, New York, p 2449–2452 12. Flaschka HA, Barnard AJ Jr (eds) (1976) Chelates in analytical chemistry. Marcel Dekker, New York

5

Solubility Equilibria

The precipitation of sparingly soluble compounds, especially salts, is intimately connected with the development of scientific chemistry. This started during the tenebrous times of alchemy. It was realized very early on that precipitation offers the possibility to separate chemical compounds or their constituents. Later, the potential of precipitation for quantitative analysis by weighing (gravimetry) and for precipitation titrations was discovered. Solubility equilibria of salts, including oxides and hydroxides, relate a solid phase to dissolved ions:
 ðMen þ Þm ðAm Þn s  mMen þ þ nAm

ðEquilibrium 5:1Þ

The braces with the index s (solidus) enclose the formula of the solid compound being a pure phase. The ions on the right-hand side of the equilibrium are dissolved in the solvent, i.e., they are present as solvated (in water, hydrated) ions. As discussed in Chap. 3 the dissociation of salts by the solvation of ions is thermodynamically favored because the overall Gibbs energy of the system is decreased. For dissolving an ionic solid phase, lattice energy has to be overcome, which can only be achieved by solvation enthalpy and entropy (the entropy multiplied by absolute temperature) of the ions. The lattice energy and Gibbs energy of solvation affect the extent of solubility: if the lattice energy is much larger than the Gibbs energy of solvation, the compound will have low solubility. Salts can also thermally dissociate, e.g., when their aqueous solution is fed into a flame: at first the solvent in the droplets evaporates and solid salt crystals are formed. Then these salt crystals melt and evaporate. During that evaporation, the salt dissociates (mainly) to form atoms, e.g., Na and Cl in the case of sodium chloride. This is what happens when sodium chloride solution is sprayed into a flame and the flame displays a yellow color—a small fraction of the sodium atoms are thermally excited and return to their ground state by the ejection of photons at two wavelengths, 588,9951 and 589,5924 nm (yellow). Because the flame does not have dipole

© Springer Nature Switzerland AG 2019 F. Scholz and H. Kahlert, Chemical Equilibria in Analytical Chemistry, https://doi.org/10.1007/978-3-030-17180-3_5

107

108

5

Solubility Equilibria

molecules to solvate ions, atoms are formed, which are in fact very reactive and will quickly form compounds again in the colder zone of the flame.

Solubility equilibria can also be called precipitation equilibria, if Equilibrium 5.1 is written in reverse, i.e., with the ions on the left-hand side and the solid phase on the right-hand side. Usually, solubility equilibria are written as follows: Mem An  mMen þ þ nAm

ðEquilibrium 5:2Þ

Since a pure phase has a molar ratio activity of 1.0 (Chap. 3) the law of mass action (LMA) contains only the concentration activities of the ions: n K s ¼ am c;Men þ ac;Am

ð5:1Þ

This equation is only valid, when the dissolved ions exist in the solution in equilibrium with the solid phase! If the solid phase is absent, the concentration of dissolved cations and anions can vary independently. (Of course, when only one salt is dissolved, the cations and anions are related according to the stoichiometry of the salt.) The equilibrium constant Ks is called the solubility product. It depends on temperature and pressure. It is dimensionless, as are all other equilibrium constants based on activities. However, solubility products based on concentrations are fre ðm þ nÞ quently used. They have the dimension mol L1 . Their numerical value approaches that of the dimensionless constant, especially when the solubility is very small and the activities of the ions approach their concentrations. The formation of a precipitate (solid phase) happens only when the activities of the ions exceed those defined by the solubility product. In fact, it is possible to prepare solutions in which the activities (concentrations) of cations and anions exceed the values set by the solubility product: such solutions are called supersaturated. They are not in equilibrium with the solid phase and can be prepared only when the solid phase is absent. Supersaturated solutions are thermodynamically unstable and can only exist for a certain period, that is, when the nucleation of the solid phase is kinetically inhibited. When a salt solution is saturated at an elevated temperature and then slowly cooled, so that the ion activities exceed the solubility product, nuclei of the solid salt may not be formed for some period of time. Normally, this needs very clean solutions, containing no dust particles. Some salts are especially suitable for the preparation of supersaturated solutions, e.g., sodium sulfate: Fig. 5.1 shows a hot saturated solution of sodium sulfate being poured into a very clean Petri dish. Even after cooling to room temperature, no crystals of Na2SO410H2O precipitate. Only when a few small crystals of the salt are added to the solution, as crystallization germs, does quick crystallization from the supersaturated solution occur.

5

Solubility Equilibria

109

Fig. 5.1 a A hot saturated solution of sodium sulfate is poured into a Petri dish. b Even after cooling to room temperature no precipitate can be detected. This solution is now supersaturated with sodium sulfate. c Addition of some small crystals of Na2SO410H2O initiates the crystallization process in the supersaturated solution and drives the system to equilibrium. d Originating from the crystallization germs, long needles of Na2SO410H2O grow, and a solubility equilibrium is established at room temperature

5.1

The Saturation Concentration

The dissolution of a compound (e.g., a salt) in a solvent happens at a constant temperature and pressure until a certain maximum activity (concentration) of the dissolved compound has been reached. This maximum is called saturation activity or saturation concentration. When the saturated solution and the solid compound (salt) are both present, dissolution still continues but the rate of precipitation, i.e., the rate at which the solute is transferred back to the solid, exactly equals the dissolution rate.

110

5

Solubility Equilibria

This is the situation for a reversible dissolution–precipitation system. In the following discussion, only salts are considered: the saturation concentration SMem An of a salt is the number of moles of the salt Mem An dissolved in one liter of saturated solution (i.e., not in one liter of solvent!). From the stoichiometry of the dissolution– precipitation equilibrium (Equilibrium 5.1), and assuming that complete dissociation of the salt happens (no formation of ion pairs in solution), it follows that: cMen þ ¼ mSMem An

ð5:2Þ

cAm ¼ nSMem An

ð5:3Þ

and

Using concentrations as good approximations of activities, it is possible to relate the equilibrium concentrations of the ions to the solubility product. Equation 5.1 now transforms to: þn Ks ¼ ðmSMem An Þm ðnSMem An Þn ¼ mm nn Sm Mem An

ð5:4Þ

This equation can be solved for the saturation concentration: SMem An ¼

rffiffiffiffiffiffiffiffiffiffiffi Ks m m nn

mþn

ð5:5Þ

This equation is only correct for a pure solution. The presence of other compounds (also salts) can considerably alter the solubility. In the case of salts, two situations must be distinguished: (1) the salt effect caused by common ion addition, and (2) the salt effect by foreign ion addition. Common ion addition means that an ion is added (in the form of a salt) which is also present in Mem An , i.e., either Men+ or Am−. As an example, the case of a saturated AgCl solution can be considered: addition of a not too diluted KCl solution to a saturated AgCl solution (with solid AgCl always present) increases the concentration of chloride ions since the solubility of KCl is much higher than that of AgCl. The equilibrium between the solid AgCl and the saturated AgCl solution is described by the chemical equilibrium: fAgClgs  Ag þ + Cl

ðEquilibrium 5:3Þ

and the solubility product: Ks;AgCl ¼ cAg þ cCl

ð5:6Þ

The equilibrium concentration of silver ions follows from Eq. 5.2: cAg þ ¼ SAgCl

ð5:7Þ

5.1 The Saturation Concentration

111

The equilibrium concentration of chloride ions cannot be calculated using Eq. 5.3, because the chloride concentration is affected by the addition of KCl. Since KCl is completely dissociated, the overall equilibrium concentration of chloride is the sum of the equilibrium concentration caused by the dissolution of AgCl and the  added KCl concentration CKCl :  cCl ¼ SAgCl þ CKCl

ð5:8Þ

Inserting this chloride concentration into the solubility product yields:     Ks;AgCl ¼ SAgCl SAgCl þ CKCl SAgCl ¼ S2AgCl þ CKCl

ð5:9Þ

The saturation concentration of AgCl follows as: SAgCl

C ¼  KCl þ 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi   2 CKCl þ Ks;AgCl 2

ð5:10Þ

This shows that a common ion addition decreases solubility. This effect is also called “salting out.” When the added common ion concentration (here of KCl) exceeds the equilibrium concentration of the sparingly soluble salt (here AgCl), it is  possible to use the approximation cCl ¼ CKCl , from which follows  SAgCl ¼ cAg þ ¼ Ks;AgCl CKCl . However, for high concentrations of KCl, a second effect is operative: silver ions can form soluble complexes with chloride ions: Ag þ + 2Cl  AgCl 2

ðEquilibrium5:4Þ

with the stability constant: b2 ¼

cAgCl2 cAg þ c2Cl

ð5:11Þ

Then, Eqs. 5.7, 5.8, 5.9, and 5.10 can no longer be used and must be replaced by: SAgCl ¼ cAg þ þ cAgCl2

ð5:12Þ

The balance equation for the charges is: cK þ þ cAg þ ¼ cCl þ cAgCl2

ð5:13Þ

The potassium ion concentration is not affected by Equilibrium 5.4, so that  cK þ ¼ CKCl . Rearranging Eq. 5.6 as:

112

5

cCl ¼

Solubility Equilibria

Ks;AgCl cAg þ

ð5:14Þ

and inserting into Eq. 5.11, leads to: cAgCl2 ¼ b2 cAg þ c2Cl ¼

2 b2 Ks;AgCl

ð5:15Þ

cAg þ

Inserting Eqs. 5.14 and 5.15 into Eq. 5.13 produces, after some rearrangement: cAg þ

SAgCl

C ¼  KCl þ 2

C ¼  KCl þ 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 CKCl 2 þ Ks;AgCl þ b2 Ks;AgCl 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2  CKCl 2 þ þ Ks;AgCl þ b2 Ks;AgCl 2



 CKCl 2

ð5:16Þ

2 b2 KL;AgCl ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r

 2  CKCl 2 þ þ K þ b K s;AgCl 2 s;AgCl 2

ð5:17Þ The resulting dependence of the AgCl saturation concentration on the added concentration of KCl is depicted in Fig. 5.2. At a very low concentration of added KCl the effect is negligible. At an added concentration of about 10−6 mol L−1, the saturation concentration drops, has a minimum at 310−3 mol L−1, and then increases with increasing added KCl concentration.

Fig. 5.2 Dependence of the solubility of AgCl (saturation concentration) on the added concentration of KCl, using the solubility product Ks;AgCl ¼ 1010 mol2 L2 and the stability constant b2 ¼ 105 L2 mol2

-2

-3

-4

logSAgCl -5

-6

-7

-8 -10

-8

-6

-4

logC

-2 O KCl

0

2

5.1 The Saturation Concentration

113

Foreign ion addition affects solubility on the basis of two phenomena: foreign ions change the ionic strength of the solution, and by doing so they affect the activity of the dissolved ions via their activity coefficients (see physical chemistry textbooks, Debye–Hückel theory, etc.). When the activity coefficients of the ions of the sparingly soluble salt are decreased by foreign ion addition, this can lead to an increase in their concentrations. This effect is called “salting in.” Foreign ions may also undergo equilibrium reactions with the cations or anions of the dissolved salt. This leads to an increase of solubility, because the products of Equilibrium 5.2 are removed and their equilibrium concentrations have to be restored.

5.2

The pH Dependence of Saturation Concentration

When metal cations Men þ or anions Am undergo protolysis reactions in aqueous solutions a pH dependence of solubility results. In the case of metal cations these are the protolysis and condensation reactions discussed in Chap. 3. Only in the case of pKa values of the metal aqua cations being smaller than 14 do protolysis reactions have to be taken into consideration. For anions, protolysis reactions need to be considered only for pKb values smaller than 14. Many anions forming insoluble salts, e.g., O2 ; OH ; S2 ; Se2 ; 3 3 2 2 2 Te2 ; CO2 3 ; PO4 ; AsO4 ; CrO4 ; MoO4 ; and WO4 , belong to this group.

5.2.1 pH Dependence of the Saturation Concentration of Metal Hydroxides, Oxide Hydroxides, and Oxides The solubility product of a metal hydroxide MeðOHÞn follows from the chemical equilibrium:
 MeðOHÞn s  Men þ þ nOH

ðEquilibrium 5:5Þ

The metal ions are present in aqueous solutions as hydrated ions, and thus a more precise description is:


MeðOHÞn

 s



n þ þ xH2 O  MeðOH2 Þx þ nOH

ðEquilibrium 5:6Þ

The LMA gives for Equilibrium 5.5: Ks;MeðOHÞn ¼ ac;Men þ anc;OH

ð5:18Þ

It is noteworthy that the solubility products of the solid phase are the same for the hydroxides, oxide hydroxides, and oxides of the same metal ion. This is obvious from the following example: when the oxide MeOn=2 is considered, instead of the hydroxide MeðOHÞn , water is the reaction partner forming hydroxide ions:

114

5


 n MeOn=2 s þ H2 O  Men þ þ nOH 2

Solubility Equilibria

ðEquilibrium 5:7Þ

The molar ratio activity of water ax;H2 O is unity for very low concentrations of dissolved oxides, and Eq. 5.18 also applies to oxides. For a metal oxide hydroxide MeOm=2 ðOHÞnm , which we can suppose to have formed from the hydroxide MeðOHÞn according to the reaction: MeðOHÞn  MeOm=2 ðOHÞnm þ

m H2 O 2

ðEquilibrium 5:8Þ

the following equilibrium can be formulated:


MeOm=2 ðOHÞnm

 s

þ

m H2 O  Men þ þ nOH 2

ðEquilibrium 5:9Þ

In this case Eq. 5.18 also describes the solubility product. Equilibria 5.5, 5.7, and 5.9 show that the solubility equilibria of metal hydroxides, metal oxide hydroxides, and metal oxides always have metal ions and hydroxide ions as products. The formation of hydroxide ions is the reason for the pH dependence of the solubility of metal hydroxides, metal oxide hydroxides, and metal oxides, because hydroxide ions participate in the autoprotolysis of water: 2H2 O  H3 O þ + OH

ðEquilibrium 5:10Þ

In the following text, only the pH dependence of solubility caused by the protonation of hydroxide ions is considered. The pH dependence caused by the acid– base reactions of metal aqua ions has already been discussed in Chap. 3. Equation 5.18 describes the solubility of MeðOHÞn , MeOn=2 , and MeOm=2 ðOHÞnm . Using concentrations instead of activities, the equation reads: Ks;MeðOHÞn ¼ cMen þ cnOH

ð5:19Þ

 ðn þ 1Þ In this case, the solubility products bear units, e.g., mol L1 for MeðOHÞn ; MeOn=2 , and MeOm=2 ðOHÞnm . The saturation concentrations are as follows: SMeðOHÞn ¼ cMen þ

ð5:20Þ

SMeOn=2 ¼ cMen þ

ð5:21Þ

SMeOm=2 ðOHÞnm ¼ cMen þ

ð5:22Þ

5.2 The pH Dependence of Saturation Concentration

115

When the metal hydroxides, oxide hydroxides, and oxides have more than one metal ion, the relations are as follows: Mez ðOHÞn :

SMez ðOHÞn ¼ 1z cMen þ

ð5:23Þ

Mez On=2 : SMez On=2 ¼ 1z cMen þ

ð5:24Þ

Mez Om=2 ðOHÞnm : SMez Om=2 ðOHÞnm ¼ 1z cMen þ

ð5:25Þ

The pH dependence of the solubility of MeðOHÞn results from the substitution of cMen þ into Eq. 5.20 by the term

Ks;MeðOHÞn cnOH

from Eq. 5.19:

SMeðOHÞn ¼ Substituting cOH with

Kw cH O þ

Ks;MeðOHÞn cnOH

ð5:26Þ

(autoprotolysis of water) yields:

3

SMeðOHÞn ¼

Ks;MeðOHÞn n cH 3 O þ Kwn

ð5:27Þ

Logarithmizing and rearranging gives: log SMeðOHÞn ¼ log Ks;MeðOHÞn  log Kwn þ log cnH3 O þ

ð5:28Þ

log SMeðOHÞn ¼ log Ks;MeðOHÞn  n log Kw þ n log cH3 O þ

ð5:29Þ

log SMeðOHÞn ¼ log Ks;MeðOHÞn þ npKw  npH

ð5:30Þ

Note that Eq. 5.30 describes a linear dependence of solubility on pH. For MeOn=2 and MeOm=2 ðOHÞnm the pH dependences are: log SMeOn=2 ¼ log Ks;MeOn=2 þ npKw  npH

ð5:31Þ

log SMeOm=2 ðOHÞnm ¼ log Ks;MeOm=2 ðOHÞnm þ npKw  npH

ð5:32Þ

Table 5.1 gives the specific forms of Eqs. 5.30, 5.31, and 5.32 for some specific stoichiometries of compounds. Table 5.2 lists the solubility products of some metal hydroxides, oxide hydroxides, and oxides.

116 Table 5.1 Equations of the pH dependences of metal hydroxides, oxide hydroxides, and oxides having specific stoichiometries. For pKw the value 14.0 was used (the exact value is 13.99 at 25 °C and 0.1 mPa). For other conditions see [1–3]

5

Solubility Equilibria

Formula

Equation describing logS as function of pH

Me(OH)2

log SMeðOHÞ2 ¼ log Ks; MeðOHÞ2 þ 2pKw  2pH log SMeðOHÞ2 ¼ log Ks; MeðOHÞ2 þ 28:0  2pH

Me(OH)3

log SMeðOHÞ3 ¼ log Ks; MeðOHÞ3 þ 3pKw  3pH log SMeðOHÞ3 ¼ log Ks; MeðOHÞ3 þ 42:0  3pH

Me(OH)4

log SMeðOHÞ4 ¼ log Ks; MeðOHÞ4 þ 4pKw  4pH log SMeðOHÞ4 ¼ log Ks; MeðOHÞ4 þ 56:0  4pH

MeO

log SMeO ¼ log Ks; MeO þ 2pKw  2pH log SMeO ¼ log Ks; MeO þ 28:0  2pH

MeO(OH) log SMeOðOHÞ ¼ log Ks; MeOðOHÞ þ 3pKw  3pH log SMeOðOHÞ ¼ log Ks; MeOðOHÞ þ 42:0  3pH MeO2

log SMeO2 ¼ log Ks; MeO2 þ 4pKW  4pH log SMeO2 ¼ log Ks; MeO2 þ 56:0  4pH

Me2 O3

log SMe2 O3 ¼ log 0:5 þ log Ks; Me2 O3 þ 6pKw  6pH log SMe2 O3 ¼ log 0:5 þ log Ks; Me2 O3 þ 84:0  6pH

5.2.2 Graphical Presentation of the Solubilities of Metal Hydroxides as a Function of pH Figure 5.3 shows the dependence of the logarithm of solubility ðlog SÞ of Mg(OH)2 , Ca(OH)2 , and Ba(OH)2 as function of equilibrium pH. Figure 5.4 gives the precipitation zones for a solution containing 102 mol L1 Mg2 þ ; Ca2 þ , and Ba2 þ . In this solution, the precipitation of Mg(OH)2 starts at pH = 9.43, that of Ca(OH)2 at pH = 12.41, and that of Ba(OH)2 at pH = 13.2. From this figure, the following insights can be deduced: (1) theoretically it should be possible to precipitate pure Mg(OH)2 in the range 9:43\pH\12:41. (2) It is impossible to precipitate Ca(OH)2 without coprecipitating Mg(OH)2 . When the precipitation of Ca(OH)2 starts, the solution still contains 1.1  10–8 mol L–1 Mg2 þ . At a pH shortly before the precipitation of Ba(OH)2 , it still contains 2.8  10–10 mol L–1 Mg2 þ . This means, that the precipitated Ca(OH)2 still contains 8.6  10–5 weight-% (w/w) Mg(OH)2 . (3) It is impossible to precipitate Ba(OH)2 without coprecipitating Mg(OH)2 and Ca(OH)2 . However, conclusions (1) to (3) are only theoretically correct in the framework of the provided simple treatment. In reality, the phenomenon of coprecipitation (Sect. 5.3) will result in a coprecipitation of calcium and barium ions with Mg(OH)2 and of barium ions with Ca(OH)2 . In Fig. 5.5 the log S of Ni(OH)2 , Fe(OH)2 , and Co(OH)2 is depicted as a function of equilibrium pH. Because of the similarity of solubility products, the lines are positioned very near to one another, and the inset in the figure gives the lines in a narrower range. Clearly, a simple separation by precipitation is impossible. In Fig. 5.6 the dependence of the log S of Fe(OH)3 , Cr(OH)3 , and Ce(OH)3 is given as a function of equilibrium pH. The positions of the lines allow the

5.2 The pH Dependence of Saturation Concentration Table 5.2 Solubility products, given as pKs ¼  log Ks , for metal hydroxides, oxide hydroxides, and oxides [4] at 25 °C. Note that solubility products are slightly sensitive to the ionic strength, which is not considered here

117

General formula

Specific compound

pKs

Me(OH)2

Ca(OH)2 Ba(OH)2 Mg(OH)2 b  Zn(OH)2 Zn(OH)2 (amorphous) b  Cd(OH)2 Cu(OH)2 Fe(OH)2 Co(OH)2 Ni(OH)2 Mn(OH)2 Ce(OH)3 Ga(OH)3 In(OH)3 Fe(OH)3 Pu(OH)4 Th(OH)4 Ti(OH)4 a  FeO(OH) c  AlO(OH) HgO PbOred SnO ZnO UO2 ZrO2 SnO2 PbO2 Fe2 O3 Sb2 O3

5.19 3.6 11.15 16.2 15.52 14.35 19.32 15.1 14.9 15.2 12.8 23.2 37 36.9 38.8 47.3 44.7 53.1 41.5 34.02 25.44 15.3 26.2 16.66 56.2 54.1 64.4 64 42.75 17.66

Me(OH)3

Me(OH)4

MeO(OH) MeO

MeO2

Me2 O3

separation of these hydroxides by precipitation (with a similar limitation to the one discussed for magnesium, calcium, and barium hydroxides). Figure 5.7 depicts the dependence of the log S of Fe(OH)2 , Fe(OH)3 , and a  FeO(OH) on equilibrium pH. From these plots, it is obvious that a  FeO(OH) (goethite) possesses the lowest solubility, and thus it is not surprising that it is one of the most frequently found minerals in the aerobic ranges of soil. It is produced by the oxidation of iron(II) in aerobic environments [5]. Wherever iron(II) aqua ions reach aerobic strata, e.g., when groundwater rises in the capillaries of soil, they are quickly oxidized to iron(III) aqua ions, which then undergo protolysis and condensation reactions (Chap. 3), which finally leads to the formation of goethite (or

118

5

Fig. 5.3 Dependence of the log S of Mg(OH)2 , Ca(OH)2 , and Ba(OH)2 on equilibrium pH. pKs;CaðOHÞ2 ¼ 5:19; pKs;BaðOHÞ2 ¼ 3:6; and pKs;MgðOHÞ2 ¼ 11:15

Solubility Equilibria

pH 0

2

4

6

8

10

12

14

0 -2 -4 -6

logS -8

Mg(OH)2 -10

Ca(OH)2 Ba(OH)2

-12 -14

Fig. 5.4 Dependence of the log S of Mg(OH)2 , Ca(OH)2 , and Ba(OH)2 on equilibrium pH, and the precipitation zones of the hydroxides for a solution containing 10–2 mol L–1 Mg2 þ , Ca2 þ , and Ba2 þ . pKs;CaðOHÞ2 ¼ 5:19; pKs;BaðOHÞ2 ¼ 3:6; and pKs;MgðOHÞ2 ¼ 11:15

pHMg(OH) start

pHCa(OH) start

2

8

10

2

pH

12

pHBa(OH) start

2

14

0 -2 -4 -6

logS -8 -10 -12 -14

Mg(OH)2 Ca(OH)2 Ba(OH)2

other iron(III) oxide hydrates). Further dehydration by elevated temperatures may even lead to Fe2 O3 . The oxidation of iron(II) to iron(III) has used about 58% of all the oxygen that has ever been formed by photosynthesis on Earth! In addition, 38% has been used for the oxidation of sulfur and sulfide to sulfate. Only 4% of the overall oxygen formed by photosynthesis has survived in our atmosphere, and this 4% now makes up the 20.9% of our present-day atmosphere. Of course, the oxygen content of the atmosphere is dynamic, as a certain part is constantly used for oxidation processes and roughly the same amount is formed by photosynthesis,

5.2 The pH Dependence of Saturation Concentration Fig. 5.5 Dependence of the log S of Ni(OH)2 , Fe(OH)2 , and Co(OH)2 on equilibrium pH. pKs;NiðOHÞ2 ¼ 15:2; pKs;FeðOHÞ2 ¼ 15:1; and pKs;CoðOHÞ2 ¼ 14:9

0

2

119

4

6

pH

8

10

0

12

14

Ni(OH)2 Fe(OH)2

-2

Co(OH)2

-4 pH

-6

6

logS

7

8

0

-8 -1

logS

-10

Fig. 5.6 Dependence of the log S of Fe(OH)3 , Cr(OH)3 , and Ce(OH)3 on equilibrium pH. pKs;CrðOHÞ3 ¼ 29:8; pKs;FeðOHÞ3 ¼ 38:8; and pKs;CeðOHÞ3 ¼ 23:2

-2

-12

-3

-14

-4

pH 0

2

4

6

8

10

12

14

0

-5

-10

logS -15

Fe(OH)3 -20

Cr(OH)3 Ce(OH)3

-25

meaning the overall concentration remains practically constant. It is estimated that about 3  1014 kg a1 of oxygen goes into and out of the atmosphere. The overall amount of O2 in the atmosphere is 1:088  1018 kg. This is a good example of dynamic equilibrium, a steady state equilibrium, and it emphasizes the importance of the element iron to our Earth. There are seasonal oscillations of the atmospheric oxygen content (about ±15 ppm) anticorrelated with CO2 oscillations, and a long-term sliding down caused by the burning of fossil fuels [6].

120

5

Fig. 5.7 Dependence of the log S of Fe(OH)2 , Fe(OH)3 , and a  FeO(OH) on equilibrium pH. pKs;FeðOHÞ2 ¼ 15:1; pKs;FeðOHÞ3 ¼ 38:8; and pKs;aFeOðOHÞ ¼ 41:5

0

2

4

6

pH

Solubility Equilibria

8

10

12

14

0 -2 -4 -6

logS

-8 -10 -12

α -FeO(OH) Fe(OH)2

-14

Fe(OH)3

-16

5.2.3 Solubility Equilibria of Metal Hydroxides in the Presence of Complexing Agents
 Figure 5.8 shows a plot of log S of the aluminum hydroxide AlðOHÞ3 s versus equilibrium pH, considering the formation of soluble hydroxo complexes. Here the hydroxide ion also acts as complexing agent. The function log cAl3 þ ¼ f ðpHÞ is given by the green line. This line results when it is assumed that fAl(OH)3 gs is in

3 þ . The calculation is made equilibrium with Al3 þ ions, i.e., with AlðOH2 Þ6 according to the previously presented theory:


AlðOHÞ3

 s

 Al3 þ þ 3OH

Ks;fAlðOHÞ g ¼ cAl3 þ c3OH 3 s cAl3 þ ¼

Ks;fAlðOHÞ g 3

c3OH

s

Ks;fAlðOHÞ g c3H3 O þ 3 s ¼ Kw3

log cAl3 þ ¼ log Ks;fAlðOHÞ g þ 3pKw  3pH 3 s

ðEquilibrium 5:11Þ ð5:33Þ

ð5:34Þ ð5:35Þ

The green line represents the solubility of fAl(OH)3 gs according to Eq. 5.35 when soluble hydroxo complexes are neglected. For the inclusion of hydroxo complexes, their stability constants must be used. The formation of hydroxo complexes can be considered either as a result of the acid–base reactions of the aqua

5.2 The pH Dependence of Saturation Concentration

121

complex or as a complex formation by ligand exchange. The formation of the first hydroxo complex is described by the following acid–base equilibrium:

2 þ

3 þ AlðOH2 Þ6 + H2 O  AlðOH2 Þ5 OH + H3 O þ

ðEquilibrium 5:12Þ

with the acidity constant: Ka; AlðOH Þ 3 þ ¼ ½ 2 6

c AlðOH Þ OH 2 þ cH3 O þ ½  2 5 c AlðOH Þ 3 þ ½ 2 6

ð5:36Þ

Alternatively, a complex formation equilibrium can be formulated:

AlðOH2 Þ6

3 þ



2 þ + OH  AlðOH2 Þ5 OH + H2 O

ðEquilibrium 5:13Þ

with the stability constant: c AlðOH Þ OH 2 þ ½  2 5 Kstab; AlðOH Þ OH 2 þ ¼ ½  2 5 c AlðOH Þ 3 þ cOH ½ 2 6

ð5:37Þ

The acidity constant and the stability constant are related to one another by the following equation: Ka; AlðOH Þ 3 þ ½ 2 6 Kstab; AlðOH Þ OH 2 þ ½  2 5

¼ cH3 O þ cOH ¼ Kw

ð5:38Þ

where Kw is the autoprotolysis constant (ionic product) of water. Usually, only one constant is given in the literature, and it is also common not to write the water molecules in the formulae, e.g.: Al3 þ þ OH  Al(OH)2 þ

ðEquilibrium 5:14Þ

The stability constant of this complex is often given as K11 , indicating that the complex contains one metal ion and one hydroxide ion. The concentration of Al(OH)2 þ in equilibrium with fAl(OH)3 gs is calculated as follows: K11 ¼

cAlðOHÞ2 þ cAl3 þ cOH

¼ 6:17  109 L mol1 ðat 25  CÞ ½7

ð5:39Þ

From this it follows that: cAlðOHÞ2 þ ¼ K11 cAl3 þ cOH

ð5:40Þ

122

5

cAlðOHÞ2 þ ¼ K11

Solubility Equilibria

Ks;fAlðOHÞ g c3H3 O þ K Ks;fAlðOHÞ g c2H3 O þ 3 s 3 s w ¼ K 11 Kw3 cH3 O þ Kw2

ð5:41Þ

and finally: log cAlðOHÞ2 þ ¼ log K11 þ log Ks;fAlðOHÞ g þ 2pKw  2pH 3

ð5:42Þ

s

The black line in Fig. 5.8 gives the dependence of log cAlðOHÞ2 þ on pH.

The concentration of Al(OH)2þ in equilibrium with fAl(OH)3 gs can be calculated as follows: Al3 þ þ 2OH  Al(OH)2þ K12 ¼

cAlðOHÞ2þ cAl3 þ c2OH

ðEquilibrium 5:15Þ 

¼ 1:0  1019 L2 mol2 ðat 25 CÞ; ½7

cAlðOHÞ2þ ¼ K12 cAl3 þ c2OH cAlðOHÞ2þ

ð5:43Þ ð5:44Þ

Ks;fAlðOHÞ g c3H3 O þ K 2 Ks;fAlðOHÞ g cH3 O þ 3 s 3 s w ¼ K12 ¼ K 12 Kw3 Kw c2H3 O þ

ð5:45Þ

log cAlðOHÞ2þ ¼ log K12 þ log Ks;fAlðOHÞ g þ pKw  pH 3 s

ð5:46Þ

The orange line in Fig. 5.8 represents the dependence of log cAlðOHÞ2þ on pH.

The concentration of Al(OH)0 3 in equilibrium with fAl(OH)3 gs can be calculated as follows: Al3 þ þ 3OH  Al(OH)0 3 K13 ¼

cAlðOHÞ0 3

cAl3 þ c3OH

ðEquilibrium 5:16Þ

¼ 1:0  1027 L3 mol3 ðat 25 CÞ; ½7

cAlðOHÞ0 ¼ K13 cAl3 þ c3OH 3

cAlðOHÞ0 3

Ks;fAlðOHÞ g c3H3 O þ K 3 3 s w ¼ K13 ¼ K13 Ks;fAlðOHÞ g 3 s Kw3 c3H3 O þ log cAlðOHÞ0 ¼ log K13 þ log Ks;fAlðOHÞ g 3 3 s

ð5:47Þ ð5:48Þ

ð5:49Þ ð5:50Þ

5.2 The pH Dependence of Saturation Concentration

123

The blue line in Fig. 5.8 shows the pH dependence of log cAlðOHÞ0 . Note that 3

Al(OH)0 is a dissolved neutral complex, more precisely it is the complex 3

0 AlðOH2 Þ3 ðOHÞ3 . It is interesting that the concentration of Al(OH)0 3 in equilibrium with fAl(OH)3 gs is independent of pH, because Equilibria 5.11 and 5.16 can be summarized as follows: fAl(OH)3 gs  Al(OH)0 3

ðEquilibrium 5:17Þ

In this equilibrium the dissolved OH ions do not participate and hence there is no pH dependence. The pink line in Fig. 5.8 gives the pH dependence of log cAlðOHÞ4 in equilibrium with fAl(OH)3 gs , which is accessible according to the following equations: Al3 þ þ 4OH  Al(OH) 4 K14 ¼

ðEquilibrium 5:18Þ

cAlðOHÞ4 ¼ 3:98  1032 L4 mol4 ðat 25 CÞ; ½7 cAl3 þ c4OH cAlðOHÞ4 ¼ K14 cAl3 þ c4OH

c

AlðOHÞ 4

ð5:51Þ ð5:52Þ

Ks;fAlðOHÞ g c3H3 O þ K 4 Kw 3 s w ¼ K14 ¼ K14 Ks;fAlðOHÞ g 3 s c Kw3 c4H3 O þ H3 O þ

ð5:53Þ

log cAlðOHÞ4 ¼ log K14 þ log Ks;fAlðOHÞ g  pKw þ pH

ð5:54Þ

3

s

The overall solubility of fAl(OH)3 gs , i.e., the solubility taking account of all described equilibria, is the sum of all the dissolved species of aluminum:

 log SfAlðOHÞ g ¼ log cAl3 þ þ cAlðOHÞ2 þ þ cAlðOHÞ2þ þ cAlðOHÞ0 þ cAlðOHÞ4 ð5:55Þ 3

3

s

log SfAlðOHÞ g ¼ log Ks;fAlðOHÞ g þ log 3 s 3 s

c3H3 O þ Kw3

þ K11

c2H3 O þ Kw2

c þ Kw þ K12 H3 O þ K13 þ K14 Kw c H3 O þ

!

ð5:56Þ Figure 5.8 clearly shows that the overall solubility is, at all pH values, determined by the species with the highest solubility: for pH values lower than 4 this is

3 þ

0 AlðOH2 Þ6 ; for pH values around 8 this is AlðOH2 Þ3 ðOHÞ3 ; and for pH values higher than 10 this is Al(OH) 4 . From this it follows that Eq. 5.35 is a good approximation for the overall solubility for pH values below 4, and Eq. 5.54 is a good approximation for pH values higher than 10.

