Binding Phenomena: General Description and Analytical Applications (Physical Chemistry in Action) [1st ed. 2023] 3031397355, 9783031397356

Table of contents :
Preface
Contents
Abbreviations
1 Introduction
Reference
2 Binding to Simple Substrates with One Binding Site
2.1 Acid–Base Reactions. A Simple Example of Proton Binding
2.1.1 The Average Relative Number of Moles of Bound Protons and the Degree of Protonation
2.1.2 The Average Relative Number of Moles of Dissociated Protons and the Degree of Dissociation
2.2 Complexation. A Simple Example of Complexation Without Competitive Binding
2.3 Adsorption. Simple Adsorption Equilibrium
2.4 Electron Binding. A Simple Case of Redox Reactions
2.5 The General Binding Process and the Binding Driving Force
2.6 About the Fulfilment of the Langmuir Equation for Different Processes
References
3 One Substrate with Two Different Binding Sites. Competitive Binding. Two Different Binding Species. Two Different Binding Substrates
3.1 Introduction
3.2 One Substrate with Two Different Binding Sites
3.3 Two Different Binding Species on the Same Substrate
3.3.1 Competitive Adsorption
3.3.2 Competitive Binding Between Protonation and Complexation
3.4 Electron Binding to Two Redox Couples
3.5 Two Different Binding Sites. A Simple Example of Proton Binding in Ampholytes
3.6 Formation of Zwitterions
References
4 Titration of Simple Substrates
4.1 Introduction
4.2 Simple Examples
4.2.1 Titration of a Strong Monoprotic Acid
4.2.2 Titration of a Weak Polyprotic Acid
4.3 About the Additivity of Titration Curves
4.4 Titration of an Arbitrary Mixture of Acids with an Arbitrary Mixture of Bases
4.5 A Simple Example of Titration with Complex Formation
4.6 Titration of a Simple Ampholyte
4.7 Redox Titrations
4.8 Electron Titration
4.9 Titration of Zwitterions
4.9.1 The Neutral Form Alone
4.9.2 The Zwitterion Form Alone
4.9.3 The Two Forms in Equilibrium
References
5 Continuous Distribution Functions. Cumulative and Density Distribution Functions. Known Examples
5.1 Introduction
5.2 Continuous Distribution Functions Cumulative and Density Probability Distribution Functions [1, 2]
5.2.1 Uniform Density Function
5.2.2 Dirac Delta Distribution
5.2.3 Gaussian Function
5.3 The Relation Between Binding Problems and Distribution Functions
References
6 Elements of Adsorption on Heterogeneous Substrates
6.1 Introduction
6.2 Adsorption on Heterogeneous Substrates. The Distribution Function for the Adsorption Energy
6.3 Theoretical Binding Isotherms in the Presence of Interaction Between the Bound Species
6.4 Statistical Deduction of the Langmuir Isotherm
6.5 The Ising Model
6.6 The Mean Field Approximation or Bragg–Williams Approximation
References
7 Theoretical Basis About Solid Polymers, Gels and Single Chains in Solution Related to the Titration of Macromolecules
7.1 Introduction
7.2 The State of Macromolecules
7.3 Some Physical Properties of Polymers
7.3.1 Deformation Term of a Single Macromolecule
7.4 The State of Macromolecules in Solution
7.5 Polyelectrolytes in Solution
7.5.1 The End-To-End Distance
7.6 Interactions of Polyelectrolytes with Other Species Present in the Solution
7.7 Electrostatic Interactions
7.7.1 Electrostatic Interaction for Charges Spheres
7.7.2 Electrostatic Interactions for Charged Cylinders
7.8 The Interaction of Polymers with the Solvent. Flory−Huggins Theory of Polymers in Solution
7.9 Swelling of Polymer (Non-Polyelectrolytic) Gels
7.10 The Swelling Equilibrium of Polyelectrolyte Gels
7.11 Electrostatic Interactions in Polyelectrolyte Gels
7.12 Deformation and Electrostatic Interactions and Binding Equilibria in Single Dissolved Macromolecules
7.13 The Equation for the pH Change During the Course of a Proton TC
References
8 Acid–Base Equilibria and Complexes at Complex Substrates. Polyacids and Polybases
8.1 Introduction
8.2 Polyacids. Average Number of Bound Protons
8.2.1 Average Number of Moles of Dissociated Protons
8.3 Polybases
8.4 Complexation and Competitive Binding in Multi-ligand Complexes
8.4.1 Ligands Without Hydrolysis
8.4.2 Ligands Undergoing Hydrolysis. Generalization of Competitive Binding
References
9 Acid–Base Titration of Complex Substrates. Binding Constant Distribution
9.1 Introduction
9.2 Titration of Polyacids and Polybases
9.2.1 Titration of Polyacids: Polymethacrylic Acid
9.2.2 Titration of Polybases. Polyvinylamine (PVA)
References
10 The Acid–Base Behaviour of Polyampholytes. The Case of Colloidal Oxides
10.1 Introduction
10.2 The Binding Polynomial for a Colloidal Oxide
10.3 Titration of Colloidal Oxides
10.4 Summary of Models on the Oxide Solution Interface
10.5 Analysis of Some Experimental Results with the Constant Capacitance Model
References
11 Titration of Polyampholytes. Poly-zwitterions and Other Examples
11.1 Titration of Polyampholytes
11.2 Titration of Copolymers of PVP and PMA
11.3 A Simple Model for Zwitterion Formation in Polyelectrolytes of Different Composition Assuming Ka y Kb Independent of 〈rH〉 and Its Titration
11.4 Acid Base Titration of Humic Acids
11.5 Titration of Proteins: Tanford’s Method
11.5.1 Stoichiometry
11.5.2 Theoretical Interpretation
References
12 Electron Titrations of Electrochemically Active Macromolecules
12.1 Introduction
12.2 Change of Volume as a Function of the Potential
12.3 Change of the Binding Species Concentration in the External Solution as a Function of the Oxidation Fraction
12.4 Electron Binding to Polyaniline
References
13 Appendices
13.1 Appendix 1. Macroconstants and Microconstants
13.1.1 Macroconstants and Microconstants
13.1.2 The Example of Ciprofloxacin
13.2 Appendix 2. Statistical Factors
13.3 Appendix 3. Elements of Statistical Thermodynamics
13.3.1 Introduction
13.3.2 The Example of an Ideal Gas
13.3.3 Subsystems
13.4 Appendix 4. The Binding Polynomial as the Partition Function of the Bound Species. Ghost−Site Binding Constants
13.5 Appendix 5. The Gibbs Adsorption Isotherm. Two-Dimensional State Equations
13.5.1 The Gibbs–Duhem Equation
13.5.2 The Gibbs Adsorption Isotherm
13.6 The General Binding Process and the Binding Driving Force
References
Index

Citation preview

Physical Chemistry in Action

Waldemar A. Marmisollé Dionisio Posadas

Binding Phenomena General Description and Analytical Applications

Physical Chemistry in Action

Physical Chemistry in Action presents volumes which outline essential physicochemical principles and techniques needed for areas of interdisciplinary research. The scope and coverage includes all areas of research permeated by physical chemistry: organic and inorganic chemistry; biophysics, biochemistry and the life sciences; the pharmaceutical sciences; crystallography; materials sciences; and many more. This series is aimed at students, researchers and academics who require a fundamental knowledge of physical chemistry for working in their particular research field. The series publishes edited volumes, authored monographs and textbooks, and encourages contributions from field experts working in all of the various disciplines.

Waldemar A. Marmisollé · Dionisio Posadas

Binding Phenomena General Description and Analytical Applications

Waldemar A. Marmisollé Departamento de Química Facultad de Ciencias Exactas Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas Universidad Nacional de La Plata La Plata, Argentina

Dionisio Posadas Departamento de Química Facultad de Ciencias Exactas Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas Universidad Nacional de La Plata La Plata, Argentina

ISSN 2197-4349 ISSN 2197-4357 (electronic) Physical Chemistry in Action ISBN 978-3-031-39735-6 ISBN 978-3-031-39736-3 (eBook) https://doi.org/10.1007/978-3-031-39736-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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

To my sons Galo and Lucio (W.A.M)

Preface

Generalizations are always an advantage for students to widen their vision and gain a more comprehensive view of a subject. It also may be helpful for teachers in the sense that more compactness may be reached in the subject presentation. The idea of searching and advancing on a common description of binding phenomena for different process very familiar in chemistry began about the ends of the first decade of this century when one of us was lecturing statistical thermodynamics for students of physical chemistry at the last year of the career and the other was attending the course. Inspired by the book of T. L. Hill Introduction to Statistical Thermodynamics we became interested in the subject. It happens that at that time one of us was working in searching different physical– chemical properties of electrochemically active polymers, that is, polymers that can be reversibly oxidized and reduced. Since these polymers are also polyelectrolytes in the sense that they can bind ions (mostly protons) from the solution, they show all the characteristics of the acid–base properties of polymers in solution, namely besides proton binding and dissociation they also show a coupling between the processes of deformation, binding and the screening of the charged sites by small ions present in solution. Since due to their electrochemical activity they may also win or lose electrons, it was clear to us that the latter process must be also coupled with the others. Later on, with the collaboration of other members of the group, we prove both experimentally and based on simple statistical mechanical models this was the case. This book is addressed to undergraduate students of chemistry, although we consider a special course for graduate students oriented to analytical, physical and biological chemistry may be of interest to them mainly in the aspects of describing the binding problems associated to macromolecules and its titration. One of the main purposes of this book is to emphasize that binding problems can be described mathematically in the same way for very different chemical processes. Thus adsorption, acid–base, complexation and electrochemical reactions are treated in exactly the same way. However, it is always convenient to start with the simpler systems so there, we may enquire what the reasons for the common description are. Then we advance over more vii

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complex systems, that is, natural and synthetic substances that have multi-binding sites like synthetic and natural macromolecular ones, metallic oxide colloids, humic acids and the like. In doing so, we soon find that the binding constants are not true constants but are distributed properties over certain energy intervals. Moreover, very often this distribution may be modelled by statistical mechanical models. Another fact that arises from the analysis of the titration curves of polymer solutions is the occurrence of what Katchalsky, around the middle of last century, called mechanochemical effects referring to the coupling between binding and deformation of the polymer. More recently, it was demonstrated the existence of other types of couplings for instance between the redox potential binding and deformation of macromolecules. La Plata, Argentina

Waldemar A. Marmisollé Dionisio Posadas

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 5

2

Binding to Simple Substrates with One Binding Site . . . . . . . . . . . . . . 2.1 Acid–Base Reactions. A Simple Example of Proton Binding . . . . 2.1.1 The Average Relative Number of Moles of Bound Protons and the Degree of Protonation . . . . . . . . . . . . . . . 2.1.2 The Average Relative Number of Moles of Dissociated Protons and the Degree of Dissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Complexation. A Simple Example of Complexation Without Competitive Binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Adsorption. Simple Adsorption Equilibrium . . . . . . . . . . . . . . . . . . 2.4 Electron Binding. A Simple Case of Redox Reactions . . . . . . . . . 2.5 The General Binding Process and the Binding Driving Force . . . 2.6 About the Fulfilment of the Langmuir Equation for Different Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7

3

One Substrate with Two Different Binding Sites. Competitive Binding. Two Different Binding Species. Two Different Binding Substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 One Substrate with Two Different Binding Sites . . . . . . . . . . . . . . 3.3 Two Different Binding Species on the Same Substrate . . . . . . . . . 3.3.1 Competitive Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Competitive Binding Between Protonation and Complexation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Electron Binding to Two Redox Couples . . . . . . . . . . . . . . . . . . . . .

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23 23 23 24 25 25 27

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3.5

Two Different Binding Sites. A Simple Example of Proton Binding in Ampholytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Formation of Zwitterions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5

6

Titration of Simple Substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Simple Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Titration of a Strong Monoprotic Acid . . . . . . . . . . . . . . . 4.2.2 Titration of a Weak Polyprotic Acid . . . . . . . . . . . . . . . . . 4.3 About the Additivity of Titration Curves . . . . . . . . . . . . . . . . . . . . . 4.4 Titration of an Arbitrary Mixture of Acids with an Arbitrary Mixture of Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 A Simple Example of Titration with Complex Formation . . . . . . . 4.6 Titration of a Simple Ampholyte . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Redox Titrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Electron Titration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Titration of Zwitterions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.1 The Neutral Form Alone . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.2 The Zwitterion Form Alone . . . . . . . . . . . . . . . . . . . . . . . . 4.9.3 The Two Forms in Equilibrium . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Distribution Functions. Cumulative and Density Distribution Functions. Known Examples . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Continuous Distribution Functions Cumulative and Density Probability Distribution Functions [1, 2] . . . . . . . . . . . . . . . . . . . . . 5.2.1 Uniform Density Function . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Dirac Delta Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Gaussian Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Relation Between Binding Problems and Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elements of Adsorption on Heterogeneous Substrates . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Adsorption on Heterogeneous Substrates. The Distribution Function for the Adsorption Energy . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Theoretical Binding Isotherms in the Presence of Interaction Between the Bound Species . . . . . . . . . . . . . . . . . . . 6.4 Statistical Deduction of the Langmuir Isotherm . . . . . . . . . . . . . . . 6.5 The Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 The Mean Field Approximation or Bragg–Williams Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7

8

9

Theoretical Basis About Solid Polymers, Gels and Single Chains in Solution Related to the Titration of Macromolecules . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The State of Macromolecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Some Physical Properties of Polymers . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Deformation Term of a Single Macromolecule . . . . . . . . 7.4 The State of Macromolecules in Solution . . . . . . . . . . . . . . . . . . . . 7.5 Polyelectrolytes in Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 The End-To-End Distance . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Interactions of Polyelectrolytes with Other Species Present in the Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Electrostatic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.1 Electrostatic Interaction for Charges Spheres . . . . . . . . . 7.7.2 Electrostatic Interactions for Charged Cylinders . . . . . . . 7.8 The Interaction of Polymers with the Solvent. Flory−Huggins Theory of Polymers in Solution . . . . . . . . . . . . . . 7.9 Swelling of Polymer (Non-Polyelectrolytic) Gels . . . . . . . . . . . . . 7.10 The Swelling Equilibrium of Polyelectrolyte Gels . . . . . . . . . . . . . 7.11 Electrostatic Interactions in Polyelectrolyte Gels . . . . . . . . . . . . . . 7.12 Deformation and Electrostatic Interactions and Binding Equilibria in Single Dissolved Macromolecules . . . . . . . . . . . . . . . 7.13 The Equation for the pH Change During the Course of a Proton TC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acid–Base Equilibria and Complexes at Complex Substrates. Polyacids and Polybases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Polyacids. Average Number of Bound Protons . . . . . . . . . . . . . . . . 8.2.1 Average Number of Moles of Dissociated Protons . . . . . 8.3 Polybases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Complexation and Competitive Binding in Multi-ligand Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Ligands Without Hydrolysis . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Ligands Undergoing Hydrolysis. Generalization of Competitive Binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acid–Base Titration of Complex Substrates. Binding Constant Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Titration of Polyacids and Polybases . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Titration of Polyacids: Polymethacrylic Acid . . . . . . . . . 9.2.2 Titration of Polybases. Polyvinylamine (PVA) . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

10 The Acid–Base Behaviour of Polyampholytes. The Case of Colloidal Oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Binding Polynomial for a Colloidal Oxide . . . . . . . . . . . . . . . 10.3 Titration of Colloidal Oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Summary of Models on the Oxide Solution Interface . . . . . . . . . . 10.5 Analysis of Some Experimental Results with the Constant Capacitance Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Titration of Polyampholytes. Poly-zwitterions and Other Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Titration of Polyampholytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Titration of Copolymers of PVP and PMA . . . . . . . . . . . . . . . . . . . 11.3 A Simple Model for Zwitterion Formation in Polyelectrolytes of Different Composition Assuming K a y K b Independent of and Its Titration . . . . . . . 11.4 Acid Base Titration of Humic Acids . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Titration of Proteins: Tanford’s Method . . . . . . . . . . . . . . . . . . . . . . 11.5.1 Stoichiometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Theoretical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Electron Titrations of Electrochemically Active Macromolecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Change of Volume as a Function of the Potential . . . . . . . . . . . . . . 12.3 Change of the Binding Species Concentration in the External Solution as a Function of the Oxidation Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Electron Binding to Polyaniline . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Appendix 1. Macroconstants and Microconstants . . . . . . . . . . . . . 13.1.1 Macroconstants and Microconstants . . . . . . . . . . . . . . . . . 13.1.2 The Example of Ciprofloxacin . . . . . . . . . . . . . . . . . . . . . . 13.2 Appendix 2. Statistical Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Appendix 3. Elements of Statistical Thermodynamics . . . . . . . . . 13.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 The Example of an Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . 13.3.3 Subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Appendix 4. The Binding Polynomial as the Partition Function of the Bound Species. Ghost−Site Binding Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 117 119 124 126 128 134 135 135 136

137 141 142 144 144 145 147 147 151

152 153 161 163 163 163 166 168 170 170 172 174

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Contents

13.5 Appendix 5. The Gibbs Adsorption Isotherm. Two-Dimensional State Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.1 The Gibbs–Duhem Equation . . . . . . . . . . . . . . . . . . . . . . . 13.5.2 The Gibbs Adsorption Isotherm . . . . . . . . . . . . . . . . . . . . . 13.6 The General Binding Process and the Binding Driving Force . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

176 176 177 178 181

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Abbreviations

B BI BP B-W E EAM EDTA ES LE Pani PMA POT PVA PVP S SHE TC VW

Binding species Binding isotherm Binding polynomial Bragg–Williams Emeraldine Electrochemically active macromolecules Ethylene diamine tetra acetate Equation of state Leucoemeraldine Polyaniline Polymethacrylic acid Poly (o-toluidine) Polyvynilamine Polyvynilpyrie Substrate Standard hydrogen electrode Titration curve van der Waals

xv

Chapter 1

Introduction

There is a variety of chemical phenomena that can be described under the heading of binding problems. We understand binding as the attachment of a binding species (B), which might be an ion, an electron, a neutral atom or a molecule, to another chemical species or groups of atoms or molecules such as other single ion, a molecule, a macromolecule, a colloidal particle, a solid surface, etc. We will call this other part, the substrate (S). In general, we can say that the substrate has a site for the union of the binding species, and this site can be either empty or occupied. Let us consider some simple examples. If we consider acids and bases in the Brönsted concept, that is, acids are all substances capable of delivering protons in solutions and bases are all those capable of accepting them, an example would be the binding of a proton (the binding species) to a simple base (the substrate). This class of binding processes includes all the acid–base chemistry. Let us consider the acetic acid, CH3 COOH. In this case, the carboxylic group of acetate (–COO− ) would be the site where a proton can be bound. If we instead consider the Lewis concept of acids and bases, that is acids are those substances capable of accepting pairs of electrons, then these binding process includes also all the coordination chemistry. In this regard, a ligand (the binding species) binds to a metallic ion centre (the substrate). As another example let us consider the binding species are electrons and the substrate some oxidized species in solution capable of accepting this electron to become a reduced species. This also is a binding process and includes all kinds of redox and electrochemical reactions. In this regard, Fe3+ ions in solution (substrate) can bind an electron (binding species) to form Fe2+ . A final example comes from the field of adsorption. There, a binding species, usually in the gas phase or in solution, binds to the surface of the substrate, usually a solid. This kind of binding may be physical or chemical in nature. That is, the bonding may occur by a chemical reaction or simply by intermolecular attractive forces as the van der Waals forces. As a simple example, let us consider the adsorption of argon (binding species) to graphite (substrate).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. A. Marmisollé and D. Posadas, Binding Phenomena, Physical Chemistry in Action, https://doi.org/10.1007/978-3-031-39736-3_1