124

5

Solubility Equilibria

pH 0

2

4

6

8

10

12

14

0

0

-2

-2

-4

-4

-6

-6

-8

-8

-10

-10

-12

-12

-14

-14

logci

logS


 Fig. 5.8 Plot of the log S of hydroxide AlðOHÞ3 s versus equilibrium pH, considering the formation of soluble hydroxo complexes [3]

The bold blue line in Fig. 5.8 shows the overall pH dependence of the solubility of fAl(OH)3 gs , taking into account the formation of hydroxo complexes. Although Fig. 5.8 already depicts a rather complex model, it is most probably still distant from reality. This figure has been constructed for didactic reasons. To depict the real system, at the very least the following complexes should be included: Al2 ðOHÞ42 þ , Al3 ðOHÞ54 þ , and Al13 ðOHÞ732þ . The nature of all hydroxo complexes of aluminum has not yet been fully elucidated. A common method for studying such systems is

3 þ to perform a titration of the aqua complex, here AlðOH2 Þ6 , with hydroxide ions and record the equilibrium pH as a function of hydroxide addition. Then, the resulting titration curves are simulated based on a reasonable reaction (equilibrium) model [8] and variation of the acidity and complex stability constant of the assumed (!) species. In some cases, ions are assumed to exist in solution, which have been identified in solid precipitates (as in the case of Al13 ðOHÞ732þ ) by X-ray diffraction,

5.2 The pH Dependence of Saturation Concentration

125

or which have been identified in solution with the help of spectroscopic techniques (as in the case of aluminum by 27 Al NMR). A very serious problem with such an acid–base titration is the fact that the condensation reactions of the hydroxo complexes and also of solid phases may be very slow, so that one cannot always be sure that the titration curve represents, at all pH values, the chemical equilibrium of all species. Sometimes, different reaction schemes can be used to simulate the titration curves with a very similar fitting quality, so that a decision between the different schemes is hard to make. These problems are well beyond the scope of this introductory text, and the interested reader may consult [9] and [10]. In [5] the iron (III) hydroxide system is discussed, similar to Fig. 5.8. The authors consider the formation of some iron(III) minerals, e.g., goethite, hematite, lepidocrocite, and magnetite, as resulting from acid–base reactions, condensations, and precipitations. This paper is an excellent example of the importance of such calculations. If redox equilibria are additionally operative, they must be included as well. Here we suggest the books of Butler [11] and Šůcha and Kotrlý [12].

5.2.4 pH Dependence of the Solubility of Metal Sulfide The pH dependence of the solubility of metal sulfides is presented here as a good example of sparingly soluble salts possessing a dianion. To this group belong, among others, carbonates, oxalates, sulfates, selenides, and tellurides. The dianions of these salts are relative strong bases, so that protonation reactions can play a role in aqueous solutions. In the case of sulfides, the following solubility Equilibria must be considered: fMeSgs  Me2 þ + S2

ðEquilibrium 5:19Þ

fMe2 S3 gs  2Me3 þ + 3S2

ðEquilibrium 5:20Þ

(here are given only the solubility equilibria of metal ions having oxidation states of +2 and +3). The acid–base equilibria involving sulfide S2 , hydrosulfide HS , and hydrogen sulfide H2 S are: H2 S + H2 O  HS + H3 O þ

ðEquilibrium 5:21Þ

HS + H2 O  S2 + H3 O þ

ðEquilibrium 5:22Þ

H2 S has two acidity constants Ka1 and Ka2 . When a solid metal sulfide fMeSgs is in contact with an aqueous solution, the sulfide ions S2 are protonated and the sum of all sulfide species is equal to the equilibrium concentration of metal ions Me2 þ : ð5:57Þ cMe2 þ ¼ cH2 S þ cHS þ cS2

126

5

Solubility Equilibria

c This is true only when H2 S cannot leave the solution (or only in such negligible amounts that the partition between the solution and gas atmosphere is negligible). Furthermore, it is assumed that the metal ions do not undergo complex formation or acid–base reactions. In real cases, such equilibria must be included in most cases.

Equilibrium 5.19 is characterized by the solubility constant: Ks;MeS ¼ cMe2 þ cS2

ð5:58Þ

The solubility (saturation concentration) of fMeSgs is: SfMeSgs ¼ cMe2 þ

ð5:59Þ

With Eq. 5.57 it follows that: SfMeSgs ¼ cH2 S þ cHS þ cS2 ¼

SfMeSgs ¼ cS2

SfMeSgs

Ks;MeS ¼ cMe2 þ

cS2 c2H3 O þ Ka1 Ka2 c2H3 O þ

Ka1 Ka2

S2fMeSgs

SfMeSgs ¼

¼ Ks;MeS

cS2 cH3 O þ þ cS2 Ka2

cH O þ þ 3 þ1 Ka2

c2H3 O þ Ka1 Ka2

SfMeSgs cMe2 þ ¼ Ks;MeS

þ

Ka1 Ka2

c2H3 O þ Ka1 Ka2

! ð5:61Þ

cH O þ þ 3 þ1 Ka2

c2H3 O þ

ð5:60Þ

!

cH O þ þ 3 þ1 Ka2

cH O þ þ 3 þ1 Ka2

ð5:62Þ ! ð5:63Þ

!

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ks;MeS 2 Ks;MeS c þþ c þ þ Ks;MeS Ka1 Ka2 H3 O Ka2 H3 O

ð5:64Þ

ð5:65Þ

It is easy to see that the first term below the square root is due to the formation of H2 S, the second term is due to HS , and the third term is due to S2 . For the pH ranges, in which one of the sulfide forms dominate, approximation equations can be used:

5.2 The pH Dependence of Saturation Concentration

127

(a) pH \ pKa1 ðH2 S dominatesÞ

SfMeSgs

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ks;MeS 2  c þ Ka1 Ka2 H3 O

1 1 1 log SfMeSgs ¼ log Ks;MeS  log Ka1  log Ka2 þ log cH3 O þ 2 2 2 1 1 1 log SfMeSgs ¼  pKs;MeS þ pKa1 þ pKa2  pH 2 2 2

ð5:66Þ ð5:67Þ ð5:68Þ

(b) pKa1 \ pH \ pKa2 ðHS dominatesÞ

SfMeSgs

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ks;MeS ¼ c þ Ka2 H3 O

ð5:69Þ

1 1 1 log SfMeSgs ¼ log Ks;MeS  log Ka2 þ log cH3 O þ 2 2 2

ð5:70Þ

1 1 1 log SfMeSgs ¼  pKs;MeS þ pKa2  pH 2 2 2

ð5:71Þ

  (c) pKa2 \ pH S2 dominates

SfMeSgs ¼

pffiffiffiffiffiffiffiffiffiffiffiffi Ks;MeS

ð5:72Þ

1 log SfMeSgs ¼ log Ks;MeS 2

ð5:73Þ

1 log SfMeSgs ¼  pKs;MeS 2

ð5:74Þ

Figure 5.9 shows a plot of the pH dependence of the solubility ðlog SPbS Þ of PbS. In the case of Me3 þ metal ions forming sulfides Me2 S3 , the pH dependence is calculated as follows:

128

5

Fig. 5.9 pH dependence of the solubility ðlog SPbS Þ of PbS, based on the following constants: pKs;PbS ¼ 27:45; pKa1;H2 S ¼ 7:02; and pKa2;H2 S ¼ 13:9

Solubility Equilibria

pH 0

2

4

6

8

10

12

14

-2

-4

-6

logSPbS

-8

-10

-12

-14

cMe3 þ

Ks;Me2 S3 ¼ c2Me3 þ c3S2

ð5:75Þ

2 cMe3 þ ¼ ðcH2 S þ cHS þ cS2 Þ 3

ð5:76Þ

2 c 2 cH3 O þ 2 cS2 cH3 O þ ¼ þ S þ cS2 3 Ka1 Ka2 Ka2

cMe3 þ

c2H3 O þ cH O þ 2 ¼ cS2 þ 3 þ1 3 Ka1 Ka2 Ka2 cS2

cMe3 þ

cMe3 þ

5 3

2 ¼ 3

! ð5:77Þ

! ð5:78Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ks;Me S ¼ 3 2 2 3 cMe3 þ

ð5:79Þ

! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cH3 O þ cH 3 O þ 3 Ks;Me2 S3 þ þ1 Ka2 c2Me3 þ Ka1 Ka2

2 cH O þ 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cH3 O þ ¼ cMe3 3 þ 3 Ks;Me2 S3 þ 3 þ1 3 Ka1 Ka2 Ka2

cMe3 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2H3 O þ cH O þ 2p 3 Ks;Me2 S3 ¼ þ 3 þ1 3 Ka1 Ka2 Ka2

ð5:80Þ ! ð5:81Þ

! ð5:82Þ

5.2 The pH Dependence of Saturation Concentration



2SfMe2 S3 gs

5 3

53

5 3

2 SfMe2 S3 g

s

5 3

2 cH O þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cH3 O þ ¼ 3 Ks;Me2 S3 þ 3 þ1 3 Ka1 Ka2 Ka2

2 cH O þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cH3 O þ ¼ 3 Ks;Me2 S3 þ 3 þ1 3 Ka1 Ka2 Ka2

SfMe2 S3 g ¼ 2

53

s

S5fMe2 S3 gs

129

! ð5:83Þ

!

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2H3 O þ cH O þ 2p 3 Ks; Me2 S3 þ 3 þ1 3 Ka1 Ka2 Ka2

ð5:84Þ !

!3  3 c2H3 O þ cH3 O þ 2 ¼2 Ks;Me2 S3 þ þ1 3 Ka1 Ka2 Ka2 5

S5fMe2 S3 gs

SfMe2 S3 gs

c2H3 O þ cH O þ 1 ¼ 2 3 Ks;Me2 S3 þ 3 þ1 23 Ka1 Ka2 Ka2

ð5:85Þ

ð5:86Þ

!3

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !3 u u c2H3 O þ 5 cH 3 O þ 1 t ¼ Ks;Me2 S3 þ þ1 22 33 Ka1 Ka2 Ka2

ð5:87Þ

ð5:88Þ

In Fig. 5.10 the pH dependence of the log SMeS of various metal sulfides MeS is shown. In Fig. 5.11 the pH dependence of the log SMe2 S3 of various metal sulfides Me2 S3 is depicted. The calculations, on which Figs. 5.10 and 5.11 are based, were made under the assumption that solid sulfide phases are in equilibrium with the solutions of varying pH, and that an escape of hydrogen sulfide gas is impossible. If an understanding of the well-known hydrogen sulfide separation procedure is aimed for, another model needs consideration: in this modela constant overall concentration of hydrogen

sulfide

CH 2 S ¼ cH2 S þ cHS þ cS2

is supposed to exist, which is equal to the

saturation concentration of H2 S gas

 in water at 20 °C, and which amounts to 1  log CH2 S ¼ 0:93 . In Fig. 5.12 the pH dependence of the loga0:117 mol L rithm of solubilities is given for sulfides of lead, antimony, bismuth, and   manganese Under pKs;PbS ¼ 27:45; pKs;Sb2 S3 ¼ 92:8; pKs;Bi2 S3 ¼ 71:8; pKs;MnS ¼ 15:5 . these conditions the solubilities of bismuth, antimony, and lead sulfide are such that precipitation happens at all pH values, even down to 0, for metal concentrations above 103 mol L1 , whereas manganese remains dissolved and precipitates only in the ammonium sulfide group (pH values between 5 and 8).

130

5

Solubility Equilibria

pH 0

2

4

6

8

10

12

14

0

MnS

-5

FeS -10

PbS

logSMeS

CdS

-15

CuS

-20

HgS

-25

Fig. 5.10 pH dependence of log SMeS of the sulfides MnS, FeS, PbS, CdS, CuS, and HgS. pKs;PbS ¼ 27:45; pKs;MnS ¼ 15:5; pKs;FeS ¼ 17:3; pKs;CdS ¼ 29:0; pKs;CuS ¼ 44:1; pKs;HgS ¼ 44:1 ; pKa1;H2 S ¼ 7:02; and pKa2;H2 S ¼ 13:9

pH 0

2

4

6

8

10

12

14

0 -2

As2S3

-4 -6

logS

-8 -10

PbS

-12 -14

Bi2S3

-16

Fig. 5.11 pH dependence of log S of As2S3, PbS, and Bi2S3. pKs;PbS ¼ 27:45; pKs;As2 S3 ¼ 28:4; pKs;Bi2 S3 ¼ 71:8; pKa1;H2 S ¼ 7:02; and pKa2;H2 S ¼ 13:9

5.3 Coprecipitation

131 pH 0

2

4

6

8

10

12

14

0

-10

logcMe

MnS

-20

PbS -30

Bi2S3 Sb2S3

-40

Fig. 5.12 pH dependence of the log cMe of MnS, PbS Bi2S3, Sb2S3 in  solutions with an overall   of 0:117 mol L1 concentration of hydrogen sulfide CH2 S ¼ cH2 S þ cHS þ cS2

 log CH 2 S ¼ 0:93 pKs;PbS ¼ 27:45; pKs;MnS ¼ 15:5; pKs;Bi2 S3 ¼ 71:8; pKs;Sb2 S3 ¼ 92:8; pKa1;H2 S ¼ 7:02; and pKa2;H2 S ¼ 13:9

5.3

Coprecipitation

The term coprecipitation is specifically used for cases when a minor component B is precipitated together with a major component A, because of specific interactions between these two components. (The same term is also used for the case that two components are precipitated together because they precipitate independently under certain conditions. Such cases can be found in the foregoing part of this chapter.) In 1932 Kolthoff1 precisely defined the coprecipitation of a minor component with a major component as being due to mutual interaction—he gave three possible reasons for coprecipitation [13]: formation of mixed crystals, adsorption, and occlusion. Formation of solid solutions (mixed crystals) between two components A and B Solid solutions can be formed when the radii of two metal ions Menð1Þþ and Menð2Þþ differ by less than 15%, provided both form salts with a common anion. The same n holds true for two anions An ð1Þ and Að2Þ with a common cation. This rule bears the name of William Hume-Rothery,2 who discovered it [14]. Provided the radii are so 1

Izaak Maurits Kolthoff (1894–1993) was a Dutch–US analytical chemist. William Hume-Rothery (1899–1968) was an English metallurgist and materials scientist.

2

132

5

Solubility Equilibria

Fig. 5.13 Mixed crystals of KðClO4 Þ1x ðMnO4 Þx . From left to right: x = 0.0000; 0.0004; 0.0030; and 0.0100 [3]

similar, they can randomly substitute each other, either in the cation sublattice or in the anion sublattice. This is even possible when the charges of the ions differ; however, then charge neutrality leads to defects (missing ions) in the sublattice of the common ion. Generally, the charges of the mutually substituting ions should be the same or differ only by one unit. Furthermore, the crystallographic structures of the two pure solid components should be very similar, the coordination numbers should be equal, and the electronegativities should be very similar. The formation of solid solutions is controlled by thermodynamics, more specifically by mixed phase thermodynamics, and can be described by equilibrium constants. As in the case of liquid solutions, some systems are not completely miscible, but exhibit miscibility gaps. An impressive example of mixed crystal formation is the system potassium perchlorate–potassium permanganate. Here the permanganate and perchlorate anions can substitute one another in crystals having potassium ions as the common cation [15]. Pure potassium perchlorate is colorless, and the mixed crystals show an increasingly pink color with increasing permanganate content (cf. Fig. 5.13). The formation of mixed crystals (solid solutions) is not only an issue in analytical chemistry but plays an important role in many technologies too, including roles in the metallurgy and semiconductor industries. Mixed crystals can be made up of ions as well as of molecules; inorganic and organic constituents are possible. For example, mixed crystals of urea and thiourea can exist. In analytical chemistry, mixed crystals consisting of LaF3 as the major constituent and small concentrations of Eu2 þ in the cation sublattice are used as membranes for fluoride sensitive electrodes, because charge neutrality affords that the Eu2 þ ions create vacations in the fluoride sublattice, and these vacations result in a slight increase of ionic conductivity of the membrane—a prerequisite for the application of the membrane in potentiometric measurements.

5.3 Coprecipitation

133

Oxygen sensors (lambda sensors), used for the control of oxygen content in exhaust gases of combustion engines in automobiles, have a solid electrolyte based on ZrO2 with Y3 þ ions in a cation sublattice. Also here, charge neutrality affords that the presence of Y3 þ ions creates vacancies in the anion (O2 ) sublattice, which guarantees a sufficient ionic conductivity at temperatures of 550–700 °C. The solid electrolyte separates two noble metal electrodes, one in the exhaust gas and one in the atmosphere, and the voltage between the two electrodes is a measure of the oxygen partial pressures in both gases. Adsorption of component B on the surface of A Normally, precipitates possess high surface areas because of their small particles; sometimes these surface areas can be very large. The surfaces of precipitates, especially of metal oxides, hydroxides, and oxide hydrates, are prone to bind metal ions by surface complexation (Chap. 4). Such adsorptive binding is another reason for coprecipitation. It is controlled by thermodynamics and characterized by sorption equilibrium constants. Occlusion of solution droplets containing component B in the precipitate of A The inclusion of solution droplets in a precipitate is called occlusion. This is more or less a random phenomenon, not controlled by thermodynamics (although the wettability, a thermodynamic property, of the surface by the solution may have effects). Occlusion can be minimized by choosing the right precipitation conditions, especially the right precipitation rate: fast precipitation favors occlusion and slow precipitation can minimize it. When a precipitation agent, e.g., a sodium hydroxide solution, is dropped into an iron(III) chloride solution, the locally very high hydroxide concentration may easily lead to the occlusion of the solution (sodium and chloride ions) and the precipitate may, even after washing, still contain sodium and chloride ions. A much better approach is so-called homogeneous precipitation: adding hexamethylenetetramine (also called methenamine or urotropin) to the cold solution does not lead to any precipitate; however, when the solution is slowly heated, hexamethylenetetramine hydrolyzes and forms ammonia and formaldehyde, and the ammonia reacts with the water to provide ammonium ions and hydroxide ions. The latter precipitate iron(III) oxide hydrate. This precipitation takes place from an homogeneous solution, and so occlusion is minimal.

References 1. Bandura AV, Lvov SN (2006) J Phys Chem Ref Data 35:15–30 2. Kotrlý S, Šůcha L (1985) Handbook of chemical equilibria in analytical chemistry. Ellis Horwood Ltd, Chichester 3. Scholz F, Kahlert H (2015) ChemTexts 1:7 4. Harris DC (2007) Quantitative chemical analysis, 7th edn. Freeman and Company, New York 5. Jolivet J-P, Chanéac C, Tronc E (2004) Chem Commun 481–487 6. Petsch ST (2003) The global oxygen cycle. In: Schlesinger WH, Holland HD, Turekian KK (eds) Treatise on geochemistry, vol 8. Elsevier

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7. Martell AE, Motekaitis RJ (1992) Determination and use of stability constants, 2nd edn. Wiley-VCH, New York 8. Sabatini A, Vacca A, Gans P (1992) Coord Chem Rev 120:389–405 9. Brown PL, Sylva RN, Batley GE, Ellis J (1985) J Chem Soc Dalton Trans 1967–1970 10. Perry CC, Shafran KL (2001) J Inorg Biochem 87:115–124 11. Butler JN (1998) Ionic equilibrium: solubility and pH calculations. Wiley, New York 12. Šůcha L, Kotrlý S (1972) Solution equilibria in analytical chemistry. Van Nostrand Reinhold, New York 13. Kolthoff IM (1932) J Phys Chem 36:860–881 14. Raynor GV (1969) Biogr Mems Fell R Soc 15:109–139 15. Johnson GK (1979) J Chem Educ 56:618–619

6

Redox Equilibria

Redox reactions are chemical reactions in which the oxidation state of elements is changing. The oxidation state can only change by accepting or donating electrons. Oxidation means donation of electrons, and reduction means accepting electrons. The oxidation state is identified by the oxidation number, which has only a formal meaning when covalent bonds are involved. In the case of the ionic compound CaCl2, the calcium ion has the oxidation number +2, corresponding to its charge of 2+. The chloride ions have the oxidation number −1 and a charge of 1−. In the case of oxidation numbers, the sign precedes the number, whereas in the case of charges the sign follows the number. In methanol (CH3OH), all hydrogen atoms possess the formal oxidation number +1, oxygen −2, and carbon −2. Formal oxidation numbers of atoms in chemical compounds with covalent bonds are useful to balance the stoichiometry of transferred electrons in redox reactions. However, they should not be regarded as the charges of the atoms. Oxidation numbers are given to the atoms of a compound on the assumption of completely ionic bonds, which is strictly true only for ionic compounds. For components having covalent bonds, it is a useful tool to balance redox equations. In reality (see general chemistry text books) completely ionic and completely covalent bonds are just theoretical limits, and real bonds always have a certain percentage of both types. For calculating oxidation numbers, the following rules hold: (a) Elements in the elementary state always have an oxidation state of 0. This also holds for di- or polyatomic molecules made up of the same element atoms, e.g., O2 and S8. (b) Monoatomic element ions have an oxidation number which is equal to their charge (only that the position of sign changes; see above). (c) Hydrogen has, in all compounds, an oxidation number of +1, with the only exception being in metal hydrides, where it has a number of −1. (d) Oxygen has an oxidation number of −2, except in peroxides −1, superoxides −1/2, OF2 +2, and O2F2 +1. (e) Fluorine has, in all compounds, an oxidation number of −1. © Springer Nature Switzerland AG 2019 F. Scholz and H. Kahlert, Chemical Equilibria in Analytical Chemistry, https://doi.org/10.1007/978-3-030-17180-3_6

135

136

6

Redox Equilibria

Table 6.1 Analogies between redox and Brønsted–Lowry acid–base reactions

Exchanged particle Donator Acceptor Reaction equilibrium Theoretical breakdown of the overall equilibrium in single equilibria

Brønsted–Lowry acid–base reactions

Redox reactions

Proton (H+) Acid (e.g., HB1 ) Base (e.g., B 2)  HB1 + B 2  B1 + HB2

Electron (e–) Reducing agent (reduced form “Red”) Oxidizing agent (oxidized form “Ox”) m1 Red1 þ m2 Ox2  m1 Ox1 þ m2 Red2 (charges are neglected here) (a) Red1  Ox1 þ n1 e (b) Ox2 þ n2 e  Red2 ðm1 n1 ¼ m2 n2 ¼ zÞ (charges are neglected here)

(a) HB1  H þ + B 1 (b) H þ + B 2  HB2

(f) The halogens chlorine, bromine, iodine, and astatine have, when in compounds, an oxidation number of −1, except in compounds with oxygen and with other halogens, which have a higher atomic number. (g) In neutral compounds, the sum of the mathematical products of oxidation numbers and corresponding stoichiometric coefficients is 0, and in ions this sum corresponds to the charge of the ion. (h) In the case of organic compounds, the oxidation numbers of carbon atoms are calculated for each carbon atom individually. (i) Many elements can be present in one compound in different oxidation states. (j) In compounds having covalent bonds, the electrons are assigned to the more electronegative element. Equally, electronegative elements share electrons so that each element gets the same number of electrons. The electron is an elementary particle possessing a charge of –1, equal to −1.602176565(35)
10−19 C (Coulombs). Formally, a so-called “classical” radius of 2.818
10−15 m has been calculated; however, currently it is assumed that an electron is a point charge with 10−18 m supposed as the upper limit of its radius. The calculated radius is of a similar size to that of a proton (0.875
10−15 m), also considered to be uncertain, not to mention the fact that in any case, protons, like atoms, have no strictly defined spatial limits. Hence, electrons and protons are the smallest particles exchanged in chemical reactions. Only in a vacuum can they exist freely, and in all matter they are bonded to other particles. This brings about a number of analogies between redox and Brønsted–Lowry acid–base reactions, as listed in Table 6.1.

6.1 Quantitative Treatment of Redox Equilibria

6.1

137

Quantitative Treatment of Redox Equilibria

Like all other chemical equilibria, redox equilibria are characterized by equilibrium constants, which are based on activities. However, it became customary to use electric potentials (voltages) instead, because—in certain cases—redox equilibria can be realized in galvanic half-cells, where they establish a Galvani potential difference D/1;2 , which can be related to the Galvani potential difference of a second half-cell, so that the difference between the two Galvani potential differences can be measured as the voltage of the galvanic cell. A galvanic half-cell is built up of two adjacent phases, one being an electron conductor (e.g., silver metal), and the other an ion conductor (e.g., an aqueous solution of silver nitrate) (Fig. 6.1). Between these two phases a potential difference is established, which, despite temperature and pressure, depends only on the activities of the oxidized (e.g., Ag+) and reduced forms (e.g., Agmetal) of the redox couple.1 When based on molar ratios, the activity of a pure phase (like silver metal) is unity, the Galvani potential difference of the electrode depends only on the molar activity of the potentialdetermining dissolved species. In the case of the Ag+/Agmetal electrode, these are the silver ions. When the reduced form is also a dissolved species (e.g., Fe2+ ions), the molar activity of the reduced form is also responsible for the Galvani potential difference. In textbooks on physical chemistry and electrochemistry, the so-called Nernst equation (Eq. 6.1) is derived and presented as follows: D/ ¼ D/— þ

RT aOx ln nF aRed

ð6:1Þ

where D/ is the Galvani potential difference; D/— is the standard Galvani potential difference; R is the universal gas constant (8.3144 J mol–1 K–1); T is the absolute temperature in K; n is the number of electrons exchanged between the reduced and oxidized forms; F is the Faraday constant (charge of one mole of electrons, 96485.33289(59) C mol–1); aOx is the activity of the oxidized form; and aRed is the activity of the reduced form. Nernst derived this equation for a metal electrode in contact with a solution of ions of the same metal. Peters published this equation for the case of a dissolved redox pair [1].2 The potential difference DEcell (also called the electromotive force, emf) of the galvanic cell can be measured when the electron conducting phases of the galvanic cell are connected by a high-impedance voltmeter (Fig. 6.1). The high-impedance is

1

A Galvani potential difference is established across all possible interfaces between two phases. This cannot be discussed here in detail but we refer the reader to the paragraph about glass electrodes in Chap. 7.5.3.1. 2 In the case of the Nernst equation, a problem occurs that is similar to the problem of defining pH: since at least one of the two forms of Ox or Red are ionic, and because single ion activities are, for principle reasons, inaccessible and only approximately determinable, the Galvani potential differences of half-cells and the potential differences of Galvani cells are not precisely defined, which, fortunately, is not a serious issue for applications.

138

6

Redox Equilibria

Fig. 6.1 Scheme of a galvanic half-cell and a galvanic cell

necessary in order to minimize the current flowing through the voltmeter, so that the activities of the oxidized and reduced forms will not be changed by faradaic reactions. The Galvani potential difference of a half-cell is principally inaccessible, as its measurement would introduce additional interfaces with connecting phases, which would produce additional Galvani potential differences. In order to characterize galvanic half-cells by their Galvani potential differences, it is necessary to relate all these differences to one and the same reference half-cell. For aqueous solutions, the so-called standard hydrogen electrode (SHE) with the redox system H3 O þ =H2 is chosen for that purpose. This redox partners “hydrated protons” and hydrogen are in a reversible equilibrium at the interface between the aqueous solution and platinum, more specifically, platinum black, which is a finely dispersed form of platinum, deposited on compact platinum metal.3 The SHE consists of a platinum plate covered by platinum black and inserted in an aqueous acid solution with an activity aH3 O þ ¼ 1. The Galvani potential difference established at the aqueous solution | platinum interface is strictly unknown, and arbitrarily set to zero volts at all temperatures and pressures, which is suitable as a convention but strictly speaking wrong. It is possible to estimate the real Galvani potential difference of the SHE to be –4.44 (±0.02) V [2]. However, for building up a thermodynamic scale, allowing the measurement or calculation of the potential difference of galvanic cells, this value is meaningless, because all half-cells refer to the SHE, and so the absolute value of 3

Platinum black is highly active in catalyzing the oxidation of hydrogen gas with air; a property used by the German chemist Wolfgang Döbereiner (1780–1849) in constructing a lighter (“Döbereiner’s lamp” or “Döbereiner’s tinderbox”).

6.1 Quantitative Treatment of Redox Equilibria

139

the SHE cancels.4 Combining the SHE as one half-cell with any other half-cell, and measuring the voltage between the two, yields the conventional electrode potential of the other half-cell, with the convention that ESHE ¼ D/SHE ¼ 0 V. The physical meaning of this measurement is that the two redox pairs of the two half-cells are compared with respect to their reducing and oxidizing power. The combination of . the Ag|Ag+ electrode with the SHE is symbolically written as follows: Pt|H2 .. Ag+| Ag. The vertical line | represents the interface between the two phases, the vertical . interrupted line .. symbolizes the contact between two electrolyte solutions, the so-called liquid junction. There are possibilities to avoid the liquid junction in special cells, where the electrolyte of both half-cells is identical. In the case of a liquid junction, an additional potential drop builds up across this junction, the so-called liquid-junction potential. It is of kinetic origin, because it is caused by the different diffusion rates of ions constituting the electrolytes. The liquid-junction potential can be minimized using so-called salt bridges, especially those containing a saturated KCl solution. The reaction of a galvanic cell is written so that the spontaneous electron flow goes from the left-hand side to the right-hand side. In the case of the . example Pt|H2 .. Ag+|Ag it follows: 1 H2 þ Ag þ þ H2 O  H3 O þ þ Ag 2

ðEquilibrium 6:1Þ

On the left-hand side, the electrons are with hydrogen, whereas on the right-hand side they are with silver. The convention that the reaction must be written in the direction of electron flow for the oxidized species of the second half-cell, is the so-called Stockholm convention from 1953. This means that all electrode reactions of the second half-cells are written as reductions (here the reduction of Ag+ ions). By definition, the potential difference of the galvanic cell (the emf, or the voltage of the cell) is the potential of the right half-cell minus the potential of the left half-cell: DEcell ¼ D/right halfcell  D/left halfcell ¼ D/Ag þ =Ag  D/SHE

ð6:2Þ

The consequence of the Stockholm convention is that all oxidizing agents that are able to oxidize hydrogen in spontaneous reactions have positive standard potentials (see subsequent text), and oxidizing agents which are incapable of this, have negative standard potentials. Since by convention it holds that D/SWE ¼ ESWE ¼ 0 V, it follows that DEcell ¼ DEAg þ =Ag . Thus, the cell potential defined by Eq. 6.2 is a measure of the oxidizing power of the oxidized species of 4

Strictly speaking, the combination of half-cells will only then give thermodynamically based potential differences, when both half-cells operate reversibly, i.e., reduction and oxidation need to be unhindered. In reality, this is rare. See subsequent discussion on redox systems coupled to acid– base equilibria and also the discussion on redox titrations in Chap. 7.

140

6

Redox Equilibria

the second half-cell for the oxidation of hydrogen. (In the past, another convention —the so-called American convention—was used, where the oxidation reaction of the second half-cell was combined with SHE [3]. This yields opposite signs for the second half-cell.) The Nernst equation for the cell potential of a redox system in combination with the SHE is now: —  EOx=Red ¼ EOx=Red þ

RT aOx ln nF aRed

ð6:3Þ

—  Realizing the condition aOx ¼ aRed yields: EOx=Red ¼ EOx=Red . Under this condition, —  the cell potential is equal to the standard potential EOx=Red of the redox system. If the redox system is coupled to an acid–base equilibrium, the activity aH3 O þ also has to be unity for defining the standard potential. Now it is possible to arrange the standard potentials of all redox systems in a series of increasing values. This is called the galvanic series of electrode reactions (Table 6.2).

c For a correct interpretation of standard potentials, one should remember that the standard potential is a measure of the oxidizing power of Ox+ in the reaction:

1 H2 þ Ox þ þ H2 O  H3 O þ þ Red 2

ðEquilibrium 6:2Þ

c Of course, the oxidized form can have other charges or no charge at all— this requires a reformulation of the latter equilibrium.

Without derivation and with reference to textbooks on physical chemistry and electrochemistry, it is now necessary to give the relation between cell potential E, the Gibbs Energy G, and the activities of the oxidized and reduced forms: DG ¼ nFE ¼ RT ln

aOx aRed

ð6:4Þ

The same relation exists for standard values and the equilibrium constant KOx=Red : — —  ¼ nFEOx=Red ¼ RT ln KOx=Red DGOx=Red

ð6:5Þ

This relation shows that, as mentioned in the introduction of this chapter, standard potentials (and generally cell potentials) can be used instead of equilibrium constants because they are related to one another: —  EOx=Red ¼

RT ln KOx=Red nF

ð6:6Þ

6.1 Quantitative Treatment of Redox Equilibria

141

Table 6.2 Galvanic series of electrode reactions (or half-cell reactions)a Half-cell

Electrode reaction

E— (V vs. SWE)

Li+|Li

Li þ þ e  Li

−3.040

þ

+



−2.924

Rb þ e  Rb K þ þ e  K Cs þ þ e  Cs

Rb |Rb K+|K Cs+|Cs Ca2+|Ca

−2.924 −2.923 −2.76

Ca2 þ þ 2e  Ca Na þ þ e  Na

+

Na |Na Mg2+|Mg

Mg

2þ

−2.713 −2.375



þ 2e  Mg

Al3+|Al

Al3 þ þ 3e  Al

−1.706

Zn2+|Zn

Zn2 þ þ 2e  Zn

−0.7628

Cr3+/Cr2+|Ptb

Cr3 þ þ e  Cr2 þ

−0.41

2+

2þ

−0.409



þ 2e  Fe

Fe |Fe

Fe

Cd2+|Cd

Cd2 þ þ 2e  Cd

−0.4026

Ni2+|Ni

Ni2 þ þ 2e  Ni

−0.23

Pb2+|Pb

Pb2 þ þ 2e  Pb 2H þ þ 2e  H2

−0.1263

H2/H+aq|Pt 2+ +

0.0000 +0.167

Cu /Cu |Pt

Cu2 þ þ e  Cu þ

Cu2+|Cu

Cu2 þ þ 2e  Cu
3
4 þ e  FeðCNÞ6 FeðCNÞ6
3
4 þ e  WðCNÞ8 WðCNÞ8
3
4 þ e  MoðCNÞ8 MoðCNÞ8

[Fe(CN)6]3−/[Fe(CN)6]4−|Pt [W(CN)8]3−/[W(CN)8]4−|Pt 3−

4−

[Mo(CN)8] /[Mo(CN)8] | Pt Ag+|Ag Hg22 þ |Hg O2, H+/H2O|Pt 3+ + Cr2 O2 7 /Cr , H |Pt +

Au |Au + 2+ MnO 4 , H /Mn |Pt

Ag þ þ e  Ag 1=2O2 þ 2H þ þ 2e  H2 O

Au

+0.457 +0.725 +0.7996 +0.7961

Hg22 þ þ 2e  2Hg þ Cr2 O2 7 þ 14H þ 

+0.3402 +0.356



þ 6e  2Cr

3þ

þ 7H2 O

þ e  Au

þ  2þ MnO þ 4H2 O 4 þ 8H þ 5e  Mn

+1.229 +1.36 +1.83 +1.491

+2.42 H4XeO6, H+/XeO3, |Pt H4 XeO6 þ 2H þ þ 2e  XeO3 þ 3H2 O F2/F−|Pt F2 þ 2e  2F +2.866 a Some half-cells are given with Pt as inert electrode material. Here, Pt is just given as an example of an inert electrode material, and the standard potentials must be understood as being independent of the electrode material b For a pair of dissolved redox species, the half-cell is written as follows: Ox/Red | “inert metal”

where KOx=Red is the equilibrium constant of Equilibrium 6.2. c It is important not to forget that the standard potentials given in Table 6.2 all relate to water being used as the solvent. In other solvents, other data needs to be used—even the order of the half-cells may be inverted in other solvents. This was understood for the first time by the German physical chemist Karl

142

6

Redox Equilibria

Fredenhagen (1877–1949), who was a student of Wilhelm Ostwald and Walther Nernst. Furthermore, it is important to indicate the forms of the redox species as —  : it makes a big difference indices of the potentials, e.g., EOx=Red and EOx=Red
3 þ
3 whether Ox is FeðH2 OÞ6 or FeðCNÞ6 and whether Red is
2 þ
4 or FeðCNÞ6 . FeðH2 OÞ6

6.2

Calculating the Equilibrium Constants of Redox Equilibria

Although Eq. 6.6 allows the calculation of the equilibrium constant of Equilibrium 6.2, it fails to allow the calculation of the equilibrium constant of Equilibrium 6.3: m1 Red1 þ m2 Ox2  m1 Ox1 þ m2 Red2

ðEquilibrium 6:3Þ

However, this can be easily achieved when the latter equilibrium is formally split into the following two equilibria: Ox1 þ n1 e  Red1

ðEquilibrium 6:4Þ

Ox2 þ n2 e  Red2

ðEquilibrium 6:5Þ

The respective Nernst equations are: —  EOx1 =Red1 ¼ EOx þ 1 =Red1

RT aOx1 ln n1 F aRed1

ð6:7Þ

—  EOx2 =Red2 ¼ EOx þ 2 =Red2

RT aOx2 ln n2 F aRed2

ð6:8Þ

The standard potentials and equilibrium constants are related as follows: —  n1 FEOx ¼ RT ln K1 1 =Red1

ð6:9Þ

—  n2 FEOx ¼ RT ln K2 2 =Red2

ð6:10Þ

6.2 Calculating the Equilibrium Constants of Redox Equilibria

143

The Equilibria 6.4 and 6.5 both relate to the reaction of oxidized forms with hydrogen, and hence Eqs. 6.9 and 6.10 give the equilibrium constants of these two equilibria. The stoichiometry needs to fulfill relation m1 n1 ¼ m2 n2 , with the product given the symbol z: m 1 n1 ¼ m 2 n2 ¼ z

ð6:11Þ

Then, the equilibrium constant of Equilibrium 6.3 follows as: K¼

2 amRed am 1 K m2 2 Ox1 ¼ 2m1 m2 m1 aOx2 aRed1 K1

ð6:12Þ

The following rearrangements yield an equation for calculating any redox equilibrium: ln K ¼ m2 ln K2  m1 ln K1 ln K ¼ m2

n2 F — n1 F — E E  m1 RT Ox2 =Red2 RT Ox1 =Red1

ln K ¼ log K ¼ z

ð6:13Þ

 zF  — —  EOx2 =Red2  EOx 1 =Red1 RT

ð6:14Þ ð6:15Þ

  F —  —  EOx  E =Red Ox =Red 2 2 1 1 2:303RT

ð6:16Þ

 16:9  —  —
z EOx2 =Red2  EOx 1 =Red1 ½V

ð6:17Þ

For 25 °C it follows that5: log K ¼

 K ¼ 10

16:9
z ½V

 —  EOx

2 =Red2

 — EOx

1 =Red1

ð6:18Þ

The last two equations allow us to understand that the equilibrium constants of most redox equilibria are either very large or very small, since even for rather small differences between the standard potentials of the two redox systems, the multiplier 16:9
z produces equilibrium constants which strongly deviate from unity. ½V —  c For using Eq. 6.17 and Eq. 6.18 it is important to make sure that EOx is 2 =Red2 the standard potential of the oxidant of Equilibrium 6.3!

5

2:303RT=F equals 0.05916 V at 25 °C. For simplicity we use here 0.059 V.

144

6

Redox Equilibria

The following examples show how the equilibrium constants can be calcu
2 þ lated (i) for the oxidation of FeðH2 OÞ6 with Cr2 O2 7 , and (ii) for the
3 þ
3 þ with FeðH2 OÞ6 : oxidation of CrðH2 OÞ6
2 þ
3 þ þ  2 CrðH2 OÞ6 Cr2 O2 7 þ 14H3 O þ 6 FeðH2 OÞ6
3 þ þ 6 FeðH2 OÞ6 þ 9H2 O

ðiÞ

ðEquilibrium 6:6Þ





3 þ þ 6 FeðH2 OÞ6 þ 9H2 O  Cr2 O2 7
 2þ þ 14H3 O þ þ 6 FeðH2 OÞ6

ðiiÞ

CrðH2 OÞ6

3 þ

ðEquilibrium 6:7Þ

The following standard potentials need to be used:
3 þ þ  þ 9H2 O ðaÞ Cr2 O2 7 þ 14H3 O þ 6e  2 CrðH2 OÞ6

standard potential: E—

Cr2 O2 7 =½CrðH2 OÞ6 

ðbÞ




FeðH2 OÞ6

3 þ

3þ

ðEquilibrium 6:8Þ

¼ þ 1:36 V


2 þ þ e  FeðH2 OÞ6

ðEquilibrium 6:9Þ

standard potential: E— 3þ 2 þ ¼ 0:77 V ½FeðH2 OÞ6  =½FeðH2 OÞ6  Applying Eq. 6.17 yields:   16:9 — 
6 E— 2  E 3þ 3þ 2þ Cr2 O7 =½CrðH2 OÞ6  ½V  ½FeðH2 OÞ6  =½FeðH2 OÞ6  101:4 ð1:36½V  0:77½VÞ  59:8 ¼ ½V 

ðiÞ log K ¼

ð6:19Þ K ¼ 1059:826

ð6:20Þ

6.2 Calculating the Equilibrium Constants of Redox Equilibria

145

  16:9 —  — 
6 E ðiiÞ log K ¼ 3þ 2þ  E 3þ Cr2 O2 ½V ½FeðH2 OÞ6  =½FeðH2 OÞ6  7 =½CrðH2 OÞ6  101:4 ð0:77½V  1:36½VÞ  59:8 ¼ ½V ð6:21Þ K ¼ 1059:826

ð6:22Þ


2 þ These results show that the oxidation of FeðH2 OÞ6 with Cr2 O2 7 is possible, since its equilibrium constant is very large, whereas the oxidation of
3 þ
3 þ CrðH2 OÞ6 with FeðH2 OÞ6 is impossible.

6.3

Formal Potentials

In the last paragraph, the calculation of equilibrium constants was based on standard potentials, which means that all participating chemical species are in the state for which the standard potentials have been defined (e.g., in the case given earlier
2 þ
3 þ
3 þ Cr2 O2 , CrðH2 OÞ6 , and FeðH2 OÞ6 ). However, in the 7 , FeðH2 OÞ6 majority of real cases, the reacting species are present in different forms, iron(III) may perhaps be in the form of chloro complexes or hydroxo complexes (Chap. 3), and the activity of H3 O þ ions is not unity, as is assumed for the standard potential of Equilibrium 6.8. To account for all such deviations, conditional equilibrium constants need to be considered. In the case of redox equilibria, these are so-called 0 —  formal potentials (also called real potentials) Ec;Ox=Red , with the index c representing conditional. In principle, formal potentials can be calculated provided all side reactions (side equilibria) are known together with their equilibrium constants. In many cases, the latter is illusive because not all chemical equilibria are known, and even when they are known, their equilibrium constants may be unknown. c As already discussed in the context of the conditional stability constants of complexes, the philosophy here is the same: with the help of experimental data, equilibria are treated and calculated without detailed knowledge of the chemistry and thermodynamics of all involved equilibria. Finally, the formal potential of an oxidant expresses its oxidation power under very well-defined and controlled chemical conditions (sic.!).

146

6

Redox Equilibria


3 þ
2 þ The redox couple FeðH2 OÞ6 = FeðH2 OÞ6 is well-suited for discussing the definition of formal potentials on different levels of accuracy in describing a real situation. The standard potential E— 3þ 2 þ = 0.77 V is defined for iron(III) ½FeðH2 OÞ6  =½FeðH2 OÞ6  and iron(II) ions which exclusively have water as ligands, so that the following equilibrium is operative:


Fe(H2 OÞ6

3 þ


2 þ þ e  Fe(H2 OÞ6

ðEquilibrium 6:10Þ

Using the activities of hexaaqua ions, the Nernst equation follows in the form: a FeðH OÞ 3 þ RT ½ 2 6 E FeðH OÞ 3 þ = FeðH OÞ 2 þ ¼ E— ln þ 3þ 2þ ½ 2 6 ½ 2 6 F a FeðH OÞ 2 þ ½FeðH2 OÞ6  =½FeðH2 OÞ6  ½ 2 6

ð6:23Þ

It is now possible to define different formal potentials, depending on what is included, i.e., activity coefficients only or side reaction coefficients also.

6.3.1 Formal Potentials That Include Activity Coefficients To account for activity coefficients deviating from unity, instead of Eq. 6.23 one may write: 3þ f c FeðH OÞ 3 þ RT ½FeðH2 OÞ6  RT ½ 2 6 ln ln E FeðH OÞ 3 þ = FeðH OÞ 2 þ ¼ E— þ 3þ 2þ þ ½ 2 6 ½ 2 6 F f FeðH OÞ 2 þ F c FeðH OÞ 2 þ ½FeðH2 OÞ6  =½FeðH2 OÞ6  ½ 2 6 ½ 2 6

ð6:24Þ Since activity coefficients are constant for a certain solution composition (but possibly not unity), a formal potential can be defined which only considers these deviations from unity. The sum of the standard potential and the term comprising the activity coefficients defines this formal potential: 3þ f 0 RT ½FeðH2 OÞ6  ln E— þ ¼ E— ð6:25Þ 3þ 2þ 3þ 2þ c; ½FeðH2 OÞ6  =½FeðH2 OÞ6  F f FeðH OÞ 2 þ ½FeðH2 OÞ6  =½FeðH2 OÞ6  ½ 2 6

and the Nernst equation can be written as follows: c FeðH OÞ 3 þ 0 RT ½ 2 6 ln E FeðH OÞ 3 þ = FeðH OÞ 2 þ ¼ E— ð6:26Þ 3þ 2þ þ ½ 2 6 ½ 2 6 c; ½FeðH2 OÞ6  =½FeðH2 OÞ6  F c FeðH OÞ 2 þ ½ 2 6

6.3 Formal Potentials

147

The deviations of these formal potentials from the standard potentials are usually small, since activity coefficients do not vary very much.

6.3.2 Formal Potentials That Include Activity Coefficients and Side Reaction Coefficients
3 þ When side reactions are involved it is not only FeðH2 OÞ6 that is present, but
2 þ
1 þ
0 , FeðH2 OÞ4 ðOHÞ2 , FeðH2 OÞ3 ðOHÞ3 , and many also FeðH2 OÞ5 OH other, perhaps even unknown, complexes. The same is true for iron(II), although the formation of hydroxo complexes is less pronounced (Chap. 3). As in the case of complex formation equilibria, side reaction coefficients aFeðIIIÞ and aFeðIIÞ can be defined for iron(III) and iron(II): aFeðIIIÞ ¼

¼

c FeðH OÞ 3 þ þ c FeðH OÞ OH 2 þ ½ 2 6 ½ 2 5 

c FeðH OÞ 3 þ ½ 2 6 þ c FeðH OÞ ðOHÞ 1 þ þ c FeðH OÞ ðOHÞ 0 þ


½ 2 4 ½ 2 3 2 3

c FeðH OÞ 3 þ ½ 2 6 c0FeðIIIÞ ð6:27Þ

aFeðIIÞ ¼

¼

c FeðH OÞ 2 þ þ c FeðH OÞ OH 1 þ ½ 2 6 ½ 2 5 

c FeðH OÞ 2 þ ½ 2 6 þ c FeðH OÞ ðOHÞ 0 þ c FeðH OÞ ðOHÞ 1 þ


½ 2 4 ½ 2 3 2 3

c FeðH OÞ 2 þ ½ 2 6 c0FeðIIÞ ð6:28Þ

The sum of the concentrations of all iron(III) species is c0FeðIIIÞ , and that of all iron (II) species is denoted by c0FeðIIÞ . For the equilibrium concentrations c FeðH OÞ 3 þ and ½ 2 6 c FeðH OÞ 2 þ , the following relations are used in Eq. 6.26: ½ 2 6 c FeðH OÞ 3 þ ¼ aFeðIIIÞ c0FeðIIIÞ ½ 2 6

ð6:29Þ

c FeðH OÞ 2 þ ¼ aFeðIIÞ c0FeðIIÞ ½ 2 6

ð6:30Þ

148

6

Redox Equilibria

This leads to: 0

0 RT aFeðIIIÞ cFeðIIIÞ ln E FeðH OÞ 3 þ = FeðH OÞ 2 þ ¼ E— ð6:31Þ 3þ 2þ þ ½ 2 6 ½ 2 6 c;½FeðH2 OÞ6  =½FeðH2 OÞ6  F aFeðIIÞ c0FeðIIÞ

0 0 RT aFeðIIIÞ RT cFeðIIIÞ ln ln þ þ E FeðH OÞ 3 þ = FeðH OÞ 2 þ ¼ E— 3þ 2þ ½ 2 6 ½ 2 6 c;½FeðH2 OÞ6  =½FeðH2 OÞ6  F F aFeðIIÞ c0FeðIIÞ

ð6:32Þ Since the side reaction coefficients are constants for certain defined solution conditions, the respective terms can be summed up with those of the activity coefficients in order to define an extended formal potential: E FeðH OÞ 3 þ = FeðH OÞ 2 þ ½ 2 6 ½ 2 6

0 RT cFeðIIIÞ ln 0 ¼ Ec;FeðIIIÞ=FeðIIÞ þ F cFeðIIÞ 0  —

ð6:33Þ

This extended formal potential is: 3þ f 0 RT ½FeðH2 OÞ6  RT aFeðIIIÞ —  ln ln ¼ E— þ þ Ec;FeðIIIÞ=FeðIIÞ 3þ 2þ F f FeðH OÞ 2 þ F aFeðIIÞ ½FeðH2 OÞ6  =½FeðH2 OÞ6  ½ 2 6 ð6:34Þ

The following examples are given to show the importance of formal potentials for the calculation of equilibrium constants. Let us consider the
2 þ oxidation of FeðH2 OÞ6 by Cr2 O2 7 . In the first stage, only the side reaction coefficients of iron ions are taken into account; later the pH dependence of the chromium(VI)/chromium(III) system will be included. When iron(II) is titrated with Cr2 O2 7 , at pH = 3 with a 0.01 M EDTA solution, the following side reaction coefficients are given in the literature [4]: log aFeðIIIÞ;pH ¼ 3; 0:01 M EDTA ¼ 12:0 log aFeðIIÞ;pH ¼ 3; 0:01 M EDTA ¼ 1:8 Assuming that the activity coefficients are unity, the formal potential of iron(III)/iron(II) is, at 25 °C:

6.3 Formal Potentials

149

0 1012:0 —  —  Ec;FeðIIIÞ=FeðIIÞ;pH ¼ E þ 0:059½V] log 3 þ 2 þ ¼ 3; 0:01 M EDTA ½FeðH2 OÞ6  =½FeðH2 OÞ6  101:8 ð6:35Þ

0

—  Ec;FeðIIIÞ=FeðIIÞ;pH ¼ 3; 0:01 M EDTA ¼ 0:77½V] þ 0:059½V] log

¼ 0:1682 [V]

1012:0 101:8

ð6:36Þ

The formal potential of chromium(VI)/chromium(III) is affected both by the side reaction coefficient of chromium(III) and the pH dependence of the chromate system: aH O þ RT aCr2 O2 ln 2 7 3 þ 6F a 3þ ½CrðH2 OÞ6  14

ECr

ECr

½CrðH2 OÞ6 

2 2 O7 =

2 2 O7 =

3þ

½CrðH2 OÞ6 

3þ

¼E

— 

Cr2 O2 7 =½CrðH2 OÞ6 

¼ E—

Cr2 O2 7 =

3þ

½CrðH2 OÞ6 

3þ

þ

ð6:37Þ

aCr2 O2 RT RT 7 ln a14 ln 2 H3 O þ þ 6F 6F a ½CrðH2 OÞ6  ð6:38Þ

The side reaction coefficient of chromium(III) in a 0.01 M ethylenediaminetetraacetic acid (EDTA) solution is, at pH = 3: log aCrðIIIÞ;pH ¼ 3; 0:01 M EDTA ¼ 10:5. From this follows the formal potential of chromium(VI)/chromium(III): 0

 — Ec;ChromðVIÞ=ChromðIIIÞ ¼ E—

Cr2 O2 7 =½Cr ðH2 OÞ6 

3þ

þ

RT RT 1 ln a14 ln H3 O þ þ 6F 6F a2 ½CrðH2 OÞ6  ð6:39Þ

For 25 °C it follows that: 0

  0:059 ½V]
14 6  0:059 1 3 [V] log 21
log 10 þ 6 10

—  Ec;ChromðVIÞ=ChromðIIIÞ ¼ 1:36½V] þ

ð6:40Þ

0

—  Ec;ChromðVIÞ=ChromðIIIÞ ¼ 1:36½V]  0:413½V] þ 0:2065½V] ¼ 1:1535 [V]

ð6:41Þ

150

6

Redox Equilibria

The equilibrium constant of oxidation of Fe(II) with dichromate is as follows:  0 16:9  —0  —
6 Ec;chromiumðVIÞ=chromiumðIIIÞ  Ec;FeðIIIÞ=FeðIIÞ;pH ¼ 3; 0:01 M EDTA ½V 16:9
6ð1:1535½V]  0:1682½V]Þ  99:91 ¼ ½V

log K ¼

ð6:42Þ Comparison of this result with that of Eq. 6.20 reveals that under these conditions the equilibrium constant is larger by about 40 orders of magnitude. This is the result of strong complexation of both Fe(III) and Cr(III) by EDTA. Since Fe(III) and Cr(III) are the reaction products, this complexation strongly favors product formation.

c The discussion of formal potentials is given in order to make clear the following essentials:

1. Under real conditions, it is mandatory to use formal potentials instead of the tabulated standard potentials. 2. Formal potentials may even reverse the equilibrium, making possible reactions that are impossible under standard conditions, or vice versa.

6.4

pH-Dependent Redox Potentials

Two cases need to be distinguished: (1) redox equilibria coupled to acid–base equilibria, that can be experimentally separated, and (2) redox equilibria with inherently coupled acid–base equilibria. An example of case (1) is the quinone/ hydroquinone system; an example of case (2) is the dichromate/chromium(III) system. Both examples will be explained in detail.

6.4.1 Redox Equilibria with Coupled Acid–Base Equilibria That Can Be Experimentally Separated The redox equilibrium quinone/hydroquinone Since many different quinones are known, we refer here to 1,4-benzoquinone and 1,4-dihydroxybenzene (hydroquinone) (Fig. 6.2).

6.4 pH-Dependent Redox Potentials

151

O

OH

O

OH

C6 H 4 ( OH )2 ≡ H 2Q

C6 H 4 O 2 ≡ Q

Fig. 6.2 1,4-benzoquinone and 1,4-dihydroxybenzene (hydroquinone)

Hydroquinone is a diphenol and thus a diprotic acid: H2 Q + H2 O  H3 O þ + HQ

ðEquilibrium 6:11Þ

HQ þ H2 O  H3 O þ þ Q2

ðEquilibrium 6:12Þ

The corresponding acidity constants are pKa1 ¼ 9:8 and pKa2 ¼ 11:4. Above pH 11.4 the completely deprotonated form Q2 dominates in aqueous solution. This means that above pH 11.4 only the pure redox equilibrium: Q + 2e  Q2

ðEquilibrium 6:13Þ

 — with the standard potential EQ=Q 2 ¼ 0:074 V needs to be considered. Below that

pH value, the protonated forms HQ and H2 Q are also present and need to be accounted for when calculating the redox potential, because their presence decreases the equilibrium concentration of Q2 . Most importantly, the oxidized form 1,2-benzoquinone and the protonated reduced form 1,2-dihydroxybenzene form a very stable charge-transfer complex called quinhydrone. This is a sparingly soluble crystalline dark green compound which forms, in aqueous solution, equal and small concentrations of both oxidation states:  cOx ¼ CRed

ð6:43Þ

 where CRed denotes the overall concentration of all reduced forms:  ¼ cH2 Q þ cHQ þ cQ2 CRed

ð6:44Þ

152

6

Redox Equilibria

Since cOx ¼ cQ , it follows that:  cQ ¼ cH2 Q þ cHQ þ cQ2 ¼ CRed

ð6:45Þ

The Nernst equation for the redox Equilibrium 6.13 is:  — EQ=Q2 ¼ EQ=Q 2 þ

RT aQ ln nF aQ2

ð6:46Þ

Because of the low solubility of quinhydrone in water it is possible to use concentrations instead of activities (provided that the aqueous solution does not contain higher electrolyte concentrations): —  EQ=Q2 ¼ EQ=Q 2 þ

RT cQ ln nF cQ2

ð6:47Þ

The concentration cQ2 can be calculated using Eq. 6.44, with the help of pKa values: c2H O þ cQ2 cH O þ cQ2  ¼ cH2 Q þ cHQ þ cQ2 ¼ 3 þ 3 þ cQ2 ð6:48Þ CRed Ka1 Ka2 Ka2  CRed

¼ cQ2

cQ2 ¼ 

c2H3 O þ

Ka1 Ka2

cH O þ þ 3 þ1 Ka2

 CRed c2 þ H3 O

Ka1 Ka2

þ

cH O þ 3 Ka2

! ð6:49Þ



ð6:50Þ

þ1

This allows Eq. 6.47 to be written as follows:   — EQ=Q2 ¼ EQ=Q 2 þ

RT ln nF

cQ

c2H

3O

þ

Ka1 Ka2

þ

cH O þ 3 Ka2

 þ1 ð6:51Þ

 CRed

 , the last equation simplifies to: According to Eq. 6.45 cQ ¼ CRed

EQ=Q2

c2H3 O þ cH O þ RT —  ln ¼ EQ=Q þ 3 þ1 2 þ nF Ka1 Ka2 Ka2

! ð6:52Þ

6.4 pH-Dependent Redox Potentials

153

pH 0

2

4

6

8

10

12

14

0 -2

H2Q

-4 -6

logci -8 -10 -12 -14

Fig. 6.3 pH-logci diagram of hydroquinone

c2H3 O þ

cH3 O þ from the Ka1 Ka2 Ka2 presence of HQ , and term 1 from Q2 . For the three distinguishable pH ranges in which one of the three reduced forms dominates (see the pH-logci diagram in Fig. 6.3) it is possible to derive approximative equations: where the term

results from the presence of H2 Q, the term

(a) pH \ pKa1 , i.e., pH \ 9:8 (H2 Q dominates). 2 RT cH3 O þ ln nF Ka1 Ka2

ð6:53Þ

RT 1 RT ln ln c2H3 O þ þ nF Ka1 Ka2 nF

ð6:54Þ

—  EQ=Q2 ¼ EQ=Q 2 þ

 — EQ=Q2 ¼ EQ=Q 2 þ

—  The sum EQ=Q 2 þ

RT nF

0

—  ln Ka11Ka2 can be taken as the formal potential Ec;Q=Q , 2 ;pH\pK

which is only valid for the condition pH \ pKa1 : 0

—  EQ=Q2 ¼ Ec;Q=Q þ 2 ;pH\pK

a1

RT ln c2H3 O þ nF

a1

ð6:55Þ

154

6

Redox Equilibria

For 25 °C it follows that: 0

 — þ EQ=Q2 ¼ Ec;Q=Q 2 ;pH\pK

a1

0:059 ½V log c2H3 O þ 2

0

—  EQ=Q2 ¼ Ec;Q=Q þ 0:059½V log cH3 O þ 2 ;pH\pK

a1

0

—  EQ=Q2 ¼ Ec;Q=Q  0:059½VpH ¼ 0:6994½V  0:059½VpH 2 ;pH\pK

a1

ð6:56Þ ð6:57Þ ð6:58Þ

The latter equation demonstrates that the redox potential EQ=Q2 is, in this pH range, a linear function of pH with a slope of 0:059 V per pH unit. (b) In the small range pKa1 \ pH \ pKa1 , i.e., 9:8 \ pH \ 11:4 (HQ dominates). RT cH3 O þ ln nF Ka2

ð6:59Þ

RT 1 RT ln ln cH3 O þ þ 2F Ka2 2F

ð6:60Þ

—  EQ=Q2 ¼ EQ=Q 2 þ

—  EQ=Q2 ¼ EQ=Q 2 þ

RT 1 ln can also be taken as a formal potential, which is now 2F Ka2 0 —  . At 25 °C the redox potential is described by: assigned as Ec;Q=Q 2 ;pK \pH\pK —  The sum EQ=Q 2 þ

a1

a2

0

—  EQ=Q2 ¼ Ec;Q=Q 2 ;pK

0

—  EQ=Q2 ¼ Ec;Q=Q 2 ;pK

a1 \pH\pKa2

a1 \pH\pKa2



þ

0:059 ½V log cH3 O þ 2

ð6:61Þ

0:059 0:059 ½VpH ¼ 0:4103½V  ½VpH 2 2 ð6:62Þ

Formally, in this range, EQ=Q2 also depends linearly on pH, however, the slope is 0:059 ½V per pH unit. In reality, the slope is not linear because of the proximity  2 of pKa1 and pKa2 . In that range, all three forms (H2Q, HQ−, and Q2−) are present in comparable concentrations and all three should be considered.

6.4 pH-Dependent Redox Potentials

155

Fig. 6.4 Dependence of EQ=Q2 on pH

pH 0

2

4

6

8

10

12

14

0.8 0.7 0.6 0.5

E/V 0.4 0.3 0.2 0.1 0.0

(c) In the range pKa2 \ pH , i.e., 11:4 \ pH (Q2 dominates). Here, the following simple relation holds:  — EQ=Q2 ¼ EQ=Q 2 þ

RT ln 1 nF

ð6:63Þ

and thus the potential is independent of pH: —  EQ=Q2 ¼ EQ=Q 2

ð6:64Þ

Figure 6.3 shows the pH-logci diagram of hydroquinone and Fig. 6.4 depicts the dependence of EQ=Q2 on pH in the range 0–14. The redox potential EQ=Q2 of quinhydrone can be easily measured using a platinum wire dipped in an aqueous solution containing some dispersed quinhydrone, using any suitable reference electrode. This is possible because the redox system of quinone/hydroquinone imprints its reversible potential to a platinum electrode. Most of the redox systems do not exhibit this favorable property. The arrangement of a “platinum electrode in an aqueous solution with dispersed quinhydrone” is commonly called the “quinhydrone electrode.” It was introduced in 1921 by the Danish chemist Christian Saxtrop Biilmann (1873–1946). In contrast to the glass electrodes (Chap. 7), this electrode can be used in fluoride-containing solutions, which strongly corrode glass. Since the dispersion of quinhydrone in the analyte solution contaminates it, this kind of quinhydrone electrode is not very

156

6

Redox Equilibria

user-friendly—a composite electrode has been developed which consists of surface modified graphite, an organic binder, and quinhydrone [5, 6]. The surface modification is necessary to render quinhydrone reversible on graphite.

6.4.2 Redox Equilibria with Inherently Coupled Acid–Base Equilibria The redox equilibrium permanganate/manganese(II)
2 þ Permanganate ions MnO , 4 can be reduced in an acidic solution to MnðH2 OÞ6 in a neutral solution to solid manganese(IV) oxide MnO2 (as a mineral called pyrolusite), and in a strongly alkaline solution to MnO2 4 . The redox system 2þ MnO =Mn formally involves the following equilibrium: 4
2 þ  þ þ 6H2 O MnO 4 þ 5e þ 8H3 O  MnðH2 OÞ6 possessing the standard potential E—

MnO 4 =½MnðH2 OÞ6 

2þ

ðEquilibrium 6:14Þ

¼ þ 1:491 V. Manganese

(VII) exists in aqueous solutions only in the form MnO 4 because manganese(VII)aqua ions cannot exist (Chap. 3). However, a reduction of MnO 4 is unavoidably accompanied by protonations of the O2 ions. There is no way to experimentally separate the proton transfer from the electron transfer equilibria. Therefore, Equilibrium 6.14 must be considered as an inherent coupling of a redox equilibrium with acid–base equilibria, and the respective Nernst equation reads: EMnO = MnðH OÞ 2 6 4 ½

2þ

¼E

 —

2þ

MnO 4 =½MnðH2 OÞ6 

8 RT aMnO4 aH3 O þ ln þ 5F a MnðH OÞ 2 þ ½ 2 6

ð6:65Þ

Analogously, the following equilibria are also inherently coupled redox and acid– base equilibria:
3 þ þ  Cr2 O2 þ 9H2 O 7 þ 14H3 O þ 6e  2 CrðH2 OÞ6

ðEquilibrium 6:15Þ

þ   6BrO 3 þ 6H3 O þ 6e  Br þ 9H2 O

ðEquilibrium 6:16Þ

þ   ClO 4 þ 8H3 O þ 8e  Cl þ 12H2 O

ðEquilibrium 6:17Þ


4 þ UO22 þ þ 4H3 O þ þ 2e  UðH2 OÞ6

ðEquilibrium 6:18Þ

þ  NO 3 þ 4H3 O þ 3e  NO þ 6H2 O

ðEquilibrium 6:19Þ

6.4 pH-Dependent Redox Potentials

157

In all these systems, the oxidation power of the oxidant is strongly affected by the pH. This can be easily shown using permanganate as an example—rearrangement of Eq. 6.65 yields: aMnO4 RT RT EMnO = MnðH OÞ 2 þ ¼ E—  ln ln a8H3 O þ þ 2þ þ 2 MnO4 =½MnðH2 OÞ6  6 5F a MnðH OÞ 2 þ 5F 4 ½ ½ 2 6 ð6:66Þ At 25 °C this gives: EMnO = MnðH OÞ 2 þ ¼ E—  2þ þ 2 MnO4 =½MnðH2 OÞ6  6 4 ½

aMnO4 0:059 8
0:059 ½V log ½V log aH3 O þ þ a MnðH OÞ 2 þ 5 5 ½ 2 6

ð6:67Þ EMnO = MnðH OÞ 2 þ ¼ E—  2þ þ 2 MnO4 =½MnðH2 OÞ6  6 4 ½ 

8
0:059 ½VpH 5

aMnO4 0:059 ½V log 5 a MnðH OÞ 2 þ ½ 2 6 ð6:68Þ

For a solution containing 10−1 mol L−1 permanganate and 10−4 mol L−1
2 þ
2 þ þ MnðH2 OÞ6 the redox potential of the system MnO 4 ; H3 O = MnðH2 OÞ6 is at pH 0 and pH 3: pH ¼ 0

EMnO = MnðH OÞ 2 þ ¼ 1:491½V] þ 2 6 4 ½

0:059 101 ½V log 4 ¼ 1:5264 V 5 10

pH ¼ 3 EMnO = MnðH OÞ 2 þ ¼ 1:491½V] þ 2 6 4 ½

0:059 101 3
8
0:059 ½V log 4  ½V¼ 1:2432 V 5 5 10

Generally, for an equilibrium of the form: Oxk þ þ mH3 O þ þ ne  Hm Redðk þ mnÞ þ þ mH2 O

ðEquilibrium 6:20Þ

the Nernst equation is:  — EOxk þ =Hm Redðk þ mnÞ þ ¼ EOx k þ =H

ðk þ mnÞ þ m Red

þ

aOxk þ am RT H3 O þ ln nF aHm Redðk þ mnÞ þ

ð6:69Þ

158

6

—  EOxk þ =Hm Redðk þ mnÞ þ ¼ EOx k þ =H

m Red

ðk þ mnÞ þ

þ

Redox Equilibria

RT RT aOxk þ ln am ln H3 O þ þ nF nF aHm Redðk þ mnÞ þ ð6:70Þ

For 25 °C one can write: —  EOxk þ =Hm Redðk þ mnÞ þ ¼ EOx k þ =H

m Red

ðk þ mnÞ þ

þ

m 0:059 aOxk þ 0:059½V] log aH3 O þ þ ½V log aHm Redðk þ mnÞ þ n n

ð6:71Þ c The latter equation shows that the ratio mn is most important for the effect of pH on the redox equilibrium.