1

2

1 Introduction

As it will become apparent below, at equilibrium, these apparently unrelated phenomena have a common basic description: they are binding phenomena. Moreover, the models employed to theoretically describe them also have a common basic description. It is clear that much can be gained by recognizing binding reactions from the start and to apply this concept to whatever kind of binding equilibrium we wish. Here we will not be interested in the nature of the binding forces but rather in the form of the equation describing the binding equilibrium. That is, the equations relating the activity of the species to be attached and the activity of the bound species. This is called the binding isotherm (BI). On the other hand, as we will see, there are special cases of binding, such as competitive binding and binding to heterogeneous or complex substrates that are treated in essentially the same way. Here we understand heterogeneous substrates as those in which the binding energy depends on the amount of bound species. In other words, the binding energy depends on the extent of the binding reaction. Examples of heterogeneous substrates are real surfaces and macromolecules. This concept comes from the field of adsorption. In this regard, a surface that has all the binding sites equivalent and with the same binding energy is called a homogeneous surface. In contrast to an adsorbent that has sites of different binding energy is called heterogeneous. From the binding viewpoint homogeneous adsorption is equivalent to a dilute solution of binding molecules and heterogeneous adsorption equivalent to multisite binding or binding to a macromolecule. So, it becomes reasonable to speak of binding substrates in terms of homogeneous and heterogeneous. Moreover, in many cases heterogeneity comes from interaction between the bound species; therefore since the mathematical description of binding in all the different situations is the same, it would be correct to call the presence of interactions as heterogeneous binding. Furthermore, sometimes binding sites that have identical binding constant are referred as having the same affinity, and the presence of interactions is called cooperativity although this term is not employed in the different branches of chemistry where binding problems are present. The problem of binding to a heterogeneous substance or material naturally leads to the idea of distributed thermodynamic quantities to describe complex systems. This is a very useful idea that has been employed in different fields of physical and analytical chemistry to understand the binding behaviour of complex molecules and materials. Even more, nowadays, when new materials are frequently discovered, it is necessary to describe them accordingly. Sometimes, as it happens in the field of adsorption, it becomes impossible to measure the activity of the bound species. In these cases, the BI relates the activity of the species to become attached with the concentration of the bound species. In the field of adsorption, the concentration of the bound species is usually defined as the number of adsorbed moles per unit area, ┌. However, it is also expressed as the coverage, θ, which is either, when the adsorption is monomolecular, as the quotient ┌ / n or ┌ / ┌ max , where n is the maximum number of moles of sites at the surface and ┌ max the maximum number of moles bound to the surface. The surface coverage accounts for the number of adsorbed molecules on a surface divided by the number of molecules in a filled monolayer on that surface. This concept can be used to describe

1 Introduction

3

the fraction of binding sites that are occupied in other binding problems, such as acid–base, complexation and redox equilibria. In the context of proton binding, the fraction of protonated carboxylic moieties in polyacrylic acid can be denoted as θ H , as it represents the fraction of proton binding sites that are occupied. In some cases, this proton coverage can be also thought as the degree of protonation (see Chap. 2). Due to its historical significance, adsorption was one of the first binding processes to receive significant attention. As a result, much of the terminology used in the study of binding phenomena is derived from the field of adsorption. It is worth noting that adsorption can be classified as mobile or fixed, depending on the mobility of the adsorbed molecules. When adsorption is due to the formation of a chemical bond with the substrate, it is classified as fixed or localized. The behaviour of a pure adsorbed substance may be considered as a phase and be represented by an equation of state (ES) in the same way as in pure phases. If the adsorbed phase behaves as an ideal gas the ES is similar to pV = RT for one mol. However, instead of p, we must employ the two-dimensional pressure, π, the area, A, and the surface concentration. In this case the BI has the simple form: ┌ = K a, Where K is the binding constant. This is known as the Henry’s isotherm. For localized adsorption, the BI results (see Chap. 6 for a statistical thermodynamic deduction of this equation): θ =Ka 1−θ This is known as the Langmuir’s isotherm. As mentioned earlier, in cases where the system does not behave ideally due to interactions between the adsorbed species, the activity of the adsorbed species cannot be directly measured. To address this issue, an approximate equation of state such as the van der Waals equation is often used, in place of the ideal gas equation of state. In this book, to emphasize the binding processes, we will write all the equations representing chemical process as binding reactions. Sometimes, this will be opposite to the common usage in different branches of chemistry. In those cases, we will revert to the common usage to keep with it. Thus, in the case of acid–base reactions, we will call this process, proton binding, and also we will speak of a protonation reaction or an association reaction. The opposite of that reaction is the acid dissociation, though in the case of proton dissociation, for the reasons given by Albert and Serjeant [1], we will prefer the term ionization reaction. Other term which may be conflictive is that employed in general chemistry for designing acids that may dissociate more than one proton as polyacids, as it is the case of phosphoric or citric acids. Within this book, when speaking of polymeric

4

1 Introduction

substances, we will use the terms polyacids or polybases to mean polymers or macromolecules that have many positions able to bind. In these cases, we will employ also the term multi-acid or multi-base. When considering the equilibrium of processes as those mentioned above, equilibrium constants will arise. In general, we will employ the same term for that constant as we employed for the processes. However, we will employ lower case, k, to designate binding constants and capital K to designate dissociation or ionization constants. In Chaps. 2 and 3, we will concentrate our attention in the common description of binding problems in chemistry as applied to simple or homogeneous substrates, as opposed to the case of heterogeneous substrates, simple substrates are those for which there is only one value of the binding energy, the same for all the substrate species. Also, we will consider different cases of competitive binding for simple substrates. In Chap. 2 we present the different binding on substrates that have only one kind of binding sites. There we introduce the concept of average relative number of bound protons, and similarly for proton dissociation, we will define the average relative number of dissociated protons. These quantities are related to θ H and the degree of dissociation (α), respectively. These apply also to polyelectrolytes (Chap. 3) and multi-acids and bases (Chap. 7). Also they are generalized to other type of processes (complexation, redox reactions, etc.). Also there we discuss the problem of competitive binding and the binding to amphoteric substances. In Chap. 3 we consider the corresponding titrations. Titrations are the major application of the binding concept in chemistry. Although this subject has been extensively treated, we believe there are still some new applications stemming from the generalized concept of binding. Titrations allow to obtain the average relative number of bound species, whatever the process we are considering. In some cases, for instance in amphoteric substances, only differences of the corresponding average relative number of bound species may be obtained. The average relative number of bound species allows to determine the apparent equilibrium constant of the process. In the case of multi-site substances this results in a distribution of binding constants. In Chaps. 5, 6 and 7 we make a diversion to briefly describe distribution functions (Chap. 5), heterogeneous adsorption (Chap. 6) and some aspects of the physical chemistry of macromolecules (Chap.7). In Chap. 5 we discuss the concept of distribution functions mentioned just above. In Chap. 6 we introduce the Ising and the Bragg Williams isotherms, two very useful theoretical isotherms to describe binding in the presence of interactions between bound species. Later on (Chap. 9), we will apply the Ising model to describe the titration of polyvinyl amine. All the concepts presented in Chaps. 5–7 are very convenient to keep in mind when it comes to interpret the experimental results of the titration of the different multi-site systems. In Chap. 8 we show the application of the concept of average relative number of bound species to acid–base equilibria and complexes at complex substrates.

Reference

5

In Chap. 9 we will present examples of the acid–base titration of some synthetic polymers. In Chap. 10 we show the acid–base titration of colloidal oxides. Owing to their surface chemical structure these substances, in the solid state, may behave as acid or as basis according to the pH of the solution they are in contact with. For the sake of completeness, we show in Chap. 11 the titration response of a synthetic polyampholyte, humic compounds and proteins. In Chap. 12 we consider the electron titration of a polymer and show the wide variety of information that may be obtained from the analysis of the results. For several reasons we tried to avoid the inclusion in the main book of topics that were rather involved though not very fundamental or that involved long deductions. However, we also though that they should be present in it, so that we gather them together in a chapter. This is Chap. 13 in the form of appendices.

Reference 1. Albert A, Serjeant EP (1962) Ionization constants of acids and bases. Methuen & Co., London

Chapter 2

Binding to Simple Substrates with One Binding Site

2.1 Acid–Base Reactions. A Simple Example of Proton Binding We start with this kind of processes as they are more familiar to science students. This type of reactions has been treated extensively before, and there are plenty of excellent books and articles about the subject (see for instance [1–3]). Let us consider an acid–base reaction in solution, in the sense of Brönsted (see, for instance [4]), by which an acid HA exchange one proton with a base B− to give the corresponding conjugate acid, HB, and a new base, A− . HA + B− ⇌ A− + HB.

(2.1)

This reaction is composed of two partial reactions. One, in which a proton is detached from HA: HA ⇌ A− + H+ ,

(2.2)

and other one, in which a proton is bound to B− . B− + H+ ⇌ HB.

(2.3)

These reactions can be compared to redox reactions (see below) where an electron is gained by an oxidized species and one electron is lost by a reduced one. Reaction 2.2 is the acid dissociation reaction, whereas reaction 2.3 is the acid association reaction, for which k a is defined the corresponding operational association constant. Here the subscript “a” indicates the acid–base character of the considered process.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. A. Marmisollé and D. Posadas, Binding Phenomena, Physical Chemistry in Action, https://doi.org/10.1007/978-3-031-39736-3_2

7

8

2 Binding to Simple Substrates with One Binding Site

2.1.1 The Average Relative Number of Moles of Bound Protons and the Degree of Protonation We will consider an acid AH in solution, where the following association reaction takes place: A− + H+ ⇌ AH,

(2.4)

with its proton association constant defined as: [AH] , [A− ][H+ ] [ ( )]]1/2 { −W . β = 1 − 4θ (1 − θ ) 1 − exp kB T ka =

(2.5)

Here, we can define the degree of protonation or average relative number of moles of bound protons, , as: =

mols of bound H+ . mols of Acid

(2.6)

Customarily, except in the case of adsorption where the use of θ is recommended, α is generally employed when considering proton binding, even for the cases of several sites. However, when we come to multisite substrates, it is better to use the average relative number of moles of bound protons, since it is an average quantity. So, we decided to employ this notation from the start. The analytical or formal concentration of the acid, C A , is the mass balance over the different A species, and it is defined as: CA = [AH] + [A− ].

(2.7)

Then, C A is the total concentration of A species, and the concentration of associated protons is simply [HA]; so becomes: =

[AH] [AH] . = [ −] CA A + [AH]

(2.8)

Replacing the equilibrium concentrations in terms of k a and [H+ ], it results: [ ] ka H+ [ ]. = 1 + ka H+

(2.9)

2.1 Acid–Base Reactions. A Simple Example of Proton Binding

9

It can be seen from Eq. 2.9 that, if we know the protonation constant and the pH (here considered as pH = − log[H+ ]), we can calculate the fraction . Furthermore, this fraction depends only on the product k a [H+ ]. This means that independently of the nature of the acid (k a value), all acids have the same degree of protonation at the same value of the product k a [H+ ]. It is clear that acids with a higher association constant (lower dissociation constant) will reach, say, a value of = 0.5 at lower [H+ ] than those with a lower association constant (higher dissociation constant) (Fig. 1.1). However, they will have the same value of at the same value of k a [H+ ]. As we will see, these equations have the same mathematical form as the Langmuir Isotherm employed to describe the adsorption on a homogeneous surface [5]. The functional relationship is represented in Fig. 2.1. Furthermore, we will see that this also means that the cumulative distribution function of binding energies as a function of is a step function (see Chap. 5).

1.0

0.8 0.6 0.4 0.2 0.0 0

2

4

6

8

10

12

Ka

ka

10-3 10-5 10-7

10 5 10 7 10

pH 1.0

0.8 0.6 0.4 0.2 0.0 -4

-2

0

2

3

4

+

log (ka [H ]) Fig. 2.1 Average relative number of bound protons as a function of the pH (top) and as a function of the product k a [H+ ] (bottom) for the dissociation of acids with different constants

10

2 Binding to Simple Substrates with One Binding Site

2.1.2 The Average Relative Number of Moles of Dissociated Protons and the Degree of Dissociation We could have written Eq. 2.4 as occurring in the reverse way, as shown in Eq. 2.2. In that sense, a dissociation reaction is defined. Dissociation reactions are much more commonly employed than association reactions when studying acid–base equilibrium. In this case, it is defined the dissociation constant as: Ka =

[ − ][ + ] A H [AH]

(2.10)

Note that association and dissociation constants are related by K a = 1/k a . Frequently, dissociation constants are expressed in terms of pK a = − log K a . Then, strong acids are those with low pK a values and weak acids correspond to those with high pK a values. It is also important to note that along this book we will employ capital K for dissociation constants and lower case k for the association ones. To emphasize the concept of binding we will mostly consider association reactions. But, it is interesting to remark that the dissociation ones can be treated equally, by applying the same methodology. Moreover, for simplicity, we will use throughout this work, operational constants. This is equivalent to consider constant activity coefficients. Sometimes the name formal is employed for these constants, especially in the case redox reactions (e.g. formal redox potential). Concerning the equations, operational constants can be expresses as a quotient of concentrations whereas thermodynamic constants are written in terms of activities. We now define the average relative number of moles of dissociated protons, , as: =

mols of dissociated H+ . mols of acid

(2.11)

That is: [ −] A [ ]. = [AH] + A−

(2.12)

2.1 Acid–Base Reactions. A Simple Example of Proton Binding

11

Proceeding as before we find: =

[ ]−1 K a H+ [ ]−1 . 1 + K a H+

(2.13)

From their definitions, it is easy to show that: + = 1. For the specific case of dissociation equilibria, the degree of dissociation is often denoted by α. There, for the case of simple acids, the following relationships can be written as: [ −] A [ ] = = 1 − . (2.14) α= [AH] + A− The degree of dissociation is often employed for describing the speciation of polyprotic acids (such as phosphoric acid) as a function of pH. Note that the product k a [H+ ] is dimensionless as association constants units are reciprocal of concentration units. Then, this product can be considered a dimensionless concentration. In the case of proton binding, it is related with pH and pK a , by ([ ] ) ( [ ]) log ka H+ = log H+ /K a = pK a − pH.

(2.15)

These relationships allow defining K a as the proton concentration at which = 0.5. At this condition, k a [H+ ] = 1 and pH = pK a . For any acid, almost complete protonation, let us say higher than 0.99, takes place for log(k a [H+ ]) ≈ 2, whereas almost complete dissociation could be considered for lower than 0.01, which corresponds to log(k a [H+ ]) ≈ − 2. Then, acid equilibrium becomes important for − 2 < log(k a [H+ ]) < 2, which corresponds to the pH range pK a − 2 < pH < pK a + 2. This pH region can be observed in Fig. 2.1. Relative number, binding degree and coverage For the case of processes described as a chemical reaction, the degree of reaction can be computed as α = ξ /ξ max , where the extent of reaction (ξ ) is divided by the maximum extent of reaction, when the process is completed. For specific reactions, terms such as “degree of dissociation“, “degree of ionization”, “degree of oxidation” are then commonly used. In this regard, the name degree of protonation means the degree of reaction for the protonation process as described by reaction (2.4). However, it is also possible to define the degree of dissociation taking into account the reverse reaction (2.2). They

12

2 Binding to Simple Substrates with One Binding Site

both measure the extent of binding. As the degree of dissociation is commonly used to describe acid–base equilibrium in the first courses of analytical chemistry, we keep the symbol α for the extent of the dissociation reaction. On the contrary, and because of the analogy with adsorption process, we use the symbol θ for describing the fraction of occupied proton binding sites. In the particular case of protonation, the symbol θ H is then used. Although it could numerically coincide with the degree of association related to reaction (2.4), the use of θ H emphasizes the independence of the nature of the process leading to a protonation state. In this regard, when pH is 4.76, half of acetate anions are bound to protons (pK a of acetic acid is 4.76) independently of the way the solution was prepared (by adding HCl to sodium acetate or adding NaOH to acetic acid). Both average relative number of moles of bound protons, , and θ H can be used to quantify the extent of the proton binding process. Moreover, they do numerically coincide for the case of proton binding to substrates with a single site as that described by reaction (2.4). However, in the case of molecules with multiple sites, they are different (see Sects. 2.5 and 3.2).

2.2 Complexation. A Simple Example of Complexation Without Competitive Binding In the previous section we considered Brönsted acids and bases. We may now consider the concept of Lewis acids and bases [4]: acids are those substances capable of accepting electron pairs, and bases are those capable of donating them. With these ideas in mind we may consider complexation reactions as an extension of the subject that we developed in the previous section. We will assume here that A− comes from a salt of a strong acid in water, so we do not have to worry about the hydrolysis of A− . Thus, the formation of a metal complex through the reaction of a metallic cation, M+ (the Lewis acid), with a suitable Lewis base, A− , yields a metal complex, MA: M+ + A− ⇌ MA.

(2.16)

The operational complexation constant or formation constant can be written as: [MA] kC = [ + ][ − ] . M A The analytic concentration of A− and M+ is C A and C M , respectively.

(2.17)

2.3 Adsorption. Simple Adsorption Equilibrium

13

[ ] CM = M+ + [MA],

(2.18)

[ ] CA = A− + [MA].

(2.19)

The degree of bound ligand or ligand number, [6] is =

moles of bound A− [MA] , = [ +] moles of M species M + [MA]

(2.20)

which results [ ] kC A− [ ]. = 1 + kC A−

(2.21)

This equation is the isotherm for the complexation binding. An example of this case could be the formation of the soluble complex between divalent cations and EDTA (Y). In this case, the complexation equilibrium is M2+ + Y4− ↔ MY2−

(2.22)

With the corresponding constant ] [ MY2− kC = [ 2+ ][ 4− ] . M Y

(2.23)

Figure 2.2 shows the mean number of EDTA molecules bound per divalent cation, , as a function of the EDTA concentration in solution (free EDTA in solution). The dimensionless concentration for this equilibrium can be defined as k c [A− ]. Then, in terms of the dimensionless concentration, the binding behaviour for the different ions considered are the same (Fig. 2.2, bottom). Numerically, k c corresponds to the ligand concentration at which half of the metal cation is bound. Also, due to the functional form of Eq. 2.21, the dependence of on log(k c [A− ]) is the same as that shown for the case of proton binding.

2.3 Adsorption. Simple Adsorption Equilibrium The phenomenon of adsorption concerns the binding of one species (adsorbate) present in a gaseous or solution phase (the external phase), onto suitable sites (nodes) of a surface (thes adsorbent) [5, 7]. We will first consider the nodes are arranged in an ideal two-dimensional lattice. In practice, it is possible to obtain very nearly ideal surfaces by cleaving or cutting conveniently single crystals. The adsorption

14

1.0 0.8 0.6

Fig. 2.2 Degree of bound EDTA, , as a function of the EDTA concentration (top) and as a function of the dimensionless concentration (bottom) for the complexation of different divalent cations. Complex formation constants are indicated in the plot [1]

2 Binding to Simple Substrates with One Binding Site

0.4 0.2 0.0 -18

-16

-14

-12

-10

-8

-6

-4

-2

log ([EDTA]/M) 1.0 0.8

0.6

kC / M-1

0.4

Mg

0.2 0.0 -4

-2

2+

8

6.2 10 Ca2+ 4.5 1010 Cr 2+ 4.0 1013

0

2

4

log (kC [EDTA])

equilibrium can be represented as S + A ⇌ SA,

(2.24)

where S means the free sites, A is the adsorbate in gas or solution and SA means the occupied site. We will also consider that the surface has a total of M adsorption sites and that there are N adsorbed molecules [5]. A convenient variable for the surface concentrations of occupied and empty sites are the fractions for the occupied sites, θ = N / M, and (1 − θ ) for the empty ones. In adsorption literature, θ is referred to as coverage degree. Then, we can write the equilibrium constant as: ' kad =

θ (1−θ )[A]

(2.25)

where [A] is the concentration of the adsorbate in the external phase. It is interesting to note that, since we are considering an ideal lattice, all the sites are equivalent. That

2.4 Electron Binding. A Simple Case of Redox Reactions

15

is, the energy put into play for the adsorption on each site is the same, independently if other sites are occupied or not. These assumptions are the basis of the Langmuir model. Customarily, Langmuir equation is written as: θ=

' kad [A] . ' 1 + kad [A]

(2.26)

In the case of gas molecules, it is usual to employ pressure instead of concentrations. For ideal gases (or real gases at low pressures), molar concentrations and pressures are equivalent at the same temperature as [A] = p/RT. As we are employing operational constants, the units for the adsorption constants depend on the experimental variable considered. Therefore, sometimes the Langmuir isotherm for gases ' /RT . adsorption can be found in literature as: θ = k ad p /(1 + k ad p), where kad = kad Note that Eq. 2.26 is the same expression as those employed for acid association and complexation without competitive binding. These problems are then equivalent: we are always dealing with the binding of species onto equivalent sites, considering concentrations (or pressure) instead of activities for the species in the bulk phase. Figure 2.3 shows the dependence of the coverage degree calculated from the experimental constants determined by Langmuir for the adsorption of different gases on mica at 90 K [8]. Note also that, as k ad units are reciprocal of pressure, the product (k ad p) becomes a dimensionless pressure. As shown in Fig. 2.3, the coverage degree has a unique dependence the dimensionless pressure for all the gases considered. Also, the coverage degree is 0.5 for p = 1/k ad .