6.5

Relations Between the Standard Potentials of an Element Having Various Oxidation States: Luther’s Rule

Many elements exist in various oxidation states. One example is chromium, pos
2 þ sessing the oxidation number 0 as metal, +2, e.g., in CrðH2 OÞ6 , +3, e.g., in
3 þ 2 2 , and +6, e.g., in CrO4 und Cr2 O7 . To find the relations between CrðH2 OÞ6 the standard potentials of the different redox systems, it is necessary to remember that it is not the standard potentials that are additive, but the Gibbs free energies, as the latter is a state function. For a metal existing in oxidation states 0, +n1, and +(n1 + n2) the following redox equilibria must be considered: Me  Men1 þ þ n1 e Men1 þ  Meðn1 þ n2 Þ þ þ n2 e Me  Meðn1 þ n2 Þ þ þ ðn1 þ n2 Þe

Equilibrium 6:21 Equilibrium 6:22 Equilibrium 6:23

and the relation of the respective Gibbs free energies is: DG—Me=Men1 þ þ DG—

Men1 þ =Meðn1 þ n2 Þ þ

¼ DG—

Me=Meðn1 þ n2 Þ þ

ð6:72Þ

Since the relation between the Gibbs free energy and the standard potential of a  — redox system is DG—Ox=Red ¼ nFEOx=Red (Eq. 6.5), the standard potentials are related by:

6.5 Relations Between the Standard Potentials of an Element … —  —  n1 EMe=Me n1 þ þ n 2 E

Men1 þ =Meðn1 þ n2 Þ þ

159

¼ ðn1 þ n2 ÞE—

ð6:73Þ

Me=Meðn1 þ n2 Þ þ

This equation was derived by Luther [7].6 The following examples illustrate Luther’s rule for the case of oxygen: O2 þ 2H3 O þ þ 2e  H2 O2 þ 2H2 O

EO—2 =H2 O2 ¼ 0:695 V

H2 O2 þ 2H3 O þ þ 2e  4H2 O

EH—2 O2 =H2 O ¼ 1:7635 V

O2 þ 4H3 O þ þ 4e  6H2 O

EO—2 =H2 O ¼ 1:229 V

H2 O2 þ H3 O þ þ e  HO þ 2H2 O

EH—2 O2 =HO
¼ 0:714 V

HO þ H3 O þ þ e  2H2 O

—  EOH =H2 O ¼ 2:813 V

The Gibbs free energies are related by: DG—O2 =H2 O2 þ DG—H2 O2 =H2 O ¼ DG—O2 =H2 O

ð6:74Þ

DG—H2 O2 =OH þ DG—OH =H2 O ¼ DG—H2 O2 =H2 O

ð6:75Þ

They give the following relations between the standard potentials: 2EO—2 =H2 O2 þ 2EH—2 O2 =H2 O ¼ 4EO—2 =H2 O

ð6:76Þ

—  —  EH—2 O2 =OH þ EOH =H2 O ¼ 2EH2 O2 =H2 O

ð6:77Þ

EO—2 =H2 O ¼

0:695½V þ 1:7635½V ¼ 1:229 V 2

EH—2 O2 =H2 O ¼

0:714½V þ 2:813½V ¼ 1:7635 V 2

—  EOH =H2 O ¼ 2
1:7635 [V]  0:714 [V] ¼ 2:813 V

ð6:78Þ ð6:79Þ ð6:80Þ

It is remarkable to see that hydrogen peroxide is a stronger oxidant than oxygen, although the oxygen in hydrogen peroxide has a lower oxidation state (i.e., −1) than in molecular oxygen (i.e., 0). Even more surprising is that the standard potential for the reduction of the hydroxyl radical HO to water

6

Robert Thomas Dietrich Luther (1868–1945) was a German physical chemist, coworker of Wilhelm Ostwald, and distant descendant of the German church reformer Martin Luther.

160

6

Redox Equilibria

Fig. 6.5 Latimer diagram of oxygen, hydrogen peroxide, and water

is 1.1 V more positive than that of the reduction of hydrogen peroxide to water, although oxygen has in the hydroxyl radical and in hydrogen peroxide the same oxidation state (−1). Clearly, this is the result of the additivity of Gibbs free energies and the stoichiometry of reactions. This makes the hydroxyl radical one of the strongest oxidants, and since it is a common metabolic by-product, living organisms have developed strategies to destroy it by so-called radical scavengers. The relations between the standard potentials of an element are frequently presented in Latimer diagrams7: the standard potentials are given above lines connecting the different oxidation states. For oxygen, hydrogen peroxide, and water the Latimer diagram is given in Fig. 6.5.

6.6

Biochemical Standard Potentials

When redox equilibria are coupled with acid–base equilibria, and hence the respective Nernst equation has also a term for the activity of protons, the standard potentials always relate to aH3 O þ ¼ 1. Equation 6.65 is an example for this, as, only when aMnO4 ¼ a MnðH OÞ 2 þ ¼ aH3 O þ ¼ 1 does the term ½ 2 6

RT 5F

aMnO a8

þ

ln a 4 H3 O2 þ follow ½MnðH2 OÞ6   — EMnO = MnðH OÞ 2 þ ¼ E  2 þ . This means that all standard potentials of 2 MnO4 =½MnðH2 OÞ6  6 4 ½ systems in which protons participate are related to the condition aH3 O þ ¼ 1, i.e., pH = 0. For biochemical and all biologically relevant redox systems, pH = 0 is normally an unrealistic condition. Therefore, it became customary to define so-called biochemical standard potentials E0 which are related to pH = 7. As demonstrated in Sect. 6.4.2, for a redox system coupled with an acid–base equilibrium: Equilibrium 6:20 Oxk þ þ mH3 O þ þ ne  Hm Redðk þ mnÞ þ þ mH2 O

7

These diagrams are named in honor of the US physical chemist Wendell Mitchell Latimer (1893– 1955).

6.6 Biochemical Standard Potentials

161

the Nernst equation is: EOxk þ =Hm Redðk þ mnÞ þ ¼ EOxk þ =Hm Redðk þ mnÞ þ þ

aOxk þ am RT H3 O þ ln nF aHm Redðk þ mnÞ þ

ð6:69Þ

At 25 °C it follows that (using the exact value of 2:303RT=F = 0.05916 V): —  EOxk þ =Hm Redðk þ mnÞ þ ¼ EOx k þ =H

m Red

þ

ðk þ mnÞ þ

m 0:05916 aOxk þ 0:05916½V] log aH3 O þ þ ½V log n n aHm Redðk þ mnÞ þ

ð6:81Þ —  The first two terms on the right-hand side, i.e., EOx k þ =H

m þ n 0:05916½V] log aH3 O

m Red

ðk þ mnÞ þ

þ

already define a formal potential. The formal potential relating to pH = 7 is now called the biochemical standard potential: 0

—  E0 ¼ Ec;Ox k þ =H

m Red

ðk þ mnÞ þ

0

—  E0 ¼ Ec;Ox k þ =H

ðk þ mnÞ þ m Red

—  ¼ EOx k þ =H

—  ¼ EOx k þ =H

—  E0 ¼ EOx k þ =H

m Red

ðk þ mnÞ þ

ðk þ mnÞ þ m Red

ðk þ mnÞ þ m Red



þ

þ

m 0:05916½V] log aH3 O þ n ð6:82Þ

m 0:05916½V] log 107 ð6:83Þ n

m
0:41412 ½V] n

ð6:84Þ

c In biochemical and biological literature one needs to be very attentive as to whether the author means the (chemical) standard potential —  EOx or the biochemical standard potential E0 , as this is not k þ =H Redðk þ mnÞ þ m

always clearly indicated!

6.7

Redox Potentials in Non-aqueous Solvents

In non-aqueous solvents other standard reactions than the oxidation of hydrogen are used. There is not yet an agreement about one reference system. Such a reference system should fulfill the following criteria: 1. It should be a one-electron reaction. 2. The reduced form should be a neutral molecule and the oxidized form a cation. 3. Both components of the reference redox systems should be sufficiently large species, so that the solvation enthalpies are low enough to make it solvent independent.

162

6

Redox Equilibria

Fig. 6.6 Ferrocene, ecliptic conformation

4. The equilibrium of the reference redox system should be established very quickly on an electrode. 5. The standard potential of the reference system should be low enough to prevent an oxidation of the solvent. 6. The electron transfer should not provoke any significant change of the structure of the two components of the reference redox system. Most literature data on standard potentials in non-aqueous solvents and solvent mixtures relate to the following reference system: bis(biphenyl)chromium(I)/bis (biphenyl)chromium(0). Frequently data are also given with respect to ferrocenium/ ferrocene (bis(η5-cyclopentadienyl)iron(III)/bis(η5-cyclopentadienyl)iron(II)) (Fig. 6.6) or DMFC+/DMFC (DMFC = decamethylferrocene). These redox systems fulfill the abovementioned criteria quite well [8]. However, potentials measured versus these redox systems in non-aqueous systems cannot easily be compared to data for aqueous solutions, and a recalculation versus the standard hydrogen electrode for aqueous solutions is problematic (although possible with rather good accuracy when reasonable non-thermodynamic assumptions are made). These problems resemble those relating acidity constants between different solvents. Similar to the treatment of the chemical potential of protons in different solvents, Krossing et al. defined an absolute redox potential for all solvents [9]. As for protons, they defined the zero point of the scale as relating to the chemical potential of the electron in an ideal electron gas at 1 bar and 25 °C, which they set to zero. The absolute chemical potential of an electron in a solvent labs;e gives the difference between the chemical potential of the solvated electron lsolv;e and that of the electron in the electron gas (under the defined standard conditions, i.e., 1 bar and 25 °C) le ;ideal : labs;e ¼ lsolv;e  le ;ideal

ð6:85Þ

6.7 Redox Potentials in Non-aqueous Solvents

163

The absolute potential can also be related to the oxidation of hydrogen in the different solvents; however, then the different chemical potentials of protons in the different solvents must be considered as well. With knowledge of the solvation energies of the redox pairs in a solvent, the absolute redox potentials can be calculated.

6.8

Graphical Presentation of Redox Equilibria

Similar to the presentation of acid–base equilibria in pH-logci diagrams, redox equilibria can be depicted in E-logci diagrams. Figure 6.7 shows the redox equi
3 þ
2 þ librium FeðH2 OÞ6 = FeðH2 OÞ6 as a function of the redox potential E of the solution, assuming that the latter is equal to EFe3 þ =Fe2 þ , i.e., that other redox systems are without effect. In Fig. 6.7 the functions log cFe3 þ ¼ f ðEÞ and log cFe2 þ ¼ f ðEÞ are given. For simplicity only the redox state of iron is given, but it is meant that both iron(II) and iron(III) are present as hexaaqua ions. At 25 °C the Nernst equation for the iron system is:  — EFe3 þ =Fe2 þ ¼ EFe 3 þ =Fe2 þ þ 0:059½V log

cFe3 þ cFe2 þ

ð6:86Þ

E/V 0.0

0.5

1.0

1.5

2.0

0

-5

logci -10

-15

Fe3+

Fe2+

-20


3 þ
2 þ Fig. 6.7 E-logci diagram of the redox equilibrium FeðH2 OÞ6 = FeðH2 OÞ6 as a function of the redox potential E of the solution at 25 °C

164

6

Redox Equilibria

 Since the overall concentration of iron is Ciron ¼ cFe3 þ þ cFe2 þ , the following equations result:

cFe3 þ   c 3þ Ciron Fe

ð6:87Þ

cFe3 þ  cFe3 þ

ð6:88Þ

—  EFe3 þ =Fe2 þ ¼ EFe 3 þ =Fe2 þ þ 0:059½V log

 — EFe3 þ =Fe2 þ  EFe 3 þ =Fe2 þ

0:059[V] cFe3 þ   c 3 þ ¼ 10 Ciron Fe

E

 Ciron  cFe3 þ ¼ 10 cFe3 þ

¼ log

 Ciron

— E Fe3 þ =Fe2 þ 0:059½V

Fe3 þ =Fe2 þ

ð6:89Þ

— 

E 3þ E Fe =Fe2 þ Fe3 þ =Fe2 þ 0:059½V

ð6:90Þ

 Ciron

cFe3 þ ¼

ð6:91Þ

 —

1 þ 10



E 3þ E Fe =Fe2 þ Fe3 þ =Fe2 þ 0:059½V

— 

log cFe3 þ ¼

 log Ciron

 log 1 þ 10



E 3þ E Fe =Fe2 þ Fe3 þ =Fe2 þ 0:059½V

! ð6:92Þ

Analogously, one can derive the equation for log cFe2 þ ¼ f ðEÞ: E

log cFe2 þ ¼

 log Ciron

 log 1 þ 10

— E Fe3 þ =Fe2 þ 0:059½V

Fe3 þ =Fe2 þ

! ð6:93Þ

As in the case of acid–base pH-logci diagrams, the asymptotes of the true functions log cFe3 þ ¼ f ðEÞ and log cFe2 þ ¼ f ðEÞ can be drawn in the diagram when the two ranges E \ E— and E [ E— are considered separately: E \ E—  log cFe3 þ ¼ log Ciron   log cFe2 þ ¼ log Ciron

E \ E— — E 3þ

Fe

=Fe2 þ

0:059½V

þ

EFe3 þ =Fe2 þ 0:059½V

 log cFe3 þ ¼ log Ciron  log cFe2 þ ¼ log CEisen þ

— E 3þ

Fe

=Fe2 þ

0:059½V



EFe3 þ =Fe2 þ 0;059½V

Frequently, redox equilibria include the participation of solid phases, e.g., metallic iron and iron oxides. For such cases, so-called predominance diagrams are very useful. In the case of redox equilibria, they are called Pourbaix diagrams,8 8

Marcel Pourbaix (1904–1998) Belgian physical chemist.

6.8 Graphical Presentation of Redox Equilibria Fig. 6.8 Pourbaix diagram for iron, calculated for a solution concentration of 1 mol L−1

165

2.0 1.6

FeO24

1.2

Fe3+

0.8

Fe2O3·nH2O

0.4

E/ V

Fe

0.0

2+

-0.4

Fe3O4 Fe(OH)2

-0.8

HFeO2

Fe

-1.2 -1.6 0

2

4

6

8

10

12

14

pH

because Pourbaix proposed them for the first time in 1938. In these diagrams, lines are depicted at which neighboring species possess the same activity. Figure 6.8 shows one possible diagram for iron. The lines separating the domains of predominance can be calculated with the help of Nernst equations, if necessary, also considering acid–base equilibria. In Fig. 6.8 the most simple to calculate line is that which separates dissolved Fe2+ c from Fe3+ at low pH values: at this line, the condition cFe3 þ ¼ cFe2 þ , i.e., cFe32 þþ ¼ 1 Fe

—  2+ from holds, and thus EFe3 þ =Fe2 þ ¼ EFe 3 þ =Fe2 þ . For the line separating dissolved Fe metallic iron, the iron(II) concentration (specifically its activity) has to be fixed. Since metallic iron has an activity of 1, the Nernst equation is:  — 0:059 2þ EFe2 þ =Fe ¼ EFe 2 þ =Fe þ 2 ½V log cFe . This line depends on the iron(II) concentration. Frequently, diagrams are drawn for 1 mol L−1. Pourbaix diagrams are often used to consider the equilibria in the case of the corrosion of metals (and also of some redox active minerals). The line separating dissolved Fe3+ from dissolved FeO2 4 can be calculated, when the equilibrium:

 þ FeO2  Fe3 þ þ 12H2 O 4 þ 3e þ 8H3 O

ðEquilibrium 6:24Þ

is described using the respective Nernst equation: cFeO2 cH3 O þ 0:059 4 ½V log 3 cFe3 þ 8

—  EFeO2 þ 3þ ¼ E FeO2 =Fe3 þ 4 =Fe

4

ð6:94Þ

166

6

Redox Equilibria

For cFeO2 ¼ cFe3 þ it follows that: 4 8 —  0:059½VpH EFeO2 3þ ¼ E 3þ  FeO2 4 =Fe 4 =Fe 3

ð6:95Þ

The following points always need to be remembered when Pourbaix diagrams are discussed: (1) The diagrams are based on thermodynamic data (standard potentials, acidity constants, stability constants, and solubility constants), i.e., they are describing equilibrium situations. Such conditions are rarely met! (2) The diagrams are, even for equilibrium conditions, only as good as the underlying models. If, as has been done for Fig. 6.8, the acid–base equilibria of iron aqua complexes are ignored, the diagram cannot show the reality of conditions where hydroxo complexes and their condensation products are formed. (3) The diagrams are only as good as the thermodynamic data used—such data are not always sufficiently accurate.

6.9

Kinetic Aspects of Redox Equilibria

Two types of redox equilibria have to be distinguished: homogeneous and heterogeneous. A homogeneous redox equilibrium exists, when the participating species are in one phase, normally in a solution phase. An example is
2 þ
3 þ
3 þ þ Cr2 O2  2 CrðH2 OÞ6 þ 6 FeðH2 OÞ6 þ 9H2 O 7 þ 14H3 O þ 6 FeðH2 OÞ6

ðEquilibrium 6:6Þ Since a single collision of 21 ions (left-hand side of the reaction equilibrium) is completely impossible, the forward reaction must go via a number of single steps. This means that the reaction mechanism is completely different from the thermodynamically correct equilibrium equation. The many single reaction steps lead in many cases to a slow, sometimes extremely slow establishment of equilibrium. An example of an extremely slow establishment of equilibrium is the oxidation of arsenite ions ðAsO2 3 Þ with cerium(IV) in an acidic solution. When solutions are very pure, the reactants are so stable that no reaction can be observed over long time periods. However, when certain catalysts are added (e.g., iodide ions), the rate of reaction (establishment of equilibrium) is measurable and is a function of catalyst concentration. The analytical procedure based on measuring the rate of a reaction as a function of catalyst concentration is called catalymetry, or simply kinetic analysis [10]. Another example of a catalyzed reaction is the oxidation of oxalate ions in an

6.9 Kinetic Aspects of Redox Equilibria

167

acidic solution with permanganate. Here the catalyst is Mn2+, the reduction product of permanganate: the reaction is autocatalytic because the reaction product acts as a catalyst. If a fast reaction is desirable from the very beginning, as, e.g., in a titration of oxalate, it makes sense to add some manganese(II) salt at the beginning. Kinetic methods of analysis are especially important in biochemical analysis, aimed to determine enzyme activities or substrate concentrations [11]. A heterogeneous equilibrium is established in a two-phase system, e.g., in electrochemistry at the electrode(s). An example is the equilibrium between silver(I) ions and silver metal. This equilibrium is quickly established from both sides, and therefore it is called a reversible equilibrium. However, most of the heterogeneous equilibria between a metal and a solution containing metal ions are slow, i.e., irreversible. Also, most of the redox equilibria of dissolved redox couples are irreversible at so-called “inert” electrodes. The frequently discussed example:
2 þ  þ MnO þ 6H2 O 4 þ 5e þ 8H3 O  MnðH2 OÞ6

ðEquilibrium 6:14Þ

is typical for this kind of system. At a platinum or gold electrode, the redox potential calculated with the Nernst equation for that equilibrium cannot be expected to be the measured value. However, for unknown reasons, the measured redox potential is in this case surprisingly near to the calculated value. This is just a funny caprice of nature, since the mechanism of potential establishment comprises several stepwise reduction reactions of permanganate, and the anodic counterpart can even be the oxidation of water! Thus, the established potential is a so-called mixed potential, because two different redox systems are taking part, and the mixed potential establishes, where the kinetically fixed reduction and oxidation currents compensate one another. The slowness of establishing equilibrium of a redox reaction can even be considered the basis for life on Earth: it is only because the reaction between the reduced carbon compounds (that make up the organic matter of life) and the oxygen in our atmosphere is so slow, that life is possible in an world containing oxygen. The entire living world is not in equilibrium with the environment. This disequilibrium was established, and is ongoing, as a result of the energy flux from the sun, via photosynthesis.

References 1. Scholz F (2017) J Solid State Electrochem 21:1859–1874 2. Bockris JO’M, Khan ShUM (1993) Surface electrochemistry. A molecular level approach. Plenum Press, New York, pp 75 3. Latimer WM (1952) The oxidation states of the elements and their potentials in aqueous solutions, 2nd edn. Prentice-Hall, New York 4. Kragten J (1978) Atlas of metal-ligand equilibria in aqueous solution. Ellis Horwood Chichester 5. Düssel H, Komorsky-Lovrić Š, Scholz F (1995) Electroanalysis 7:889–894 6. Scholz F, Steinhardt T, Kahlert H, Pörksen JR, Behnert J (2005) J Chem Educ 82:782–786

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7. (a) Luther R (1899) Z Phys Chem (Leipzig) 30:628–680; (b) Luther R, Wilson R (1900) Z Phys Chem (Leipzig) 34:489–494 8. Gritzner G, Kůta J (1984) Pure Appl Chem 56:461 9. Radtke V, Himmel D, Pütz K, Goll SK, Krossing I (2014) Chem Eur J 20:4194 10. Kopanica M, Stará V (1983) Kinetic method in chemical analysis. In: Svehla G (ed) Comprehensive analytical chemistry, vol 18. Elsevier, Amsterdam 11. Bergmeyer HU (ed) (1984) Methods of enzymatic analysis. Verlag Chemie, Weinheim, Deerfield Beach, Basel

7

Titrations

This chapter covers only the basics of titrations in solutions, i.e., gas titrations, which are also possible, are not considered. Also excluded are the so-called biological titrations, in which the contents of bacteria or viruses are determined, and biochemical assays, which work according to the titration principle. Due to lack of space, coulometric titrations cannot be presented and reference is made to textbooks of instrumental analysis [1, 2]. Detailed practical instructions are outside the scope of this book and can be found in the relevant textbooks [3].

7.1

The History of Titrations

Today, titrations, together with gravimetric methods, are among the classical methods of analysis. It is difficult to imagine that titrations, as they were developed in the nineteenth century, represented a revolution in analytical chemistry and were not welcomed by all chemists because gravimetric analysis was considered the only reliable method. The main driving force behind their development and establishment were the requirements of the emerging chemical industry, i.e., the need for faster and simpler methods of quality control than gravimetric methods. It is not possible to identify the exact date titrations were introduced, as there were several precursors. For example, the first approaches to titrimetric determinations were made by Robert Boyle1 and Claude Joseph Geoffroy2 [4, 5]: a reagent, e.g., solid potassium carbonate, was added to an acid solution for as long as there was carbon dioxide formation. From the weighed mass of solid potassium carbonate the acid concentration was calculated. Francis Home3 (Fig. 7.1) was

1

Chapter 3. Claude Joseph Geoffroy (1685–1752) French naturalist. 3 Francis Home (1719–1781) Scottish chemist and physician. 2

© Springer Nature Switzerland AG 2019 F. Scholz and H. Kahlert, Chemical Equilibria in Analytical Chemistry, https://doi.org/10.1007/978-3-030-17180-3_7

169

170

7

Titrations

Fig. 7.1 Francis Home, © Wellcome Collection

probably the first to use solutions as reagents for titrating, and the first to measure the volume of a reagent solution required, even if he used a teaspoon to do so: … Exp. 13. In order to discover what effect acids would have on these ashes, and what quantity of the former the latter would destroy; from which I might be able to form some judgement of the quantity and strength of the salt they contained; I took a drachm of blue pearl ashes, and poured on it a mixture of one part spirit of nitre, and six parts water; which I shall always afterwards use, and call the acid mixture. An effervescence arose, and, before it was finished, 12 teaspoonfuls of the mixture were required. This effervescence with each spoonful of the acid mixture was violent, but did not last long… [6].

In 1767 William Lewis4 published a detailed analytical procedure to determine the carbonate content of eight different potashes. Lewis was the first to suggest the use of a color indicator for the determination of the endpoint of the procedure [7]. For the exact determination of the amount of acid consumed, he conducted a differential weighing of a suitable glass vessel, from which the acid was gradually taken. For this, he mostly used hydrochloric acid, the concentration of which he had previously determined with pure potassium carbonate. François Antoine Henri Descroizilles5 is considered the inventor of the burette, which he introduced in 1791 under the French name “alcalimètre” (Fig. 7.2) for the determination of chlorine in bleaching solutions.

4

William Lewis (1708–1781), English chemist. François Antoine Henri Descroizilles (1751–1825), French pharmacist and chemist.

5

7.1 The History of Titrations

171

Fig. 7.2 Alcalimeter according to François Antoine Henri Descroizilles (Museo Galileo, Firenze)

The introduction of titrimetric methods also goes back to Joseph Louis Gay-Lussac.6 The terms “burette,”7 “pipette,”8 and “titrate”9 were introduced by him and are still used today. The most well-known titration elaborated by Gay-Lussac is the very exact determination of silver ions by a precipitation titration. In 1855 Friedrich Mohr10 (Fig. 7.3) published his famous textbook Lehrbuch Der Chemischanalytischen Titrirmethode [8, 9], in which he collected, for the first time, the then sporadically published procedures and presented them systematically according to his own experience. Above all, he standardized the preparation of reagent solutions and introduced improved measuring apparatus, e.g., the pinchcock burette (Fig. 7.4). In subsequent years, there were only a few fundamental innovations in the field of titrimetry. Methods already described by Mohr were applied to a larger spectrum of samples and were modified. The applications were completely empirical—theoretical fundamentals were missing. The use of color indicators in volumetric analysis, especially in the determination of acids and bases, increased rapidly after 1870 with the development of new synthetic dyes. It had already been recognized 6

Joseph Louis Gay-Lussac (1778–1850), French chemist and physicist. French: la burette = cannikin. 8 French: la pipette = suction tube, whistle, diminutive of pipe. 9 Derived from the French le titre = fineness. 10 Karl Friedrich Mohr (1806–1876), German chemist and pharmacist. 7

172 Fig. 7.3 Friedrich Mohr, © Austrian National Library/INTERFOTO

Fig. 7.4 Pinchcock burette after Mohr (from [10])

7

Titrations

7.1 The History of Titrations

173

that the equivalence point of an acid–base titration did not always coincide with a color change in an indicator, i.e., not all indicators were equally well suited for different titrations. It was not until 1894 that Wilhelm Ostwald provided an explanation for the color change of acid–base indicators on the basis of ionic equilibria. A short historical outline of the development of titrations can be found in the 8th edition of Mohr’s titration methods, published by Beckurts [5]. A still useful compilation of titration methods is Kolthoff’s and Stenger’s three volume work entitled Volumetric Analysis [11]. Nowadays, instrumental analytical methods dominate in laboratories. However, classic titrations are indispensable in everyday laboratory life. The main reason is that measurements with modern methods can be performed in the lower concentration range in which classical methods fail, but are limited in the upper concentration range (approximately between 10−3 and 1 mol L−1)—in terms of precision they are inferior to the classical methods. In addition, instrumental methods must be calibrated, usually with standard solutions of known concentration. The exact concentrations of such standard solutions are determined in most cases with classical titration methods. This, and the fact that physical measuring methods like conductometry, potentiometry, amperometry, or photometry can be applied to follow classical titrations shows the interrelation of classical and modern methods. Often, the terms “potentiometric,” “amperometric,” or “conductometric” titrations are used, even though a classic titration is actually performed, i.e., the standard solution is added to the analyte, and the change of the corresponding physical quantity is recorded instrumentally. Therefore, it is better to use the terms “potentiometric,” “conductometric,” or “amperometric” indication. Nowadays, automated titration systems dominate in practice [12, 13]. Mostly, the reagent solution is added to the sample solution by an electronically controlled pump. There are also flow systems, either using segmented flow or unsegmented flow injection.

7.2

General Theory of Titrations

In chemistry, titrimetric analysis (titrations) or volumetric analysis comprise analytical methods in which the amount n
A of a substance A (the sample, analyte, or titrand) is determined by measuring the volume of a solution containing substance B (the standard solution, titrator, or titrant) of known concentration, necessary for a complete (stoichiometric) reaction of A with B [14]. Often, the term titrand is confused with the term titrator. To avoid this, it should be noted that the Latin suffix *tor always designates the active part, e.g., terminator, gladiator, navigator, operator, senator. Since standard solution is added to the analyte it can be regarded as the active part, although in a chemical reaction of course, both partners are reacting.

174

7

Titrations

A fundamental prerequisite for the reaction underlying a titration is that it proceeds according to a defined stoichiometry, e.g.: A þ B ! AB

ðEquilibrium 7:1Þ

A þ 2B ! AB2

ðEquilibrium 7:2Þ

A þ 3B ! AB3

ðEquilibrium 7:3Þ

or

or

Another requirement is that the chemical reaction proceeds quantitatively. In general, these are equilibrium reactions, which is why it would be better to use equilibrium arrows, i.e., A þ B  AB, A þ 2B  AB2 , and A þ 3B  AB3 . However, the notation with only one arrow indicates that the equilibrium lies very far on the side of the products (the equilibrium constant for the corresponding reaction is very large). It is also important that the equilibration is rapid and not kinetically inhibited. According to IUPAC recommendations [9], the equivalence point is the point where the amount of added titrator n
;EP corresponds exactly to the amount of B substance to be titrated. This means that at the equivalence point, to n mol of A have been added exactly n mol B (after Equilibrium 7.1), 2n mol B (after Equilibrium 7.2), and 3n mol B (after Equilibrium 7.3), depending on the stoichiometry of the reaction. If we assume a general reaction equation with the stoichiometric coefficients x and y: xA þ yB ! Ax By

ðEquilibrium7:4Þ

the ratio of the amounts at the equivalence point is: n
A

n
;EP B

¼

x y

ð7:1Þ

In this chapter the symbols x, y, etc., are used for the stoichiometric coefficients instead of the Greek letter m (nu) to prevent confusion with the italic Latin letter v (vee) as a symbol for volume. Since a volume is measured during titration, it can also be said that in titrations the volume of the titrator B is determined, which contains exactly this required amount B. The analytical concentration of substance A can then be calculated using simple stoichiometric equations. This shows another prerequisite for a volumetric determination: it must be possible to prepare a reagent solution containing substance B in a defined concentration (CB
), or the concentration of B must have been exactly determined by an independent analysis.

7.2 General Theory of Titrations

175

The amount of substance A (n
A ) can be calculated with the concentration of titrator B (CB
) and the volume of the titrator needed to reach the equivalence point (vEP B ), taking into account that the amount of B consumed up to the equivalence
EP point (n
;EP B ) is given by concentration CB and volume vB according to: n
;EP ¼ CB
vEP B B

ð7:2Þ

x x ¼ CB
vEP n
A ¼ n
;EP B B y y

ð7:3Þ

The concentration of sample A can be calculated using the volume of the sample measured exactly using a pipette (vA): CA
¼

x CB
vEP B y vA

ð7:4Þ

Since not all reactions fulfil the desired conditions, in practice various titration strategies have been developed in order to be able to quantify substances or substance mixtures by means of titration.

7.3

Titration Methods

7.3.1 Direct Titration Direct titration means that a definite sample volume A, measured with a volumetric pipette (vA), is transferred into a titration vessel (e.g., an Erlenmeyer flask) and optionally diluted. Depending on the titration and indication type, conditions are set accordingly (e.g., adjustment of the sample pH by adding a buffer, adding an indicator, or heating). The burette is filled with the solution of titrator B with concentration CB
, and the titrator is added dropwise to the sample solution while constantly swirling the flask or stirring the sample solution by means of a magnetic stirrer (cf. Fig. 7.5). When color indicators are used, the titrator is added to the sample solution until the color of the indicator changes, and the consumed volume (vEP B ) of the standard solution can be read on the burette (Sect. 7.5). The titration is often continued beyond the equivalence point in order to facilitate the detection of the equivalence point (determination of vEP B by extrapolation; Sect. 7.5).

7.3.2 Inverse Titration Sometimes it is not possible to place a sample in an Erlenmeyer flask, adjust the conditions, and titrate directly with a standard solution. One example is the

176

7

Titrations

Fig. 7.5 Schematic representation of a direct titration

permanganometric determination of nitrite ions. This titration must be performed in a heated acidic solution—at room temperature the reaction is too slow. Under these conditions, nitrous acid is decomposed and can also be lost by evaporation. Here, direct titration is reversed, i.e., a definite amount of titrator (n
B ¼ CB
vB ) is placed in an Erlenmeyer flask and a burette is filled with sample solution (cf. Fig. 7.6). The volume of sample solution A consumed, until reaching the equivalence point, is read on the burette (vEP A ), and the concentration of the sample solution can be calculated according to: CA
¼

x CB
vB y vEP A

ð7:5Þ

7.3.3 Back Titration Back titrations are always useful if no suitable indication procedure is available for direct titration or if the reaction between sample and standard solution is very slow. An exact volume of the sample (vA) is placed in an Erlenmeyer flask and the

7.3 Titration Methods

177

Fig. 7.6 Schematic representation of an inverse titration

reaction conditions adjusted (pH, indicator, temperature, etc.). To this solution an exact volume of standard solution B (vB) is added that contains an excess amount of titrator (more than is needed to reach the equivalence point in a direct titration, n
B ¼ n
;EP þ nExc B ). When the reaction is completed (the equilibrium is established), B the excess amount of titrator (nExc B ) is determined by means of direct titration with a second standard solution C (cf. Fig. 7.7). An equilibrium is established between B and C according to: ðEquilibrium 7:5Þ

rB þ wC ! Cw Br

The amount of sample A equals the difference between the added and excess amounts of titrator B: n
A ¼

 x

nB  nExc B y

ð7:6Þ


;EP The excess amount of titrator (nExc B ) is determined by the amount nC :

nExc B ¼

r
;EP n w C

ð7:7Þ

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Titrations

Fig. 7.7 Schematic representation of a back titration

The concentration of sample A is calculated as follows: CA
¼

 x 
r CB vB  CC
vEP C yvA w

ð7:8Þ

The following must be taken into consideration when choosing the second standard solution (C) for back titration: 1. If the back reaction in Equilibrium 7.4 is kinetically inhibited (very slow back reaction), then it is practically unaffected by the presence of the reaction product AxBy. A good example is the complexometric determination of Cr(III). The Cr (III)–ethylenediaminetetraacetic acid (EDTA) complex is kinetically so inert that a back titration can be performed even in a strongly acidic solution with a metal ion which forms a more stable complex with EDTA than Cr(III), like Fe(III). 2. If the back reaction in Equilibrium 7.4 is very fast (not kinetically inhibited), then the equilibrium constant for Equilibrium 7.5 must be smaller than that for Equilibrium 7.4, because otherwise the following reaction would occur in the range of the equivalence point: ywC þ rAx By  yCw Bv þ rxA

ðEquilibrium 7:6Þ

7.3 Titration Methods

179

7.3.4 Substitution Titration Substitution titrations are also called displacement titrations and can be used like back titrations if a direct titration is not feasible. The sample solution is placed in an Erlenmeyer flask and an excess of substance CwBv is added. The analyte displaces a reagent C: rxA þ yCw Br  rAx By þ ywC

ðEquilibrium 7:7Þ

The amount of displaced reagent C is determined by an appropriate titration with standard solution B (Fig. 7.8). Equilibrium 7.5 is established, and the amount of sample A can be calculated according to: n
A ¼

rx
rx w
;EP x
;EP nC ¼ n ¼ nB yw yw r B y

ð7:9Þ

The concentration of A can be calculated with Eq. 7.4. Here, too, when choosing substance CwBr, it should be noted that the equilibrium constant of Equilibrium 7.4 has to be much larger than that of Equilibrium 7.5, so that Equilibrium 7.7 is very far on the right side.

Fig. 7.8 Schematic representation of a substitution (displacement) titration

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7

Titrations

Fig. 7.9 Schematic representation of an indirect titration

7.3.5 Indirect Titration “Indirect titration” means that analyte A is converted chemically before titration and is not titrated on its own. An exact volume of the sample is measured and submitted to the titration flask. An accurately measured volume of a solution of substance B, with which analyte A forms the stoichiometric reaction product AxBy, is also given to the titration flask. B must be added in excess. When equilibrium between A and B is established, the excess of B is titrated with an appropriate standard solution C in a direct titration (Fig. 7.9). An example is the determination of sulfate ions. A sample solution is mixed with an excess of barium chloride solution. A precipitation of barium sulfate is formed, and the excess barium ions determined with diethylenetriaminepentaacetic acid (DTPA) solution and Erio T as an indicator (Sect. 7.4.2). Calculation of the amount or concentration of analyte is performed as in the case of back titrations (cf. Eq. 7.8).

7.4

Theoretical Considerations and Graphical Representation of Titration Curves

In practice, one has to find a suitable method to determine the equivalence point of a titration. In all cases, there should be a marked change in a property of the solution near the equivalence point. The point at which the strongest property change of a

7.4 Theoretical Considerations and Graphical Representation of Titration Curves

181

solution is registered is defined as the end point of a titration. The equivalence point corresponds to the true amount of sample which has to be determined by a titration, while the end point gives the experimentally determined concentration, which is subjected to an uncertainty caused by systematic and random errors. To visualize the progress of a titration, either the change in concentration of a solution component (more often the logarithm of the concentration of a solution component) or the change of a physical property of the solution (e.g., conductivity, potential, or absorption) is plotted versus the added volume of the titrator. First, the stoichiometry of the chemical reaction between analyte and titrator must be formulated. To make titration curves comparable, degree of titration is a very useful quantity. It was introduced to normalize the course of titration, by calculating the ratio of currently added amount of
;EP titrator n
;s B , to the amount needed to reach the equivalence point nB : s¼

currently added amount of titrator amount of titrator which is necessary to reach the equivalence point

ð7:10Þ

The degree of titration can assume values between zero and infinity. At the equivalence point it is unity (1.00). Titration curves are often plotted in the range of s = 0 to s = 2 to also show conditions after the equivalence point, or in order to better visualize the equivalence point as an inflection point. A degree of titration of s = 2 means that one has again added as much titrator as was necessary to reach the equivalence point, or, in other words, it means a 100% overtitration. For mixtures of substances and for substances which may undergo more than one reaction with a standard solution (e.g., polybasic acids), the problem arises of how to define the “amount of titrator necessary to reach the equivalence point.” There is no common rule for this, but it makes sense to equate this amount with, e.g., the total amount of titratable protons. This means that in the case of the titration of phosphoric acid with sodium hydroxide solution, at s = 1/3 the first proton is titrated (yielding a solution mainly containing H2 PO 4 ), at s = 2/3 the second proton is titrated (yielding a solution mainly containing HPO2 4 ), and at s = 3/3 = 1 the third proton is titrated (yielding a solution mainly containing PO3 4 ). Of course, one can also define the degree of titration in such a way that at s = 1 the first proton is titrated, at s = 2 the second proton is titrated, and at s = 3 the third proton is titrated. In such a case, the titration curve should be plotted in the range 0  s  4 so that the final concentration of sodium hydroxide (in moles per liter) equals the starting concentration of phosphoric acid. Titration curves may look quite different depending on the parameters shown on the y-axis and can usually be theoretically calculated or simply constructed using information on the underlying chemical equilibria. Such a calculation and graphical representation is very useful to estimate the feasibility of a titration. For constructing or calculating titration curves, the following idea is helpful: a titration is a synthesis, i.e., by titrating sample A with titrator B a new substance Ax By is formed. At values of s between 0 and 1, this synthesis is not yet complete, i.e., there is still a significant proportion of A present in the solution in addition to the newly formed

182

7

Titrations

substance Ax By . Substance B is almost completely consumed by the reaction before reaching the equivalence point because the equilibrium is far on the side of the products. When s = 1 is reached, the synthesis is 100% complete, providing a solution of substance Ax By . For reversible equilibria, e.g., the titration of acetic acid with sodium hydroxide solution, there is a pure sodium acetate solution. Of course, because of protolysis, small amounts of non-protolyzed acetic acid and small amounts of “sodium hydroxide” (of course in the form of Na þ and OH − ) are still present. At s > 1, substance Ax By and an excess of substance B are present in the titration mixture. To construct titration curves, one has to consider how the composition of the solution affects the measured parameter.

7.4.1 Acid–Base Titrations In acid–base titrations a proton transfer takes place between the reactants. Either an acid HB to be determined reacts with the solution of a very strong base of known concentration, or the concentration of a base B− is determined by adding a standard solution containing a strong acid in a known concentration. If one follows the idea of synthesis, one can say that the acid HB is converted into its corresponding base by adding the very strong base, and vice versa. As an example, by titrating hydrochloric acid solution with sodium hydroxide solution, at s = 0.5 (50% conversion or 50% of the synthesis) the solution contains equal amounts of hydrochloric acid and sodium chloride (of course, both are dissociated or protolyzed). At s = 1, one has synthesized a pure sodium chloride solution. At s = 1.5, all the H3 O+ ions of the hydrochloric acid have been converted to water, i.e., an equal amount of dissolved sodium chloride has formed, and the solution contains an excess of 50% sodium hydroxide (of course also dissociated). Hence it makes sense to plot the pH of the titration mixture versus the degree of titration to follow the course of titration. Such titration curves of acid–base titrations can be easily constructed from a pH-logci diagram. The latter has to be rotated by 90°, and then its mirror image (with respect to the horizontal line at pH = 7) has to be produced (Fig. 7.10a–c). Then, the titration diagram can be arranged in a way that both pH scales are parallel to one another (Fig. 7.10d). The construction of titration curves in pH-s coordinates is particularly useful since the pH is easily accessible by means of potentiometry (cf. Sect. 7.5). Consider first the titration of a very strong acid with a very strong base. Since very strong acids and bases are leveled in aqueous solutions (Chap. 3), only the following reaction takes place: H3 O þ þ OH  2 H2 O

ðEquilibrium 7:8Þ

The equilibrium constant of this reaction is the reciprocal of the autoprotolysis constant of water, i.e., 1014 (at 25 °C), indicative of a practically complete reaction. This also means that each added amount of OH− ions consumes an equal amount of

7.4 Theoretical Considerations and Graphical Representation of Titration Curves

183

Fig. 7.10 a pH-logci diagram; b pH-logci diagram rotated by 90°; c pH-logci diagram resulting in (b) and mirrored at its middle axis (pH = 7); and d pH-logci diagram together with the titration diagram, so that both pH scales are parallel to one another

H3O+ ions. The pH value of the solutions at each moment of titration is easily predictable: before the equivalence point is reached it holds: nsH3 O þ ¼ n
H3 O þ  n
;s OH

ð7:11Þ

where nsH3 O þ is the amount of hydronium ions at a distinct degree of titration s; n
H3 O þ is the initial amount of acid; and n
;s OH is the amount of base added up to the degree of titration s.
;EP
With n
;s OH ¼ s  nOH ¼ s  nH3 O þ it follows that: nsH3 O þ ¼ n
H3 O þ  s  n
H3 O þ ¼ ð1  sÞn
H3 O þ

ð7:12Þ


 If this value is divided by the volume of the titration mixture vG ¼ vA þ vsOH (with vA being equal to the volume of sample A and vsOH equal to the volume of the base added up to the degree of titration s), the initial amount of the acid is substituted by n
H3 O þ ¼ CH
3 O þ vA (CH
3 O þ equals the initial concentration of the acid), and the negative logarithm is introduced, the pH value of the titration mixture before the equivalence point is reached can be calculated as follows: pHs ¼  log csH3 O þ ¼  log CH
3 O þ  logð1  sÞ  log


vA  vA þ vsOH

ð7:13Þ

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7

Titrations

The last term in Eq. 7.13 is the result of dilution of the titration mixture by the added volume of the base solution. vA is also known as the dilution factor. When one can choose a large volume ðvA þ vsOH Þ of vA compared with vsOH (i.e., the concentration of the base is much larger than the concentration of the acid to be titrated), the dilution factor may be negligible.

The term

Eventually, the dilution is negligible if a relatively large concentration of the base is chosen, and the following simplified relation is obtained: pHs ¼  log csH3 O þ ¼  log CH
3 O þ  logð1  sÞ

ð7:14Þ

At s = 1, enough OH− ions have been added to complete the reaction with the H O+ ions of the solution. As already mentioned, the result is a corresponding salt solution. When hydrochloric acid is titrated with sodium hydroxide, a pure sodium chloride solution is synthesized, which would also be obtained if one dissolved the same amount of sodium chloride in water. This means that in the case of a titration of a strong acid with a strong base, at the equivalence point the relation cEP ¼ cEP OH H3 O þ holds, and with the autoprotolysis constant of water at 25 °C the pH value is 7. After reaching the equivalence point, an excess of OH− ions nExc OH is added to the salt solution, i.e.: 3


;s
nExc OH ¼ nOH  nH3 O þ

ð7:15Þ


;EP
and with the relation n
;s OH ¼ s  nOH ¼ s  nH3 O þ it follows that:
nExc OH ¼ ðs  1ÞnH3 O þ

pHs ¼ pKW þ log cExc OH ¼ 14 þ log CH3 O þ þ logðs  1Þ þ log

ð7:16Þ vA  vA þ vsOH

ð7:17Þ Finally, when the dilution can be neglected, it follows that:
pHs ¼ pKW þ log cExc OH ¼ 14 þ log CH3 O þ þ logðs  1Þ

ð7:18Þ

Figure 7.11 displays a number of pH-logci diagrams for different initial concentrations of hydrochloric acid and corresponding titration curves. The titration diagram shows a characteristic s-shaped (sigmoidal) curve. First, the pH increases slowly with the addition of the base, followed by a steep increase. The slope has a maximum at the equivalence point, before the increase in pH becomes slower. This means that the curve shows an inflection point where the change in pH with the addition of the basic solution is greatest. Analogous considerations apply to the titration of a very strong base with a very strong acid.

7.4 Theoretical Considerations and Graphical Representation of Titration Curves

185

logci -14

-12

-10

-8

-6

-4

-2

0

H3O

14

14

12

12

10

10

8

8 pH

Cl

OH

6

6

4

4

2

2

0

0

a) b) c)

0.0

0.5

1.0

1.5

2.0

Fig. 7.11 Titration curve of hydrochloric acid with different initial concentrations and sodium hydroxide


solution: a log Chydrochloric acid ¼ 3; b log Chydrochloric acid ¼ 2; and c log Chydrochloric acid ¼ 1

If a moderately strong or weak acid is titrated with a strong base, the following titration reaction takes place: HB þ OH  B þ H2 O

ðEquilibrium 7:9Þ

Equilibrium 7.4 is the reversed reaction of the corresponding base and water, i.e., the equilibrium constant is the reciprocal of Kb. Since a moderately strong or weak acid corresponds to a weak or moderately strong base (i.e., Kb is small), one can regard this reaction as also being quantitative (complete) after addition of the required amount of OH− ions. Before the addition of OH− ions (s = 0), the solution is that of the moderately strong or weak acid with the initial amount n
HB , and the pH value can be predicted with the volume of the sample vA by using known equations (cf. Chap. 3). If OH− ions are added, the ratio between HB and B− continuously changes as B− is formed according to Equilibrium 7.9, and before the − s s equivalence point is reached it holds: n
;s OH ¼ nB (where nB is the amount of B at degree of titration s). Since at any moment of titration the following relation must also be valid: n
HB ¼ nsHB þ nsB (where nsHB equals the amount of HB at degree of titration s), it holds that: nsHB ¼ n
HB  n
;s OH . Likewise, at any moment of titration the law of mass action (LMA) must be satisfied: Ka ¼

csB csH3 O þ csHB

¼

csH3 O þ





 s n
;s OH vA þ vOH 
 vA þ vsOH n
HB  n
;s OH

ð7:19Þ


;EP
Using the relationship n
;s OH ¼ s  nOH ¼ s  nHB , rearranging Eq. 7.19, and introducing the negative logarithm leads to the following equation for 0 < s < 1:

186

7

pHs ¼  log csH3 O þ ¼ pKa þ log s  logð1  sÞ

Titrations

ð7:20Þ

One important conclusion from Eq. 7.20 is that in a broad range before reaching the equivalence point, the titration curve of a moderately strong or weak acid with a strong base is almost independent of the initial concentration of the acid. This is because a buffer solution (Chap. 3) is synthesized, as both the acid and the corresponding base are present in the solution. It can be seen from the titration curve that as the addition of a strong base progresses, the pH initially changes more sharply, but then slowly, before the slope increases again, as has already been described in Sect. 3.7 “Buffer solutions.” At s = 0.5 the amount of acid equals the amount of corresponding base, i.e., pH = pKa. At this point the buffer solution has its maximum buffer capacity, i.e., the slope of the titration curve has a minimum. At the equivalence point the amount of OH− added to the solution exactly equals the initial amount of acid n
HB . This solution is the same as a solution that would be obtained when dissolving the same amount of a salt with a corresponding base, since the anion and cation are the same as the cation of the strong base used as the titrator (e.g., Na+) in the volume of the titration mixture. In other words, one must calculate the pH of a solution of a weak base (Chap. 3). c The pH value of the equivalence point of a titration of a moderately strong to weak acid with a very strong base will therefore always be in the alkaline range (pH > 7) at the equivalence point.

The higher the initial concentration of the acid to be titrated and the greater its pKa value, the higher the pH at the equivalence point. c Analogously, the pH value at the equivalence point of a titration of a weak to moderately strong base with a strong acid will always be in the acidic range (pH < 7).

In the titration of an acid HB, an excess of OH− ions nExc OH is added to the salt solution after reaching the equivalence point, and the considerations here are the same as for the titration of a strong acid with a strong base after reaching the equivalence point, i.e.:

pHs ¼ pKW þ log cExc OH ¼ 14 þ log CHB þ logðs  1Þ þ log

vA  ð7:21Þ vA þ vsOH

or, if dilution can be neglected:
pHs ¼ pKW þ log cExc OH ¼ 14 þ log CHB þ logðs  1Þ

ð7:22Þ

The construction of a titration curve for the titration of acetic acid with a strong base is displayed in Fig. 7.12.

7.4 Theoretical Considerations and Graphical Representation of Titration Curves

187

logci -14

-12

H3O

-10

-8

-6

HAc

-4

-2

0 14

14

12

12

10

10

8

8

P6

P5 P4 P3

pH 6

6

4

4

2

2

0

0

OH Ac

P2 P1

0.0

0.5

1.0

1.5

2.0


Fig. 7.12 Titration diagram of acetic acid (log CHB ¼ 1, pKa = 4.75) with sodium hydroxide


Fig. 7.13 Titration diagram of alanine (log Calanine ¼ 1, pKa1 ¼ 2:3, pKa2 ¼ 9:9) with sodium hydroxide

For the calculation of titration curves of polybasic acids with a very strong base or polyacidic bases with a very strong acid, similar considerations have to be made. This is explained by the example of the titration of the amino acid alanine. Amino acids are usually commercially available as so-called hydrochlorides, i.e., as salts in the form H2 A þ Cl . In the pH-logci diagram (Fig. 7.13) the following three forms of amino acid have been coded as follows:

188

7

O

O

R

O

R +

OH

Titrations

R +

NH 3

NH 3

H2A+

HA+/−

O

O NH 2

A−

Of course, the neutral form HA also exists, however, its concentration is so small that it is not discussed here (Chap. 3). At the beginning of titration (s = 0) a solution of alanine hydrochloride is present. Since the chloride ion is a very weak base, it is only the protolysis of the cation that determines the pH of the solution, according to: H2 A þ þ H2 O  HA þ = þ H3 O þ

ðEquilibrium 7:10Þ

The pH of the solution can be calculated with the simplified equation derived for a moderately strong acid, or it can be taken from the pH-logci diagram (Fig. 7.13) and transferred into the titration diagram (P1 in Fig. 7.13, cH3 O þ  cHA þ = ). In the range 0 < s < 0.5, the solution contains a buffer consisting of H2A+ and HA+/−. The pH of such a solution is calculable at any moment of titration in that
;EP
range by considering the relation n
;s OH ¼ s  nOH ¼ s  2nH2 A þ : pHs ¼  log csH3 O þ ¼ pKa1 þ log 2s  logð1  2sÞ

ð7:23Þ

At s = 0.25 (P2 in Fig. 7.13) the amount of H2A+ equals the amount of HA+/−, i.e., the maximum buffer capacity is reached and pH = pKa1. At the first equivalence point (s = 0.5), the cation is completely converted to HA+/−. HA+/− can react both as an acid and a base: HA þ = þ H2 O  A þ H3 O þ

ðEquilibrium 7:11Þ

HA þ = þ H2 O  H2 A þ þ OH

ðEquilibrium 7:12Þ

The pH of this solution is given by the simplified equation: 1 pHs¼0;5 ¼ ðpKa1 þ pKa2 Þ 2

ð7:24Þ

or can be taken from the pH-logci diagram (P3 in Fig. 7.13, cH2 A þ  cA ) and transferred to the titration diagram. In the range 0.5 < s < 1, the solution again contains a buffer, now consisting of HA+/− and A−. For amounts, the following relations hold:
nsA ¼ n
;s OH  nH2 A þ

ð7:25Þ

7.4 Theoretical Considerations and Graphical Representation of Titration Curves

nsHA þ = ¼ 2n
H2 A þ  n
;s OH

189

ð7:26Þ

Insertion into the LMA for the second protolysis step, and taking account of the
;EP
relation n
;s OH ¼ s  nOH ¼ s  2nH2 A þ , leads to: Ka2 ¼ ¼

 
n
;s csH3 O þ   n þ OH H2 A 2n
H2 A þ  n
;s OH

ð2s 

  2sn
H2 A þ  n
H2 A þ csH3 O þ  ¼  2n
H2 A þ  2sn
H2 A þ

ð7:27Þ

1ÞcsH3 O þ

ð2  2sÞ

By rearranging and taking the logarithm it follows that: pHs ¼  log csH3 O þ ¼ pKa2 þ logð2s  1Þ  logð2  2sÞ

ð7:28Þ

At s = 0.75 (P4 in Fig. 7.13) one half of HA+/− has been converted to A−, i.e., the second point with a maximum buffer capacity is reached (pH = pKa2). At the second equivalence point (s = 1), HA+/− has been completely converted to A−, i.e., the added amount of sodium hydroxide exactly equals the titratable protons of the dibasic acid. The result is a salt solution of NaA and NaCl. The anion A− can for the solution of this weak undergo protonation, and the concentration of cs¼1 H3 O þ acid can be calculated using the simplified equation CA
 ¼ pHs¼1 ¼  log cs¼1 H3 O þ ¼

n
H

2A

þ

ðvA þ vs¼1 OH Þ

pKw  pKa2  log CA
 2

: ð7:29Þ

or taken from the pH-logci diagram (P5 in Fig. 7.13, cHA þ =  cOH ) and transferred to the titration diagram. At s = 1.5 (P6 in Fig. 7.13, cA  cOH ) the solution contains the salts NaA, NaCl, and NaOH, each at concentration of 0.1 mol L−1 (neglecting dilution). The amount of excess sodium hydroxide can be calculated according to:
;s


nExc OH ¼ nOH  2nH2 A þ ¼ 2nH2 A þ ðs  1Þ ¼ nH2 A þ

ð7:30Þ


Neglecting dilution, it holds that cOH ¼ Calanine ¼ 0:1 mol L1 . The pH effect − due to the protonation of A can be disregarded.

7.4.2 Complexometric Titrations Chelating ligands are well suited for the titrimetric determination of metal ions, because the latter form very stable chelates. One of the most used ligands is the

190

7

Titrations

Fig. 7.14 Gerold Schwarzenbach (Copyright: Helv. Chim. Acta. Reproduced from Helv. Chim Acta 61 (1978) 1949–1961)

anion of EDTA (already considered in Chap. 4), since many metal ions, regardless of their charge, form 1:1 complexes with ethylenediaminetetraacetate. This compound was first synthesized by Ferdinand Münz [15] in the 1930s. Ferdinand Münz (1888–1969) worked from 1927 in the main laboratory of I.G. Farben in Leverkusen. He patented a synthesis of EDTA from monochloroacetic acid and ethylenediamine to replace citric acid with EDTA in the textile industry. In early 1945 he was sent to the concentration camp at Theresienstadt—fortunately he survived.

The application of chelate ligands, especially EDTA, for complexometric titrations was first developed by Gerold Schwarzenbach (Fig. 7.14) in the 1940s at ETH Zurich. Gerold Schwarzenbach (1904–1978) studied at ETH Zurich, where he earned his doctorate in 1928. In 1929 he was a Ramsey Fellow at University College London, then senior assistant to Paul Karrer at the University of Zurich. In 1930 he completed habilitation; in 1936 he was appointed titular professor; and in 1937–1938 he was a Rockefeller Fellow with Leonor Michaelis in New York and with Linus Pauling at Caltech in Pasadena. In 1943 he was appointed extraordinary professor and in 1947 full professor of analytical chemistry at the University of Zurich. In 1955 he became an ordinary professor and head of the laboratory of inorganic chemistry at ETH Zurich.

7.4 Theoretical Considerations and Graphical Representation of Titration Curves

191

Schwarzenbach was the author of the first monograph on complexometric titrations [16]. More extensive treatises were later published [17, 18]. 1945 can be regarded as the year of birth of complexometric titrations as an analytical method. They are thus the most recent “classical titrations,” following acid–base, precipitation, and redox titrations. At the meeting of the Swiss Chemical Society on February 25, 1945 in Bern in his lecture “Acids, Bases and Complexing Agents,” Schwarzenbach presented for the first time complexometric titrations of metal ions with the alkali metal salts of nitrilotriacetic acid and ethylenediaminetetraacetic acid. Due to the high precision of the titrations and the usually simple stoichiometric ratios, a large number of indirect titrations, substitution and back titrations have been developed with which it is now possible to determine cations as well as anions by volumetric analysis. Numerous selective determination methods without previous elaborate separation steps are possible with the choice of suitable complexing agents, auxiliary complexing agents, indicators, and buffers. The progression of complexometric titrations is best represented as a dependence of the negative decadic logarithm of the metal ion concentration  log cMen þ ¼ pMe on the titration degree s . Here, the degree of titration is defined as follows: s¼

added amount of ligand (n
;s L Þ

amount of ligand added until equivalence point is reached (n
;EP L Þ

ð7:31Þ

The calculation of selected points of the titration curve is done here neglecting dilution, which can easily be taken into account when necessary (see remarks on dilution in Sect. 7.4.1). For a complex formation of Me2 þ and L4 ions according to: Me2 þ þ L4  ½MeL2

ðEquilibrium 7:13Þ

the pMe at s ¼ 0, s ¼ 1:0, and s ¼ 2:0 is as follows:

(a) s ¼ 0: pMes¼0 ¼  log CMe , where CMe is the initial concentration of the metal ions. (b) s ¼ 1:0: At the equivalence point it holds that:

Kstab ¼

c½MeL2 cMe2 þ cL4

¼


CMe c2Me2 þ

ð7:32Þ

because ns¼1 ¼ ns¼1 and thus also cs¼1 ¼ cs¼1 . If the value of the stability L4 Me2 þ L4 Me2 þ

constant is large, the relation cs¼1  C holds, and hence c½MeL2  CMe . Me2 þ Me2 þ Then it follows that: cs¼1 Me2 þ

rffiffiffiffiffiffiffiffiffi
CMe ¼ Kstab

ð7:33Þ

192

7

Titrations

1 1
pMes¼1 ¼  log CMe  pKstab 2 2

ð7:34Þ

(c) s ¼ 2:0: It now holds that n
;s¼2 ¼ 2n
;EP , i.e., the solution contains L4 L4 s¼2 s¼2
c½MeL2 ¼ cL4  CMe . This leads to: Kstab ¼ cs¼2 Me2 þ 

cs¼2 ½MeL2 cs¼2 cs¼2 Me2 þ L4




CMe
cMe2 þ CMe

ð7:35Þ

1 ; i:e: pMes¼2  pKstab : Kstab

ð7:36Þ

For the progress of titration in the range 0  s  1 it holds that csMe2 þ ¼ Introducing the relationship it follows that:

n
;s L4

¼s

n
;EP L4

¼s

n
Me2 þ

n


Me2 þ

n
;s4

vG

L

.

and neglecting the dilution


pMes ¼  log CMe  logð1  sÞ

ð7:37Þ


After reaching the equivalence point (1\s), the relation cs½MeL2  CMe applies, n
;s

n



and hence csL4 ¼ vLG4  MevG2 þ ¼ ðs  1ÞCMe 2 þ . Inserting these relationships into the stability constant and rearranging gives: cs

2

Kstab ¼ cs ½MeLc s Me2 þ

L4


CMe
ð s1 ÞCMe Me2 þ s

 cs

csMe2 þ  ðs11ÞKstab ; i:e:

pMe  logðs  1Þ þ log Kstab

:

ð7:38Þ

Figure 7.15 shows the titration diagram for the titration of Mg2+ with EDTA,
neglecting dilution (log CMg 2 þ ¼ 2; log Keff;MgEDTA ¼ 8:19). For real examples it is of course necessary to use the effective stability constant Keff instead of the constant Kstab , since this constant determines log cMe2 þ at s ¼ 1:0 and s ¼ 2:0. Because the initial concentration of the metal ions plays an important role at the beginning of titration, both parameters have an influence on the course of the
titration diagram. The larger Keff and CMg 2 þ , the steeper the titration curve near the equivalence point. And thus, as for all other titrations, the random titration errors are smallest (cf. Sect. 7.5.2 “Titration errors”). An important finding here is that at large values of Keff , titrations exhibit very steep slopes on their titration curves even
at relatively small values of CMe n þ , i.e., they can be performed with small random titration errors. In Chap. 4 it was stated that the side reaction coefficients of EDTA strongly depend on the pH of the solution. The larger the pH of a solution, the less the

7.4 Theoretical Considerations and Graphical Representation of Titration Curves

193

10

8

pMe

6

4

2

0 0.0

0.5

1.0

1.5

2.0


Fig. 7.15 Titration diagram of Mg2+ ions with EDTA, neglecting dilution (log CMg 2 þ ¼ 2; log Keff;MgEDTA ¼ 8:19)

EDTA is protonated and the more the side reaction coefficient of EDTA approaches 1. However, the pH of the titration mixture cannot be arbitrarily shifted into the alkaline range, because many metal ions form sparingly soluble metal hydroxides and thus the side reaction coefficient of the metal ions can become very small. In these cases, auxiliary complexing agents must be added. An example is the titration of zinc ions at pH  10 in an ammonia buffer. Ammonia forms complexes with zinc ions. This prevents the formation of zinc hydroxide precipitates. In this case, ammonia is referred to as an auxiliary ligand. The following equilibria are established in a solution containing zinc ions and ammonia: Zn2 þ þ NH3  Zn(NH3 Þ2 þ ;

log K1 ¼ 2:18

ðEquilibrium 7:14Þ

Zn(NH3 Þ2 þ þ NH3  Zn(NH3 Þ22 þ ;

log K2 ¼ 2:25

ðEquilibrium 7:15Þ

Zn(NH3 Þ22 þ þ NH3  Zn(NH3 Þ23 þ ;

log K3 ¼ 2:31

ðEquilibrium 7:16Þ

Zn(NH3 Þ23 þ þ NH3  Zn(NH3 Þ24 þ ;

log K4 ¼ 1:96

ðEquilibrium 7:17Þ

The side reaction coefficient of the zinc ions can be calculated according to: aZn2 þ ¼

cZn2 þ cZn

2þ

þ cZnðNH

3Þ

2þ

þ cZnðNH

2þ 3 Þ2

þ cZnðNH

2þ 3 Þ3

þ cZnðNH

2þ 3 Þ4

ð7:39Þ

194

7

Titrations

With the relationships: cZnðNH cZnðNH cZnðNH cZnðNH

2þ 3 Þ2

2þ 3 Þ3

2þ 3 Þ4

¼ K2 cZnðNH

¼ K3 cZnðNH

¼ K4 cZnðNH

¼ K1 cZn2 þ cNH3 ¼ b1 cZn2 þ cNH3

ð7:40Þ

cNH3 ¼ K1 K2 cZn2 þ c2NH3 ¼ b2 cZn2 þ c2NH3

ð7:41Þ

cNH3 ¼ K1 K2 K3 cZn2 þ c3NH3 ¼ b3 cZn2 þ c3NH3

ð7:42Þ

cNH3 ¼ K1 K2 K3 K4 cZn2 þ c4NH3 ¼ b4 cZn2 þ c4NH3

ð7:43Þ

2þ 3Þ

3Þ

2þ

2þ 3 Þ2

2þ 3 Þ3

the side reaction coefficient can be expressed as a function of the cumulative formation constants: aZn2 þ ¼

1 1 þ b1 cNH3 þ b2 c2NH3

þ b3 c3NH3 þ b4 c4NH3

ð7:44Þ

Considering the values of the stability constants of the individual zinc-ammine complexes, and taking into account that the concentration of ammonia in the titration mixture is quite large in order to allow sufficient buffering (normally the concentrations are between 0.1 mol L−1 and 0.5 mol L−1), it is clear that the side reaction coefficient is small and that only a few zinc ions are free in the solution. Nevertheless, this concentration is sufficient for a titration. Titration of zinc ions in the presence of ammonia The titration mixture −3
contains an initial concentration of zinc ions (CZn mol L−1, and an 2 þ ) of 10 þ equilibrium concentration (also in equilibrium with NH4 ) of NH3 (cNH3 ) of 0.1 mol L−1. This solution should be titrated with EDTA solution with a concentration of 0.1 mol L−1, i.e., dilution can be neglected. The side reaction coefficient of the zinc ion under these conditions is aZn2 þ ¼ 1:8  105 . At pH = 10 (the usual value for the titration of zinc ions in the alkaline range) the side reaction coefficient of the ligand is aL4 ¼ 0:36 (cf. Chap. 4). The effective stability constant of the zinc EDTA complex is then: Keff;Zn ammine ¼ aZn2 þ aL4 KStab;Zn2 þ ¼ 2:05  1011 L mol1

ð7:45Þ

It is now possible once again to calculate and construct the corresponding titration curve for each point of the titration:
(a) s ¼ 0: pZns¼0 ¼  log CZn 2 þ  log aZn2 þ ¼ 7:74. s
(b) 0  s  1: pZn ¼  log CZn 2 þ  log aZn2 þ  logð1  sÞ. (c) s ¼ 1:0:

7.4 Theoretical Considerations and Graphical Representation of Titration Curves

Keff;Zn ammine ¼

195

c

c½ZnL2

¼

c0Zn2 þ c0L4


Zn2 þ
0 CZn 2þ  a CZn 2þ  c Zn2 þ Zn2 þ ¼  2  2 c 2 þ Zn c0Zn2 þ a 2þ

ð7:46Þ

Zn

cZn2 þ ¼ 

pZn

s¼1

aZn2 þ 2Keff;Zn ammine

¼  log 

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2Zn2 þ a2Zn2 þ CZn 2þ þ þ 2 4Keff;Zn ammine Keff;Zn ammine

aZn2 þ 2Keff;Zn ammine

ð7:47Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
a2Zn2 þ a2Zn2 þ CZn 2þ þ þ : 2 4Keff;Zn ammine Keff;Zn ammine ð7:48Þ

(d) 1\s: Nearly all zinc ions are bonded to EDTA, and the following
and csL4 ¼ approximations can be applied: cs½ZnL2  CZn 2þ
ðs  1ÞCZn 2 þ . It follows that:

pZns  logðs  1Þ þ log Keff;Zn ammine  log aZn2 þ :

ð7:49Þ

2+

In Fig. 7.16, the titration diagram for the titration of Zn ions under such conditions is displayed. The blue curve shows the presence of ammonia in an

16

pZn

14

12

0.5 M NH 3

10

0.1 M NH 3 8 0.0

0.5

1.0

1.5

2.0

Fig. 7.16 Titration diagram for the titration of Zn2+ with EDTA (dilution is neglected) at pH = 10 in the presence of two different equilibrium concentrations of ammonia
(log CZn 2 þ ¼ 2; log KStab;Zn2 þ ¼ 16:5; aL4 ¼ 0:36)

196

7

Titrations

equilibrium concentration of 0.5 mol L−1. Obviously, the larger the equilibrium concentration of the auxiliary ligand, the lower the change in the equilibrium concentration of the zinc ions near the equivalence point. Since ammonia also acts as a buffer base and buffer capacity plays an important role, in practice, a concentration range for the addition of ammonia must be very carefully estimated.

7.4.3 Precipitation Titrations Precipitation titrations are applicable to analytes that form a sparingly soluble precipitate with a standard solution in a stoichiometrical uniform, relatively rapid reaction. These requirements, and the fact that the recognition of the equivalence point is sometimes very difficult, mean that the number of practically usable precipitation titrations is quite limited. Nevertheless, they play an important role in analytical chemistry, as they are mainly used for the determination of widespread halides with silver nitrate standard solutions (argentometry) or for the determination of silver ions with a halide or thiocyanate standard solution. For titration curves, it is expedient to represent the negative logarithm of the equilibrium concentration of the analyte as a function of the degree of titration. As an example, the titration of chloride ions with a silver nitrate standard solution is discussed here. The reaction must be formulated as: Cl þ Ag þ  fAgClgs

ðEquilibrium 7:18Þ

Equilibrium 7.18 is the reversal of the solubility equilibrium of silver chloride, i.e., the equilibrium constant is the reciprocal value of the solubility product of silver chloride. Because this product is very small (pKs,AgCl = 9.8), Equilibrium 7.18 is far to the right side. Hence, each aliquot of Ag+ almost completely reacts with Cl−. The degree of titration s is defined here as follows: s¼

added amount of silver ions (n
;s Þ Ag þ amount of silver ions needed to reach the equivalence point (n
;EP Þ Ag þ ð7:50Þ

The amount of silver ions needed to reach the equivalence point n
;EP Ag þ
equals the initial amount of chloride ions nCl (according to the stoichiometry of the reaction).