2.4 Electron Binding. A Simple Case of Redox Reactions In a general homogenous redox reaction there is an electron exchange between a substance 1 in a reduced state (Red1 ) and another, 2, in an oxidized state (Ox2 ) to give a substance 1 in an oxidized state (Ox1 ) and another, 2, in a reduced one (Red2 ) [9]: Red1 + Ox2 ⇌ Red2 + Ox1 .

(2.27)

This reaction is composed of two half-reactions, in one of them the species Red1 loses one electron to give the oxidized species Ox1 ; and in the other the species Ox2 gains one electron to give the species Red2 . Red1 ⇌ Ox1 + e−

(2.28)

16

2 Binding to Simple Substrates with One Binding Site 1.0 0.8 0.6

k ad / bar -1

0.4 0.2 0.0 0

20

40

60

80

Ar CH 4

0.065 0.107

CO

0.650

100 120 140 160

p / bar 1.0 0.8 0.6 0.4 0.2 0.0 -2

-1

0

1

2

log ( k ad p ) Fig. 2.3 Surface coverage degree, θ, of mica surfaces as a function of the gas pressure for the gases indicated, at 90 K (top). Universal Langmuir plots of the coverage as a function of the driving force (bottom) calculated from the constant values reported for this system [8]

and Ox2 + e− ⇌ Red2

(2.29)

Each of this reaction pair is called a redox couple. These reactions can be visualized as electron binding or electron detachment processes; exactly in the same way as we considered proton binding to a convenient base, particles binding onto a surface and so on. These half-reactions can be ordered in a scale of increasing tendency to lose electrons. When both constituents of the redox couple are in their standard states, this scale is the scale of standard electrode potentials, E 0 .

2.4 Electron Binding. A Simple Case of Redox Reactions

17

Reaction 2.27 can occur irreversibly in solution, by mixing the reactants Red1 and Ox2 , let us say for example, a Fe2+ salt and a Ce4+ salt. Else, it can be carried out in a galvanic cell by suitably connecting an electrode containing the Fe2+ /Fe3+ couple to another electrode containing the Ce3+ /Ce4+ couple. The electric potential developed by each one of the redox couples, when their constituents are at an arbitrary activity is given by the Nernst equation [10]: ε = ε0 −

RT ared ln . n F aox

(2.30)

For simplicity, we will use, as before, concentrations instead of activities (approximation that is only valid in dilute solutions) and then for the general redox couple Red ↔ Ox + e− , we will define1 : ε = ε0' −

RT [Red] ln , F [Ox]

(2.31)

where ε0' is the apparent formal Galvani electric potential [10]. Since the total concentration [Ox] + [Red] = C T , is constant, we may write the fraction of bounded electrons, or degree of reduction, as the fraction of redox sites that are occupied. This is θe =

[Red] , CT

(2.32)

where the subscript “e” emphasizes the idea of electron binding. Then, the fraction of free sites, or degree of oxidation, as 1 − θe =

[Ox] . CT

(2.33)

In such a way that the Nernst equation now reads: ) ( θe RT . ln ε=ε − F 1 − θe 0'

(2.34)

Individual electrode potentials, ε, cannot be determined experimentally. Instead, voltage differences between pairs of electrodes (forming a galvanic cell), E, are measurable. However, the individual electrode potential of the standard hydrogen Following the usage, we will employ the symbol ε; to denote the Galvani electric potential difference ε = φ metal − φ solution at an isolated electrode [10], and E to denote the emf of a galvanic cell.

1

18

2 Binding to Simple Substrates with One Binding Site

electrode (SHE) is arbitrarily set as zero (εSHE = 0), which allows assigning the voltage difference of the cell formed by a given electrode and the SHE as the electrode potential, E = ε − εSHE . ) ( θe RT ln E=E − F 1−θe 0'

(2.35)

This equation can be rearranged to give: ] ] [ [ θe F(E − E 0' ) FE = kel exp − , = exp − 1 − θe RT RT

(2.36)

( ) where we have used the relation kel = exp F E 0' /RT . ] [ kel exp − FRTE ]. [ θe = 1 + kel exp − FRTE

(2.37)

Again, this expression has the form of the Langmuir isotherm. It is equal to those already described for acid protonation and adsorption. However, it should be noted that instead of the concentration term multiplying the constant k el , here we have the exponential of an electrical potential. It is useful to define the electron activity in the electron conducting phase, ae , in terms of this potential as, RT ln ae = −F E. So that, the former equation can be written as: θe =

kel ae . 1 + kel ae

(2.38)

This equation directly resembles a typical binding equation. This proves that redox reactions can be viewed as an electron binding process and that, at equilibrium, the same type of equation can be employed to describe it. Analogously to pH, the pe is sometimes employed for describing the redox processes: pe = −log ae , which in terms of the electrode potential becomes, pe =

F ε. 2.303RT

(2.39)

Finally, the product k el ae behaves as a normalized activity. In this regard, the term log(k el ae ) can be visualized as a dimensionless potential, as log(kel ae ) = −

F (ε − ε0 ) = −( pe − pe0 ). 2.303RT

(2.40)

2.4 Electron Binding. A Simple Case of Redox Reactions

19

In terms of this dimensionless potential, all plots in Fig. 2.4 do coincide. Furthermore, they show the same functional relationship as those presented for the acid–base, complexation and adsorption equilibrium. The advantages of considering redox equilibrium as a binding process are now evident: once a binding problem has been solved, the solution for other binding problems is the same. Thus we could have written directly Eq. 1.38 from Eq. 1.15 and the definition of k el . 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

E/V 1.0 0.8 0.6

E0/ V

0.4

Fe3+/2+ Fe(CN)63−/4−

0.2 0.0

Cr3+/2+ -4

-2

0

2

4

log (kel ae) = −(E−E 0)F/2.303RT Fig. 2.4 Reduced fraction, θ, as a function of the potential, E, for three different redox couples in aqueous solutions at 298 K (top). Reduced fraction as a function of the driving force (Eq. 2.35) (bottom). The redox couples are shown in the figure. E 0 (Fe3+ /Fe2+ ) = 0.77 V, E 0 (Fe(CN)6 3− / Fe(CN)6 4− ) = 0.36 V, E 0 (Cr3+ /Cr2+ ) = − 0.41 V

20

2 Binding to Simple Substrates with One Binding Site

2.5 The General Binding Process and the Binding Driving Force Let us consider a general binding process: S∗ + B ⇌ SB,

(2.41)

where S* (the substrate) is a species with unbound sites (unoccupied sites), SB means the bound state (occupied sites) and B means the free binding species. We will analyse this substrate-binding species equilibrium for the case of 1:1 binding. From a chemical equilibrium point of view, at T and p constant, binding proceeds if it leads to a decrease of the Gibbs free energy, ∆G. Then, binding takes place if the final state (SB) has lower free energy than the initial state (S* + B). The driving force is defined as the Affinity, A, of the reaction [11]. In turn, the affinity is defined as: A = −(dG/dξ ), where ξ is the extent of the reaction. Thus if the driving force is positive reaction 2.41 will proceed to the right and vice versa. In Chap. 13, Sect. 13.6, we consider a short analysis showing the relation between ∆G and A for a process like that of Eq. 2.41.

2.6 About the Fulfilment of the Langmuir Equation for Different Processes It might be surprising why the equilibrium of processes as different as the protonation of a base (Eq. 2.9), the attachment of a ligand (Eq. 2.21), the loss of an electron (Eq. 2.38) and the adsorption of a particle (Eq. 2.26) may be all represented by the same functional relationship, a Langmuir-type isotherm. This is a model based on the following assumptions: (1) the number of free sites is bounded (not infinity); in this way we ensure that the degree of binding remains finite; (2) only one particle per site is admitted (this is equivalent to the assumption of monolayer adsorption for physical adsorption); in this way we set the limits of the degree of binding between zero and unity; (3) both free and occupied sites behave ideally, that is, there are no interaction among them, so that their activity factors do not depend on the fraction of occupied sites; (4) both free and occupied sites are indistinguishable; this is important for the statistical mechanic deduction of the Langmuir equation (see Chap. 6, Sect. 6.3). All these considerations are implicitly taken as fulfilled when a stoichiometric equation is used to describe the binding process. On the other hand, it might be surprising that in the adsorption problem, the sites are fixed in a reticule whereas in the protonation case the “adsorption sites”, that is the basic sites, are able to move in space. A moment of thought will reveal that this is immaterial to the purpose of the binding equilibrium. We are only interested in the binding equilibrium system independently of what happens with the rest. In fact, from the point of view of the statistical thermodynamics, this can be circumvented

References

21

by defining the basic sites as subsystems of the whole ensemble (see [5], Sect. 7.3 and Appendix 1, Sect. 3). This is implicitly taken into account when chemical thermodynamic considerations are applied to the chemical reactions used to describe the binding processes.

References 1. Harris DC (2015) Quantitative chemical analysis, 9th edn. W. H. Freeman 2. Skoog DA, West DM, Holler FJ (1995) Analytical chemistry, 6th edn. McGraw Hill, New York 3. Butler JN (1964) Solubility and pH calculations. Principles of chemistry series. Addison−Wesley Publishing Company, Reading 4. Gould ES (1962) Inorganic reactions and structure. Holt, Rinehard and Winston, New York 5. Hill TL (1960) An introduction to statistical thermodynamics. Addison−Wesley Pub Co. 6. Rossotti FJC, Rossotti H (1961) The determination of stability constants and other equilibrium constants in solution. McGraw−Hill Book Company, Inc., New York 7. Clark AC (1970) Adsorption and catalysis. Academic Press, New York 8. Langmuir I (1918) The adsorption of gases on plane surface of glass, mica and platinum. J Am Chem Soc 40:1361–1403 9. Bard AJ, Faulkner LF (1980) Electrochemical methods. Wiley, New York 10. Conway BE (1965) Theory and principles of electrode processes. The Ronald Press, New York 11. Muller P (1994) Glossary of terms used in physical organic chemistry. Pure Appl Chem 66:1077–1184

Chapter 3

One Substrate with Two Different Binding Sites. Competitive Binding. Two Different Binding Species. Two Different Binding Substrates

3.1 Introduction In this chapter we will consider one substrate with two different binding sites, different binding species on the same substrate and two different binding species on different sites of a given substrate.

3.2 One Substrate with Two Different Binding Sites Let us consider an anion with two different binding sites as, for instance A2− . It could bind two protons according to: A2− + H+  HA− .

(3.1)

With a protonation constant k a1 equal to: ka1

 − HA =  2−  +  . A H

(3.2)

And HA− + H+  H2 A,

(3.3)

[H2 A] ka2 =  −  +  . AH H

(3.4)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. A. Marmisollé and D. Posadas, Binding Phenomena, Physical Chemistry in Action, https://doi.org/10.1007/978-3-031-39736-3_3

23

24

3 One Substrate with Two Different Binding Sites. Competitive Binding. … 2.5 2.0

1.5 1.0 0.5 0.0

-4

-2

0

2

4

6

8

10

12

pH Fig. 3.1 Average relative number of bound protons vH  corresponding to the case of one substrate with two different binding sites, k a2 = 104 , k a1 = 100

According to the definition of the average relative number of bound proton, vH , of Chap. 2 (see Eq. 3.6), and replacing [HA− ] and [H2 A] from Eqs. 2.2 and 2.4 it results: v H  =

ka1 [H+ ] + 2ka1 ka2 [H+ ]2 . (1 + ka1 [H+ ] + ka1 ka2 [H+ ]2 )

(3.5)

There are terms of the type ka1 ka2 [H+ ]2 both in the numerator and the denominator. A plot of vH  versus pH is shown in Fig. 3.1. Two features can be seen immediately from Fig. 3.1. First, the maximum vH  value is two. This is a consequence of a base having two sites. The protonation degree would be in this case vH  / n, where n is the number of sites. Second, in the case k a2 > k a1 the curve vH  versus pH has two inflexion points.

3.3 Two Different Binding Species on the Same Substrate In this section we will consider several different examples: firstly, two neutral adsorbates on the same adsorbent (competitive adsorption) (see for instance [1]) and the equilibrium of two cations (protons and an ionic metal, M+ ) with the same anionic base, A− (competitive binding between protonation and complexation).

3.3 Two Different Binding Species on the Same Substrate

25

3.3.1 Competitive Adsorption In this case we are dealing with two substances A and B, competing for the same active sites on the surface. The adsorption reactions are written as: A + empty site  site occupied with A.

(3.6)

We will call the coverage of sites with the substance A, θ A . And B + empty site  site occupied with B.

(3.7)

Since the concentration of free sites is now (1 − θ A − θ B ), the corresponding equilibrium constants for reactions 3.6 and 3.7, are defined as: kA =

θA , (1 − θA − θB ) CA

(3.8)

kB =

θB . (1 − θA − θB ) CB

(3.9)

and

Equations 3.15 and 3.16 can be written as: θA =

kA C A 1 + kA C A + kB C B

(3.10)

θB =

kB C B . 1 + kA C A + kB C B

(3.11)

and

These expressions can be compared with those shown below, Eqs. 3.18 and 3.19, for the case of competition between complexation and protonation and Eqs. 3.32 and 3.33 for ampholytes.

3.3.2 Competitive Binding Between Protonation and Complexation Now we will consider the competition between two cations say H+ and M+ for a base A− . This case is similar to the case of two adsorbates competing for sites on the same substrate (Sect. 3.3.1). The case of complexation of a metal with a ligand

26

3 One Substrate with Two Different Binding Sites. Competitive Binding. …

that may also bind more than one proton will be considered in the next chapter. We will consider an acid AH in solution, where the following association reaction takes place: A− + H+  AH

(3.12)

with its proton association constant defined as: [AH] ka =  −  +  . A H

(3.13)

Also there is as complexation reaction: M+ + A−  MA.

(3.14)

The complexation constant is: kc =

[MA] , [M+ ][A− ]

(3.15)

The mass balance condition for A species results: CA = [A− ] + [MA] + [HA].

(3.16)

And the mass balance for M is: CM = [M+ ] + [MA].

(3.17)

The average number of metal ion bound is vMA  = [MA]/C A , so by applying Eqs. 3.15 and 3.17, it results: vMA  =

kc [M+ ] . 1 + ka [H+ ] + kc [M+ ]

(3.18)

And the average number of proton ion bound, vMH  = [HA]/C A , so that applying Eqs. 3.13, 3.16 and 3.17, it results: vHA  =

  ka H+ . 1 + ka [H+ ] + kc [M+ ]

(3.19)

Equations 3.18 and 3.19 can be converted to: vHA  = ka [H+ ]. 1 − vMA  − vHA 

(3.20)

3.4 Electron Binding to Two Redox Couples 1.2 1.0 0.8

Fig. 3.2 Semi-logarithmic plot according to Eq. 3.18 for three different pH values. k a = 1e5, k c = 1e5. Continuous line: pH = 7; long-dashed line: pH = 6; short-dashed line: pH 5

27

0.6 0.4 0.2 0.0 -12

-10

-8

-6

-4

-2

0

+

log([M ])

And vMA  = kc [M+ ]. 1 − vMA  − vHA 

(3.21)

Note that Eqs. 3.18 and 3.19 are entirely similar to Eqs. 3.10 and 3.11. They are the Langmuir isotherms for the case of competitive adsorption. These describe the competition of H+ and M+ for the base A− . A semi-logarithmic plot of Eq. 3.18 is shown in Fig. 3.2 for three different values of the pH. Plots similar to those of Fig. 2.1 are obtained. Note that decreasing the pH has an effect similar to increasing the metal binding constant.

3.4 Electron Binding to Two Redox Couples We will consider now the presence of two redox couples in one electrode. Let these couples be: Ox1 + e−  Red1 ,

(3.22)

Ox2 + e−  Red2 .

(3.23)

and

With standard redox potentials given by E 0 1 and E 0 2 . In this case if the redox potential of the couple 2 is higher than the redox potential of the couple 1 (one may think here of couple 1 being Fe2+ /Fe3+ and couple 2 as being Ce3+ /Ce4+ ), most of the Ce4+ will react with the Fe2+ to form Ce3+ and Fe3+ .

28

3 One Substrate with Two Different Binding Sites. Competitive Binding. …

This situation is equivalent to the case analysed in the preceding paragraph. If pe < E 0 1 all Fe will be in the Fe2+ form and also all the Ce will be in the Ce3+ form. At pe = E 0 1 , [Fe2+ ] = [Fe3+ ]. If E 0 1 < pe < E 0 2 there will be only Fe3+ and Ce3+ . Finally, if pe > E 0 2 there will exist only Fe3+ and Ce4+ . Because k 1 = exp ( − E 0 1 / RT ) and k 2 = exp ( − E 0 2 /RT ). Therefore, by analogy with Eqs. 3.10 and 3.11, the average number reduced 1 species is:  E  k1 exp − RT  E   E  , θ1 =  1 + k1 exp − RT + k2 exp − RT

(3.24)

and the average number of reduced 2 species is:  E  k2 exp − RT  E   E  . θ2 =  1 + k1 exp − RT + k2 exp − RT

(3.25)

3.5 Two Different Binding Sites. A Simple Example of Proton Binding in Ampholytes The term ampholyte is employed to describe substances that have both an acid and a basic group. Simple examples of these substances are o-aminophenol (I), picolinic acid (II), glycine (III) and δ-aminovaleric acid (IV). A general discussion of the acid–base properties of these substances may be found in reference [2]. CO2H

NH2 OH

(I) H2N-CH2-CO2H (III)

N (II) H2N-CH2-CH2-CH2-CH2-CO2H (IV)

Let us consider, for instance, o-aminophenol which we will represent as H2 N– ϕ–OH. It has two pK values, 4.17 (K 1 = 10−4.17 ) and 9.87 (K 2 = 10−9.87 ) [2]. At pH highly acid the basic group is completely protonated and the acidic group is not ionized; at pH = 4.17, the basic group is only half ionized. At intermediate pH values neither group is ionized, and at pH = 9.87 the acidic group is half ionized.

3.5 Two Different Binding Sites. A Simple Example of Proton Binding … Fig. 3.3 Calculated vH  (continuous line) and rH  (dashed line) versus pH for o-aminophenol

1.2 1.0

29 H2N- -O-

H3N+- -OH

,

0.8 0.6



0.4 0.2 0.0 0

2

4

6

8

10

12

14

16

pH

The distribution of the different species as a function of pH is shown in Fig. 3.3. These statements mean that in acid media the basic group protonates according to: H2 N−ϕ−OH + H+  (H3 N+ −ϕ−OH).

(3.26)

And in basic media the acid group loses H+ according to: H2 N−ϕ−OH  (H2 N−ϕ−O− ) + H+ ,

(3.27)

where ϕ stands for the phenyl ring. Equations 3.26 and 3.27 are the equilibrium reactions, and the corresponding equilibrium constants will be, respectively: k1 =

[H3 N+ −ϕ−OH] , [H2 N−ϕ−OH][H+ ]

(3.28)

K2 =

[H+ ][H2 N−ϕ−O− ] . [H2 N−ϕ−OH]

(3.29)

and

The analytical concentration of the ampholyte, C a is written as: Ca = [H2 N−ϕ−OH] + [H3 N+ −ϕ−OH] + [H2 N−ϕ−O− ].

(3.30)

Or replacing (H3 N+ −ϕ−OH) from 3.28 and (H2 N−ϕ−O− ) from 3.29: Ca = [H2 N−ϕ−OH] (1 + k1 [H+ ] + K 2 ([H+ ])−1 ).

(3.31)

30

3 One Substrate with Two Different Binding Sites. Competitive Binding. …

The average amount of protons bound to the basic group (H2 N–), vH  , is: vH  =

[H3 N+ −ϕ−OH] , Ca

(3.32)

and the average amount of protons ionized from the acid group (–O− ), is: [H2 N−ϕ−O− ] . Ca

(3.33)

K 2 ([H+ ])−1 . {1 + k1 [H+ ] + K 2 ([H+ ])−1 }

(3.34)

k1 [H+ ] . 1 + k1 [H ] + K 2 ([H+ ])−1

(3.35)

rH  = It is easy to show that: vH  = And that r H  = 

+

Equations 3.34 and 3.35 can be written in the form: vH  = k1 [H+ ], 1 − vH  − rH 

(3.36)

rH  = (K 2 [H+ ])−1 . 1 − vH  − rH 

(3.37)

Again, Eqs. 3.36 and 3.37 are equivalent to the Langmuir isotherms for competitive adsorption (compare with Eqs. 3.10 and 3.11).