7.4 Theoretical Considerations and Graphical Representation of Titration Curves

197

Similar to titration diagrams for complexometric determinations, one can first calculate selected points of the titration curve, e.g., at s ¼ 0, s ¼ 1:0, and s ¼ 2:0 (neglecting dilution; Sect. 7.4.1 “Acid–base titrations”):

(a) s ¼ 0: pCls¼0 ¼  log CCl  , where C  is the initial concentration of the Cl chloride ions. (b) s ¼ 1:0: The added amount of silver ions equals the initial amount of chloride ions n
Cl . Solid silver chloride forms in equilibrium with the solution (solubility equilibrium of silver chloride). Due to stoichiometry, the relationship cs¼1 ¼ cs¼1 Cl applies in the solution. Then it follows that: Ag þ

cs¼1 Cl ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Ks;AgCl ; i: e: pCls¼1 ¼ pKs;AgCl 2

ð7:51Þ

(c) s ¼ 2:0: Here, the concentration of Ag+ in the solution is almost completely determined by the amount of silver ions added after reaching the
equivalence point. It holds that cs¼2 ¼ CCl  , and hence: Ag þ cs¼2 Cl ¼

Ks;AgCl
: ; i: e: pCls¼2 ¼ pKs;AgCl þ log CCl
CCl 

ð7:52Þ

In the range 0  s  1 the following relation holds (vtotal is the volume of the titration mixture): csCl

¼


 CCl



n
;s Ag þ vtotal

ð7:53Þ

¼ sn
Cl it follows that: and with the relation n
;s Ag þ
s
csCl ¼ ð1  sÞCCl  ; i: e: pCl ¼  logð1  sÞ  log C  Cl

ð7:54Þ

After reaching the equivalence point, the equilibrium concentration of the silver ions can be calculated according to: csAg þ ¼

n
;s  n
Cl Ag þ

and from this it follows that:

vtotal


¼ ðs  1ÞCCl 

ð7:55Þ

198

7

Titrations

10

8

pCl

6

4

2

0 0.0

0.5

1.0

1.5

2.0


Fig. 7.17 Titration diagram for the titration of Cl− with Ag+ (log CCl  ¼ 1, pKs,AgCl = 9.8)

csCl ¼

Ks;AgCl s

; i: e: pCl ¼ pKs;AgCl þ logðs  1Þ þ log CCl ð7:56Þ ðs  1ÞCCl 

See Fig. 7.17. Also interesting is the titration of a solution of two ions, both of which give a slightly soluble precipitate with the titrator. Often, such a titration is referred to as simultaneous titration, although the precipitations proceed of course consecutively. The less soluble precipitate is formed first. If both solubility products are sufficiently different, the first precipitate will precipitate almost completely before the precipitation of the second precipitate begins. As an example, the titration curve for a solution that contains equal concentrations of chloride and iodide will be discussed here. The progress of such a titration is followed by means of the concentration of silver ions,

1 depicting both curves in one diagram. When CCl  ¼ C  ¼ 10 mol L1 , it I makes sense to define the degree of titration as follows:

7.4 Theoretical Considerations and Graphical Representation of Titration Curves

s¼

n
;s Ag þ n
Cl þ n
I

199

ð7:57Þ

where n
Cl is the initial amount of chloride ions and n
I is the initial amount of iodide ions. This means that the equivalence point for iodide is reached at s ¼ 0:5, and the equivalence point for chloride is at s ¼ 1:0, because silver iodide will precipitate first as it is less soluble than silver chloride. For the calculation of the function log cAg þ ¼ f ðsÞ the solubility products are needed: Ks;AgCl ¼ 1010 mol2 L2 and Ks;AgI ¼ 1016 mol2 L2 (these are good approximative data ). As with all other titration curves discussed here, the dilution by the addition of the titrator is neglected: (a) s ¼ 0: This concentration should be 0 before adding silver ions, but one can calculate which Ag+ ion concentration would be in equilibrium with iodide ions before AgI, the most sparingly soluble compound, precipitates. K For cI ¼ 101 mol L1 it follows that: cs¼0 ¼ Cs;AgI ¼ 1015 mol L1 .
Ag þ I pffiffiffiffiffiffiffiffiffiffiffi ¼ cs¼0:5 ¼ Ks;AgI ¼ (b) s ¼ 0:5 (equivalence point for iodide): cs¼0:5 I Ag þ 108 mol L1 . (c) The question now is at which silver ion concentration the precipitation of AgCl starts? This is apparently the case when the solubility product of
1  ¼ 10 AgCl is exceeded. With CCl mol L1 , such is the case at Ks;AgCl 1 s¼? 9 cAg þ ¼ C
 ¼ 10 mol L . This means one cannot reach the equivaCl

lence point of iodide without precipitating AgCl! The size of the resulting error is calculated below. pffiffiffiffiffiffiffiffiffiffiffiffiffi (d) s ¼ 1:0 (equivalence point for chloride): cs¼1:0 ¼ cs¼1:0 Ks;AgCl ¼ Cl ¼ Ag þ 105 mol L1 . (e) s ¼ 2:0 (100% overtitrated with respect to chloride and iodide ions):

1 cs¼2:0 ¼ CCl  þ C  ¼ 2  10 mol L1 . I Ag þ

Figure 7.18 displays the titration diagram log cAg þ ¼ f ðsÞ. It consists of two independent curved branches, one for iodide (red) and one for chloride (black). Theoretically, the curves intersect at the point where the precipitation of silver chloride starts. If the silver ion concentration is measured experimentally by means of potentiometry (Sect. 7.5.3.1) a curvature is obtained around the intersection point. This is due to the fact that mixed crystals are formed on a small scale and that the establishment of the respective equilibria is slow. To calculate the error resulting from the fact that the start of AgCl precipitation is before the equivalence point of iodide, it must first be understood that the intersection of the two curves is the only possible point for indexing

200

7

Titrations

16 14

AgI

12

pAg

10 8 6

AgCl

4 2 0.0

0.5

1.0

1.5

2.0

Fig. 7.18 Titration diagram for a solution containing equal concentrations of iodide and

1  ¼ C  ¼ 10 mol L1 ; Ks;AgCl ¼ 1010 mol2 L2 and Ks;AgI ¼ 1016 choride: CCl I 2 2 mol L . Curved branch for the precipitation of silver iodide given in red; curved branch of the precipitation of silver chloride given in black

iodide content. Since this point (although not quite as clear as in Fig. 7.18) is very easy to determine, one has to calculate iodide concentration at this point. In (c), the concentration of silver ions cAg þ ¼ 109 mol L1 was calculated for the incipient silver chloride precipitation. It follows that K cI ¼ 9 s;AgI 1 ¼ 107 mol L1 . The absolute error is Fabs ¼ cAg þ  cI , 10 mol L since only at the equivalence point the concentrations of silver and iodide ions are equal. At the intersection point of the two curved branches it follows that Fabs ¼ 109 mol L1  107 mol L1  107 mol L1 , and that the relative error is Frel ¼ Fabs =CI
 ¼ 107 mol L1 =101 mol L1 ¼ 106 , or –10–4%. This is a completely negligible error. Choosing a much higher concentration of chloride than iodide, or titrating chloride together with bromide, can lead to significant errors, as one can easily see when performing analogous calculations. In the case of precipitation titrations, the initial concentration of the analyte and the solubility constant Ks of the resulting precipitate determine the steepness of the titration curve: the higher the initial concentration and the smaller the solubility constant, the steeper the titration curve and the smaller the expected random error.

7.4 Theoretical Considerations and Graphical Representation of Titration Curves

201

7.4.4 Redox Titrations As mentioned in Chap. 6, redox equilibria can be quantitatively described using the Nernst equation. It therefore makes sense to display the progress of a redox titration by plotting the potential of the titration mixture versus the degree of titration. This is particularly useful because with potentiometry (Sect. 7.5.3.1) a measuring technique is available with which the calculated curves can also be compared, if a reversible electrode is available. But even in cases where the redox system on an electrode is irreversible, the measured titration curves may be (accidently) similar to those calculated. First, the simple case of cerimetric determination of iron(II) ions is described: Fe2 þ þ Ce4 þ  Fe3 þ þ Ce3 þ

ðEquilibrium 7:19Þ

The degree of titration can be defined here as follows: s¼

n
;s added amount of Ce4 þ Ce4 þ ¼ amount of Ce4 þ needed to reach the equivalence point n
;EP Ce4 þ

ð7:58Þ

For the two partial equilibria the reactions can be written as: Fe3 þ þ e þ  Fe2 þ

ðEquilibrium 7:20Þ

Ce4 þ þ e  Ce3 þ

ðEquilibrium 7:21Þ

The respective Nernst equations are: EFe3 þ =Fe2 þ ¼ EFe þ 3þ =Fe2 þ

RT aFe3 þ ln ; F aFe2 þ

ECe4 þ =Ce3 þ ¼ ECe þ 4þ =Ce3 þ

with EFe ¼ 0:77 V 3þ =Fe2 þ

RT aCe4 þ ln ; with ECe ¼ 1:61 V 4þ =Ce3 þ F aCe3 þ

ð7:59Þ ð7:60Þ

The equilibrium constant for Equilibrium 7.19 can be calculated according to Eq. 6.18 (Chap. 6):  K ¼ 10

16:9 ½V

 E 4 þ Ce

=Ce3 þ

E 3 þ Fe

=Fe2 þ

¼ 1014:2

ð7:61Þ

This means that the equilibrium is almost completely on the right side. When both redox systems are present in a solution at the same time, the solution of course has only ONE redox potential (just as a solution containing multiple acid–base pairs can only have ONE pH value). The ratios Ox1 =Red1 and Ox2 =Red2 , in the examples of Fe(III)=Fe(II) and Ce(IV)=Ce(III) above, are such that Eqs. 7.59 and 7.60 are EFe3 þ =Fe2 þ ¼ ECe4 þ =Ce3 þ ¼ Esolution .

202

7

Titrations

c This potential is always determined by the redox system, which is present in the largest total concentration. The reason for this is that a system always follows the difference in Gibbs energies, which corresponds to the maximum effective work—this is larger in a system having a higher total concentration, since the redox potential of a solution is a measure of its ability to do work. Electrical work is the product of charge and potential difference (DqDE). One can most easily estimate the possible electrical work, if the recoverable work by means of the oxidation of a third partner, e.g., hydrogen, in a solution containing both redox systems, Fe(III)=Fe(II) and Ce(IV)=Ce(III), is considered. Since the concentration of Ce(IV) is very small before reaching the equivalence point (resulting from Eq. 7.61), the product DqFeðIIIÞ!FeðIIÞ DEðOx=Red;H2 =H þ Þ is much larger than the product DqCeðIVÞ!CeðIIIÞ DEðOx=Red;H2 =H þ Þ , having the same difference of redox potential as the solution and the potential of the standard hydrogen electrode (DEðOx=Red;H2 =H þ Þ ). Thus, the ability to work for virtual hydrogen oxidation is only determined by the system Fe(III)=Fe(II). After reaching the equivalence point it is impossible to oxidize hydrogen with Fe(III) present in the solution, since the formed Fe(II) would be oxidized immediately by Ce(IV). This means that it is only the system Ce(IV)=Ce(III) that determines workability after reaching the equivalence point, and not the system Fe(III)=Fe(II). If both redox systems are present in equal concentrations (assuming a 1:1 stoichiometry of the redox reaction), i.e., at the equivalence point of such a titration, both equations must be taken into consideration to calculate the potential of the resulting solution. Figure 7.19 schematically represents the areas in which the redox systems Ox1 =Red1 and Ox2 =Red2 determine the redox potential of the solution.

E

2

EOx 2 /Red2

x

Ox2/Red2

E/V

Ox1/Red1 and Ox2/Red2

x Ox1/Red1

E

=0.5

EEP

EOx1 / Red1 +EOx 2 /Red 2 2

EOx1 /Red1

x

0.0

0.5

1.0

1.5

2.0

Fig. 7.19 Schematic representation of the areas, in which redox systems Ox1 =Red1 and Ox2 =Red2 determine the potential of the solution (assuming a 1:1 stoichiometry of the redox reaction)

7.4 Theoretical Considerations and Graphical Representation of Titration Curves

203

A special feature of redox titration curves is that the redox potential of the solution at the beginning of the titration (at s = 0) is determined by more or less unknown side reactions. A pure solution of iron(II) ions with absolutely no iron(III) ions would have, according to Eq. 7.59, a potential of 1, which makes no sense and indeed never exists, because by adjustment of the equilibrium: 1 Fe2 þ þ H3 O þ  Fe3 þ þ H2 þ H2 O 2

ðEquilibrium 7:22Þ

there is always a certain concentration of Fe3+. Before the equivalence point is reached (0  s  1), the added cerium(IV) ions are consumed to form equal amounts of iron(III). To calculate the potential of the titration mixture, the ratio of the concentrations of iron(III) and iron(II) has to be considered. By analogy to acid–base equilibria, this is referred to as a redox buffer solution, i.e., a solution whose redox potential changes only a little by adding a small amount of an oxidizing or reducing agent. The following relations apply: n
;s ¼ nsFe3 þ , n
;s ¼ sn
Fe2 þ , and nsFe2 þ ¼ n
Fe2 þ  n
;s . This gives the potential Ce4 þ Ce4 þ Ce4 þ of the titration mixture as: RT asFe3 þ RT nsFe3 þ ln s ln ¼ EFe 3þ 2þ þ =Fe F aFe2 þ F nsFe2 þ RT s ln þ F ð 1  sÞ

E ¼ EFe þ 3þ =Fe2 þ ¼ EFe 3þ =Fe2 þ

ð7:62Þ

The curve is of sigmoidal shape with an inflection point at s = 0.5. As with acid– base equilibria, there is a point of maximum buffering at which the potential of the solution is equal to the standard potential EFe , since the logarithmic term in 3þ =Fe2 þ

Eq. 7.62 becomes zero. At s = 1.0, exactly that amount of Ce4+ has been added, which corresponds to n
Fe2 þ . In the solution, therefore, the amounts of Ce3+ and Fe3+ are equal, and very small equal amounts of Ce4+ and Fe2+ are also present (which follows from Equilibrium 7.19). The potential of the titration mixture at the equivalence points EEP ¼ EFe3 þ =Fe2 þ ¼ ECe4 þ =Ce3 þ applies. If the two Nernst equations 7.59 and 7.60 are summed, it follows that: RT aFe3 þ RT aCe4 þ ln ln þ ECe þ 4þ =Ce3 þ F aFe2 þ F aCe3 þ RT aFe3 þ aCe4 þ ln þ ECe þ 4þ =Ce3 þ F aFe2 þ aCe3 þ

2EEP ¼ EFe þ 3þ =Fe2 þ ¼ EFe 3þ =Fe2 þ

ð7:63Þ

i.e., the potential at the equivalence point of the titration is the arithmetic mean of the standard potentials: EEP ¼

EFe þ ECe 3þ 4þ =Fe2 þ =Ce3 þ

2

ð7:64Þ

204

7

Titrations

However, this only applies to cases in which the reaction proceeds according to the above stoichiometry. The case where stoichiometric coefficients are not equal is discussed using the example of Equilibrium 7.23. After reaching the equivalence point (1 < s), practically all the iron is present as Fe3+. The amount of Ce3+ corresponds to the amount of Fe3+ (and thus to the original amount of Fe2+), and the solution additionally contains Ce4+ ions added after the equivalence point. The potential of the titration solution can be calculated with the concentration ratio of Ce3+ to Ce4+, according to Eq. 7.60. With the ¼ n
;s  n
Fe2 þ , nsCe3 þ ¼ n
Fe2 þ , and n
;s ¼ sn
Fe2 þ it follows that: relations nExc Ce4 þ Ce4 þ Ce4 þ þ ECe4 þ =Ce3 þ ¼ ECe 4þ =Ce3 þ

RT aCe4 þ RT ln lnðs  1Þ ¼ ECe þ 4þ =Ce3 þ F aCe3 þ F

ð7:65Þ

In the case that s = 2.0, the logarithmic term in Eq. 7.65 equals zero, and the potential of the titration mixture is equal to the standard potential of the corresponding redox couple of the titrator. The course of the titration curve is similar to that of a weak acid with a very strong base (cf. Fig. 7.20). Looking at Eqs. 7.62, 7.64, and 7.65, it is obvious that the titration curve depends only slightly on the initial concentration of the analyte and that the potential change near the equivalence point is greater, the greater the difference of the standard potentials of the reactants. Of course, in order to calculate real titrations, formal potentials instead of standard potentials must be used. Now the general case that stoichiometric coefficients are not equal is discussed. Furthermore, it is assumed that the establishment of equilibrium depends on the pH of the solution, i.e., the electron transfer is coupled with a proton transfer, e.g., in

Fig. 7.20 Theoretical titration curve for the titration of Fe2+ with Ce4+

1.6

E/V

1.4

1.2

1.0

0.8

0.6 0.0

0.5

1.0

1.5

2.0

7.4 Theoretical Considerations and Graphical Representation of Titration Curves

205

permanganometric titrations in acidic solutions (a more detailed description is given in [19]). The equilibrium is as follows (here the hydrated protons are simply written as H+): m1 Red1 þ m2 Ox2 þ mH þ  m1 Ox1 þ m2 Red2 þ

m H2 O 2

ðEquilibrium 7:23Þ

where index 1 is used for the analyte (titrand) and index 2 for the titrator. The electron transfer at the titrator is coupled with a proton transfer. Of course, analogous considerations can be made for the opposite case. The situation is further generalized in a way that the reaction products already can be present in the analyte solution before starting the titration, i.e., the initial mixture has the following qualitative composition:
titration  Red1;in þ Ox1;in þ Red2;in . For the degree of titration the following applies: n
;s n
;s added amount of Ox2 Ox2 2 ¼ ¼ v2 Ox s¼
amount of Ox2 needed to reach the equivalence point n
;EP v1 nRed1 Ox 2

ð7:66Þ Before the equivalence point is reached, Ox2 is consumed to form Ox1 (provided that the equilibrium constant of Equilibrium 7.23 is large). The following relations v2
nsOx1 ¼ vv12 n
;s n
;s or apply: nsRed1 ¼ n
Red1  vv12 n
;s Ox2 , Ox2 þ nOx1;in , Ox2 ¼ v1 snRed1

nsRed1 ¼ n
Red1  sn
Red1 , and nsOx1 ¼ sn
Red1 þ nOx1;in . With Q1 ¼

nOx1;in n
Red ,

the potential of

1

the solution before reaching the equivalence point can be calculated according to: þ s\1 : E ¼ EOx 1 =Red1

RT asOx1 RT Q1 þ s ln s ¼ EOx ln þ 1 =Red1 m2 F aRed1 m2 F ð 1  sÞ

ð7:67Þ

Additionally, the shape of the sigmoidal curve is determined by the ratio of the amounts of the oxidized and reduced form of the analyte, especially if Q1 s. After reaching the equivalence point, the following applies to the amounts: nsOx2 ¼

v2
v2
v2
s n
;s Ox2  v1 nRed1 ¼ v1 nRed1 ðs  1Þ and nRed2 ¼ v1 nRed1 þ nRed2;in . With Q2 ¼

nRed2;in n
Red ,

the

1

potential of the solution is described by: s[1 : E ¼ EOx þ 2 =Red2 E¼

EOx 2 =Red2

þ

m

aOx2 aH þ RT m1 F ln aRed2 v2

¼ EOx þ 2 =Red2 m

v1 ðs1ÞaH þ RT m1 F ln vv2 þ Q2

v

m 2
v1 nRed1 ðs1ÞaH þ RT m1 F ln vv2 n
Red þ nRed 1

1

2;in

ð7:68Þ

1

At the equivalence point, the stoichiometrically needed amount of Ox2 is added,
s¼1 s¼1 i.e., for the amounts applies: ns¼1 Ox1 ¼ nRed1 þ nOx1;in , v2 nRed1 ¼ v1 nOx2 (very small!),

206

7

Titrations

v2
and ns¼1 Red2 ¼ v1 nRed1 þ nRed2;in . The potential of the solution at the equivalence point is then given as:

EEP ¼

aOx1 aOx2 am m2 E1 þ m1 E2 RT Hþ ln þ m1 þ m2 ðm1 þ m2 ÞF aRed1 aRed2

ð7:69Þ

With Q1 and Q2, Eq. 7.69 can be simplified: EEP ¼

ð1 þ Q1 Þam m2 E1 þ m1 E2 RT Hþ ln m1 þ m1 þ m2 ðm1 þ m2 ÞF m2 Q 2 þ 1

ð7:70Þ

In Fig. 7.21 it is obvious, that the potential change near the equivalence point decreases with increasing values of Q1 and Q2. If Q1 = Q2 = Q, the position of the equivalence point is independent of Q. However, if Q1 6¼ Q2, the equivalence point shifts up or down. The ratio of the stoichiometric coefficients, however, has a more significant influence on the position of the equivalence point (cf. Fig. 7.22). Here, the influence of Q is comparably low. In Fig. 7.23, the theoretical titration curve for the titration of Fe2+ ions with permanganate in sulfuric acid solutions with different activities of protons was calculated assuming that neither Fe3+ ions nor Mn2+ ions are present at the beginning of the titration, to illustrate the influence of proton activities. The titration curves are not symmetric; this must be taken into consideration when evaluating redox titrations. In Table 7.1 some important redox titration methods are given.

Fig. 7.21 Theoretical titration curves as a function of Q1 and Q2 (v1 ¼ v2 ¼ 1, aH þ ¼ 1, m = 1, EOx ¼ 0:000 V, 1 =Red1 EOx ¼ 1:000 V, and 2 =Red2 T = 298.15 K). The curves were calculated with Eqs. 7.67, 7.68, and 7.70

7.4 Theoretical Considerations and Graphical Representation of Titration Curves

207

Fig. 7.22 Theoretical titration curves as a function of Q1 and Q2 (v1 ¼ 5; v2 ¼ 1, aH þ ¼ 1, m = 1, EOx ¼ 0:000 V, EOx ¼ 1:000 V, and T = 298.15 K). The curves were calcu1 =Red1 2 =Red2 lated with Eqs. 7.67, 7.68, and 7.70

Fig. 7.23 Theoretical titration curve for the titration of Fe2+ with MnO in a sulfuric acid 4 0 solution with different proton activities (Q1 ¼ Q2 ¼ 0; v1 ¼ 5; v2 ¼ 1, m = 8, Ec;Fe ¼ 3þ =Fe2 þ

¼ 1:507 V, and T = 298.15 K). The curves were calculated with Eqs. 7.67, 0:68 V, EMnO  =Mn2 þ 4

7.68, and 7.70

208

7

Titrations

Table 7.1 Important redox titration methods Method

Titrator

Titrand

Reference

KMnO4, acidic

2þ Fe2+, oxalic acid/oxalate, H2O2, NO 2 , UO2

[3, 20]

KMnO4, neutral

Mn2+

[3]

Dichromatometry

K2Cr2O7

Fe2+

Bromatometry

KBrO3

AsO3 3

Iodometry

I2

AsO3 3

Cerimetry

ðNH4 Þ2 CeðSO4 Þ3

2+ 2+ AsO3 3 , Fe , Sn , H2O2

[21]

Oxidimetric determinations Manganometry

[3] ,

SbO3 3

,

SbO3 3 ,

2+

+

, Sn , Cu , 2+

Sn ,

Hg22 þ

NH4þ ,

S2 O2 3

[3, 5] [3, 5]

Reductometric determinations Iodometry

KI (Na2S2O3)

3+ 3+  Cu2+, Cr2 O2 7 , Co , MnO2, PbO2, ClO3 , Fe

[3, 5]

Ferrometry

FeSO4

þ Cr2 O2 7 , VO2

[3]

Titanometry

TiCl3

 Fe3+, ClO 3 , NO3

[5]

7.5

Indication Methods for Titrations

7.5.1 Classical Methods 7.5.1.1 Self-indicating Titrations Self-indicating titrations, i.e., titrations in which the equivalence point can be recognized without auxiliary means, include some precipitation and redox titrations. No such acid–base titrations are known. The simplest way of detecting the titration endpoint in precipitation titrations is to carry out the titration until further addition of titrator causes no turbidity in the titration solution. This type of titration is quite cumbersome and time consuming, but the determination of silver ions with a standard sodium chloride solution, proposed by Gay-Lussac as early as 1832, is one of the most accurate methods of volumetric analysis. This titration is also called the “clear-point method” because the initially formed solid silver chloride remains partially in colloidal distribution, clouding the solution and flocculating completely just before reaching the equivalence point, after which no further excess silver ions are present in the solution to stabilize the colloidal particles. Silver ions adsorbed on colloidal AgCl particles lead to a positive charge and, at relatively low total ionic strengths, to such a large Debye length for the surrounding negative charge cloud (see also DLVO theory in physical chemistry textbooks) that the attractive interactions between the particles are small enough to prevent their aggregation.

On the other hand, in the case of the determination of cyanide ions with a silver nitrate standard solution, as described by Liebig in 1851, titration is carried out until the first appearance of turbidity. Here, two equilibria must be considered, a complex formation equilibrium and a solubility equilibrium:

7.5 Indication Methods for Titrations

  Ag þ þ 2CN  AgðCNÞ2

209

ðEquilibrium 7:24Þ

with log b2 ¼ 21:1 and AgCN  Ag þ þ CN

ðEquilibrium 7:25Þ

with pKL ¼ 14:2. Upon addition of silver ions to a solution containing cyanide ions, the dicyanoargentate ion forms first. Silver cyanide cannot yet be formed because the equilibrium concentration of silver ions is insufficient to exceed the solubility product. n þ Only when the molar ratio is nAg [ 0:5 does AgCN precipitate. The titration must be
 CN

carried out very slowly shortly before reaching the equivalence point, so that equilibrium establishment is ensured. This titration is clearly complexometric but is often discussed in textbooks in the context of precipitation titrations. Some permanganometric and iodometric redox titrations can be performed without an indicator, as they are self-indicating. A permanganate solution has a deep purple color. Manganese(II) solutions are pale pink, but dilute solutions are virtually colorless. In the direct titration method (addition of MnO 4 ) titration is performed until the solution has a pale reddish-purple color. Since the permanganate ions have a relatively high molar absorption coefficient (*2.5  104 L mol–1 cm–1 at 520 nm), a very low concentration (in the range of 10−5 mol L−1) is sufficient to produce a pale reddish-purple color. In iodometry, the [I3]− complex is formed in an aqueous iodide-containing solution in the presence of iodine. Its intrinsic color is so strong that a solution with a concentration of 5  10−5 mol L−1 of the complex is still clearly recognizably yellow.

7.5.1.2 Indication with Color Indicators Color indicators are ions or molecules that also undergo an equilibrium reaction with the ion to be determined. In this case, the color or the intensity of the color of a solution changes depending on the equilibrium concentration of the substance to be determined. Instrumentally, by means of measuring the absorption of a solution, the endpoint of a titration can be determined. However, it is also possible to use indicators, which exhibit strong color changes, or the appearance or disappearance of a color perceived by the human eye in the vicinity of the equivalence point. Such color systems exist for all types of equilibrium reactions, so that many classical titrations can be indicated with these compounds. It is obvious that the reaction that causes the color change must be similar to the titration reaction, i.e., an acid–base indicator is itself a substance that undergoes an acid–base reaction, an indicator for complexometric titrations has to undergo a complex formation reaction, and so on. Some selected indicators are discussed in detail in the following material. Acid–base indicators An acid–base indicator is a weak or moderately strong acid or base, where the indicator acid has a different color and structure to the corresponding indicator base [22]. In aqueous solutions the following equilibrium is established:

210

7

HI þ H2 O  I þ H3 O þ

Titrations

ðEquilibrium 7:26Þ

where HI is the indicator acid and I− is the indicator base. Of course, other corresponding acid–base pairs also exist, e.g., H2 I/HI , HI =I2 , and HI þ =I. Equilibrium 7.26 is described by the acidity constant Ka, which is called here KI (I for indicator): KI ¼

ac;I ac;H3 O þ ac;HI

ð7:71Þ

Indicators may be monochromic (one species is colorless, one species absorbs light in the visible range), or bichromic (both forms absorb light of different wavelengths in the visible range). If Equilibrium 7.26 is established, at pH values less than pKI, the indicator acid is mainly present, i.e., the color of the solution corresponds to the color of the indicator acid. At pH values larger than pKI, the indicator base dominates and the color of the solution corresponds to the color of the indicator base. At pH ¼ pKI , the ratio of concentration activities ac;HI ac;I is equal to one, and in the vicinity of this pH a continuous change in the color of the solution, from the color of the indicator acid to the color of the indicator base, is observable. Since the human eye perceives the pure color of one component, when this is in tenfold excess to the other colored component , mixed colors are observed a 1  ac;Ic;HI  10 in the range 10 1 . According to the Henderson–Hasselbalch equation a

(buffer equation is pH ¼ pKI  log ac;Ic;HI ; Chap. 3) the color change takes place at an interval of DðpHÞ = pKI  1. This interval is also called the transition interval of the indicator [12]. This is of course only a rough estimate, provided that both colors are perceived by the human eye with the same sensitivity. The transition interval is also affected by the ionic strength of the solution, especially if the activity coefficients of the indicator acid and indicator base are affected differently by the presence of electrolytes (e.g., if one form is an ion and the other is a neutral molecule). For monochromic indicators, the color can be recognized when around 10% of the total analytical concentration is in the colored form, i.e., the pH at which the color can be detected depends essentially on the total concentration of the indicator. To intensify the color contrast, pH-independent dyes can be mixed with the indicators. An example is the indicator Tashiro. It consists of a mixture of methyl red (indicator component) and methylene blue (contrast dye). Acidic solutions (pH\pKI ) appear violet (combination of intense pink and blue), basic solutions (pH [ pKI ) appear green (combination of yellow and blue), and at pH ¼ pKI (pKI is the acid constant of methyl red) the solutions are gray (combination of yellow, intense pink, and blue). Methylene blue does not act as a pH indicator here and is only a permanent color component in this mixture. However, methylene blue can be used as a redox indicator since the reduced form is colorless. It is therefore necessary to ensure that in acid–base titrations with Tashiro no reduction of methylene blue can happen.

7.5 Indication Methods for Titrations

211

Today, more than 200 indicators are known, and the entire pH range of strongly acidic to strongly basic aqueous solutions is thus covered, i.e., a suitable color indicator is available for almost any acid–base titration [23]. Of course, acid–base titrations are not limited to aqueous solutions. For non-aqueous solutions analogous considerations apply, and a number of indicators are also in use. For aqueous solutions, dyes of the following classes of substances are most frequently used: azo dyes, phthaleins, and sulfonephthaleins. Azo dyes Most acid–base indicators with an azo group are derivatives of azobenzene (two phenyl rings linked via an N=N double bond; Scheme 7.1). In addition, the introduction of polar substituents, such as carboxyl or sulfonic acid groups, increases the solubility of the compounds. Azo dyes are indicator bases. During protonation, a cation is formed in which the positive charge is delocalized (resonance between an aromatic compound and a quinoid structure). The color changes from yellow to red (or to red–violet/intense pink) (Scheme 7.2). Methyl orange (R1: SO3Na, R2: H, R3: CH3, R4: H) and methyl red (R1: H, R2: H, R3: CH3, R4: COOH) are two prominent members of the group of azo dyes.

Scheme 7.1 General formula of azo dyes (only one resonance structure is shown)

Scheme 7.2 Schematic description of the structural changes of an azo dye during protonation/deprotonation (only one resonance structure is given for the azo base and two for the indicator acid)

212

7

Titrations

Scheme 7.3 General formula of phthaleins (only one resonance structure is shown)

Scheme 7.4 Schematic description of the structural changes of phthaleins during protonation/deprotonation (only one resonance structure is depicted for the acid and one resonance structure for the base)

Scheme 7.5 Schematic description of the structural changes of phthaleins in strong alkaline solutions (pH [ 13) during protonation/deprotonation (only one resonance structure is depicted for each species)

Phthaleins Phthaleins are triarylmethane dyes (Scheme 7.3). All phthaleins are rather insoluble in water but soluble in ethanol. The acid form of most phthaleins is colorless (lactone form), whereas in a basic medium a colored quinoid structure is formed due to the opening of the lactone ring (Scheme 7.4). Therefore, most phthaleins are monochromic indicators. In strongly alkaline solutions (pH [ 13) the colored quinoid structure is transformed into the colorless trianion (Scheme 7.5).

7.5 Indication Methods for Titrations

213

Scheme 7.6 Phenolphthalein (only one resonance structure is depicted for each species)

Scheme 7.7 Structure of the neutral biprotonated form of sulfonephthaleins (only one resonance structure is depicted for each species)

Scheme 7.8 Two-step dissociation scheme of sulfonephthaleins (only one resonance structure is depicted for each species)

The most prominent phthaleine is phenolphthalein (Scheme 7.6, R1 = R2 = H). Phenolphthalein is colorless in protonated form, and at pH values above 8.2 a solution containing phenolphthalein has a pink color. Sulfonephthaleins Sulfonephthaleins are closely related to phthaleins (Scheme 7.7). They are sparingly soluble in water in biprotonated form, whereas the monosodium salts are soluble in water. Their structure contains the less stable sulfone ring, which opens in acidic solutions. This leads to the formation of a quinoid structure, and the sulfonephthaleins are colored in acidic form, too. Sulfonephthaleins can be protonated in two steps, and all three species in Scheme 7.8 are colored [24]. Bromothymol blue (R1: CH(CH3)2, R2: Br, R3: CH3) and Bromocresol green 1 (R : H, R2: Br, R3: CH3), e.g., belong to the group of sulfonephthaleins.