3.6 Formation of Zwitterions Under certain circumstances, the acidic part of an ampholyte dissociates to protonate the basic function of the ampholyte and form a different species that it is characterized by having both the acid and the basic groups charged. These compounds are called zwitterions. The criteria to detect the formation of these substances are given in Ref. [2]. As a consequence of the charges very often they have pK a values that differ considerably from those of their neutral analogues. When the zwitterion is titrated with an acid, a proton is added to the anionic group, and when it is titrated with an alkali, a proton will be removed from the cationic group. Let us consider

3.6 Formation of Zwitterions

31

glycine, H2 NCH2 COOH (pK a 2.2 and 9.9) [2]. In very acid media the stable form is H3 N+ CH2 COOH. As the pH increases, this becomes in equilibrium not with the neutral form but with the zwitterion: H3 N+ CH2 COOH  H3 N+ CH2 COO− + H+ .

(3.38)

Albert and Serjeant [2], called this process proton gained. The ionization constant, K 1Z = 40, is given by: K 1Z =

[H3 N+ CH2 COO− ][H+ ] . [H3 N+ CH2 COOH]

(3.39)

We will define the average number of protonated zwitterions vHZ  as: vHZ  =

[H3 N+ CH2 COOH] . CaZ

(3.40)

As the pH is increased further the zwitterion becomes in equilibrium with the anionic form (proton loss): H3 N+ CH2 COO−  H2 NCH2 COO− + H+ . K 2Z =

[H2 NCH2 COO− ][H+ ] . [H3 N+ CH2 COO− ]

(3.41) (3.42)

We will define the average number of ionized zwitterions rHZ  as: rHZ  =

[H2 NCH2 COO− ] , CaZ

(3.43)

where CaZ is the total concentration of zwitterions. CaZ = [H3 N+ CH2 COOH] + [H3 N+ CH2 COO− ] + [H2 NCH2 COOH].

(3.44)

This may be written as: CaZ = [H3 N+ CH2 COO− ] (1 +

1 + K 2Z [H+ ]−1 ) K 1Z [H+ ]−1

(3.45)

So that: vHZ  =

1 . 1 + K 1Z [H+ ]−1 + K 1Z K 2Z [H+ ]−2

(3.46)

32

3 One Substrate with Two Different Binding Sites. Competitive Binding. …

And: rHZ  =

K 1Z K 2Z [H+ ]−2 . 1 + K 1Z [H+ ]−1 + K 1Z K 2Z [H+ ]−2

(3.47)

If there were only neutral form the protonated form would deprotonate according to; H3 N+ CH2 COOH  H2 NCH2 COOH + H+ .

(3.48)

With a constant: K1 =

[H3 N+ CH2 COOH][H+ ] . [H2 NCH2 COOH]

(3.49)

And the neutral form would dissociate according to: H2 NCH2 COOH  H2 NCH2 COO− + H+ .

(3.50)

And K2 =

[H2 NCH2 COO− ][H+ ] . [H2 NCH2 COOH]

(3.51)

It is clear that this problem is entirely similar to the case of the zwitterion, therefore: vH  =

1 . + K 1 K 2 [H+ ]−2

(3.52)

K 1 K 2 [H+ ]−2 . 1 + K 1 [H+ ]−1 + K 1 K 2 [H+ ]−2

(3.53)

+ −1

1 + K 1 [H ]

And: rH  =

For the sake of comparison in Fig. 3.4 we show the two averages calculated with the constants corresponding to glycine: pK 1 = 3.35, pK 2 = 9.78, pK 1Z = 4.43 and pK 2Z = 7.7 [2], (see also [3]). It should not be surprising that the constants of the neutral and those of the zwitterion are different as in the latter the amino and acid groups are charged thus strongly influencing the deprotonation reactions [2]. It may also happen that the zwitterion is in equilibrium with the neutral form. In this case: H3 N+ CH2 COO−  H2 NCH2 COOH.

(3.54)

3.6 Formation of Zwitterions

33

Fig. 3.4 Calculated vH  , vHZ  , rH  and rHZ  for glycine

1.2 1.0

,

0.8

0.6

0.4 0.2

0.0 0

2

4

6

8

10

12

pH

With an equilibrium constant, R:   H3 N+ CH2 COO− . R= [H2 NCH2 COOH]

(3.55)

R is the equilibrium constant for the tautomerization between the zwitterion and the neutral species, and hence, it measures how stable is the zwitterionic form when compared with the neutral one. R can be written in terms of the previous constants as: K2 K 1Z = . K1 K 2Z

(3.56)

K 1Z K 2Z = K 1 K 2 .

(3.57)

R= This expression leads to

This relation is trivial because it is the equilibrium constant for the global equilibrium H3 N+ CH2 COOH  H2 NCH2 COOH + 2H+ ,

(3.58)

which, of course, does not depend on the intermediate forms. In this case, in which the species H3 N+ CH2 COOH may deprotonate to the neutral or the zwitterion forms in which there are two alternative ways, the constants are, in fact microconstants (see Chap. 13.1). In these cases, it is not possible to determine the constants from a titration.

34

3 One Substrate with Two Different Binding Sites. Competitive Binding. …

In the case where H3 N+ CH2 COOH may deprotonate to any of both: H3 N+ CH2 COO− (the zwitterion) or to the neutral form (H2 NCH2 COOH) we may define the macroscopic constants for the global processes K g1 and K g2 as: K g1 =

([H3 N+ CH2 COO− ] + [H3 NCH2 COO])[H+ ] . [H3 N+ CH2 COOH]

(3.59)

[H2 NCH2 COO− ][H+ ] . ([H3 N+ CH2 COO− ] + [H3 NCH2 COO])

(3.60)

And K g2 =

The relation between K g1 and K g2 and the microconstants, is as follows [4]: K g1 = K 1 + K 1Z .

(3.61)

1 1 1 = + . K g2 K2 K 2Z

(3.62)

And

For the sake of comparison in Fig. 3.5 we show the two averages calculated with the constants corresponding to glycine: pK 1 = 2.35, pK 2 = 9.78, pK 1Z = 4.43 and pK 2Z = 7.7 [2]. It is convenient to  find the expressions for the number of protonated zwitterions v plus neutral form, Hg and the number of ionized zwitterions plus neutral form,  rHg . vH  , Fig. 3.5 Calculated  vHZ  and vHg for glycine, and calculated with the macroconstants, respectively

1.2 1.0



0.8

0.6 0.4 0.2 0.0 0

2

4

6

8

pH

10

12

14

16

References

35

 vHg =

1 . + K g1 K g2 [H+ ]−2

(3.63)

K g1 K g2 [H+ ]−2 . 1 + K g1 [H+ ]−1 + K g1 K g2 [H+ ]−2

(3.64)

1 + K g1

[H+ ]−1

And  rHg =

We note that the expressions are the same as in the presence of the zwitterion only or of the neutral form only except that now only the macroscopic constants are vH  , vHZ  and vHg with the above data. It can involved. In Fig. 3.5 are compared be noted that vHZ  and vHg are equal. This is not surprising since in this case as the pH is increased it should form first H2 NCH2 COOH; however as soon it forms because the equilibrium H3 N+ CH2 COO− = H2 NCH2 COOH works so as to form the zwitterion and everything happens as if there were only the zwitterion.

References 1. Clark AC (1970) Adsorption and catalysis. Academic Press, New York 2. Albert A, Serjeant EP (1962) Ionization constants of acids and bases. Methuen & Co., London 3. Saroff HA (1994) Glycine: a simple zwitterion: analysis of its proton-binding isotherm. J Chem Educ 71(8):637 4. Hernández MT, Montero J (1997) Calculationg microspecies concentration of zwitterion amphoteric compounds: ciprofloxacin as example. J Chem Educ 74(11):1311

Chapter 4

Titration of Simple Substrates

4.1 Introduction One of the main applications of binding is titrations. It has been written a lot about the subject (See for instance, [1, 2]). However, one of the most important applications of the present approach is the simulation of the titration curves. A consequence of Chaps. 2 and 3 is their applications to titration problems, and a consequence of them is the appearance of distributed thermodynamic quantities that we will consider in Chap. 7. Moreover, to emphasize generality, we will apply the approach presented here to non-classical chemical titrations as are the titration of macromolecules and colloidal oxides, together with electron and redox titrations. We will use, and extend, the method employed by de Levié [3], because we think that it is clearer, easier and less cumbersome than others. We will show how this method can be applied to many complex systems with minor changes. This also leads to a unified way to state titration problems. In Chaps. 2 and 3, to emphasize the concept of binding, we employed the association constants. The common usage in analytical chemistry is to employ the dissociation constants and fractions. To treat titrations, it is more convenient to employ the average relative number of moles of dissociated protons, , instead to average relative number moles of bound protons, . Since there is no difficulty in changing from one to another, we will employ from the beginning. We will begin with some simple cases and then, we will present generalizations.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. A. Marmisollé and D. Posadas, Binding Phenomena, Physical Chemistry in Action, https://doi.org/10.1007/978-3-031-39736-3_4

37

38

4 Titration of Simple Substrates

4.2 Simple Examples 4.2.1 Titration of a Strong Monoprotic Acid As the way of an introduction let us calculate the titration curve (TC) for the very simple case of a HCl solution with NaOH. Let us consider a sample of volume, V s , of an HCl solution, of concentration, C a . This sample is to be titrated by adding volumes up to a total V t , of a strong base, say NaOH, of concentration C b (see [3]). The starting point of the calculation is the electroneutrality condition: [ +] [ +] [ −] [ −] H + Na = OH + Cl .

(4.1)

Due to the dilution effect, the concentration of [Na+ ] changes during the titration. This, it is expressed as: [Na+ ] =

Cb Vt . (Vs + Vt )

(4.2)

[ −] Cl =

Ca Vs , (Vs + Vt )

(4.3)

The concentration of Cl− is

s where (VsV+V is the dilution factor. Replacing Eqs. 4.2 and 4.3, into the electroneut) trality condition (Eq. 4.1) and solving for V t /V s = f , gives the titration fraction f:

fa =

Ca −[H+ ] + [OH− ] . Cb + [H+ ]−[OH− ]

(4.4)

A graph of the pH versus f a is shown in Fig. 4.1.

4.2.2 Titration of a Weak Polyprotic Acid Let us consider the titration of a polyprotic acid like H3 PO4 of concentration C H3PO4 with NaOH of concentration C b . The dissociation processes are: H3 PO4 ⇌ H2 PO4 − + H+ ] [ ] [ H2 PO4 − + H+ . K1 = [H3 PO4 ]

(4.5)

(4.6)

4.2 Simple Examples

39

12 10

pH

8 6 4 2 0 0.0

0.5

1.0

1.5

2.0

fa Fig. 4.1 Simulated titration curve of a HCl solution with NaOH. C a = 0.01 M, C b = 0.01 M, V s = 0.01 mL

Followed by: H2 PO4 − ⇌ HPO4 2− + H+

(4.7)

] [ ] [ HPO4 2− + H+ [ ] K2 = H2 PO4 −

(4.8)

HPO4 2− ⇌ PO4 3− + H+

(4.9)

] [ ] [ PO4 3− + H+ [ ] . K3 = HPO4 2−

(4.10)

and

The electroneutrality condition now reads: [ +] [ +] [ −] ] [ ] [ H + Na = OH + [H2 PO4 ] + 2 HPO4 2− + 3 PO4 3− .

(4.11)

< > The mean value r H3 P O4 is defined as: ] [ ] [ ] [ < > H2 PO4 − + 2 HPO4 2− + 3 PO4 3− r H3 P O4 = , C H3 P O4

(4.12)

40

4 Titration of Simple Substrates

where C H3 P O4 = [H3 PO4 ] + [H2 PO4 − ] + [HPO4 2− ] + [PO4 3− ].

(4.13)

Replacing the concentrations from 4.6, 4.8 and 4.10 into 4.12 and considering 4.13, r H3 P O4 may be obtained in terms of the constants and the pH alone. [ ] [ ]2 [ ]3 < > K 1 H+ + K 1 K 2 H+ + K 1 K 2 K 3 H+ r H3 P O4 = [ ]3 . [ ]2 [ ] H+ + K 1 H+ + K 1 K 2 H+ + K 1 K 2 K 3

(4.14)

However, in this case it is not necessary to go into much detail. The concentrations of [H2 PO4 − ] + 2 [HPO4 2− ] + 3 [PO4 3− ] in 4.11, from 4.12, are given by: [ ] [ ] [ ] < > H2 PO4 − + 2 HPO4 2− + 3 PO4 3− = rH3 PO4 CH3 PO4

(4.15)

Replacing Eqs. 4.2 and 4.15, into the electroneutrality condition (Eq. 4.11) including the corresponding dilution factors of the type V s /(V s + V t ), and solving for V t /V s = f , gives the titration fraction f : f H3 P O4

> < r H3 P O4 C H3 P O4 −[H+ ] + [OH− ] = . Cb + [H+ ]−[OH− ]

(4.16)

If we know the acid total concentration and dissociation constants, and the concentration of the base; we can calculate the course of the titration curve by giving different values to [H+ ]. The inverse problem can be solved numerically by applying the Newton–Raphson method [3]. Note that whatever the equilibrium constants are, we can obtain directly from the titration curve. In Fig. 4.2 we show the titration curve calculated for H3 PO4 with NaOH.

4.3 About the Additivity of Titration Curves When titrating complex systems such as polyampholytes (see below the titration of colloidal oxides), it is customary to carry out a titration of the blank solution (the solution without the substance to titrate; it could be just water or any other electrolyte solution), then to titrate the problem sample, (this is, the blank solution plus the substance to titrate), and, finally, to isolate the titration curve of the problem substance by making the corresponding differences, at constant pH, between volumes of titrant expended on the sample volumes and those expended on the blank solution.

4.3 About the Additivity of Titration Curves

41

14 12 10

pH

8 6 4 2 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

f Fig. 4.2 Simulated titration curve of H3 PO4 . C a = 0.01 M, C b = 0.01, K 1 = 5.9 × 10−3 , K 2 = 6.15 × 10−8 , K 3 = 4.8 × 10−13

To explore the validity of this procedure, we will consider the titration of a mixture of HCl, of concentration C HCl , plus H3 PO4 , of concentration C H3PO4 , in a volume V s . The titrant is a solution of a strong base such as NaOH, of concentration C b . We have already considered the titrations of the H3 PO4 and HCl alone in previous paragraph. We want to prove that the titration curve of the H3 PO4 alone can be obtained from the titration of the mixture and from that of HCl alone, by making the differences of the titration fractions at constant pH. The titration curve for the HCl alone is simply: f HCl =

CHCl −△ , Cb + △

(4.17)

where we have defined: ] [ ] [ △ = H+ − OH− .

(4.18)

The resulting titration curve was shown in Fig. 4.1. The titration curve of the mixture is obtained as follows. The electroneutrality condition is: [ +] [ +] [ −] [ −] [ ] [ ] [ ] H + Na = OH + Cl + H2 PO4 − + 2 HPO4 2− + 3 PO4 3− (4.19) The change of [Na+ ] during the titration is given by Eq. 4.2. [Na+ ] =

Cb Vt . (Vs + Vt )

(4.20)

42

4 Titration of Simple Substrates

The concentrations of [H2 PO4 − ] + 2 [HPO4 2− ] + 3 [PO4 3− ] are given by Eq. 4.12: [

H2 PO4

−

]

[

+ 2 HPO4

2−

]

[

+ 3 PO4

3−

]

> < r H3 P O4 C H3 P O4 Vs = , (Vs + Vt )

(4.21)

< > where r H3 P O4 is the average number of dissociated protons of H3 PO4 as defined before. Replacing in 4.19 and solving for V t /V s = f mix , it results: f mix

> < CHCl + rH3 PO4 CH3 PO4 −△ = . Cb + △

(4.22)

This is the titration curve for the mixture (see Fig. 4.3b). The titration curve for H3 PO4 alone is given by Eq. 4.16 and is plotted again in Fig. 4.3. It is clear that, if we make the differences between f mix and f HCl , at constant pH we obtain f H3 P O4 (Fig. 4.3d). Although we have considered a simple example the result is of general > < validity. If C HCl > △ and r H3 P O4 C H3 P O4 > △, it follows from Eqs. 4.16, 4.17 and 4.22 that: f H3 P O4 = f mix + f HCl

(4.23)

14 12

c,d

a

10

b

pH

8 6 4 2 0

0

1

2

3

4

f Fig. 4.3 Comparison of the calculated titration curves for, a HCl alone, b HCl + H3 PO4 , c H3 PO4 alone and d obtained by subtracting the volumes of (a)–(b)

4.5 A Simple Example of Titration with Complex Formation

43

4.4 Titration of an Arbitrary Mixture of Acids with an Arbitrary Mixture of Bases According to the procedure described above, we can write the electroneutrality condition in terms of the average fraction of dissociated protons and hydroxyls: ∑ [ +] ∑ < > [ ] H + Cbi = Caj raj + OH− i

j

(4.24)

where the subscripts ai and bj stand for the i-th acids and the j-th bases, respectively. is the average number of hydroxyls dissociated by the base j and for the average number of protons dissociated by the i-th acid. Solving, as before, for V t /V s , we obtain: < > ∑ Vt j C aj r aj −△ . (4.25) =∑ Vs i C bi + △ If the number of acids and bases together with their respective concentration and dissociation constants is known, we can calculate the titration curve. That is, V t /V s as a function of the pH. If we want to know the pH for a determined amount of added titrant, we must solve Eq. 4.25 numerically. Note that in this case it is not possible to determine separately the values and the values from a unique titration curve.

4.5 A Simple Example of Titration with Complex Formation To show an example let us consider the case in which we have a trivalent metal ion, M3+ , in the form of chloride, three complex species, MeL(n−m)+ , MeL2 (n−2m)+ and MeL3 (n−3m)+ , and the ligand salt is of the type, Na2 L, that dissociates without hydrolysis giving the ligand, L2− and Na+ ions. In this example m = + 3 and n = − 2. We will assume that we have a sample of the metal solution, of a chloride salt, MeCl3 , of concentration C M and a volume V s . This is titrated with the sodium salt of the ligand of concentration C L , by adding small volumes V t . Further we will assume the pH is kept constant by a suitable media. The titration is followed by measuring the concentration of free metal with a suitable electrode. Then we have: Me3+ + L2− ⇌ MeL+ kc1 =

[ML+ ] [M3+ ][L2− ]

(4.26) (4.27)

44

4 Titration of Simple Substrates

MeL+ + L2− ⇌ MeL2 − kc2 =

[ML2 − ] [ML+ ][L2− ]

MeL2 − + L2− ⇌ MeL3 3− kc3 =

[ML3 3− ] [ML2 − ][L2− ]

(4.28) (4.29) (4.30)

(4.31)

for the complexation reactions. As before, the starting point is the electroneutrality condition. Since we will consider the pH = 7 we will not take into account the H+ and OH− terms because they are equal and cancel each other. [ +] [ ] [ ] [ ] [ ] [ ] [ ] Na + 3 M3+ + ML+ − ML2 − + ML3 3− = 2 L2− + Cl−

(4.32)

Now we express the coefficients of the complexes in the following way: [ +] [ ] [ ] [ ] [ ] Na + 3 M3+ + (3 − 2) ML+ + (3 − 4) ML2 − + (3 − 6) ML3 3− [ ] [ ] = 2 L2− + Cl− (4.33) This can be written as: [ +] ([ ] [ ] [ ] [ ]) Na + 3 M3+ + ML+ + ML2 − + ML3 3− [ ] [ ] [ ] [ ] [ ] = 2 L2− + 2 ML+ + 4 ML2 − − 6 ML3 3− + Cl−

(4.34)

Now, according to the definition of the average ligand number, , is = number of bound ligands/total number of metal species =

[ML+ ] + 2[ML2 − ]+ 3[ML3 3− ] . [M3+ ]+[ML+ ]+[ML2 − ]+[ML3 3− ]

(4.35)

(4.36)

The underlined term of the left side of the Eq. 4.34 is the total metal concentration, C M , and the underlined term on the right is twice the numerator of Eq. 4.34. Therefore, those terms can be replaced by C M and by C M , respectively. Then, Eq. 4.34 becomes: [ +] ] [ Na + 3 CM = 2 L2− + 2CM + 3 CM

(4.37)

4.5 A Simple Example of Titration with Complex Formation

Introducing the dilution factor,

1 , (Vs +Vt )

45

it results:

] [ Vt 2C L Vt = 2 L2− + 2CM , (Vs + Vt ) (Vs + Vt )

(4.38)

solving for V t /V s = f, gives the titration fraction f: [ ] CM + L− [ ] . f = CL − L−

(4.39)

Finally, the total metal concentration is given by: [ ] [ ] [ ] [ ] CM = M3+ + ML+ + ML2 − + ML3 3− ]( [ ] [ ]2 [ ]3 ). [ = M3+ 1 + kc1 L2 − + kc1 kc2 L2 − + kc1 kc2 kc3 L2 −

(4.40)

From which [M3+ ] could be evaluated. Defining pM = − log[M3+ ], it is shown as a function of the titration fraction Fig. 4.4. At a first glance it may appear surprising to see a curve with only one step in Fig. 4.4. The reason is that being k c3 much higher than the others the complex MeL3 3− predominates. Would have they been in the opposite sequence, for instance, k c1 = 108 , k c2 = 106.5 , k c3 = 102 , three steps would have been observed. 20 16

pM

12 8 4 0 -2

0

2

4

6

8

10

f Fig. 4.4 Results for a simulated titration curve with complex formation according to Eq. 4.24. Calculated titration curve for complexometric titration of a metal capable of forming three complexes. CM = 0.05, CL = 0.05, k c1 = 107.51 , k c2 = 1013.86 , k c3 = 1018.28

46

4 Titration of Simple Substrates 12 10

pH

8 6 4 2 0 1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

f Fig. 4.5 Calculated titration curve for p-aminophenol (pK a = 4.77, pK b = 9.20). C HCl = 0.1, C amph = 0.01, C b = 0.05, V s = 20 mL)

4.6 Titration of a Simple Ampholyte If we dissolve a neutral ampholyte, by neutral we mean that neither the basic nor the acid groups are ionized, the pH of the solution should correspond to the isoelectric point. To avoid to make two titration curves, one with the acid and the other with the base, it is customary to add, say, a strong acid to the sample in order to get a pH of, say, 1 or 2. So, if C HCl is the concentration of the added acid, C amph , the concentration of the ampholyte, V s , the volume of the sample, and C b and V b the concentration and the added volume of the base, respectively, then, the titration fraction results: f =

CHCl + ( − )Camph − △ . Cb + △

(4.41)

That is from the titration curve we can only obtain − , the difference between the average number of ionized acid groups minus the average number of ionized basic groups. Only in the case the corresponding ionization constants are far apart they could be determined separately. A simulated titration curve for p-oaminophenol is shown in Fig. 4.5.