214

7 HB

pH pH pHEP

pH pKa 0

2

4

0 logci

HB

O

logCHB

Titrations

6

8

10

12

14

B

-2

-4

B logci

-6

HB

-8

-10

-12

OH

H3O

-14

Fig. 7.24 pH-logci diagram with the H3 O þ line, the OH− line, and the asymptotes of HB and B
for acetic acid, with CHB ¼ 0:1 mol L1 and pKa ¼ 4:75. Green highlight: optimal transition interval of an indicator for the titration of acetic acid at that concentration with a strong base; in the case of a two-color indicator the optimum condition is pKI ¼ pHEP

In order to find the appropriate indicator for an acid–base titration, the titration curve can be consulted: since the most intense color change of an indicator occurs at pH ¼ pKI , the pKI value of the indicator should be as close as possible to the theoretically expected pH value of the equivalence point of the titration (for a discussion of titration errors, see Sect. 7.5.2). For a rough and quick estimation of the interval in which an indicator changes its color, a pH-logci diagram alone is sufficient, as it is already possible to read the pH at the equivalence point of the titration (Fig. 7.24). Complexometric indicators Common indicators for complexometric titrations are organic dyes that form chelate complexes with metal ions. These indicators are also referred to as metallochromic indicators. The metal–indicator complex must have a different color than the free indicator. Usually, these are 1:1 complexes between the metal ion and the ligand, but 1:2 or 2:1 complexes can also occur. In order to function as chelating ligands, the indicator molecules (or indicator ions) need to have multiple ligand atoms capable of coordinating a metal ion (those with lone electron pairs, such as nitrogen or oxygen). Since these also react as Brønsted bases, a number of species may be present in the equilibria during the titration of a metal ion: (a) the protonated forms of the indicator (e.g., H2 I ; HI2 , and I3 ); (b) unprotonated and protonated metal–

7.5 Indication Methods for Titrations

215

indicator complexes (e.g., MeI and MeHI); and (c) possibly also unprotonated and protonated polynuclear complexes. It is important that the establishment of the equilibrium between the metal ion and the indicator is fast, so that the color change can be recognized quickly. Of course, the effective stability constant of the metal– indicator complex (Keff;MeI ) has to be smaller than the effective stability constant of the metal–EDTA complex (Keff;MeEDTE ), so that the indicator can be released from the metal–indicator complex. However, Keff;MeI also needs to be sufficiently large, so that the color change at the equivalence point is sufficiently sharp. Similarly, as with acid–base titrations, where the negative logarithm of the acidity constant of an indicator should be close to the equivalence point pH, ideally the logarithm of the effective stability constant of the metal–indicator complex must be close to the negative logarithm of the equilibrium concentration of the metal ions at the stoichiometric equivalence point: Keff;MeI ¼ cMe ¼

cMeI c0I cMe

cMeI Keff;MeI c0I

¼

ð7:72Þ 1



cMeI c0I

Keff;MeI cMeI  log cMe ¼ log Keff;Mel  log 0 cI cMel pMe ¼ log Keff;MeI  log cI

ð7:73Þ

where cMeI is the equilibrium concentration of the metal–indicator complex; cMe is the equilibrium concentration of the free metal ions; and c0I is the sum of the equilibrium concentrations of all species of the indicator except the metal–indicator complex. The first indicator used in complexometry was murexide, the acidic ammonium salt of purpuric acid (Scheme 7.9). Later, the azo dyes were also used. Probably the most commonly used dye is Eriochrome Black T, also called Erio T, the sodium salt of 2-hydroxy-1-

Scheme 7.9 Murexide (only one resonance structure is shown)

216

7

Titrations

OH OH _

O3 S

N

N

O2 N

Scheme 7.10 Structure of the indicator Erio T (H2 I ) (only one resonance structure is shown)

(1-hydroxynaphthyl-2-azo)-6-nitronaphthalene-4-sulfonic acid (Scheme 7.10). It is present in the pH range from 7 to 12 as a negatively charged monoanion, the strongly acidic sulfone group is already deprotonated. The hydroxyl groups can be deprotonated in two steps: H 2 I þ ruby colored

H2 O

Ka1 ¼



HI2 blue

þ

H3 O þ

cHI2 cH3 O þ ¼ 106:3 mol L1 cH 2 I

ðEquilibrium7:27Þ ð7:74Þ

and HI2 blue

þ

H2 O

Ka2 ¼



I3 þ orange

H3 O þ

cI3 cH3 O þ ¼ 1011:55 mol L1 cHI2

ðEquilibrium7:28Þ ð7:75Þ

where Ka1 and Ka2 are the acid constants of the indicator Erio T. Erio T is an indicator used, for example, to determine total water hardness. Water hardness is the content of dissolved alkaline earth metal ions in water, i.e., it is essentially the content of dissolved calcium and magnesium salts, since strontium and barium ions normally play no role. Another representative of this class of substances is calconcarboxylic acid (Scheme 7.11). Calconcarboxylic acid forms a red–violet complex with calcium ions in the strongly alkaline medium (pH  13), in which magnesium ions already precipitate as hydroxide. Its shade appears redder by adsorption to the surface of the solid magnesium hydroxide—thus the transition to blue (the color of the free indicator) becomes sharper. Calconcarboxylic acid enters into the following equilibria with the associated acid constants:

7.5 Indication Methods for Titrations

217

Scheme 7.11 Calconcarboxylic acid (only one resonance structure is shown here)

Scheme 7.12 Xylenol orange (only one resonance structure is shown)

H2 I2 þ red violet

H2 O



HI3 blue

þ

H3 O þ

; Ka3 ¼ 109:3 mol L1 ðEquilibrium 7:29Þ

HI3 blue

þ

H2 O



I4 þ pale pink

H3 O þ

; Ka4 ¼ 1013:7 mol L1 ðEquilibrium 7:30Þ

Xylenol orange (Scheme 7.12) belongs to the group of sulfonephthaleins. It makes titrations possible even at relatively low pH values (frequently pH = 1 to 3, sometimes also up to pH = 5). In acidic form, xylenol orange is partially deprotonated (pKa1 < 0, pKa2 = 2.32, pKa3 = 2.85, pKa4 = 6.7, pKa5 = 10.47, pKa6 = 12.23) and with some metal ions forms pale red to reddish-violet colored complexes. The free indicator has a lemon yellow color under these conditions. In alkaline solutions, the free indicator itself is red–violet colored, which is why it is not useful as an indicator under alkaline conditions.

218

7

Titrations

Precipitation titrations For precipitation titrations, there are three ways to cause a color change in the titration solution. In the titration of halides, according to Mohr, potassium chromate serves as an indicator. The solubility of silver chromate is greater than the solubility of the silver halides and the end point of the titration is recognized by the fact that chromate ions form a red–brown precipitate after the precipitation of the silver halide with excess silver ions: 2Ag þ þ CrO2 4  fAg2 CrO4 gs

ðEquilibrium 7:31Þ

The precipitate forms as soon as the solubility product of silver chromate (Ks;Ag2 CrO4 ¼ 4  1012 mol3 L3 ) is exceeded. The titration is therefore performed up to the point where the occurrence of a brown precipitate is observable. In order to identify the equivalence point of halide determination as accurately as possible, the concentration of chromate ions necessary to form silver chromate at the equivalence point, must be calculated. If, for example, chloride ions are titrated with silver nitrate, the silver ion concentration at the equivalence point can be calculated with Eq. 7.51 from the solubility product for silver chloride (Ks;AgCl ¼ 1:58  pffiffiffiffiffiffiffiffiffiffiffiffiffi s¼1 1010 mol2 L2 ):cs¼1 Ks;AgCl ¼ 1:26  105 mol L1 . The concentraCl ¼ cAg þ ¼ tion of chromate ions in the titration solution should then ideally be 3 3 12 K cCrO4 ¼ s;Ag2 CrO42 ¼
410 mol L1 2 ¼ 0:025 mol L1 . In practice, the concentra5 1:2610 mol L cs¼1 Ag þ

tion is chosen slightly lower ( 0:005 mol L1 ), so as not to mask the color of the silver chromate by the strong intrinsic coloration of the dissolved chromate ions. The titration according to Mohr is only possible in neutral to slightly alkaline solutions, since dichromate ions are formed in acidic solutions (Chap. 3), which do not form sufficiently insoluble silver dichromate, and silver hydroxide would precipitate in alkaline solutions. The titration method described by Volhard11 for the determination of silver ions is based on the precipitation of the sparingly soluble silver thiocyanate: Ag þ þ SCN  fAgSCNgs ;

with Ks ¼ 6:84  1013 mol2 L2 ðEquilibrium 7:32Þ

Here, the equivalence point is recognized by a coloration of the supernatant solution when iron(III) ions are added to the titration mixture, which form with an excess of thiocyanate ions the soluble red-colored iron(III)–thiocyanate complex. Up to the equivalence point, the concentration of thiocyanate is still insufficient to form the complex and produce the color, i.e., the titration can be stopped on the first faint pink color of the solution.

11

Jacob Volhard (1834–1910), German chemist, nephew and student of Justus von Liebig.

7.5 Indication Methods for Titrations

219

Scheme 7.13 The adsorption indicators: a eosin and b fluorescein (only one resonance structure is shown for each species)

Adsorption indicators are used in the Fajans12 titration method. The most common adsorption indicators are eosin and fluorescein (Scheme 7.13). If a halide ion–containing solution is titrated with a standard silver nitrate solution, a precipitate of very small particles, at which excess halide ions are adsorbed, is produced before reaching the equivalence point. This results in the precipitate having a negative surface charge and the particles remaining in solution as a colloid. After reaching the equivalence point, an excess of silver ions is present in the solution— these adsorb on the surface of the precipitate. This leads to a positive surface charge of the particles. The anions of the dye adsorb on these particles, which leads to a change in the electronic structure of the dye anions and thus to a color change of the precipitate. An adsorption indicator should adsorb strongly only near the equivalence point. For the determination of bromide and iodide, eosin is suitable, however, it is not suitable for the determination of chloride ions, since eosin is strongly adsorbed by the precipitate long before reaching the equivalence point. Therefore, for the titration of chloride, fluorescein is used as the adsorption indicator. Redox titrations In general, redox indicators can be divided into two groups: (1) specific indicators that respond to an ionic species or a specific substance of the redox system, and (2) non-specific indicators that indicate the establishment of a specific redox potential. Not all reactions with a redox indicator are reversible. One of the specific indicators is starch. Although the color of the free iodine is still recognizable in fairly high dilution, the use of starch makes the endpoint of an iodometric titration much more clearly recognizable, since starch forms with iodine a deep blue compound: this is observable at iodine concentrations of 10−5 mol L−1 (an improvement of a factor of five compared with the self-coloring of the iodine solution). What is important for analytical practice is that the iodine–starch complex is dependent on the presence of iodide ions, since iodine is incorporated in the form of polyiodide chains in the starch [25]. Bromatometry uses indicators that are irreversibly oxidized by the bromine that occurs after reaching the equivalence point. These include methyl red and methyl 12

Kasimir (Kazimierz) Fajans (1887–1975), Polish American physical chemist.

220

7

Titrations

orange. Since bromate temporarily occurs at the location where the titrator drops into the titration mixture (and thus bromine is formed at this point even before reaching the equivalence point and the indicator is partially consumed), a little more indicator must be added at the end of the titration so that the end point of the titration is not indexed too early. The second group includes substances which are stable in both the oxidized and the reduced state, and which have different colors. The color change of a redox indicator can be described in a similar manner to an acid–base indicator. For the redox reaction of the redox indicator: OxI þ ne þ mH3 O þ  RedI

ðEquilibrium7:33Þ

the Nernst equation is valid: m

RT aOxI aH3 O þ ln nF aRedI m f RT OxI fH3 O þ RT RT cOxI ln ln cm ln ¼ EOx þ þ H3 O þ þ I =RedI nF nF n1 F cRedI fRedI RT cOxI 0 ln ¼ Ec;ox þ þ I =RedI ;H3 O nF cRedI

EOxI =RedI ¼ EOx þ I =RedI

ð7:76Þ

The considerations concerning the color impression and color change of the solution are the same as for acid–base and complexometry indicators; that is, at the 0 the oxidized and reduced form of the formal potential of the indicator Ec;Ox I =RedI indicator are present in equal concentrations and a mixed color is observed. In the   0 0:05916 Volt (at 25 °C) (see comments on the interval E ¼ Ec;Ox þ  n I =RedI ;H3 O

transition interval of an acid–base indicator, Equilibrium 7.26), a color transition from the color of the oxidized form to the color of the reduced form (potentials based on the standard hydrogen electrode) is observable. Hence, it is crucial that the formal potential of the indicator is in the range of the steepest slope of the titration curve. The difference between the formal potentials of the reactants must be sufficiently large so that the color change of the indicator can be clearly recognized. There are organic redox indicators, such as diphenylamine sulfonic acid (Scheme 7.14) or methylene blue (Scheme 7.15), and there are metal complexes in which the central ion (usually iron or ruthenium ions) can be oxidized or reduced (Ferroin in Scheme 7.16 and Table 7.2).

Scheme 7.14 Structure of diphenylamine sulfonate (only one resonance structure is given)

7.5 Indication Methods for Titrations

221

Scheme 7.15 Structure of methylene blue (only one resonance structure is shown)

Scheme 7.16 Structure of Ferroin (only one resonance structure is shown)

Table 7.2 Redox indicators Name

Color of the oxidized form

Color of the reduced form

Standard potential (V)

Diphenylamine sulfonic acid Methylene blue Phenosafranin Tris(1,10-phenanthroline)iron (Ferroin) Tris(2,2´-bipyridyl)ruthenium

Red–violet Blue Red Blue Pale blue

Colorless Colorless Colorless Red Yellow

0.85 0.53 0.28 1.147 1.29

7.5.2 Titration Errors in Classical Titrations When using color indicators, it is necessary to distinguish systematic and random errors resulting from the choice of indicator. The magnitude of these errors needs estimation before experimental work, in order to identify the maximum possible accuracy of the intended titration. Furthermore, error estimation allows for the fixing of ranges of analyte concentrations, in which titrations are possible.

222

7

Titrations

7.5.2.1 Systematic Titration Errors A systematic indicator error occurs when the displayed value (pH, pMe, etc.) at the indicator’s transition point (TP) does not match the theoretical value at the equivalence point (EP) of a titration. According to the IUPAC recommendation, the point indicated by an indicator is also called the end point of a titration [12]. In short, systematic errors originate from deviations between the equivalence point and the experimentally determined end point. Of course, there are a number of other sources of systematic titration errors, such as incorrectly calibrated pipettes or burettes, but these are not addressed here, where chemical equilibria are examined. A further possible source of systematic error is the so-called “indicator error.” This is the amount of titrator consumed for the reaction by the indicator itself. To minimize this error, the concentration of the indicator in the sample solution is chosen as low as possible, i.e., in the range of 10−7 to 10−6 mol L−1. In this concentration range, systematic indicator error is negligible, provided that the concentration of the titrator is higher than 10−3 mol L−1. If the endpoint of a titration does not match the equivalence point, the titration is terminated either too soon (insufficient addition of titrator, negative systematic error) or too late (addition of an excess of titrator, positive systematic error). In general, one can formulate for the absolute titration error: 0 Fabs ¼ n
;TP  n
;EP B B

ð7:77Þ

Leaving aside dilution, the absolute titration error in moles per liter is:   n
;TP  n
;EP B Fabs mol L1 ¼ B 1L

ð7:78Þ

Relating the absolute error to the starting concentration of the analyte yields the dimensionless relative error: Frel ¼

Fabs CA


ð7:79Þ

In the following, some selected examples will demonstrate how systematic errors are estimated from titration diagrams. First, consider the titration of a very strong acid with a very strong base. The endpoint of such a titration is indicated by the transition point of the color indicator. The absolute error is:
;EP
;TP 0
Fabs ¼ n
;TP OH  nOH ¼ nOH  nH3 O þ

ð7:80Þ

TP At the end point of the titration the amounts nTP OH and nH3 O þ are present, and for the absolute error in moles per liter it follows that:

7.5 Indication Methods for Titrations

223 pHTP,A = pKI,A

Fig. 7.25 Reading the relevant concentration data from the pH-logci diagram of a strong acid with log CH
3 O þ ¼ 1 to estimate the systematic titration error

0

2

4

pHTP,B = pKI,B

pH 6

8

10

12

14

0

B

-2

logci

-4 -6 -8

-10

OH

H3O

-12 -14

TP Fabs ¼ cTP OH  cH3 O þ

ð7:81Þ

The systematic error can be estimated with a pH-logci diagram, if one draws an auxiliary line at pH = pKI (pKI is the acidity constant of the indicator acid) and reads the logarithm of the concentrations of H3O+ and OH−. In Fig. 7.25, two auxiliary lines for two indicators A and B with transition points pKI,A = 5 (green auxiliary lines) and pKI,B = 8 (red auxiliary lines) are plotted on the pH-logci diagram of a strong acid ¼ 105 mol L1 and cTP;A (log CH
3 O þ ¼ 1). Indicator A gives cTP;A OH ¼ H3 O þ 109 mol L1 . The resulting absolute error is Fabs;A  105 mol L1 , and the relative systematic error is –0.01%. For indicator B, the following values are 6 ¼ 108 mol L1 , cTP;B mol L1 , Fabs;B  106 mol L1 , obtained: cTP;B OH ¼ 10 H3 O þ and Frel;B  0:001%. For the absolute error of titration of a moderately strong to weak acid with a very strong base, the following applies:
;EP
;TP 0
¼ n
;TP Fabs OH  nOH ¼ nOH  nHB

ð7:82Þ

The amount of hydroxide added until transition point is mainly consumed to form the corresponding base B−, but a small amount of OH− remains in equilibrium with B−: TP TP n
;TP OH ¼ nB þ nOH

For the amount of acid to be titrated the following balance applies:

ð7:83Þ

224

7 pHTP,A = pKI,A

Fig. 7.26 Reading the relevant concentration data from the pH-logci diagram of a weak acid with log CH
3 O þ ¼ 1 and pKa = 5 to estimate the systematic titration error

0 0

2

4

HB

pHEP,B = pKI,B

pH 6

Titrations

8

10

12

14

B

-2

logci

-4 -6 -8 -10

OH

H3O

-12 -14

TP n
HB ¼ nTP B þ nHB

ð7:84Þ

Inserting Eqs. 7.83 and 7.84 into Eq. 7.82 leads to the absolute error in moles per liter: TP Fabs ¼ cTP OH  cHB

ð7:85Þ

An estimate of the systematic error here can also be made with a pH-logci diagram if one draws an auxiliary line at pH = pKI (pKI is the acid constant of the indicator acid) and reads the logarithms of the concentrations of HB and OH− (Fig. 7.26). In the case of indicator A (green auxiliary lines, pKI,A = 6) the following data 2 8 are read: cTP;A mol L1 , cTP;A mol L1 , Fabs;A  102 mol L1 , HB ¼ 10 OH ¼ 10 and Frel;A  10%, i.e., this indicator is not suitable for the titration of a weak acid with a strong base. In the case of indicator B (red auxiliary lines, pKI,B = 8) the 4 6 following data are read: cTP;B mol L1 , cTP;B mol L1 , HB ¼ 10 OH ¼ 10 Fabs;B  104 mol L1 , and Frel;B  0:1 %. As another example, the systematic error of the determination of water hardness by complexometric titration with EDTA is considered here. Provided that the complexing agent reacts with a metal ion in a simple stoichiometry (ratio 1:1), the 0 absolute titration error Fabs is the difference between the amount of complexing
 agent added until the transition point of the indicator n
;TP is reached, and the L

 amount of metal ions to be titrated nMe :

7.5 Indication Methods for Titrations

225

0 Fabs ¼ n
;TP  n
Me L

ð7:86Þ

The amount of L added to the transition point is mainly used to form the metal–
 0;TP ligand complex nTP MeL , but a small fraction (nL ) also exists as free ligand (sum of all amounts of the protonated forms of the ligand): 0;TP n
;TP ¼ nTP L MeL þ nL

ð7:87Þ

The metal ions are either free, bound to L, or bound to the indicator. The latter amount is negligibly small, due to the small total amount of indicator: TP TP TP TP n
Me ¼ nTP Me þ nMeL þ nMeInd  nMe þ nMeL

ð7:88Þ

Inserting Eqs. 7.87 and 7.88 into Eq. 7.86 results in: 0;TP 0;TP 0 TP TP TP Fabs ¼ nTP MeL þ nL  nMe  nMeL ¼ nL  nMe

ð7:89Þ

The absolute error in moles per liter is: TP Fabs ¼ c0;TP L  cMe

ð7:90Þ

The determination of water hardness is performed in two steps. First, the sum of the amounts of calcium and magnesium is determined at pH = 10 (ammonia buffer solution) with EDTA as titrator and Erio T as indicator. Assuming that only calcium ions or only magnesium ions in a concentration of 2 mmol L−1 are present in a water sample (hardness range 2, “medium”), the titration curves can be theoretically calculated using the method described in Sect. 7.4.2 (with the effective stability constants: Keff;CaL ¼ 1010:2 L mol1 and Keff;MgL ¼ 108:19 L mol1 ). These curves are shown in Fig. 7.27a, b. For the figure, photographs of the titration solutions were taken during the titration and assembled into a color map. With magnesium and calcium ions, 1:1 complexes are formed with Erio T: Mg2 þ þ I3  ½MgI ; KMgl ¼ Ca2 þ þ I3  ½CaI ;

KCaI ¼

cMgl ¼ 107 L mol1 cI3 cMg2 þ

ðEquilibrium 7:34Þ

cCaI ¼ 105:4 L mol1 ðEquilibrium 7:35Þ cI3 cCa2 þ

As described in Sect. 7.5.1.2, Erio T is an acid, i.e., protolysis reactions have to be considered as side reactions in the formulation of the effective stability constants of the metal–indicator complexes at pH 10 (at this pH value the dominating species is HI2−):

226

7

Titrations

Fig. 7.27 Titration diagrams for a magnesium ions and b calcium ions with a starting concentration of 2  10−3 mol L−1 with EDTA, disregarding dilution. The red and blue lines mark the lower and the upper limit of the transition interval of Erio T, respectively (providing that the concentration ratio is c0I c0I 1 10 cMeI ¼ 10 and cMeI ¼ 1 , respectively)

Keff;MgI ¼

cMgI 0 cI cMg2 þ

ð7:91Þ

Keff;CaI ¼

cCaI 0 cI cCa2 þ

ð7:92Þ

with c0I ¼ cH2 I þ cHI2 þ cI3

ð7:93Þ

With the acidity constants of Erio T (pKa1 = 6.3 and pKa2 = 11.55), the following effective stability constants of metal–indicator complexes result:

7.5 Indication Methods for Titrations

227

log Keff;MgI ¼ 5:44 log Keff;CaI ¼ 3:84 The equilibrium concentrations of the metal ions at the end point of the titration (transition point of the indicator, cMeI ¼ c0I ) are cTP ¼ 3:63  106 mol L1 and Mg2 þ ¼ 1:45  104 mol L1 . Rearranging the effective stability constant of the cTP Ca2 þ

calcium and magnesium EDTA complexes to express c0;TP L , and replacing the TP
concentration of the formed metal EDTA complex by cMeL ¼ CMe  cTP Me , leads to the following, provided that only magnesium ions are present: c0;TP ¼ L


TP CMg 2þ  c Mg2 þ

cTP K Mg2 þ eff;MgL

¼ 3:55  106 mol L1

ð7:94Þ

and the relative error is: Frel;Mg

2þ

TP c0;TP Fabs L  cMg2 þ ¼
¼ ¼ 4  105
CMg2 þ CMg 2þ

ð7:95Þ

In Fig. 7.27a it can be clearly seen that the systematic error is very small, because the color change is very close to the steepest rise of the titration curve. In the case that only calcium ions are present in a solution, one obtains: c0;TP ¼ L


TP CCa 2þ  c Ca2 þ ¼ 8  1010 mol L1 TP cCa2 þ Keff;CaL

ð7:96Þ

and the relative error is: Frel;Ca2 þ ¼

TP c0;TP Fabs L  cCa2 þ ¼ ¼ 0:072

CCa2 þ CCa2 þ

ð7:97Þ

If only calcium ions are present in the solution, the color transition is too early (negative systematic error, under the conditions mentioned it is indeed −7%), and under such conditions the color change is very slow (Ds for the range of the color change is very large). Therefore, in these cases, a defined amount of magnesium is added before the titration, because then the equivalence point is quite easily recognizable (Ds is relatively small), if one wants to perform a direct titration (the negative systematic error under the given conditions for Mg2+ is only −0,004%). Thus, the sum of the amounts of both ions is determined—then the amount of magnesium ions is subtracted from this sum and the amount of calcium ions is obtained. Alternatively, a substitution titration is possible, i.e., a defined amount of magnesium or zinc–EDTA complex is added to the aqueous solution and the

228

7

Titrations

liberated amount of magnesium or zinc ions is titrated with EDTA and Erio T as an indicator. In the second step of the water hardness titration, the amount of calcium is determined separately. For this purpose, the pH of the aqueous solution is adjusted to a value above 13. Then, magnesium ions precipitate as magnesium hydroxide. Calconcarboxylic acid, an azo dye similar to Erio T (Sect. 7.5.1.2), is used as an indicator, and the following equilibrium can be formulated for the formation of the metal–indicator complex: Ca2 þ þ I4  ½CaI2 ;

KCaI2 ¼ 105:85 L mol1

ðEquilibrium7:36Þ

With the acidity constants of calconcarboxylic acid (Equilibrium 7.29 and 7.30), the concentration of calcium ions at the end point of the titration can be calculated: this amounts to cTP ¼ 1:4  106 mol L1 , and with Keff;CaL ¼ 1010:69 L mol1 Ca2 þ

follows c0;TP ¼ 3  108 mol L1 and thus a relative error of −0.07%. L Similarly, systematic titration errors of precipitation titrations and redox titrations can be estimated. With the help of such calculations it is possible to judge in which concentration range a titration can be carried out and which indicator should be used to ensure a certain maximum systematic error is not exceeded (see also [26]).

7.5.2.2 Random Titration Errors Caused by the Indicator The indicated endpoint of a titration is subject to randomly scattered readings of the used volumes at the end points of repeated titrations. This scattering can be characterized by the standard deviation of the results. In addition to experimental sources of error which contribute to the random error, the width of the transition intervals of the indicators also contribute. In the case of repeated titrations, the measured values (e.g., pH values) identified as the transition point are randomly scattered in this transition interval. The magnitude of the random error is determined by the slope of the titration curve near the transition point of the indicator (which should differ as little as possible from the equivalence point) and the width of the transition interval. If the slope of the titration curve in the transition interval is very steep, this will only lead to a small scattering of the end point volumes (of titrator) and thus of the s values (degree of titration). For the same width of transition interval (e.g., DpH), the random scattering Ds is smaller, the steeper the slope of the titration curve. With a small slope on the titration curve, Ds is larger (Fig. 7.28). Another source of random error is the color contrast of the different indicator species. The lower the contrast between colors (e.g., the contrast between orange and yellow), the harder it is to decide whether the transition point has been reached. A detailed description of common acid–base indicators can be found in [26].

7.5 Indication Methods for Titrations 10

Fig. 7.28 Dependence of the random distribution of Ds on the slope of the titration curve with bromothymol blue as indicator (transition interval 6.0–7.6). a Steep titration curve; b flat titration curve

229

(a)

9

pH

8

7 Bromothymol blue

6

5 0.990

10

0.995

1.000

1.005

1.010

(b)

9

pH

8

7 Bromothymol blue

6

5 0.990

0.995

1.000

1.005

1.010

7.5.3 Instrumental Methods of Indication for Titrations Developments in instrumental analysis techniques also led to major advances in the indication of titrations in the second half of the twentieth century [27]. The indication methods presented in Sect. 7.5.1 are all based on the use of chemical indicators. The application of instrumental indication methods makes it in most cases possible to completely avoid the use of chemical indicators. Even when chemical

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indicators are used, the use of spectrophotometry allows optical detection of a color change with far greater precision (lower standard deviations) than the naked eye. The most common instrumental methods are: (a) potentiometry (most frequently used), (b) VIS spectrophotometry, and (c) conductometry. Of course, there are also examples of almost all other instrumental techniques, e.g., oscillometry, amperometry, and thermometry, but they are used very rarely and are usually limited to research.

7.5.3.1 Potentiometric Indication of Titrations In 1893 Anton Behrend (1856–1926) had already demonstrated that a precipitation titration can be traced potentiometrically. Wilhelm Böttger (1871–1949) followed in 1897 with potentiometric acid–base titrations. Both were students of Wilhelm Ostwald in Leipzig. In 1900, Friedrich Crotogino (1878–1947) showed that redox titrations can also be indicated potentiometrically [28]. Initially, however, these early works found few applications, because at that time a potentiometric measurement was very laborious and time consuming, because one had to use a Poggendorff compensation circuit. It was only through the development of high-impedance potentiometric measuring devices that simple and fast measurements were possible, especially with glass and other membrane electrodes (ion-sensitive electrodes). In 1934, Beckman described, in US Patent No. 4,823,364. 2,058,761, the first modern measuring device for electromotive force, based on vacuum tubes [29]. We now simply call these meters “pH meters,” and of course, they no longer have vacuum tubes, but integrated semiconductor circuits. Beckman unleashed a veritable revolution in metrology, and the potentiometric tracking of titrations became routine, followed later by instrumental recording of the titration curves, and finally, as computers developed, data acquisition, storage, and mathematical evaluation turned into an automated measurement routine. Potentiometry is the measurement of electrode potentials, i.e., always of potential differences. Using an electrode of the first kind, i.e., a metal in contact with its salt solution (e.g., Ag in contact with an Ag+ solution) the potential of that electrode is measured against that of a reference electrode. This is also done with an electrode of the second kind (e.g., an Ag/AgCl electrode). In the case of membrane electrodes the potential difference between two (usually identical) reference electrodes between which a membrane is located is measured. In all cases, these are heterogeneous equilibria. Here, the glass electrode because of its great importance for pH indications of acid–base titrations (and of course generally for pH measurements), the silver electrode because of its good applicability to indicate precipitation titrations of the halides, and the redox electrodes with “inert” noble metal electrodes are exemplarily presented. The glass electrode A glass electrode is an arrangement of one reference electrode on each side of a thin membrane made of a special glass, which has a not too large electric resistance. On

7.5 Indication Methods for Titrations

231

reference electrode 1

reference electrode 2

analyte solution

inner buffer solution

SiOH SiOH SiOH SiO

SiO SiOH

Fig. 7.29 Schematic description of a glass electrode

inner solution

glass membrane

outer (analyte) solution

sol(a)

sol(i)|sol(a)

g|sol(a)

sol(i) g|sol(i)

g

Fig. 7.30 Schematic description of the potential differences at a glass membrane

both sides of the membrane the following heterogeneous equilibrium are established (Figs. 7.29 and 7.30): f Si  OHgg þ H2 O  f Si  O gg þ H3 O þ

ðEquilibrium 7:37Þ

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Since this equilibrium describes a charge transfer at a phase boundary [30], it is an electrode reaction. The establishment of this equilibrium changes the galvanic potential difference across the two phase boundaries D/solðiÞjg ¼ /solðiÞ  /g , where g is the glass phase, sol(i) is the inner solution phase (solution in the glass bulb), sol(a) is the phase of the outer solution (analyte solution), and / denotes the corresponding internal potentials. Using two identical reference electrodes, the difference between the internal potentials of the inner and outer solution phases DEmembrane ¼ D/gjsolðiÞ  D/gjsolðaÞ can be measured. The equilibria at the two phase boundaries are: sol(i)|g: f Si  OHgg þ ðH2 OÞi  f Si  O gg þ ðH3 O þ Þi

ðEquilibrium 7:38Þ

sol(a)|g: f Si  OHgg þ ðH2 OÞa  f Si  O gg þ ðH3 O þ Þa

ðEquilibrium 7:39Þ

D/gjsolðiÞ ¼ D/ gjsol þ

RT af SiO gg aH3 O þ ;ðiÞ ln F af SiOHgg

ð7:98Þ

D/gjsolðaÞ ¼ D/ gjsol þ

RT af SiO gg aH3 O þ ;ðaÞ ln F af SiOHgg

ð7:99Þ

For DEmembrane applies: DEmembrane ¼ D/gjsolðiÞ  D/gjsolðaÞ RT af SiO gg aH3 O þ ;ðiÞ ln ¼ D/  D/ gjsol þ gjsol F af SiOHgg  þ a a f SiO gg H3 O ;ðaÞ RT ln  F af SiOHgg

ð7:100Þ

and hence: DEmembrane ¼ D/gjsolðiÞ  D/gjsolðaÞ ¼ ¼

RT aH3 O þ ;ðiÞ ln F aH3 O þ ;ðaÞ

RT af SiOHgg af SiO gg aH3 O þ ;ðiÞ ln F af SiO gg aH3 O þ ;ðaÞ af SiOHgg

ð7:101Þ  The values for D/ gjsol , af SiOHgg , and af SiO gg , while unknown, should ideally be the same on both sides of the glass membrane. This is not completely fulfilled in the real case, so that a so-called asymmetry potential always occurs, which plays a role in the calibration of the glass electrodes. Equation 7.101 shows that the potential drop across the membrane is proportional to the logarithm of the

7.5 Indication Methods for Titrations

233

ratio of proton activities and thus depends only on the pH values of the two solutions: DEmembrane ¼ D/gjsolðiÞ  D/gjsolðaÞ ¼

RT aH3 O þ ;ðiÞ ln F aH3 O þ ;ðaÞ

RT ðlog aH3 O þ ;ðiÞ  log aH3 O þ ;ðaÞ Þ F RT ðpHðaÞ  pHðiÞ Þ ¼ 2:303 F

¼ 2:303

ð7:102Þ

Since the pH of the inner solution has a constant value, it follows that: RT RT pHðaÞ  2:303 pHðiÞ F F RT pHðaÞ ¼ const: þ 2:303 F

DEmembrane ¼ 2:303

ð7:103Þ

With the help of a glass electrode, thanks to the development of modern pH meters, the titration curves calculated in Sect. 7.4.1 can be measured and recorded. It should be noted that a variety of other membrane electrodes are used as ion-sensitive electrodes: if the membrane consists of a Eu-doped LaF3 single crystal, fluoride ion activities can be measured. With PVC membranes, which contain so-called ion carriers and additionally an organic salt, one can measure, for example, alkali metal and alkaline earth metal ions. Valinomycin is a crown ether that is very selective for potassium ions and therefore used as an ion carrier. Practically all membrane electrodes are now produced as so-called combination electrodes (single-rod electrodes) in which the two reference electrodes are integrated. Figure 7.31 shows the schematic structure of a combination electrode. They play an even more important role as sensors than in the indication of titrations [31, 32]. The glass electrode was originally introduced by the physiologist Max Cremer (Fig. 7.32) in the search for the causes of potential differences across cell membranes. Fritz Haber (Fig. 7.33) and Zygmunt Klemensiewisz (Fig. 7.34) developed the glass electrode into an analytically useful instrument. The theory of the glass electrode presented here was developed by Friedrich G. K. Baucke (Fig. 7.35) [33]. The silver electrode The silver electrode consists of a silver wire inserted into an Ag+-containing solution. The potential is calculated using the Nernst equation: E ¼ EAg þ þ =Ag

RT ln aAg þ F

ð7:104Þ

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h

g

(a) ionsensitive glass membrane (b) inner buffer solution (c) inner reference electrode (d) diaphragm

f e

(e) outer reference solution (f) outer reference electrode (g) refilling neck

d

(h) electric contact

c b a Fig. 7.31 Schematic description of a bulb-shaped glass electrode as a combination electrode

Fig. 7.32 Max Cremer (1865–1935) (provided by Th. Cremer, Munich)

If the potential of a silver wire is measured against a commercial reference electrode during the titration of a chloride, bromide, or iodide solution, the curves discussed in Sect. 7.4.3 and Fig. 7.18 are recorded because E ln aAg þ .