4.7 Redox Titrations Much the same as in an acid–base titration in which we add a substance capable of subtract protons, in order to titrate an Ox species, we must add a substance capable of donating electrons to it. Therefore, redox titrations consist in adding a suitable

4.7 Redox Titrations

47

reducing agent to an oxidized analyte or an oxidizing agent to a reduced analyte. The extend of the redox reaction is followed by measuring the electromotive force (emf ) of a galvanic cell formed by one electrode that contains the analyte and the titrating agent and a reference electrode. We will call the electrode containing the analyte and the titrating solution the working or indicator electrode. The electrode potential of the redox couple of the species which is going to act as the titrant must be more positive than the electrode potential of the Ox species to be titrated. The converse would be true if we were to titrate a Red species with an Ox species. For the sake of simplicity, we will consider that we employ a reference electrode that is a standard hydrogen electrode. Both electrodes are connected by a salt bridge, and it is assumed that the salt bridge completely suppress contact potentials between the solutions of both electrodes. Let us consider, for the sake of clarity, that the reduced species is Fe2+ and that the titrating agent is Ce4+ . As it is customary, and for the sake of simplicity, we will employ, instead of the activities, the concentration of the ionic species involved in the reaction. As pointed out in Chap. 1 this is not correct from a thermodynamic viewpoint. Despite of this, here we will employ the thermodynamic standard potentials, E. At the working electrode compartment there occurs a chemical reaction: Fe2+ + Ce4+ ⇌ Ce3+ + Fe3+ .

(4.42)

After each addition of the titrating reactant this reaction reaches a state of chemical equilibrium. We also assume here that the reaction is fast enough to being capable of reaching equilibrium in a reasonable time. That is △G = 0. [[ ][ ] ] Ce3+ e Fe3+ e △G = △G + RT ln [ 2+ ] [ 4+ ] = 0. Fe e Ce e o

(4.43)

Or, what is the same: [[ ][ ] ] Ce3+ e Fe3+ e −RT ln K = △G = RT ln [ 2+ ] [ 4+ ] , Fe e Ce e o

(4.44)

where [Fe2+ ]e , etc., are the concentration of products and reactants at equilibrium. Note that from the first equality of Eq. 4.44, considering △G = − nFE, it can be written as: K = nF where n = 1 in this case.

o o − E Ce ) (E Fe , RT

(4.45)

48

4 Titration of Simple Substrates

On the other hand, the reaction at the galvanic cell is: 1/2H2 (g) + 1/2Fe2 (SO4 )3 (aq) + Ce(SO4 )2 (aq) ⇌ 1/2Ce2 (SO4 )3 (aq) + FeSO4 (aq) + 1/2H2 SO4 (aq)

(4.46)

Note that the concentrations are those corresponding to the equilibrium of reaction 4.46. The emf of this galvanic cell is: RT ln E=E − F

(

1/2

)

1/2

aFe(SO4 ) aCe2 (SO4 )3 aH2 SO4

o

1/2

aFe2 (SO4 )3 aCe(SO4 )2 ( pH2 / p o )1/2

,

(4.47)

where the a are the activities, E o = E o Fe + E o Ce and we have assumed that we can employ the pressure instead the fugacity. Since we have supposed the hydrogen electrode is the SHE, aH2 SO4 = 1 and pH2 = p o = 1. In those conditions we may write the emf of the cell as: RT E = Eo − ln F

(

1/2

CFe(SO4 ) CCe2 (SO4 )3

) .

1/2

CFe2 (SO4 )3 CCe(SO4 )2

(4.48)

Let us consider the example of the titration of a volume V s of a solution that has Fe+2 in concentration C 0 Fe2+ with a solution containing Ce+4 of concentration C 0 Ce4+ . Then each aggregate of V t of Ce4+ will make that an amount α C 0 Ce4+ of Ce3+ and Fe3+ are formed and amounts of (1 − α)C 0 Ce4+ and C 0 Fe2+ − (1 − α)C 0 Ce4+ remains at equilibrium. Thus the concentration of reactants and products at equilibrium, taking into account the dilution factors, are: 0 [Ce3+ ]e = αCCe 4+

Vt , Vs + Vt

(4.49)

0 [Fe3+ ]e = αCCe 4+

Vs , Vs + Vt

(4.50)

[Fe2+ ]e =

0 0 αCFe V 2+ Vs − (1 − α)C Ce4+ t

Vs + Vt

0 [Ce4+ ]e = (1 − α)CCe 4+

,

Vt . Vs + Vt

(4.51) (4.52)

Replacing these concentrations into Eq. (4.44), α can be determined for each addition of the titrating agent and thus for all the concentrations at equilibrium. Next, we will analyse what happens in the galvanic cell. Actually, there are two cells in one. In one cell the reaction is: 1/2H2 (g) + Fe3+ (aq) → Fe2+ (aq) + H+ (aq)

(4.53)

4.7 Redox Titrations

49

whereas the reaction in the other cell is: 1/2H2 (g) + Ce4+ (aq) → Ce3+ (aq) + H+ (aq)

(4.54)

Because we have assumed that the reference electrode is the SHE, the activity of H+ is equal to unity and the partial pressure of H2 (g) is also equal to unity. Then the emf in the first cell is: E Fe =

o E Fe

( 2+ ) [Fe ] RT , ln − F [Fe3+ ]

(4.55)

whereas the emf at the second cell is: E Ce =

o E Ce

( ) [Ce3+ ] RT ln . − F [Ce4+ ]

(4.56)

It should be noted that the number of electrons exchanged in each galvanic cell is nFe = 1 and nCe = 1. Actually both reduction reactions occur on the same inert metallic electrode as symbolized in the following reaction: H2 (g) + Fe3+ (aq) + Ce4+ (aq) → Fe2+ (aq) + Ce3+ (aq) + 2H+ (aq)

(4.57)

It can be noted that the total number of electrons exchanged in reaction (4.42) is now nt = 2. The total free energy change in reaction (4.42), △G, is: n t △G = n Fe △G Fe + n Ce △G Ce

(4.58)

Replacing the values of △Gi by − nFE i , it results: n t E = n Fe E Fe + n Ce E Ce

(4.59)

Since these two cells work at the same electrode, their emf values must be added. Then: ( 2+ ) [Fe ][Ce3+ ] RT o o ln . (4.60) 2E = E Fe + E Ce − F [Fe3+ ][Ce4+ ] And finally, it results: ( 2+ ) o o RT [Fe ][Ce3+ ] + E Ce E Fe − ln . E= 2 2F [Fe3+ ][Ce4+ ]

(4.61)

50

4 Titration of Simple Substrates 2.0

E/V

1.5

1.0

0.5

0.0 0

10

20

30

40

Vt / ml Fig. 4.6 Calculated potentiometric titration curve for 20 cm3 of FeSO4 0.01 M titrated with a Ce(SO4 )2 0.01 M solution

As the concentrations at equilibrium can be determined from the knowledge of α, the initial concentrations and the volume of the sample; then E can be calculated as a function of the added volume of titrant. This is shown in Fig. 4.6 for the system just discussed.

4.8 Electron Titration The problem can be stated as follows: A volume V s of solution contains a concentration C 0 ox of the oxidized species of a redox couple capable of undergoing the following half-reaction: Ox + e− ↔ Red

(4.62)

we want to titrate the species Ox with electrons provided by an inert electrode. To make reaction 1 to occur, we have to force the circulation of a cathodic current in the external circuit. We will assume that the charge transfer reaction is fast enough for the fraction of the reaction (4.62) to be completed at that potential and considered at equilibrium. Then the Nernst equation is applicable. ( ) RT COx ln . E=E + nF CRed o

(4.63)

We can measure this potential employing a suitable galvanic cell. We will make the half-reaction to occur by imposing a cathodic current, I, on the inert electrode.

4.8 Electron Titration

51

During the course of the titration C ox will diminish and C red will increase and these changes will be reflected by the potential, E. As a charge q has circulated through the cell we have converted an amount of Ox into Red, say △C red , that is given by the Faraday law: q = n F△Cred Vs

(4.64)

Clearly △C red = − △C ox . To completely convert Ox in Red we need to pass a charge, which according to Faraday’s law, is: qmax = n FCox V

(4.65)

So we see that q here plays the role of the added titrant volume and the ratio q/qmax means the titration fraction. During the course of the titration the concentration of Ox is C 0 ox − △C ox and that of C red = − △C red . Replacing them into the Nernst equation (Eq. 4.63) we get the change in the potential during the titration as: E = Eo +

) ( COx − △COx RT , ln nF △CRed

(4.66)

which after replacement of the concentrations in terms of the charges we get: E = Eo +

[ ] 1 − (q/qmax ) RT ln . nF (q/qmax )

(4.67)

A representation of q/qmax as a function of E is shown in Fig. 4.7. 1.2 1.0

q/q max

0.8 0.6 0.4 0.2 0.0

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

E/V Fig. 4.7 Fraction of reactant titrated, q/qmax as a function of the electrode potential. E o = 0.0 V

52

4 Titration of Simple Substrates

The charge is obtained measuring the time of flowing a constant current flowing through the cell. It is advisable to employ a reference electrode to control the potential in order to see if the current is due to the process we want to measure. If the reactants are soluble it is convenient to employ a thin layer cell (TLC) [4], so to use small volumes of solution.

4.9 Titration of Zwitterions In this paragraph we will consider the titration of both a zwitterion alone, the comparison with the titration of the hypothetical neutral form alone and the titration of the neutral form in equilibrium with the zwitterion form. As example we will consider the case of glycine already discussed in Chap. 2.

4.9.1 The Neutral Form Alone Let us consider now a titration of V a ml of a solution of H2 NCH2 COOH of concentration C a with a NaOH solution of concentration C b . At any point of the titration is valid the electroneutrality condition [

] [ ] [ ] [ ] ] [ Na+ + H+ + H3 N+ CH2 COOH = H3 N+ CH2 COO− + OH−

(4.68)

Defining [Na+ ] =

Cb Vt . (Vs + Vt )

(4.69)

Employing the expressions of and (Eqs. 2.42 and 2.43), and defining △ = [H+ ] − [OH− ]. fs =

Vt Ca (1 + − ) − △ . = Vs Cb + △

(4.70)

4.9.2 The Zwitterion Form Alone Similarly, in this case we obtain (see Eqs. 2.41 and 2.44): fz =

Vt Ca (1 + − ) − △ . = Vs Cb + △

(4.71)

4.9 Titration of Zwitterions

53

Fig. 4.8 Simulated titration curves for the case of the neutral form alone (black) and the zwitterion alone (dash). Data for glycine (see Chap. 3)

16 14 12

pH

10 8 6 4 2 0 -1

0

1

2

3

4

titration fraction, f

The corresponding titration curves for f s and f z are shown in Fig. 4.8.

4.9.3 The Two Forms in Equilibrium This case is a continuation of that considered at Sect. 3.2. The equation > < > is equal < basic to Eq. (4.71), but we have to replace and by vHg and rHg from Eqs. (3.63) and (3.64). It can be seen in Fig. 4.9 that in the two cases the TC is the same, as advanced in Chap. 2. For this reason, it is only possible to obtain the macroconstants from a titration curve alone. 4 3 2

pH

Fig. 4.9 Simulated titration curves for the case of zwitterion alone (continuous line) and the two forms in equilibrium (dashed line). Data for glycine (see Chap. 3.2)

1 0 -1 0

2

4

6

8

10

titration fraction, f

12

14

16

54

4 Titration of Simple Substrates

References 1. 2. 3. 4.

Skoog DA, West DM, Holler FJ (1995) Analytical chemistry, 6th edn. McGraw Hill, New York Harris DC (2015) Quantitative chemical analysis, 9th edn. WH Freeman, New York de Levie R (1999) A general simulator for acid–base titrations. J Chem Educ 76:987–991 Bard AJ, Faulkner LF (1980) Electrochemical methods. Wiley, New York

Chapter 5

Continuous Distribution Functions. Cumulative and Density Distribution Functions. Known Examples

5.1 Introduction In the forthcoming chapters we will see that the binding energy, and consequently the binding constant, depends on the extension of the binding. We say that any of these quantities is distributed among the different sites. Therefore, it seems convenient to briefly discuss the nature and types of these distributions. At the end of the chapter we will see more about the connection between mathematical distribution functions and energy distribution functions.

5.2 Continuous Distribution Functions Cumulative and Density Probability Distribution Functions [1, 2] A distribution function is a mathematical way of describing how a property is distributed over a certain interval of the variable on which it depends. According to the nature of the problem, there may be discrete or continuous distribution functions. In our case, since we will consider materials with a big number of binding sites, we will be concerned with continuous distribution functions. Distribution functions are employed in the mathematical theory of probabilities. Since these are of little interest here, we will present a very simple approach to these functions. The interested reader may consult this topic in more advanced textbooks. To introduce the concept of distribution functions let us consider a metal bar of length l and cross section A. Let us further assume that the density of the material, ρ, is uniform. One of the ends of the bar is at the origin of the coordinate system as shown in Fig. 5.1.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. A. Marmisollé and D. Posadas, Binding Phenomena, Physical Chemistry in Action, https://doi.org/10.1007/978-3-031-39736-3_5

55

56

5 Continuous Distribution Functions. Cumulative and Density …

Fig. 5.1 Metal bar of length l and cross section A

We are interested in a function that gives the amount of mass, m, in the differential interval comprised between x 0 and x 0 + dx, that is the differential of mass dm(x) (shadowed part in Fig. 5.1). Since the density of the material, ρ, is constant, dm(x), in this case, is: dm (x) = ρ A dx = f (x) dx,

(5.1)

1.2

1.2

1.0

1.0

0.8

0.8

0.6

0.6

F(x)

f(x)

where f (x) = dm(x)/dx is the density function and represents the differential mass in a length interval dx. Its representation as a function of x is shown in Fig. 5.2. f (x) is, in this case, the so-called uniform density function. Strictly speaking from a mathematical viewpoint, a continuous density function is not defined at one point but in a neighbourhood of measure not null. For this reason, a density function has physical meaning only inside an integral. Even though, we will describe some density functions. Very often, loosely speaking, we will employ the word distribution to design the fraction of the total of a particular property in one interval of the variables on which that property depends. We will show the distribution of a chemical species as a function of the pH. In the standard textbooks of physical chemistry, we are shown the Maxwell–Boltzmann molecular velocities’ probability distribution in an ideal gas, we employ the Gauss distribution or the Students’ t distribution function to obtain the accidental error of an ensemble of experimental data and so on. These also are density functions.

0.4

0.4

0.2

0.2

0.0

0.0

-1.0

-0.5

0.0

x

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

x

Fig. 5.2 Plot of the uniform density, f (x) (left) and cumulative, F(x) (right) distribution functions defined by Eqs. 5.2 and 5.3

5.2 Continuous Distribution Functions Cumulative and Density Probability …

57

Probability density functions are normalized to unity in the interval of interest, say (a, b): {b f (x) dx = 1.

(5.2)

a

It is sometimes convenient to define a cumulative distribution function, F(x), which is the amount of the given property (probability, mass, etc.) from the beginning of the x interval, say x = a, up to a particular point x. This is done by integrating: {x F(x) =

f (x) dx.

(5.3)

a

In the case of the example of Fig. 5.1 the function F(x) would give the total mass of the bar from the origin up to the point x. Next, we will discuss some density functions that are related to what we are discussing here.

5.2.1 Uniform Density Function This is the one we just discussed in the example of Fig. 5.1. It is defined as: f (x) = c for a < x < b,

(5.4)

where c is a constant and the cumulative function as: F(x) = c (x − a).

(5.5)

5.2.2 Dirac Delta Distribution Dirac delta distribution represents the unity of some magnitude concentrated in a point. It has the following properties [3]. δ(x) = 0 δ(x) = ∞

for x /= 0 for x = 0

58

5 Continuous Distribution Functions. Cumulative and Density …

Fig. 5.3 Dirac delta distribution, δ(x) (left) and its cumulative, H(x), the Heaviside function (right)

so that: { H (x) =

δ(x) dx = 1 for x ≥ 0

(5.6)

or H (x) = 0 for x < 0.

(5.7)

The D(x) function is also called Heaviside function. Heaviside function is the formal integral of the Dirac delta function. It is represented in Fig. 5.3. In Fig. 5.3, △x tends to zero and the highness, △y, tends to infinity in a way that Eq. 5.6 is fulfilled. The corresponding cumulative function, H(x), is a step function. If we consider a unity mass concentrated at the origin, the cumulative function would be H(x) = 0 between − ∞ < x < 0 and 1 from de x ≥ 0, H(x) = 1 (Fig. 5.3). An important property of the δ(x) function is that, if g(x) is a function continuous at the origin: {∞ δ(x) g(x)dx = g(0).

(5.8)

−∞

The Dirac delta distribution could also be cantered at x 0 by writing δ(x − x0 ) so that now {∞ δ(x − x0 ) g(x)dx = g(x0 ). −∞

(5.9)

5.3 The Relation Between Binding Problems and Distribution Functions

59 1.2

0.006

1.0 0.8 0.6 0.002

0.4

G(x)

g(x)

0.004

0.2

0.000

0.0 -6

-4

-2

0

2

4

6

-6

-4

-2

x

0

2

4

6

x

Fig. 5.4 Gaussian density, g(x) (left) and cumulative, G(x) distribution functions (right)

5.2.3 Gaussian Function This is quite well known to all chemists. The density function is defined by: ] [ (x − μ)2 , g(x) = (2π )−1/2 σ −1 exp − 2σ 2

(5.10)

μ = .

(5.11)

< > (x − μ)2 = σ 2

(5.12)

where

is the mean value of x, and,

is the mean square deviation of the random variable x. In Fig. 5.4 the density function (Eq. 5.10) and the corresponding cumulative function, G(x), are shown. In the forthcoming chapters we will consider other, different, distribution functions.

5.3 The Relation Between Binding Problems and Distribution Functions As a leading case of binding we will consider the problem of adsorption on a surface. Let us consider a heterogeneous surface. Because the heterogeneity that comes from a rough surface is clear there will be not a unique value of the binding energy but rather there will be a distribution of them. The surface energy will be distributed

60

5 Continuous Distribution Functions. Cumulative and Density …

around certain value. Real adsorbents have very often these type of surfaces. Many times they are represented by the Gaussian distribution for the adsorption energy. Let us consider the distribution of binding energy in the Langmuir model. For doing this, let us firstly consider a particular heterogeneous substrate with many adsorption sites. In this substrate the binding energy varies according to the different sites, leading to different binding constants for different sites. However, the binding for each type of site (sites with similar energy) can be described by a Langmuir isotherm. Let g(ka ) be the fraction of binding sites with binding constant value between k a and k a + dk a . The coverage of any type of sites can be computed by a local isotherm (a function that is valid for this type of sites). In this case, we assume that the local isotherm for any type of sites is a Langmuir isotherm: θ (ka ) =

ka [B] . 1 + ka [B]

(5.13)

The function g(ka ) is the distribution of adsorption constants for this substrate. Then, the total coverage can be obtained by integration over all the possible values of k a : { ka [B] dka . (5.14) θ = g(ka ) 1 + ka [B] k

In the case of a homogenous substrate, all the sites have the same constant (k a 0 ) (same energy) and the Langmuir isotherm for the total coverage results: θ=

ka 0 [B] . 1 + ka 0 [B]

(5.15)

From (5.13), this is compatible with the distribution of adsorption constants being a Dirac delta distribution cantered at k a 0 . g(ka ) = δ(ka − ka 0 ).

(5.16)

We will see later on that very often the equilibrium binding constant may be obtained from the analysis of the titration curve and therefore the dependence with the coverage. Also we will see that different dependences of the binding constant (sometimes in the form of apparent pK a values or apparent formal redox potentials) as a function of the coverage have been obtained. For instance, it is quite common to find a linear dependence of ln K (or equivalently, the apparent pK a [4], or the apparent formal redox potential [5]) on the coverage of the type shown in Fig. 5.2 for F(x), at least in a wide range of coverage. In forthcoming chapters, we will consider other, theoretical and experimental distribution functions.

References

61

References 1. Weisstein EW (n.d.) Distribution function. From MathWorld, a Wolfram Web Resource. https:// mathworld.wolfram.com/DistributionFunction.html 2. Kemeny JG, Mirkil H, Snell JL, Thompson GL (1959) Finite mathematical structures. Prentice Hall, New Jersey 3. Levine IN (1974) Quantum chemistry. Allyn and Bacon, Boston 4. Marmisollé WA, Florit MI, Posadas D (2014) Acid–base equilibrium in conducting polymers. The case of reduced polyaniline. J Electroanal Chem 734:10 5. Marmisollé WA, Florit MI, Posadas D (2013) Coupling between proton binding and redox potential in electrochemically active macromolecules. The example of Polyaniline. J Electroanal Chem 707:43

Chapter 6

Elements of Adsorption on Heterogeneous Substrates

6.1 Introduction In order to interpret the TCs in complex systems, it is necessary to work out first a number of matters. Among them there is the question of the different types of binding. Considering the type of bonding between the adsorbate and the atoms of the substrate we may differentiate those in which the forces involved in bonding are of the van der Waals type and those in which a true chemical bond is established. The first type is usually called physical adsorption whereas the second chemical adsorption. Several differences exist between the two cases: in the first case the adsorption extent increases as the temperature decreases, while in the second it increases as the temperature increases. The second type of adsorption involves much higher adsorption energies than the first type. Physical adsorption allows the adsorbate to move more or less freely on the surface, while in the second type, the adsorbate particles are fixed in one position. Due to the wide range of substrates involved, such as macromolecules, oxides, proteins and humic substances, the subject of adsorption is vast. Therefore, we will attempt to limit our discussion to the scope of this book. Our initial focus will be on adsorption on heterogeneous substrates and special isotherms. Heterogeneous substrates refer to cases in which the adsorption energy varies depending on the coverage fraction. These concepts are applicable to other types of binding, as described in Chap. 1. Keeping in mind that we are using small k to describe association reactions and capital K for desorption or dissociation reactions, although in this chapter, we will also use k B to represent the Boltzmann constant.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. A. Marmisollé and D. Posadas, Binding Phenomena, Physical Chemistry in Action, https://doi.org/10.1007/978-3-031-39736-3_6

63

64

6 Elements of Adsorption on Heterogeneous Substrates

6.2 Adsorption on Heterogeneous Substrates. The Distribution Function for the Adsorption Energy At the introduction we defined heterogeneous substrates on as those whose binding energy depends on the coverage degree. Now we would like to explore the different systems and causes. We will call intrinsic heterogeneity as that due to the atomic irregularities and induced heterogeneity as that caused by the presence of the adsorbate (for instance interactions among the adsorbed species). This latter is the more relevant to the type of systems considered here, mainly lateral interactions among the bound species. It is interesting to note that the form of the isotherm in the presence of heterogeneity is similar no matter its origin. Previously, we have defined an ideal surface as that in which its adsorption energy is uniform in all its extension (homogenous substrate from the point of view of binding). Although this is a rather idealized concept, we may approach an ideal surface as a cleaved or cut plane of a single crystal. Also different planes of some metals single crystals may be prepared, although actually they are not completely free of atomic irregularities. Thus, we could speak of Au (111) or Au (110) and so on. In general, real surfaces have atomic and other defects that make its adsorption energy to change from site to site. Let us now consider an adsorption process on a real surface where there is a distribution of adsorption energies, E a , or what is equivalent, a distribution of binding constants. As we defined in a previous chapter, let g(k a ) be the distribution of adsorption constants and let us consider the coverage on the type of sites having a constant value around k a . If we call f (k a , ai , T ) to the local isotherm of this sites (being ai the activity of adsorbate in the external phase and T the temperature), the coverage of this type of sites becomes: θi (ka , ai , T ) = g(ka ) f (ka , ai , T ).

(6.1)

Note that in Chap. 5 we have used a Langmuir isotherm instead of the more general expression f (k a , ai , T ). Obviously, the total coverage, θ (ai , T ), will be the integral over all the binding energies, that is: { θ (ai , T ) =

g(ka ) f (ka , ai , T )dka ,

(6.2)

where the integration includes the whole range of adsorption constants for all the sites in the substrate. The experimentally accessible quantity is θ (ai , T ), and from it we would be need to calculate g(k a ) and f (k a , ai , T ) from Eq. (6.2). Unfortunately, this is an impossible task unless some simplifying assumptions are made. For instance, Ross and Olivier [1] assumed a Gaussian distribution for g (ln k a ). Other approaches have been considered (see, for instance: [2]). Also, many models have been developed by researchers working in the field of soil colloids (see for instance [3], and references therein).

6.4 Statistical Deduction of the Langmuir Isotherm

65

It is important to note that the local adsorption energy can change depending on the coverage of the binding species. In the case of ion binding, for instance, there may be repulsive forces between the bound species, resulting in a smaller adsorption energy at certain coverages compared to the naked substrate (i.e. all sites empty). We would like to emphasize that varying binding energies correspond to different binding constants, meaning that a distribution of binding energies is equivalent to a distribution of binding constants. This concept will be applied in the forthcoming chapters.

6.3 Theoretical Binding Isotherms in the Presence of Interaction Between the Bound Species The problem at hand is a part of what is commonly known as lattice statistics, which can be further explored in the works of Hill [4] (Chaps. 7 and 14), or Clark [5]. Binding isotherms refer to the equilibrium between the binding species in the external phase and empty substrate sites, which may be a solid surface or a polymer chain. In this section, we will assume that the external phase is a dilute gas and that the concentration variable is the gas pressure. However, if the external phase were a solution, we would employ the activity or concentration as the concentration variable. Lattice statistics simplifies the picture by treating the binding sites on a solid as nodes arranged in an ideal geometrical lattice. The lattice can have one, two or three dimensions, with a one-dimensional lattice being a linear array of nodes, which may or may not be equally spaced depending on the chosen model for the lattice. In the most commonly used scheme, the adsorbed particles interact with their closest neighbours, although more complex interaction schemes may be chosen. For a twodimensional lattice, the nodes can be arranged in a triangular, square, hexagonal or other pattern. Interestingly, in the field of physical adsorption, deviations from ideal models are accounted for by modifying the mathematical form of the binding isotherm, rather than by introducing an activity factor for the bound species. As a result, there are a variety of binding isotherms, many of which are listed in the book by Ross and Olivier [1] (also see Adamson [6]).

6.4 Statistical Deduction of the Langmuir Isotherm Let us consider a two-dimensional square lattice with M nodes, each of which can potentially bind a molecule. There are N indistinguishable, motionless, bound molecules, and the number of nodes and molecules are statistically independent.

66

6 Elements of Adsorption on Heterogeneous Substrates

In this problem, the number of nodes plays a similar role to volume in a threedimensional problem. If each node occupies an area α, then the occupied area is A. To count the number of possible states obtained by placing N molecules on M nodes, we can start with the fact that the first molecule can bind in M different ways, the second in M − 1 ways (since it cannot occupy the same node as the first molecule), and so on, until the Nth molecule, which can bind in M − (N − 1) ways. Thus, there are M!/(M − N)! possible configurations. However, exchanging the molecules among each other does not give a new configuration, so we must divide by N! to account for the N! equivalent ways of arranging N indistinguishable objects. Therefore, the number of configurations (states) is g(N, M) = M!/(M − N)!N!. The partition function can then be calculated as follows: ( Q=

) M! q N q (M−N) , (M − N )!N ! N M

(6.3)

where qN and qM are the partition functions for an occupied and a free site, respectively (usually it is considered qM = 1). The chemical potential is that given by Eq. (13.49); in this case also keeping M constant. Employing the Stirling approximation (ln N! ≈ N ln N − N) and carrying out the derivative, the chemical potential results: { μad = kB T ln

} N /M . [1 − (N /M)]qN

(6.4)

Defining the degree of coverage, θ = N/M, it results: ] θ . μad = kB T ln (1 − θ )qN [

(6.5)

Let us consider that the adsorbed gas is in equilibrium with an external gas at pressure p. The condition of equilibrium reads: μad = μgas .

(6.6)

Or, what is the same [ μad = kB T ln

] ) ( θ = μ0gas + kB T ln p/ p 0 , (1 − θ )qN

(6.7)

where p is the pressure, assuming the external phase is a dilute gas. We will use the definition of the absolute activity λ so that μ = k B T ln λ. Let us define the standard

6.4 Statistical Deduction of the Langmuir Isotherm

67

state of the adsorbed gas as θ = 0.5 in the absence of interactions. This is usual in the case of immobile adsorption. Then, the standard chemical potential results: μad 0 = −kB T ln qN .

(6.8)

So that (

μad = μad

0

θ + kB T ln 1−θ

)

) ( = μgas 0 + kB T ln p/ p 0

(6.9)

Or ) ( ( ( ) ) μ0 − μgas 0 θ = p/ p 0 exp − ad = k p/ p 0 , 1−θ kB T

(6.10)

where we have defined ) ( μ0ad − μgas 0 . k = exp − kB T

(6.11)

Note that all the thermodynamic information is included in the binding constant. Equation 6.10 is the Langmuir adsorption isotherm. In general, we will write the adsorption isotherms as: ( y = qλ =

) θ . 1−θ

(6.12)

( ) Note that in this case, qλ = k p/ p 0 . Equation (6.9) can be integrated to obtain the corresponding change in the Helmholtz free energy, △Aad . { [ △Aad = kB T M ln(1 − θ ) + θ ln

θ q(1 − θ )

]} ,

(6.13)

where M is the total number of binding sites and θ = N/M, the fraction of bound sites, N being the total number of occupied sites. For the Langmuir case the two-dimensional pressure, π (see Sect. 13.5), is: (

π ┌max

∂ ln Q = kB T ∂M

) = −kB T ln(1 − θ ). N

(6.14)

68

6 Elements of Adsorption on Heterogeneous Substrates

6.5 The Ising Model The Ising problem refers to the study of the behaviour of interacting spins on a lattice [4, 5]. In two or more dimensions, the problem becomes difficult to solve exactly, except for the special case of equal probabilities for the spins to point up or down (i.e. θ = 0.5), known as the critical point. However, in one dimension, the Ising problem can be solved exactly. The Ising model for a linear lattice is the simplest model of interacting spins, and it has been extensively studied. The exact solution for this model was first found by Ernst Ising in 1925. When interactions are present, the binding constant of a system becomes dependent on the bound fraction, giving rise to the appearance of a distribution of binding constants. This means that the probability of a molecule binding to a specific site on the lattice is not constant, but varies according to the distribution of binding constants. The presence of a distribution of binding constants makes the analysis of the system more complex. Despite the difficulty of solving the Ising problem for lattices of dimension two or higher, the Ising model remains an important and widely used model in statistical physics and condensed matter physics, as it provides insights into the behaviour of systems with interacting particles. The Ising problem as applied to adsorption may be stated as follows: Consider a linear chain of M nodes all at the same distance. On these nodes there are N particles adsorbed at random (N < M). Now we are interested in the number of pairs N 11 in which there are two contiguous sites both occupied, the number of pairs N 01 in which there is one occupied and one unoccupied and the number of pairs N 00 in which no sites are unoccupied. These quantities may be calculated by drawing lines from an occupied site towards its neighbours. If two neighbour sites are occupied there will be a double line; if there are one occupied and one free, only one line. We may write the following equations: 2N = 2N11 + N01 ,

(6.15)

2(M − N ) = 2N00 + N01 .

(6.16)

This shows that of the five variables (M, N, N 00 , N 01 and N 11 ), only three are independent. The total potential energy of the system W, is given by: W = N11 w = (N − N01 /2)w,

(6.17)

where w is the interaction energy between a single pair of occupied sites. The partition function is now: Q(N , M) = SN01 q N g(N , M, N01 ) exp(−N01 w/kB T ).

(6.18)

6.5 The Ising Model

69

There are several methods to find the partition function. Hill [4] employs the maximum term of the sum, which results . The adsorption isotherm for the one-dimensional case results: [( ln y = ln

1−θ θ

)(

θ −α 1−θ −α

)] ,

(6.19)

where: ) ( w , y = qλ exp − kB T

(6.20)

where, as said before, λ is the absolute activity, which is linked to the chemical potential. Using (6.7) it is easy to see that for a gas without interactions (w = 0), y = k (p/p0 ), and α=

2θ (1 − θ ) = , 2M β +1

(6.21)

where is the most probable number of neighbours, one empty and one occupied. The parameter β is given by: )]}1/2 { [ ( −W β = 1 − 4θ (1 − θ ) 1 − exp . kB T

(6.22)

Eliminating α between (6.19) and (6.21), it results: ) β − 1 + 2θ . ln y = ln β + 1 − 2θ (

(6.23)

The calculated values of the one-dimensional Ising isotherm are shown as θ versus ln y, in Fig. 6.1, for different values of w. For w = 0 the Ising isotherm reduces to the Langmuir isotherm. Although y is proportional to the pressure, it is interesting to plot θ as a function of (kp) in the same way as plotted in Fig. 2.3 (bottom). This plot is shown in Fig. 6.2. In this case of repulsive forces, it is seen that a higher concentration is needed to reach the same value of coverage. It is interesting to enquire how changes α with θ for the settings of Fig. 6.2. This is shown in Fig. 6.3. It is seen in Fig. 6.3 that the number of 01 pairs increases as the occupancy increases. It is reasonable to expect that the number of pairs 01 increases with θ. However, at θ = 0.5, the parameter α starts decreasing because the number of pairs starts increasing. While our present model only considers pair interactions, it is worth noting that we could extend it to include triplets, quartets and beyond. However, it’s important to

70

6 Elements of Adsorption on Heterogeneous Substrates 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -10

-8

-6

-4

-2

0

2

4

6

8

10

ln y Fig. 6.1 Representation of the Ising isotherm for a one-dimensional lattice (6.23), for different values of the interaction parameter; w/k B T = 0 (continuous line), 2 (short-dashed line), −4 (longdashed line)

Fig. 6.2 Representation of the Ising isotherm for a one-dimensional lattice, for different values of the interaction parameter, w/k B T = 1.5 (continuous line), 0 (dashed line)

keep in mind that the energy involved in these higher-order interactions is typically very small and they tend to form only at relatively high coverages. In fact, going beyond triplets is often unnecessary for most practical purposes. That being said, it is worth mentioning that the formation of triplets has been invoked to explain the titration curves of certain polybases, which we will explore further in Chap. 9. In the context of biochemistry literature, the presence of interactions is often referred to as cooperativity [7–9]. However, it’s worth noting that this term has a broader meaning in biochemistry, and for the sake of specificity, we will continue

6.5 The Ising Model

71

Fig. 6.3 Values of α as a function of θ according to Eq. (6.19) for the parameters of Fig. 6.2, w/k B T = 1.5 (continuous line), 0 (dashed line)

to use the term “interactions” in our discussion. While both terms convey a similar idea, we believe that using “interactions” will help us to focus more specifically on the type of phenomena we’re exploring here. In Chap. 8 we will apply the Ising model to the titration of polyvinylamine. To that end is convenient to employ a slightly different notation. As the titration is considered to happen in solution, we will consider the activity of protons in solution, aH+ , as the concentration variable and instead of θ for the adsorbed amount, we will employ the average relative number of bound protons, . Firstly, let us consider the left hand side of Eq. 6.23, which can be written as ln y = ln q − as, in solution, In λ =

) ( μ0 + β − 1 + 2 w , + ln aH+ + H = ln kB T kB T β + 1 − 2 μ0H+ kB T

(6.24) μ0

+ ln aH+ . Also note that, as defined before, ln q = − kBadT μ0 −μ0

and in this case, it can be defined ln k = − adkB T H+ = − ln K . As previously, small k is used to refer to binding constant (adsorption) and K constants refer to the dissociation process (desorption). So, converting to decimal logarithms for using the pH definition, Eq. (6.24) now reads: pK − pH =

) ( β − 1 + 2 1 w . + ln 2.303kB T 2.303 β + 1 − 2

(6.25)

A graphic representation of Eq. (6.25) for different values of w is shown in Fig. 6.4. Also note that the plots are symmetric with respect to the point = 0.5 and that, at fixed , the pH depends only on the values of w. Now, let us define: △pK =

w . 2.303kB T

(6.26)

72

6 Elements of Adsorption on Heterogeneous Substrates 2 1

pH-pK

0 -1 -2 -3 -4 0.0

0.2

0.4

0.6

0.8

1.0

Fig. 6.4 Calculated curve pH − pK versus according to Eq. (6.25), for different values of the interaction parameter: w/k B T = 0 (continuous line), 1.5 (dashed line), 3 (dash-dotted line)

Then, it can be shown from (6.25) that: pH1/2 = pK − △pK ,

(6.27)

where pH½ is the pH at the midpoint of the titration ( = 0.5). As it will be discussed in Chap. 8, this allows to estimate w from the titration curve. Also this may be verified in Fig. 6.4.

6.6 The Mean Field Approximation or Bragg–Williams Approximation A useful approximation to this problem, valid for any dimension, is the so-called Bragg–Williams (B–W) approximation, in which it is assumed that the occupied sites are evenly distributed (this is also called mean field approximation). Therefore, the number of occupied sites neighbours to one occupied site is proportional to N/M. That is: N11 = c(N /M)(N /2),

(6.28)

where c is the coordination number of the lattice. Then, the number of pairs of the type one occupied—one free becomes: N01 = c(1 − N /M)N .

(6.29)

6.6 The Mean Field Approximation or Bragg–Williams Approximation

73

Replacing those values in the partition function, Eq. (6.18), the resulting expression for the chemical potential for the bound species is: ) θ . + cwθ + kB T ln 1−θ (

μb = μb

0

(6.30)

As the bound species must be in equilibrium with the same species in an external solution, their chemical potentials must be equal. Here, we will assume that the activity coefficients of the species in solution are constant and we include them into the standard chemical potential of the species in solution. Thus, at equilibrium, μb = μsol .