7.5 Indication Methods for Titrations

235

Fig. 7.33 Fritz Haber (1868–1934) (© archive of the Max-Planck Society, Berlin-Dahlem)

Fig. 7.34 Zygmunt Aleksander Klemensiewicz (1886–1963) (middle) (© archive of the Max-Planck Society, Berlin-Dahlem)

Silver electrodes are also of particular importance because they are the most commonly used reference electrodes for potentiometric measurements. In Chap. 6, the standard hydrogen electrode was introduced, which, however, is unsuitable for most potentiometric applications and must therefore be replaced by an easy-to-use reference electrode. If a silver wire is covered with a porous layer of AgCl (this can be done by anodic (electrochemical) oxidation of the silver in a chloride solution)

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Fig. 7.35 Friedrich G. K. Baucke (1930–2013) (J. Solid State Electrochem. (2011) 15: 3–4)

Fig. 7.36 Typical construction of an Ag/AgCl reference electrode

Ag

AgCl

KCl solution

and inserted in a saturated KCl solution (Fig. 7.36), the silver ion activity at the silver surface is determined by the solubility product of the AgCl and the chloride ion concentration cCl of the saturated KCl:

7.5 Indication Methods for Titrations

237

RT ln aAg þ F

ð7:105Þ

RT Ks;AgCl ln  F cCl

ð7:106Þ

RT RT ln Ks;AgCl  ln cCl F F

ð7:107Þ

E ¼ EAg þ þ =Ag þ E ¼ EAg þ =Ag

þ E ¼ EAg þ =Ag

It is common to refer to the first two terms of the right-hand side of this equation as the standard potential of the silver–silver chloride electrode: EAg=AgCl ¼ EAg þ þ =Ag

RT ln Ks;AgCl F

ð7:108Þ

Metal electrodes, whose potential is controlled by the solubility product of a sparingly soluble salt, are called electrodes of the second kind. Other examples are the calomel electrode (Hg/Hg2Cl2) and the mercury–mercury oxide electrode (Hg/HgO). The potentials of reference electrodes are tabulated [34], so that measured potentials can be easily converted to values related to the standard hydrogen electrode. Of course, the temperature and possible pressure dependencies must also be taken into account, as well as the influence of activity coefficients, when evaluating the potential of a reference electrode. Redox electrodes with “inert” noble metal electrodes The potentials of dissolved redox pairs can be measured in a few cases using a noble metal electrode (or graphite: see quinhydrone electrode, Sect. 6.4.1). The titration curves calculated in Sect. 7.4.4 can only be measured as calculated if an inert electron conductor, i.e., a metal or graphite, can be found for which the redox couples are reversible. Unfortunately, there are only a few reversible systems, e.g., uranyl/uranium (UO22 þ /U4+), hexacyanoferrate(III)/hexacyanoferrate(II), and hydoquinone/quinone, on platinum. The vast majority of systems are irreversible, as was already mentioned in Sect. 6.9 “kinetic aspects of redox equilibria” for the permanganate/Mn2+ system.

7.5.3.2 Spectrophotometric Indication According to Lambert–Beer’s law [35, 36], under certain conditions there is a linear relationship between the concentration of an absorbing substance in a solution and the absorbance. The titration curves display the absorbance of the titration mixture as a function of the volume of added titrator or the titration degree. In the simplest case one obtains two straight lines that intersect at the equivalence point (Fig. 7.37): Case 1. Only the analyte is colored, not the titrator or the reaction products. Case 2. Only the titrator is colored, not the analyte or the reaction products. Case 3. Only the reaction products are colored, not the analyte or the titrator.

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Case 2

Case 1

A

A

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

Case 3

2.0

Case 4

A

0.0

Titrations

A

0.5

1.0

1.5

0.0

2.0

0.5

1.0

Case 5

A

0.0

0.5

1.0

Fig. 7.37 Possible spectrophotometric titration curves

1.5

2.0

1.5

2.0

7.5 Indication Methods for Titrations

239

Case 4. The analyte and titrator are colored, not the reaction products. Case 5. The reaction products, and to a lesser extent the titrator, are colored, not the analyte. In all cases, the evaluation of the curves can be performed by extrapolating linear branches to the intersections. The curves can be more complicated if indicators are used. In such cases one often gets a curved shape before reaching the equivalence point. If the absorption no longer changes after reaching the equivalence point, the equivalence point can also be determined by extrapolation. However, there are also curves that show a significant curvature at the equivalence point. This can be caused by, e.g., a marked solubility of sparingly soluble precipitates, a marked dissociation of complexes, too small a difference in the redox potentials of the analyte and the redox indicator, and slow establishment of the equilibrium near to the equivalence point.

7.5.3.3 Conductometric Indication The conductometric indication is based on the measurement of the electrical conductivity of a solution. Since the conductivity is a sum parameter, i.e., a function of the total content of electrolytes in a solution, the method is non-specific. Only the changes of the contributions of the reaction underlying the titration to the sum of the contributions of all ions of the solution are recorded, irrespective of whether these ions are involved in the titration reaction or not. Thus, conductometric indication can be used whenever the total concentration of ions changes (e.g., formation of sparingly soluble precipitates in precipitation titrations), or when the concentration of ions changes, having particularly high equivalent conductivities, like H3 O þ or OH−, because here the so-called Grotthuss conduction mechanism13 (see physical chemistry textbooks) occurs. The conductometric indication is particularly suitable for acid–base titrations and precipitation titrations. The conductometric indication curves can be as variable as the photometric indication curves. The course essentially depends on the relationship between the changes in the conductivity before and after the equivalence point. For instance, when a strong acid is titrated with a strong base (or vice versa), a V-shaped indication curve results, mostly exhibiting a sharp kink at the equivalence point. For the titration of hydrochloric acid with sodium hydroxide solution, the following equilibrium reaction can be formulated: H3 O þ þ Cl þ Na þ þ OH

 Cl þ Na þ þ H2 O

ðEquilibrium 7:40Þ

Figure 7.38 schematically shows the contributions individual ions make to the total conductivity and what the resulting indication curve looks like. In the presence of other electrolytes, which do not participate in the reaction, the indication curve is shifted upward, i.e., the changes in the conductivity during the titration are smaller and hence the random errors increase. 13

Theodor von Grotthuss (1785–1822), German Baltic scientist.

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0.5

1.0

1.5

Titrations

2.0

Fig. 7.38 Schematic description of the conductometric indication curve of the titration of hydrochloric acid with sodium hydroxide. Pink line, contribution of H3 O+ ions; violet line, contribution of the OH− ions; green line, contribution of Cl− ions; blue line, contribution of Na+ ions; and black line, measurable total conductivity

When titrating a weak acid with a strong base, e.g., acetic acid with sodium hydroxide solution, the conductivity of the solution at the beginning of the titration is lower than that of a strong acid, since the weak acid is only partially protolyzed: CH3 COOH þ H2 O  CH3 COO þ H3 O þ

ðEquilibrium 7:41Þ

If sodium hydroxide solution is added at the beginning of the titration, the H3 O+ ions are replaced by Na+ ions and, in addition, the dissociation of the acid is suppressed by the formed acetate ions, so that overall the conductivity decreases. Upon further addition of sodium hydroxide, the unprotolyzed acid is replaced by Na+ ions and acetate ions, therefore the conductometric indication curve passes through a minimum and then increases to the equivalence point. After reaching the equivalence point, excess OH− ions are present, which, because of their high equivalent conductivity, cause a strong increase in the total conductivity. Therefore, there is a kink in the indication curve at the equivalence point, which, however, is less sharp than in the titration of a strong acid with a strong base (Fig. 7.39).

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Fig. 7.39 Schematic representation of a conductometric indication curve for the titration of a weak acid with a strong base

0.0

0.5

1.0

1.5

2.0

References 1. Skoog DA, Holler FJ, Crouch SR (2007) Principles of instrumental analysis. Thomson Brooks/Cole, Andover 2. Harris DC, Lucy ChA (2016) Quantitative chemical analysis, 9th edn. Freeman, New York 3. Mendham J, Denney RC, Barnes JD, Thomas MJK (2000) Vogel’s textbook of quantitative inorganic analysis, 6th edn. Pearson, Harlow 4. Madsen ER (1958) The Development of Titrimetric Analysis till 1806. GEC Gad Publishers, Copenhagen 5. Beckurts H (1913) Die Methoden der Massanalyse. Vieweg und Sohn, Braunschweig, pp 1031–1045 6. Home F (1756) Experiments on Bleaching. Sands, Donaldson, Murray & Cochran, Edinburgh, p 100 7. Page FG (2001) The Birth of Titrimetry: William Lewis and the Analysis of American Potashes. Bull Hist Chem 26:66–72 8. a) Mohr F (1855) Lehrbuch der chemischen Titrirmethode. 1st edn, Viehweg und Sohn, Braunschweig, b) Mohr F, Classen A (1896) Lehrbuch der chemisch-analytischen Titrirmethode. 6th edn, Aufl. Viehweg und Sohn, Braunschweig 9. Oesper RE (1927) J Chem Educ 4:1357–1363 10. Mohr F (1874) Lehrbuch der chemisch analytischen Titrirmethode, 4th edn. Braunschweig, Vieweg und Sohn, p 4 11. Kolthoff IM, Stenger VA (1942, vol 1), (1947, vol 2), (1957 vol 3) Volumetric analysis. Interscience, New York 12. Lipták BG, Venczel K, eds (2016) Analysis and analyzers: Volume II. CRC Press, Boca Raton 13. Kolev SD, Mckelvie ID (eds) (2008) Advances in flow injection analysis and related techniques (Comprehensive Analytical Chemistry, vol 54) Elsevier, Amsterdam 14. Sandel FB, West TS (1969) Pure Appl Chem 18:427–436 15. Maier H (2015) Chemiker im “Dritten Reich”. Wiley-VCH Weinheim, pp 577–578 16. (a) Schwarzenbach G (1957) Complexometric titrations. Methuen Co, London, (b) Schwarzenbach G, Flaschka H (1969) Complexometric titrations. 2nd ed, Methuen Co, London

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17. (a) Přibil R (1982) Applied Complexometry: Pergamon Series in Analytical Chemistry, Pergamon Press, Oxford, (b) Přibil R (1960) Komplexometrie, 5 volumes, VEB Deutscher Verlag der Grundstoffindustrie, Leipzig 18. West TS (1969) Complexometry with EDTA and related reagents. BDH Chemical Ltd, Poole 19. Henrion G, Scholz F (1984) Z Chem (Leipzig) 24:180–182 20. Berka A, Vulterin J, Zýka J (1965) Newer redox titrations. Pergamon Press, Oxford 21. Petzold W (1955) Die Cerimetrie und die Anwendung der Ferroine als massanalytische Redoxindikatoren. Verlag Chemie, Weinheim 22. Kolthoff IM (1937) Acid-base indicators. The Macmillan Company, New York 23. Sabnis RW (2007) Handbook of acid-base indicators. CRC Press, Boca Raton 24. Aragoni MC, Arca M, Crisponi G, Nurchi VM, Silvagni R (1995) Talanta 42:1157–1163 25. Sänger W (1984) Naturwiss 71:31–36 26. Kahlert H, Meyer G, Albrecht A (2016) ChemTexts 2:7 27. Kraft G, Fischer J (1972) Indikation von Titrationen. De Gruyter, Berlin 28. Scholz F, Inzelt G, Stojek Z (2010) Seminal publications in electrochemistry and electroanalysis. In: Scholz F (ed) Electroanalytical methods. Springer Heidelberg, 339–342 29. Inzelt G (2005) J Solid State Electrochem 9:181–182 30. Bach H, Baucke F, Krause D (eds) (2001) Electrochemistry of glasses and glass melts, including glass electrodes. Springer, Berlin 31. Gründler P (2007) Chem Sensors. Springer, Berlin 32. Gründler P (2017) ChemTexts 3:16 33. Scholz F (2011) J Solid State Electrochem 15:5–14 34. Lewenstam A, Inzelt G, Scholz F (eds) (2013) Handbook of reference electrodes. Springer, Berlin 35. Scholz G, Scholz F (2014) ChemTexts 1:1 36. Oldham KB, Parnis JM (2017) ChemTexts 3:5

Index

A Absolute acidity, 54 Acetic acid, 37, 50 glacial, 52 Acetonitrile, 54 Acid, 26, 36 anionic, 29 corresponding, 29 medium strong, 36 monobasic, pH-logcidiagram, 62 polybasic, pH-logcidiagram, 64 polyprotic, polybasic, 38 strong, 36 very strong, 36 very weak, 36 weak, 36 Acid-base buffer, 86 Acid-base indicator, 209 Acid-base pair corresponding, 28, 37 Acid-base theories, 17 Acid-base titration, 182 Acidic solution, 35 Acidic surface group, 82 Acidity, 53 Acidity constant, 31 apparent, 41, 42 apparent, of perchloric acid, 56 macroscopic, 79 microscopic, 78, 79 Acid strength, 35 carbonic acid, 41 hydrofluoric acid, 40 inorganic acid, 38 metal aqua ion, 42 of a dissolved compound, 53 of the solvent, 53 organic acid, 50 Activity, 9

concentration, 31 molal, 34 molar ratio, 31 Activity coefficient, 31 Adsorption, 133 Adsorption indicator, 219 Affinity, 4 A-iron(III) hydroxide-oxide, 117 Alanine hydrochloride, 188 Alcohole, 50 Alkene, 50 Alkenol, 50 Alkine, 50 Alkyl group, 50 Alkynol, 50 Alumina, 26 Aluminum hydroxide, 120 Aluminum hydroxo complex, 124 Aluminum(III)-hexaquoion, 37 American convention, 140 Amino acid, 77, 103 Amino group, 77 Aminopolycarboxcylic acid, 103 Ammonia, 29, 37, 103, 193 liquid, 52 Ammonium, 29, 37 Ammonium cerium(IV) sulfate, 208 Ammonium chloride, 68, 69 Ammonium formiate, 69 Ammonium ion, 208 Amperometry, 173 Analyte, 173 Antimonate, 208 Antimony(III) oxide, 117 Antimony trisulfide, 131 Approximate equations validity areas, 70 Argentometry, 196 Arrhenius, Svante August, 20

© Springer Nature Switzerland AG 2019 F. Scholz and H. Kahlert, Chemical Equilibria in Analytical Chemistry, https://doi.org/10.1007/978-3-030-17180-3

243

244 Arsenic trisulfide, 130 Arsenite, 166, 208 Association constant, 40 Asymmetry potential, 232 Autoprotolysis, 52 Autoprotolysis constant of water, 32, 182 pressure dependency, 33 temperature dependency, 33 Autosolvolysis, 24 Auxiliary complexing agent, 193 Auxiliary ligand, 98 Azo dye, 211 B Barium hydroxide, 116, 117 Base, 26 corresponding, 29 polyacidic, 38 Base strength organic base, 50 Basicity, 55 Basicity constant, 32 Basic solution, 35 Baucke, Friedrich G. K., 233 Behrend, Anton, 230 Benzene, 54 Berthelot, Marcelin Pierre Eugène, 5, 12 Berthollet, Claude Louis, 5 b-cadmium hydroxide, 117 b-zinc hydroxide, 117 Berzelius, Jöns Jacob, 18 Bicarbonate, 29 Biilmann, Christian Saxtrop, 155 Bis(biphenyl)chromium(I)/bis(biphenyl) chromium(0), 162 Bis(η5-cyclopentadienyl)iron(III)/bis (η5-cyclopentadienyl)iron(II), 162 Bismuth(III) sulfide, 130, 131 Boiler scale, 104 Böttger, Wilhelm, 230 Boyle, Robert, 17, 169 Brønsted, Johannes Nicolaus, 20 Brønsted-Lowry Theory, 20, 52 Bromatometry, 208, 219 Bromocresol green, 213 Bromothymol blue, 213 Bromtrifluoride, 24 Buffer, 175 Buffer acid, 87 Buffer base, 87 Buffer capacity, 87, 186 Buffer curve, 87 Buffer equation, 86, 210

Index Buffer solution, 86, 186, 203 Buffer system, 90 Burette, 170, 171 C Cadmium ion, 103 Cadmium sulfide, 130 Cady, Hamilton Perkins, 24 Calcium hydroxide, 116, 117 Calcium ion, 216, 225 Calconcarboxylic acid, 216 Calomel electrode, 237 Carboanhydrase, 42 Carbonate melt, 26 Carbon dioxide, 41 Carbonic acid, 37, 41, 86 kinetic aspects, 42 Carbonyl moiety, 51 Carboxylate anion, 51 Carboxylic acid, 51 Carboxylic group, 77 Catalymetry, 7, 166 Catalysis, 13 Central ion, 93 Cerimetric determination, 201 Cerimetry, 208 Cerium(III) hydroxide, 116, 117 Cerium(IV), 166 Cerium(IV)/cerium(III), 201 Chelate, 189 Chelate effect, 11, 102, 103 Chelate forming ligand, 103 Chelation therapy, 104 Chemical potential absolute, of electrons, 162 Chemical potential of solvated protons, 54 Chloric acid, 38 Chloride, 198 Chlorous acid, 38 Chromate, 48 Chromate ion, 218 Chromium(III) hexaaqua ion, 49 Chromium(III) hydroxide, 116 Clausius, Rudolf, 10 Clay mineral, 82 Cobalt(II) hydroxide, 116, 117 Cobalt(III) ion, 208 Colloid, 50, 219 Colloidal particle, 208 Color indicator, 171, 209 Color map, 225 Colour indicator, 171 Common ion addition, 110 Concentration

Index amount, 31 analytical, 57 apparent, 41 molar, 31 Condensation reaction, 44 Conductometry, 173 Copper(I) ion, 208 Copper(II) ion, 208 copper(II) hydroxide, 117 Copper(II) sulfide, 130 Co-precipitation, 116, 131 Corrosion, 165 Cremer, Max, 233 Crotogino, Friedrich, 230 Cr(III)-EDTA complex, 178 Crystallization germs, 108 Cubic equation, 58 Cumulative stability constant, 96 Cyanidbestimmung nach Liebig, 208 D Davy, Humphry, 20 Decamethylferrocene, DMFC, 162 De Donder, Théophile Ernest, 4 Degeree of protolysis, 75 Degree of dissociation, 73 Degree of protolysis, 72 dependence on pH, 73 dependence on the overall acid concentration, 74 Degree of protonation of an indicator base, 53 Degree of titration, 181, 191, 196, 205 De Saint-Gilles, Léon Péan, 5 Descroizilles, François Antoine Henri, 170 Dicarboxylic acid, 51 Dichloromethane, 54 Dichromate, 208 Dichromate/chromium(III), 150 Dichromate ion, 48 Dichromatometry, 208 Dielectric constant, 55 Diethylene triamine, 94 Diethylenetriaminepentaacetate, 94 Diethylenetriaminepentaacetic acid (DTPA), 180 Differenciating effect, 56 Dihydrogen phosphate, 37 Dilution factor, 184 Diphenylamine sulfonic acid, 221 Disorder, 103 Dissociation, 107 of an indicator acid, 53 DLVO theory, 208

245 DMSO, 54 Döbereiner, Wolfgang, 138 Dodeka molybdato phosphate, 49 Donor atom, 93 E Electrical conductivity, 239 Electrical work, 202 Electric potential, 137 Electrode of the first kind, 230 of the second kind, 237 Electrode potential conventional, 139 Electromotoric force, 137 Electron, 136 Electron conductor, 137 Electron gas, 162 Electron-withdrawing group, 50 Electroosmotic flow, 82 Electrophoreses, 81 Electrode of the second kind, 230 Elementary charge, 136 E-logcidiagrams, 163 Elsey, Howard McKee, 24 End point of a titration, 222 Enol, 50 Enthalpy, 8 Entropy, 10, 103 decrease, 11 increase, 10 Entropy production, 4 Enzymatic analysis, 7 Eosin, 219 Equilibrium, 3 dynamic, 3 heterogeneous, 230 protolysis, 28 static, 3 Equilibrium constant, 9 conditional, 145 of zwitterion formation, 78 redox equilibrium, 142 Equivalence point, 174, 180, 181, 222 Eriochrome Black T, 215 Erio T, 180, 215, 225 Erlenmeyer flask, 175 Error absolute, 198 negative systematic, 222 positive systematic, 222 random, 181, 221, 142 relative, 198

246 systematic, 181, 221 Ethanol, 30, 50 Ethylenediamine, 94, 103 Ethylenediaminetetraacetate, 94, 99 Ethylenediaminete-traacetic acid (EDTA), 97, 148, 190, 224 Eu-doped LaF3single crystal, 233 Eutrophication, 104 Extent of reaction, 8 Extinction, 209 Extinction coeffizient molar, of permangantate ion, 209 F Fällungstitration, 196 Ferrocenium/ferrocene, 162 Ferroin, 221 Ferrometry, 208 Flood, Håkon, 26 Fluorescein, 219 Foreign ion addition, 113 Formal potential redox indicator, 220 Formiate, 29 Formic acid, 37, 50 Franklin, Edward Curtis, 24 Fredenhagen, Karl, 142 Fugacity, 7 G Gallium(III) hydroxide, 117 Galvanic cell, 137 Galvanic concentration cell, 6 Galvanic half-cell, 137 Galvanic plating, 104 Galvanic potential difference, 137 Galvanic series, 140 c-aluminum(III) hydroxide-oxide, 117 Gay-Lussac, Joseph Louis, 171 Generalized Brønsted acidity scale, 54 Geoffroy, Claude Joseph, 169 Geoffroy, Étienne François, 5 Gibbs energy, 107 Gibbs free energy, 8 molar, 4 relative changes, 8 Gibbs-Helmholtz equation, 6, 103 Gibbs, Josiah Willard, 5 Glass, 82, 83 Glass electrode, 82, 230 Glauber, Johann Rudolph, 17 Glycine, 78, 81 Glycine hydrochloride, 82 Goethite, 46, 117

Index Guldberg, Cato Maximilian, 5, 12 H Haber, Fritz, 233 Hammett acidity function, 53 Hammett, Louis P., 53 Henderson-Hasselbalch equation, 86, 210 HEPES, 90 Heteropoly anion, 49 Heteropoly metal ion, 46 Hexacyanoferrate(III)/hexacyanoferrate(II), 237 Home, Francis, 169 Homeostasis, 4, 42 Homogeneous precipitation, 133 Horstmann, August Friedrich, 5 Hume-Rothery, William, 131 Hydoquinone/quinone, 237 Hydriodic acid, 37 Hydrobromic acid, 37 Hydrocarbonate, 86 Hydrochloric acid, 29, 37, 66 Hydrocyanic acid, 37 Hydrofluoric acid, 37, 40 Hydrogen bond, 55 Hydrogen carbonate, bicarbonate, 37 Hydrogen fluoride, 54 Hydrogen oxalate, 29, 37 Hydrogen peroxide, 159, 208 Hydrogen phosphate, 37 Hydrogen sulfate, 29, 37 Hydrogen sulfide, 37, 40, 125 Hydrogen sulfide separation procedure, 129 Hydronium ion, 28, 29 Hydrophobic effect, 11 Hydrosulfide, 37 Hydroxide, 82 Hydroxide ion, 29 Hydroxo complex, 44, 98 Hydroxyl radical, 159 Hyperbola, 62 Hypochlorous acid, 38 I Iceberg water, 11 Indication conductometric, 239 instrumental, 229 potentiometric, 230 spectrophotometric, 237 Indicator, 175 bichromic, 210 monochromic, 210 Indicator acid, 209

Index Indicator base, 209 Indicator error systematic, 222 indium(III) hydroxide, 117 Inductive effect, 50 Inert complex, 98 Inner energy, 8 Inner salt, 77 Instability constant, 95 Interface, 139 Iodide, 198 Iodine, 208 Iodine-starch complex, 219 Iodometric titration, 219 Iodometry, 208, 209 Ion conductor, 137 Ion pair, 40, 98 Ion pair formation constant, 56 Ion potential, 43 Ion product of water, 32 Ion sensitive electrode, 132, 230, 233 Iron(II) hexaqua ion, 37 Iron(II) hydroxide, 116 Iron(II) ion, 208 Iron(III) ion, 208 Iron(III)/iron(II), 163, 201 Iron(II, III) oxide, 117, 118 Iron(II) sulfide, 130 Iron(III) aqua ion, 46 Iron(III) hexaqua ion, 29, 37 Iron(III) hydroxide, 117 Iron(III) oxide hydrate, 46 Irreversibility, 13 Isoelectric focusing, 81 Isoelectric point, 81, 83 Isopoly cation, 48 Isopropyl alcohol, 50 IUPAC, 174, 222 J Jander, Gerhart, 24 K Keto-enol tautomery, 50 Ketone, 50 Klarpunkttitration, 208 Klemensiewisz, Zygmunt, 233 L Lable complex, 49, 98 Lability, 49

247 Lambda sensor, 133 Lambert-Beer‘s Law, 237 Latimer diagram, 160 Latimer, Wendell Mitchell, 160 Lattice energy, 107 Laundry detergent powder, 104 Lavoisier, Antoine-Laurent, 17 Law of mass action, 5, 9, 31 Lead(II) oxide, 117 Lead(II) sulfide, 127, 130, 131 Lead(IV) oxide, 117, 208 Levelling effect of the solvent, 53 Levelling effect of water, 36 Lewis, Gilbert Newton, 7, 21 Lewis, William, 170 Ligand, 93 bidentate, 93, 94 hexadentate, 93, 94 monodentate, 93 octadentate, 94 pentadentate, 93, 94 tetradentate, 93, 94 tridentate, 93, 94 Liquid ammonia, 30 Liquid hydrogen sulfide, 30 Liquid junction, 139 Liquid junction potential, 139 Lowry, Thomas Martin, 20 Luther, Robert Thomas Dietrich, 159 Luther’s rule, 158 Lux, Hermann, 26 Lyate ion, 30, 37, 52 Lyonium ion, 30, 37, 52 M Manganese(II) hydroxide, 117 Manganese(II) ion, 208 Manganese(II) sulfide, 130 Manganese(IV) oxide, 208 Manganese sulfide, 131 Magnesium hydroxide, 116, 117, 216 Magnesium ion, 225 Manganometry, 208 Masking, 104 Mathematical approximation, 56 Maximum effective work, 202 Membrane electrode, 230 Mercury-mercury oxide electrode, 237 Mercury(I) ion, 208 Mercury(II) oxide, 117 Mercury(II) sulfide, 130 Metal aqua complex, 45, 98

248 kinetic features, 49 Metal aqua ion, 42 Metal-EDTA complex, 215 Metal hydroxo complexes, 11 Metall-indicator complex, 214 Metallochromic indicator, 214 Metal oxide, 46 Metal oxide hydrate, 46, 49 Metal-protein complex, 103 Metal sulfide, 125 Methanol, 30, 50 Methylene blue, 210, 221 Methyl orange, 211, 220 Methyl red, 210, 211, 219 Mixed crystal, 131, 198 Mohr, Friedrich, 171 Molality, 34 Molar ratio, 31 Molybdate, 48 Monobasic acids, 57 Münz, Ferdinand, 190 Murexide, 215 N Nernst equation, 6, 7, 137 Nernst, Walther, 6, 142 Neutral solution, 35 Newton, Isaac, 5 Nickel(II) hydroxide, 116, 117 Nitrate, 208 Nitric acid, 37 Nitrilotriacetate, 94 Nitrilotriacetic acid, 191 Nitrite, 208 Nitrite ion, 176 Non-specific Indikator, 219 n-th ionization potentials, 43 O Occlusion, 133 Ostwald’s law of dilution, 72, 75 Ostwald, Wilhelm, 142, 173, 230 Oxalate, 94, 166 Oxalic acid, 37 Oxalic acid/oxalate, 208 Oxidation, 135 Oxidation number, 135 Oxidation state, 135 Oxide, 82 Oxide hydrates, 82 Oxygen, 46, 159

Index P Paratungstate ion, 48 Perchlorate, 208 Perchloric acid, 37, 38 Permanganate, 48, 156, 167 Permanganate/Mn2+, 237 Permanganometry, 209 Peters, Rudolf, 7 Pfaundler von Hadermur, Leopold, 12 pH approximation equations, 64 Phenolphthalein, 213 Phenosafranin, 221 pH-logcidiagram, 182, 223 pH-logcidiagram, 59 pH meter, 230 Phosphate Buffered Salt Solution (PBS), 90 Phosphate ion, 39 Phosphoric acid, 37, 38, 40, 181 Photometry, 173 Photosynthesis, 46, 118 Phthaleins, 211, 212 Pinchcock burette, 171 Pipette, 171 PKa organic compounds, 50 PKavalue, 36, 37 Plating potential, 104 Platinum black, 138 Plutonium(IV) hydroxide, 117 Point of zero charge, 84 Polybasic acid, 58 Polyiodide chain, 219 Polyphosphate, 104 Potassium chromate, 218 Potassium dichromate, 208 Potassium iodide, 208 Potassium perchlorate-potassium permanganate, 132 Potassium permanganate, acidic, 208 Potential absolute chemical of protons, 54 biochemical standard, 160, 161 chemical, 4, 5, 9 formal, 145, 150 formal, including activity coefficients, 146 formal, including activity coefficients and side reactions, 147 mixed, 167 real, 145 redox, pH-dependent, 150

Index standard, 139 Potentiometry, 173, 182, 201, 230 Pourbaix diagram, 164 Pourbaix, Marcel, 164 Precipitate, 108 Precipitation equilibrium, 108 Precipitation titration, 171 Precipitation zone, 116 Predominance diagram, 164 Priestley, Joseph, 18 Process enforced, 10 spontaneous, 10 Propionic acid, 50 Protein, 77 Proton, 136 Pure phase, 108 PVC membrane, 233 Pyrite, 46 Pyrolusite, 156 Q Quartz, 83 Quinhydrone, 151 Quinone, 150 R Radical scavenger, 159 Rate constant, 12, 42 Reaction enthalpy, 103 Reaction rate, 86 Redox equilibrium heterogeneous, 166 homogeneous, 166 kinetic aspects, 166 Redox indicator, 219 metal complex, 219 organic, 220 Redox potential nonaqueous solvent, 161 absolute, 162 Redox reaction, 135 Redox titration methods, 206 Redox titratios, 201 Reduction, 135 Reference electrode, 155, 235, 237 Reference half-cell, 138 Resonance effect, 50 Reversibility, 13 Riesenfeld, Hermann, 6 Rouelle, Guillaume-François, 17

249 S Salt bridge, 139 Salting in effect, 113 Salting out effect, 111 Sample, 173 metal oxide-hydroxides, 114 Saturation activity, 109 Saturation concentration, 109, 110, 113 metal oxides, 113 metal hydroxides, 113 pH dependency, 113 Scheele, Carl Wilhelm, 18 Schwarzenbach, Gerold, 190 Sędziwój, Michał (Michael Sendivogius), 18 Side reaction ligand, 96 metal ion, 98 Side reaction coefficient, 96, 147 Ethylenediaminete-traacetic acid (EDTA), 192 ligand, 97 metal ion, 98, 193 pH-dependence, 99 Silberchlorid, 196 Silica gel, 83 Silicate, 83 Silver chloride, 110 Silver chromate, 218 Silver dichromate, 218 Silver electrode, 233 Silver halid, 218 Silver hydroxide, 218 Silver-silver chloride electrode, 237 Simultaneous titration, 198 Sodium acetate, 69 Sodium chloride, 69 Sodium sulfate, 108 Sodium thiosulfate, 208 Solid phase, 107 Solid solution, 131 Solubility equilibrium, 107 Solubility product, 108 Solvation, 107 Solvation enthalpie, 107 Solvation entropie, 107 Solvent amphiprotic, 52 non-aqueous, 52 protic, 28, 29 Solvoacid, 24 Solvo acid-solvobase theory, 24 Solvobase, 24

250 Sørensen, Søren Peter Lauritz, 33 Specific indicator, 219 Stability constant conditional, 96, 99 conditional (effective) pH-dependence, 99 effective, of the metal-indicator complex, 192, 215 stepwise, 95 Standard Hydrogen Electrode, 138 Standard molality, 34 Standard solution, 173 Standard solvation Gibbs energy, 54 State of equilibrium, 86 Steric effect, 55 Stockholm convention, 139 Succinic acid, 55 Sulfate ion, 38 Sulfonephthaleins, 211, 213 Sulfur dioxide, 54 Sulfuric acid, 29, 37, 38, 54 Supersaturated solution, 108 Surface charge, 84 Surface OH group, 83 T Tashiro, 210 TBS, 90 TBS-T, 90 Terpyridine, 94 Thiosulfate ion, 208 Thorium(IV) hydroxide, 117 Tin(II) ion, 208 Tin(II) oxide, 117 Tin(IV) oxide, 117 Titanium(III) chloride, 208 Titanium(IV) hydroxide, 117 Titanometry, 208 Titrand, 173 Titrant, 173 Titrate, 171 Titration, 169, 173 according to Fajans, 219 according to Volhard, 218 back, 176 complexometric, 189 direct, 175 displacement, 179 end point, 181 indirect, 180 inverse, 175 permanganometric, 205 self-indicating, 208 substitution, 179 Titration curve, 181

Index acid-base titration, 182 chloride and iodide with silver nitrate, 198 complexometric, 191 hydrochloric acid with sodium hydroxide, 184 iron(II) ions with permanganate in acidic solution, 207 magnesium ions with EDTA, 192 moderately strong or weak acid, 186 polybasic acid, 187 precipitation titration, 196 redox titration, 201 slope, 228 Titration error absolute, 222 random, 192, 228 Titration methods, 175 Titrationsgrad, 201 Titration of halides according to Mohr, 218 Titration vessel, 175 Titrator, 173 Titrimetry, 173 Total water hardness, 216 Transition interval, 228 redox indicator, 220 width, 228 Transition interval of an indicator, 210 Trichloroacetic acid, 50 Triethylene tetramine, 94 Trifluorethanol, 50 Trifluoroacetic acid, 50 TRIS, 90 TRIS-glycine, 90 Tris(1,10-phenanthroline)iron, 221 Tris(2,2´-bipyridyl)ruthenium, 221 Tungstate, 48 Two EMBED Equation.DSMT4 model, 83 2,3,22,23-dibenzo-1,4,7,10,13-pentathiatridecanoate, 94 U Uranium(IV) oxide, 117 Uranylium(IV) ion, 208 Uranyl/uranium(IV), 237 Urotropin, 133 Usanovich, Mikhail Il’ich, 26 V Vanadyl ion, 208 Van’t Hoff, Jacobus Henricus, 6, 12 Volhard, Jacob, 218 Volumetric analysis, 171, 173 Volumetric pipette, 175

Index Von Grotthuss, Theodor, 239 Von Helmholtz, Hermann, 6 Von Liebig, Justus, 20 W Waage, Peter, 5, 12 Water, 54 Water hardness, 224, 225 Wenzel, Carl Friedrich, 5 X Xylenol orange, 217

251 Z Zeolith A, 104 Zeolithe, 82 Zinc-ammine complex, 194 zinc hydroxide, amorphous, 117 Zinc ion, 193, 194 Zinc(II) oxide, 117 Zirconium dioxide, 133 Zirconium(IV) oxide, 117 Zundel complex, 35 Zwitterion, 77