(6.31)

And therefore (

μb

0

θ + kB T ln 1−θ

) = μsol 0 + kB T ln C,

(6.32)

where C is the concentration of the binding species in solution (concentrations instead of activities are being considered for the solution). The adsorption isotherm may be written as: ) ( ) ( cW θ −cW/2kB T + = ln ln y = ln qCe (2θ − 1). (6.33) 1−θ kB T So that Eq. (6.33) may be written as: k'C =

θ , 1−θ

(6.34)

where k´ = exp [−(μ0 b + cwθ − μ0 sol )/k B T ]. Note that the constant k´ includes the energy of adsorption. In the case of Langmuir k´ is independent of θ but, for instance, it is not in the case of the Bragg–Williams approximation, which results )( )] [ ( 0 θ μb − μ0sol + cwθ . k C = exp − kB T 1−θ '

(6.35)

So that the adsorption energy changes linearly with the coverage. The B–W isotherm is shown in Fig. 6.5 for c = 4 and different values of the interaction parameters. Note that there is an undulation for negative (attractive) values of the interaction parameter (dash-dotted line in Fig. 6.5). This is, in a similar way as in the van der Waals model for gases, a consequence of the assumption of evenly distributed occupied sites. This undulation is interpreted as a phase transition in which there are two coverage at equilibrium: the “gas” one and the “liquid” one. Also note that for

74

6 Elements of Adsorption on Heterogeneous Substrates

Fig. 6.5 Representation the Bragg–Williams isotherm for a square lattice for different values of the interaction parameter, w/k B T = 2 (solid line), 0 (dashed line) and −4 (dash-dotted line). Straight line: equilibrium line interpretation of the undulation

the one-dimensional Ising model (Fig. 6.1) this undulation does not appear. This is because a phase transition does not happen in one-dimensional systems. In general, it is convenient to write Eq. 6.30 as: k ' = k(θ )

(6.36)

) ( cwθ , k = kint exp − kB T

(6.37)

Or '

where k int is the “intrinsic” equilibrium constant, that is, the part of the equilibrium constant that is independent of θ. Note that this expression is independent of the nature of w. The standard free energy change for the process may be written as: △A0 = △A0int +

cwθ . kB T

(6.38)

From which it is clear that a plot of θ versus ln k´ should give a straight line of slope cw/k B T. Isotherms that show distributed binding constants often indicate a heterogeneous population of binding sites, meaning that not all sites have the same affinity for the ligand. In such cases, the relationship between the fraction of occupied sites (θ ) and the logarithm of the binding constant (ln k) may not be linear over the entire range of θ and ln k. At extreme values of the average number of ligand molecules bound per site (), the dependence of theta on ln k may deviate from linearity. This can occur

References

75

because the binding constant is not the only factor determining the occupancy of a binding site. Other factors such as steric hindrance, electrostatic interactions or cooperative effects can also affect the binding behaviour. When the relationship between theta and ln k is not linear over the entire range of θ and ln k, more complicated models are needed to describe the binding behaviour. These models can include more than one type of binding site, each with a different affinity for the ligand, or can incorporate cooperative interactions between binding sites. Such models can provide a more accurate description of the binding behaviour and help to better understand the underlying mechanisms of binding.

References 1. Ross S, Olivier JP (1964) On physical adsorption. Wiley, New York 2. Rudzinski W, Everett DH (1992) Adsorption of gases on heterogeneous surfaces. Academic Press, London 3. Molina FV (2013) Soil colloids, properties and ion binding. CRC Press, Boca Raton 4. Hill TL (1960) An introduction to statistical thermodynamics. Addison–Wesley Pub Co. 5. Clark AC (1970) Adsorption and catalysis. Academic Press, New York 6. Adamson AW (1990) Physical chemistry of surfaces, 5th edn. Wiley, New York 7. Wyman J, Gill SJ (1990) Binding and linkage. University Science Books, Mill Valley 8. Klotz IM (1997) Ligand-receptor energetics. A guide for the perplexed. Wiley, New York 9. Woodbury CP (2008) Introduction to macromolecular binding equilibria. CRC, Boca Raton

Chapter 7

Theoretical Basis About Solid Polymers, Gels and Single Chains in Solution Related to the Titration of Macromolecules

7.1 Introduction In this chapter, we will give a briefing about the factors that influence a TC of macromolecular substances. In order for a macromolecule to be acid–base titrable, it must contain ionizable groups. These kinds of macromolecules are called polyelectrolytes. We will consider mostly this kind of macromolecules. In order to be redox titrable the macromolecule there must be electrochemically active centres of some kind, it must contain groups that may be oxidized or reduced (redox centres). In the first part of the chapter we will consider qualitatively several aspects of the interaction of macromolecules with other components of a solution. In the second, we will be examining models for the interaction of small ions component of the solution with charged chains of polymers. Finally, we will consider the interaction of neutral polymers with the solvent, the interaction of polyelectrolytes gels with the solvent, the interaction of polymer gels with a solution containing binding ions and, very briefly the interaction of free chains in solution with a solution.

7.2 The State of Macromolecules In this section we will consider the possible states of macromolecules. There are two states of the macromolecules in which we will be interested: macromolecules in solution and in the solid state. A brief but juicy account of macromolecules in the solid state is given in Tabor’s book [1]. At best, macromolecules do crystalize in a partial way. A part of the solid is crystalline and other, amorphous, parts of it consisting of the loose tails of the macromolecule. At any rate there is plenty of void space that is usually filled with solvent molecules or solution.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. A. Marmisollé and D. Posadas, Binding Phenomena, Physical Chemistry in Action, https://doi.org/10.1007/978-3-031-39736-3_7

77

78

7 Theoretical Basis About Solid Polymers, Gels and Single Chains …

For a macromolecule to dissolve in a particular solvent this must be a “good” solvent, that is, one in which most of the functional groups of the macromolecules interact attractively with the solvent molecules. Thus, macromolecules with ionized groups will interact attractively with a polar solvent like water. Many times a macromolecule has both hydrophobic and hydrophilic groups.

7.3 Some Physical Properties of Polymers Many solid polymers have particular physical properties (see, for instance [2]). Among them they show an unexpected elastic behaviour like rubber. That is, they may be extended reversible several times its original size. This is due to the fact that they are formed by long hydrocarbon chains. If the chains are joined, one monomer piece to another, by single chemical bonds, these, at room temperature, cannot be stretched too much. Neither can the angles between consecutive bonds be deformed too much. However, the different consecutive groups can rotate about the bond axis. This causes departures of the polymer chain from a linear configuration. Clearly the mechanism operates at a molecular level and chains may show this behaviour even in solution. Of course this would depend on the type of solvent and of the interaction between different parts of neighbour groups (see below). The polymers that can undergo these types of changes are called flexible polymers. Then, if the free energy balance is favourable, the macromolecule will fold, that is it will undergo conformational changes, reaching a globular shape in such a way that the hydrophobic groups remain in the interior of the globule while the hydrophilic groups are at the external part of it, in a closer contact with the external aqueous phase. This phenomenon is similar to the micelle formation of fatty acids type of molecules with the polar heads pointing to the aqueous phase, if the external phase is polar; and with the hydrophobic head pointing to the interior of the micelle, if this is non-polar in character (see, for instance, [3]: Adamson, 1990). It is possible to calculate the change in free energy to deform a polymer. Assume we have a cube of side L 0 . If we apply a force perpendicular to one of their sides in one direction, the cube will become deformed and increase one of its sides to, say, L x , so to decrease its length in α x = L x /L 0 . From a thermodynamic perspective, it is evident that the only effect is to increase the orderliness of the molecules. Therefore, the effect of stretch the polymer is to decrease its entropy from S 0 to S(L x ). It can be shown [4] that the change of entropy for a free expanding cube, △S d , is: ) ) ( ( )( △Sd = S αx , αy , αz − S0 = N kB ln αx αy αz − 1/2 αx2 + αy2 + αz2 − 3 . (7.1)

7.4 The State of Macromolecules in Solution

79

7.3.1 Deformation Term of a Single Macromolecule According to Michaeli and Katchalsky [5] 1957, the free energy due to the polymer extension, F d , may be expressed in terms of the end-to-end distance (see below), , as: ( ) Fd = 3n p kB T −1 , (7.2) 0

where np is the number of polymer chains, uncharged (neutral) molecule.

0,

is, the end-to-end distance of the

7.4 The State of Macromolecules in Solution In this section we will consider the possible states of macromolecules in solution. At any rate there is plenty of void space that is usually filled with solvent molecules or solution. For a macromolecule to dissolve in a particular solvent this must be a “good” solvent, that is, one in which most of the functional groups of the macromolecules interact attractively with the solvent molecules. Thus, macromolecules with ionized groups will interact attractively with a polar solvent like water. Many times a macromolecule has both hydrophobic and hydrophilic groups. If the macromolecule is joined, one monomer piece to another, by single chemical bonds, these, at room temperature, cannot be stretched too much. Neither can the angles between consecutive bonds be deformed too much. However, the different consecutive groups can rotate about the bond axis. This causes departures of the polymer chain from a linear configuration. Of course this would depend on the type of solvent and of the interaction between different parts of neighbour groups (see below). The polymers that can undergo these types of changes are called flexible polymers. Then, if the free energy balance is favourable, the macromolecule will fold, that is it will undergo conformational changes, reaching a globular shape in such a way that the hydrophobic groups remain in the interior of the globule while the hydrophilic groups are at the external part of it, in a closer contact with the external aqueous phase. This phenomenon is similar to the micelle formation of fatty acids type of molecules with the polar heads pointing to the aqueous phase, if the external phase is polar; and with the hydrophobic head pointing to the interior of the micelle, if this is non-polar in character (see, for instance [3]).

80

7 Theoretical Basis About Solid Polymers, Gels and Single Chains …

Sometimes, in electrochemical experiments, it is very convenient to use the electrochemically active macromolecules as films covering an inert base electrode. In this case the macromolecule is a solid or a gel. Although these films may be very thin (even of a few molecules thick), we will be speaking of a different phase and not of single molecules in solution. Consequently, the models employed for describing these systems are different from that for macromolecules in solution.

7.5 Polyelectrolytes in Solution Most single macromolecules may exist in conformational states somewhere between two limiting cases: (i) completely folded, corresponding to the case of globular proteins, and (ii) totally stretched or linear macromolecules. Let us consider polyelectrolytes in solution. In these conditions, the macromolecules, besides the interaction with the solvent, may interact with other macromolecules depending on their concentration. This problem has been thoroughly studied (see, for instance [6]). However, if the solution is diluted enough in the macromolecule, we may safely disregard this type of interactions. Moreover, under this condition, we may consider, in the statistical thermodynamic sense (see [4], Chap. 7), the macromolecule as a subsystem in which we take into account only the interaction of the macromolecule with the solvent and other species present in the solution.

7.5.1 The End-To-End Distance One important aspect of polymers is the end-to-end distance, i.e. the distance between one end and the other of one macromolecule chain [1] (see Tabor’s book). In the absence of tensions, if the polymer is flexible, that is, if the monomers or groups of monomers may rotate respect of each other within the polymer, it will have a characteristic average length, l0 , that will be smaller than the length of the polymer chain if it were fully extended (stretched) (see Fig. 7.1). In fact, l0 can be much smaller than the latter. Furthermore, this distance depends on the type of solvent. Moreover, if the polymer is stretched by applying a tension, l0 will increase to a value l., ultimately reaching the full length of the polymer.

7.5 Polyelectrolytes in Solution

81

An < 2 > elementary calculation of the average of the square end-to-end distance, 0 , can be done as follows (see Fig. 6.1 in the text): < 2> 0

/ / / ∑ ∑ → → → → = 0 0 = i j , /

i

(7.3)

j

where both i and j span from 1 to σ, the number of segments, this is the statistical units of each segment. It can be shown that the average of the products of the cross-terms i j is zero [6], therefore: < 2> 0

/ =

∑

/ =σ

2 i

2 av ,

(7.4)

i

where 2av is the average (root-mean square) of the bond length in the molecule. Another useful average property of the macromolecule that can be calculated is the root-mean-square distance of the elements of the chain to its centre of √ gravity [7], , or radius of gyration, Rg [1]. The radius of gyration may be related to the volume of the actual polymer that can be directly determined from various experiments. If a chain of the polymer is considered as containing N units each of mass m, then: < 2> s = Rg2 =

∑N

m Ri2 mN

1

(7.5)

It can be shown that [7] < 2> s =

Fig. 7.1 Schematic representation of the polymer length

< 2> 0

6

(7.6)

82

7 Theoretical Basis About Solid Polymers, Gels and Single Chains …

There are several experimental methods to ascertain the shape of the polyelectrolyte chains in solution. The more important ones are: the measure of the viscosity, light scattering and birefringence [6]. These can be related to the end-to-end distance mentioned above.

7.6 Interactions of Polyelectrolytes with Other Species Present in the Solution From our viewpoint, leaving aside the interaction with the solvent molecules, there are two main types of interactions of polyelectrolytes with the components of the solution. The binding of other species presents in the solution and the electrostatic interaction with inert ionic components of the electrolyte. Protons, metal cations, anions and neutral molecules may interact specifically forming some type of bonding (we employ this terminology to differentiate the specific binding from purely electrostatic interactions) with specific sites of the polyelectrolyte. We will not consider in detail the nature of this specific interaction. However, very often they compete among them for the same binding sites in the polyelectrolyte. This binding equilibrium is usually represented by a binding isotherm as those already described. The presence of binding has important consequences in what refers to the conformation of the polyelectrolyte chains. Thus, for instance, binding of protons to a basic site as an amine site in a polyamine would make that the, otherwise neutral, i.e. free from fixed charged sites, polyelectrolyte chains become charged. In order to keep each charge as far as possible, the chains expand and thus change the conformation of the macromolecule. Note also that the binding of a proton to a site whose neighbours are already protonated would require more energy than if the neighbouring site were neutral. Furthermore, creating a pair of occupied sites would set up a repulsive force, f = (ze)2 /ε r 2 , where ε is the dielectric constant of the medium between the charges and r the distance between the sites. The other main type of interactions refers to the electrostatic interaction of, usually small, not chemically reacting, anions and cations present in the solution with the charged sites of the macromolecule in a similar way as in the ion–ion interaction according to the Debye–Hückel model. Very often, an indifferent electrolyte is added to the solution mainly with the purpose of keeping constant the activity coefficients of the ions in solution. On the other hand, in biological environments small ions are usually present in the solution. These small ions interact electrostatically with the charged sites of the molecule and screen the interaction between neighbouring sites. This screening is very important inasmuch as in its absence the interaction between the fixed sites in the polyelectrolyte chains would make the conformation to change to keep these charges as far from each other as possible. Thus the final conformation of the polyelectrolyte is likely to change notably during the course of a TC.

7.7 Electrostatic Interactions

83

7.7 Electrostatic Interactions In order to neutralize a, say, negatively charged site in a macromolecule or on a colloid surface in contact with a solution, it is necessary to bring a proton from the bulk of the solution to the proximities of the macromolecule or the colloidal particle. The electrical work, W el , expended in this process must be added to the intrinsic free energy of the process. The presence of this electrostatic field becomes very important in the titration of polyacids and bases. Following to Katchalsky et al., we may say that in polyacids, the negative field decreases the availability of protons, thus shifting the pH towards lower values. The field effect increases with the degree of ionization giving a polyelectrolyte titration curve steeper than that of the monomer. The opposite is true for the titration of polybases. The field effect can be diminished by adding a neutral salt whose ions screen off the fixed polymeric charges. Alternatively, the titration curve may be corrected by estimation of the electrostatic potential theoretically. Examples of this will be discussed when considering the analysis of titration curves of polyelectrolytes. There is no doubt that the charges involved in this process are discrete in the sense that they occupy determined volume in space. However, the mathematical aspects of the problem are considerably simplified if, instead of a discrete distribution of charges, we consider the charges as point charges and the charge distribution as a continuous one (Fig. 7.2). To calculate the electrical work, it is necessary to find the electrostatic potential, ψ, of the charge distribution due to the fixed charges and the surrounding ions that in general are considered mobile. This is a very difficult problem. In general the centres Fig. 7.2 Schematic representation of ions in solution. One ion is considered to be the central ion and the rest is the ionic cloud of opposite sign. Actually the ions are considered to be points and their distribution a continuous one

84

7 Theoretical Basis About Solid Polymers, Gels and Single Chains …

Fig. 7.3 Scheme showing the central ion, a point at a distance r from it and the corresponding electrical potential

of gravity of charges of opposite signs, q, will not coincide and the system will have a characteristic capacitance computed by C = ∂ψ/∂q. Most of the theoretical approaches are based on solving the Poisson equation for a spherical distribution for the potential (Fig. 7.3); ∇ 2 ψ(r ) =

−4πρ(r ) , ε

(7.7)

where ∇ is the Laplace operator, (∇ 2 = ∂ 2 /∂x 2 + ∂ 2 /∂y2 + ∂ 2 /∂z2 , where x, y, z are Cartesian coordinates), ρ is the charge density and ε the dielectric constant. In the case of only ions in solution one of them is considered the central one (Fig. 7.3) and Eq. (7.7) is solved in the region outside the central ion. This problem has spherical symmetry.

7.7.1 Electrostatic Interaction for Charges Spheres In the case of polyelectrolytes, when it has a globular structure, it may be assumed it is a charged sphere in which the charge is distributed uniformly either in the volume or on the surface. In the solution outside the sphere, the charge is also considered to be uniformly distributed. In the case the polyelectrolyte is linear it may be assumed it behaves as a cylindrical charge distribution. The solution is considered to be a continuous of dielectric constant εr , in which the ions move at random. Here ρ(r) is supposed to be given by the Boltzmann distribution ρ(r ) =

∑ i

ρi (r ) =

∑ i

z i en i (r ) =

∑ i

) ( z i eψ(r ) , z i en i0 exp − kB T

(7.8)

7.7 Electrostatic Interactions

85

where i refers to all the ions present in the solution, e is the elementary charge, n0 i, the concentration of the i-th ion at an infinite distance of the charged site and zi eψ = W el , is the electrostatic work to bring the charge from infinite to the point r. Combining Eqs. 7.7 and 7.8, it results: ) ( 4π ∑ z i eψ(r ) 0 . ∇ ψ(r ) = z i en i exp − ε i kB T 2

(7.9)

This is the Poisson–Boltzmann equation. In order to find the electrostatic potential, ψ, and from it calculate the electrostatic energy contribution, W el = zj eψ, to the total free energy, the procedure is similar to that usually followed when introducing the Debye–Hückel model for single ions in solution. Several drastic assumptions have to be made: (1) charge distributions are considered as continuous. (2) Ions are considered as point charges. (3) Poisson–Boltzmann is linearized. This means that the zj eψ ~ 0.5 are determined, as in the case of the Van der Waals equation, by the rule of equality of areas.

90

7 Theoretical Basis About Solid Polymers, Gels and Single Chains …

Fig. 7.6 Representation of π versus 1/ϕ 2 , for different values of the mixture parameter, χ: 0.54 (continuous line), 0. 536 (dashed line), 0.53 (dash-dotted line), 0.52 (dotted line)

7.9 Swelling of Polymer (Non-Polyelectrolytic) Gels If the macromolecules were not standing alone but, because of chain entanglement or cross-linking, they were making a network then, the solvent will enter the network generating an internal pressure within the network until it is compensated by the osmotic pressure of the solution outside. Then, the polymer undergoes a swelling equilibrium with the solvent. As a consequence of it, the polymer network swells until the developed deformation of the network equals the osmotic pressure of the solvent. This corresponds to the case of free swelling. The free energy change for this process can be written as: △A = △Am + △Ad

(7.29)

where △Ad is a new term that takes into account that, as a consequence of the osmotic equilibrium, the polymer undergoes a deformation. An expression of the deformation free energy has been given above. It may be obtained by calculating the probabilities of the deformed and non-deformed networks ([4], Chap. 21). This term is purely entropic. The term of mixture has to be modified since now, instead N 2 macromolecules, there is only one giant molecule (M = 1), so: △Am = kB T (N1 ln ϕ1 ) + χ M0 ϕ1 ϕ2

(7.30)

The free energy change, △Ad according to Flory [7], it can be written as: [ ( )] 2/3 −T △Sd = vkB T ln ϕ2 + 3 ϕ2 − 1 ,

(7.31)

where ν, according to Flory [7], is the number of cross-linked chains. Therefore, the change of free energy results:

7.10 The Swelling Equilibrium of Polyelectrolyte Gels

91

[ ( )] 2/9 △Am = kB T (N1 ln ϕ1 + N2 ln ϕ2 ) + χ M0 ϕ1 ϕ2 + vkB T ln ϕ2 + 3 ϕ2 − 1 (7.32) The swelling equilibrium results when the chemical potential of the solvent in the gel, μ1 (ϕ 2 ), becomes equal to the chemical potential of the pure solvent, μ1 (0), μ1 (ϕ2 ) − μ1 (0) =0 kB T

(7.33)

which leads to ) ( 2/3 =0 ln ϕ1 + ϕ2 + χ MN2 + χ ϕ1 ϕ2 /M + χ ϕ22 − vkB T 1 + ϕ2

(7.34)

This equation allows determining ϕ 2 at equilibrium. The four first terms are equal to the osmotic pressure of the network, π net , and the last two, to the deformation pressure of the network [Hill, 1960], pdef = − ∂△Adef /∂V. At equilibrium, obviously, πnet = − pdef . )] [ ( 2/3 πnet V1 = kB T ln ϕ1 + ϕ2 + χ ϕ22 − vkB T 1 + ϕ2

(7.35)

7.10 The Swelling Equilibrium of Polyelectrolyte Gels Let us consider a polyelectrolyte gel. This means that the network of polyelectrolyte gel has fixed charges due to the ionization (protonation) of some of the acidic (basic) groups on the monomers. This gel is in contact with an electrolyte solution containing small ions coming from the dissociation of and added salt to the solution (say Na+ and Cl− coming from added NaCl). Thus the gel not only is in equilibrium with the solvent in the external solution but also with the small ions that enter the gel in order to maintain its electrical neutrality. Thus, besides the deformation term due to the interaction of the polyelectrolyte chains with the solvent molecules, there will be an additional deformation due to the repulsion between the fixed charged sites in the same way as it happens with the single macromolecules in solution. It must be kept in mind that the number of fixed charges in the network depends on the binding equilibrium, as it was pointed out above. We will assume that fixed charges with the small ions coming from the solution. In the same way as we defined above the osmotic pressure, π, due to the interaction with the solvent, we may define an electrostatic osmotic pressure, π el , for the swelling due to the small ions. In the limit of a diluted solution we may approximate this term as it is done by Hill [4]: πel /kB T = ρ 2 /e2 l,

(7.36)

92

7 Theoretical Basis About Solid Polymers, Gels and Single Chains …

where ρ is the charge per unit volume within the network and I = ∑C i zi 2 is twice the ionic strength. This can be calculated as ρ = M N2 z i e/V

(7.37)

Since MN 2 is the number of fixed charged groups each one bearing a charge zi e/ V per unit volume. Multiplying and dividing by V 0 and bearing in mind that MN 2 / V 0 = ϕ 2 and that N/V = 1/v1 , the charge density results: ρ = z i eϕ2 /v1

(7.38)

πel /kT = (z i eϕ2 /v1 )2 /e2 l

(7.39)

So that

And the term that has to be added to Eq. (7.35) results − π el v1 / k B T = (zi ϕ 2 )2 / v1 e2 I. Then this equation results: ] [ π = πnet + πel = (kB T /v1 ) ln ϕ1 + ϕ2 + χ ϕ22 − (z i eϕ2 /v1 )2 /e2 I

(7.40)

So that we can define an “effective” χ eff = χ − z2 /v1 2 I. ] [ π = (kB T /v1 ) ln ϕ1 + ϕ2 + χeff ϕ22

(7.41)

This effect increases the swelling. It is seen that the electrostatic effect depends directly on the square of the charge on the network and inversely on the ionic strength of the external solution. Note that this is another example of the coupling just mentioned. Here it is clear that there is a coupling between deformation and electrostatic effects. Also note that there is an indirect coupling between deformation and binding because the influence of the binding on the deformation.

7.11 Electrostatic Interactions in Polyelectrolyte Gels In this section we will consider the same case of the section above but now we will include the possibility of binding of one type of ions (e.g. protons on a polyelectrolytic base) from the solution. As a consequence of the binding of charged species, the polymer becomes charged with a total charge given by: n = z ad Mθ

(7.42)

7.12 Deformation and Electrostatic Interactions and Binding Equilibria …

93

where zad is the charge of the adsorbate. Note that the charge depends on the concentration of the adsorbed species in solution through the corresponding binding isotherm. As a consequence of having charged sites, counterions coming from the external solution must ingress into the polymer to maintain the electrical neutrality. Much has been written about this contribution to △Ael (see for instance [8], Chaps. 7, 9 and 10; [14]). In order to keep the calculations as simple as possible, we will follow the very simple approach of Hill [4]. According to him, the free energy change due to this contribution can be written as: 2 △Ael = z ad Mθ 2 φ 2 /2v0 l

(7.43)

△Ael = 4π zi e2 Mθ 2 φ 2 /2v0 κ 2 εkB T ,

(7.44)

or

where, as indicated before, I = ∑C i zi 2 is twice the ionic strength, M is the number of segments—which we assumed to be equal to the number of binding sites—and C i and zi are the concentration and charge of the ion i in the external solution, respectively. We see that the coupling between binding (in the case of charged species) and the electrostatic contribution is evident: The electrostatic free energy depends on the amount of bound ions.

7.12 Deformation and Electrostatic Interactions and Binding Equilibria in Single Dissolved Macromolecules This problem is treated much in the same way as those discussed in Sect. 7.2. A brief outline is given in Hill’s book [4] (Sect. 21–5) and has also been considered by numerous workers (see, for instance, [15–18]). As can be appreciated from the foregoing discussion, there is a coupling between deformation, electrostatic interactions and binding processes. As the binding process advances the macromolecule becomes increasingly charged or neutral according to the nature of the binding process. Thus, as the charge change, both the deformation of the macromolecule and the electrostatic interactions change. Moreover, extending the polymer chain also causes a change in the electrostatic interaction since the polymer geometry changes from globular to linear. This, as will be shown in the next paragraph, change the equation for the pH from the simple expressions we will show below (see Eq. 7.45, below).

94

7 Theoretical Basis About Solid Polymers, Gels and Single Chains …

7.13 The Equation for the pH Change During the Course of a Proton TC For constant binding energy the expression for the pH has the form of a Langmuir isotherm ) ( θ (7.45) pH = pK + ln 1−θ However, in the case of macromolecular substances K, besides the intrinsic constant for the binding process, K int , depends on the influence of different contributions: (1) deformation of the macromolecule, (2) interaction between bound molecules, (3) electrostatic interactions. This is equivalent to say that the total Helmholtz free energy of the process, △Atotal is made up from different contributions. Thus: △Atotal = △Adef + △Ael + △Ab

(7.46)

where △Adef is the deformation free energy, △Ael is the electrostatic free energy and △Ab is the binding free energy. Thus, for example, if we want to correct K for electrostatic interactions we should write: ) ( Wel θ + (7.47) pH = pK int − log 1−θ 2.303RT And so on for the other contributions. In this particular case of W el , Katchalsky, Shavit and Eisenberg [19] proposed to employ electrophoretic measurements to determine the surface potential, ψ 0 against the work should be done: Wel = −eψ0

(7.48)

In general, electrophoretic measurements provide the mobility values, u (see [6], Sect. 24b); however these might be related to the ψ 0 values through: u=

εψ0 , 300ηC

(7.49)

where η is the viscosity of the solvent and here C is a constant that depends on the size and shape of the particles and on the ionic strength that can be calculated theoretically. For polymetacrylic acid the agreement between W el and − eψ 0 is very good [19].

References

95

For instance, Katchalsky and Lifson [20] and Katchalsky, Shavit and Eisenserg [19] have written the equation for the pH change of polymeric acids as follows: (

θ pH = pK − log 1−θ

)

) 0.434θ (s − s ) 6 ( i 0 κ 2 2ve2 6 ) 0 , (7.50) ( + log 1 + 2 − 6 ε κ 0 2s 1 + κ 2 0

where s denotes the number of elements per statistical element. It is an empirical parameter that depends on steric factors and on the solute–solvent interactions. As in the cases of vinyl polymers at high degrees of ionization that becomes highly solvated, s becomes dependent of α. In these cases: s = (1 − α) s0 + αsi . This expresses the fact that the non-dissociated polymer and the dissociated ones have different s values. In many cases s can be considered independent of θ, so that the preceding equation can be simplified to: ( pH = pK − log

θ 1−θ

) +

) ( 2ve2 6 log 1 + 2 ε κ 0

(7.51)

In the derivation of this equation Katchasky and Lifson [20] have assumed that: (a) The energy of interaction of all ions with their atmospheres is that given by the Debye–Hückel theory. They further considered the atmospheres as approximately spherical. (b) An additional repulsive energy exists between fixed charges, partially screened by their atmospheres. (c) The configurational entropy of the polyelectrolyte molecule is the same function of the end-to-end distance, , as that attributed to an equivalent hypothetical neutral molecule of the same chain length and solubility. It is interesting to mention that these authors considered the electrostatic energy as composed of three contributions: (a) the energy to stretch to the actual ; (b) the atmospheres are built at constant . (c) the fixed charges in the polyelectrolyte are allowed to interact and the repulsive energy is built up. It is further assumed that the length distribution is random ([4], Chap. 13). In other cases, the predominant factor is the interaction between the occupied sites as in the case of polyvinylamine (PVA). We will consider an example of a case of this kind in a forthcoming chapter.

References 1. 2. 3. 4. 5. 6. 7. 8.

Tabor D (1991) Gases, liquids and solids. Cambridge University Press Arridge RGC (1975) Mechanics of polymers. Oxford University Press, Oxford Adamson AW (1990) Physical chemistry of surfaces, 5th edn. Wiley, New York Hill TL (1960) An Introduction to statistical thermodynamics. Addison−Wesley Pub Co Michaeli I, Katchalsky A (1957) J Pol Sci 23:683 Tanford C (1961) The physical chemistry of macromolecules. Wiley, New York Flory PJ (1953) Principles of polymer chemistry. Cornell University Press, Ithaca Rice SA, Nagasawa M (1961) Polyelectrolyte solutions. Academic Press, London

96

7 Theoretical Basis About Solid Polymers, Gels and Single Chains …

9. Resibois P (1968) Electrolyte theory: an elementary introduction to a microscopic approach. Harper and Row, New York 10. Arnold R, Overbeek JTG (1950) Rec Trav Chim 69:192 11. Lifson S, Katchalsky A (1954) J Pol Sci 13:43 12. Fuoss RM, Katchalsky A, Lifson S (1951) Proc Natl Acad Sci US 37:579 13. Alfrey T, Berg PW, Morawetz H (1951) J Pol Sci 7:543 14. Oosawa F (1971) Polyelectrolytes. Marcel Dekker, New York 15. Katchalsky A (1951) J Pol Sci 7:393 16. Katchalsky A, Lifson S, Eisenberg H (1951) J Pol Sci 7:571 17. Katchalsky A, Michaeli I (1955) J Pol Sci 15:69 18. Katchalsky A, Zwick MJ (1955) J Pol Sci 16:221 19. Katchalsky A, Shavit N, Eisenberg H (1954) J Pol Sci 13:69 20. Katchalsky A, Lifson S (1953) J Pol Sci 11:409

Chapter 8

Acid–Base Equilibria and Complexes at Complex Substrates. Polyacids and Polybases

8.1 Introduction Polyacids and polybases are a particular kind of macromolecules. These can be both synthetic and natural. An example of a synthetic polyacid could be the polyacrilic acid and one of a polybase the polyvynilpyridile (PVP) or polyvynilamine (PVA). On the other hand, such substances appear in nature. Just to mention a few we may consider proteins and biological macromolecules, humic and fulvic acids, etc. The description of the protonation reactions of basis of polyprotic acids is an extension of the ideas presented in Chaps. 1 and 2. The most important parameter that describes the acid–base behaviour is the average number of bound protons. As we will see later the dependence of this parameter on pH is what determines the shape of the titration curve.

8.2 Polyacids. Average Number of Bound Protons The polyacid is considered to be of the type Hn A where n is a very high number depending on the molar mass of the macromolecule. The anion, An− , of a polyacid of the type Hn A may undergo the following association equilibria: An− + H+ ⇋ HA(n−1)− , −

(8.1) −

HA(n−1) + H+ ⇋ H2 A(n−2) ,

(8.2)

Hn−1 A− + H+ ⇋ Hn A

(8.3)

…

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. A. Marmisollé and D. Posadas, Binding Phenomena, Physical Chemistry in Action, https://doi.org/10.1007/978-3-031-39736-3_8

97

98

8 Acid–Base Equilibria and Complexes at Complex Substrates. Polyacids …

The corresponding association equilibrium constants are defined as: [ (n−1)− ] HA ka1 = [ n− ][ + ] , A H ] [ H2 A(n−2)− ka2 = [ (n−1) ][ + ] , HA H

(8.4)

(8.5)

… [Hn A] ][ ] kan = [ Hn−1 A− H+

(8.6)

The average number of moles of bound protons is defined as before as: = total number of moles of bound protons/total number of moles of A (8.7) The number is related to the degree of binding, which can be defined as the degree of reaction for the particular case of the acid–base reaction: An− + nH+ ⇋ Hn A

(8.8)

The degree of binding accounts for the reaction extent from unprotonated (empty sites) polyanion to the fully protonated (occupied sites): θH = /n

(8.9)

By considering the mass balance, the total number of moles of bound protons can be written as: ] [ ] [ ] [ n[Hn A] + (n − 1) Hn−1 A− + (n − 2) Hn−2 A2− + . . . + HA(n−1)− ∑ [ ] (8.10) j Hj A(n− j)− = j

Writing the concentrations of [H1 A(n−1)− ], [H2 A(n−2)− ],…, [Hn−1 A− ], [Hn A], in terms of the association constants, [An− ], and [H+ ], the total number of moles of bound protons results: ∑ [ ] [ ]( [ ] [ ]2 [ ]n j Hj A(n−j)− = An− ka1 H+ + 2ka1 ka2 H+ + · · · + nka1 ka2 . . . kan H+ j

] ∑ H [ + ]j [ = An− jkaj H j

where k aj H is defined as: k aj H = k a1 k a2 … k aj .

(8.11)

8.2 Polyacids. Average Number of Bound Protons

99

The total number of moles of A per volume unit is the analytical concentration of the acid, C A . It is given by the mass balance over all the A species. ] [ ] [ ] [ ] [ C A = [Hn A] + Hn−1 A− + Hn−2 A2− + · · · + HA(n−1)− + An− ∑[ ] Hj A(n−j)− (8.12) = j

Replacing the concentration in terms of the association constants and [H+ ] it results: ]( [ ] [ ]2 [ ]n [ CA = An− ka1 H+ + ka1 ka2 H+ + · · · + ka1 ka2 . . . kan H+ ⎛ ⎞ ∑ [ ]j [ n− ] = A ⎝1 + kajH H+ ⎠ (8.13) j

Dividing Eq. (8.11) by (8.13), results: [ ]j jkajH H+ = ∑ [ ]j 1 + j kajH H+ ∑

j

(8.14)

The denominator in this expression is the so-called binding polynomial (BP) (see also Chap. 13). BP = 1 +

∑

[ ]j kajH H+

(8.15)

j

The BP is the more general way to describe proton binding to a multiple acid. If all the constants k ai were equal to a single constant k a (which means equivalent binding sites), we are left again with the problem of binding i protons to n equal sites. That is the Langmuir problem. In this case, using θ H = /n, we get: [ ] θH = k a H+ 1 − θH

(8.16)

We will see that the BP plays a very important role in describing titration curves and distributed k values.

100

8 Acid–Base Equilibria and Complexes at Complex Substrates. Polyacids …

8.2.1 Average Number of Moles of Dissociated Protons In some cases it is more convenient to work with the average number of moles of dissociated protons, , in terms of the dissociation constants, K a(n–i) = k ai −1 . Let us consider a solution of the polyacid, Hn A, initially a completely protonated state. After partial dissociation to a certain degree dependent on the solution pH, the average number of dissociated protons can be defined as: = number of moles of dissociated protons/total number of moles of A (8.17) Firstly, let us consider [

[ ] [ ] ] ] ] ∑ [ [ Hn−1 A− + 2 Hn−2 A2− + · · · + (n − 1) HA(n−1)− + n An− = j Hn−j A j−

(8.18)

j

As before, we write the expressions of [Hn−1 A− ], [Hn−2 A2− ],… [HA(n−1)− ], [A ], in terms of the dissociation constants and [H+ ] n−

∑ ∑ [ ] [ ]− j j Hj A(n−j)− = [Hn A] j K ajH H+ j

(8.19)

j

where K aj H = K a1 K a2 … K aj . The total number of moles of A per unit volume is given by Eq. (8.13). In terms of the dissociation constants and [H+ ] it results: ∑[

⎤ ⎡ ∑ ] [ ] −j Hj A(n−j)− = [Hn A]⎣1 + K ajH H+ ⎦

j

(8.20)

j

Replacing Eqs. (8.19) and (8.20) in (8.17), results: [ ]− j j K ajH H+ = [ ]− j ∑ 1 + j K aj H H+ ∑

j

(8.21)

As in the case of simple acids and bases: = 1 −

(8.22)

8.3 Polybases

101

8.3 Polybases Let us consider an Arrhenius polybase. As in the case of the polyacids, for a polybasic base we can consider the following association reactions: Mn+ + OH− ⇋ M(OH)(n−1)+

(8.23)

M(OH)(n−1)+ + OH− ⇋ M(OH)(n−2)+ 2

(8.24)

… [

] M(OH)n−1+ + OH− ⇋ M(OH)n

(8.25)

with the corresponding constants: ] [ M(OH)(n−1)+ kb1 = [ n+ ][ − ] M OH [ ] M(OH)(n−2)− 2 ][ ] kb2 = [ M(OH)(n−1) OH−

(8.26)

(8.27)

… [M(OH)n ] ][ − ] kbn = [ M(OH)− n−1 OH

(8.28)

Similarly, we can define for a polybase the average number of moles of associated hydroxyls, , as: = number total of moles of associated hydroxyls /total number of moles of M

(8.29)

In a similar way to the case of polyacids, the number of associated hydroxyls is: ] ] [ ∑ [ [ ] (n−j)+ + · · · + n[M(OH)n ] = M(OH)(n−1)+ + 2 M(OH)(n−2)+ j M(OH)j 2 j

(8.30) and replacing the concentrations in terms of the constants and [OH− ] it results: [ n+ ] [ ] [ ]2 [ ]n M + kb1 OH− + 2kb1 kb2 OH− + · · · + nkb1 kb2 . . . kbn OH−

102

8 Acid–Base Equilibria and Complexes at Complex Substrates. Polyacids …

] ∑ OH [ − ]j [ = Mn+ jkbj OH

(8.31)

i

where k bj OH = k b1 k b2 …k bi . The total number of moles of M per volume unit is: ] ] [ ] [ [ + · · · + [M(OH)n ] CB = Mn+ + M(OH)(n−1)+ + M(OH)(n−2)+ 2 [ ] ∑ (n−j)+ = M(OH)j (8.32) j

Replacing the concentrations in terms of the k and [OH− ], we obtain: ]( [ ] [ ]2 [ ]n ) [ CB = Mn+ 1 + kb1 OH− + 2kb1 kb2 OH− + · · · + nkb1 kb2 . . . kbn OH− ⎛ ⎞ ∑ [ n+ ] [ ] j OH = M ⎝1 + kbj OH− ⎠ (8.33) j

So that: [ − ]j OH jkbj OH = ∑ OH [ − ]j 1 + j kbj OH ∑

j

(8.34)

As in the case of polyacids, if all the constants k bi were equal to a single constant k b ; we are left again with the problem of binding i hydroxyls to n equal sites. Defining θ OH = /n, we get: [ ] θOH = kb OH− 1 − θOH

(8.35)

Similarly, we obtain the expression for the average number of dissociated hydroxyls, : [ ]−1 [ ]−2 [ ]−n + 2K b1 K b2 OH− + · · · + nK b1 K b2 . . . K bn OH− K b1 OH− = [ ]−1 [ ]−2 [ ]−n 1 + K b1 OH− + K b1 K b2 OH− + · · · + K b1 K b2 . . . K bn OH− (8.36) [ − ]− j ∑ OH j j K bj OH = (8.37) [ ]− j ∑ 1 + j K bjOH OH− The K bj = k b(n−j) are the corresponding dissociation constants and K bj OH = K b1 K b2 …K bj .

8.4 Complexation and Competitive Binding in Multi-ligand Complexes

103

8.4 Complexation and Competitive Binding in Multi-ligand Complexes 8.4.1 Ligands Without Hydrolysis This problem is present in many substances in nature. Thus, for instance, insulin (see Chap. 11 for the acid–base titration of insulin) when isolated from living tissue always is associated with zinc. Insulin contains 4 α-carboxyl groups, 8.5 γ-carboxyl groups, 4 imidazole groups, 4 α-amino groups, 10 phenolic plus ε-amino groups and 2 guanidine groups [1]. Their intrinsic pK a values of these groups are 3.6, 4.7, 6.2, 7.3, 9.6 and 11.0. Part of these groups also binds Zn2+ ions so that there is a competition between protons and Zn2+ for binding to these groups. Also, humic acids bind protons, hydroxyls and heavy metal ions present in aquatic media (see Chap. 11) [2]. Then, there is a competition between protons (or hydroxyls) and the cations of the heavy metal. Following the same procedure as for polyacids and polybases, we will consider an ionic metal species Mm+ that can bound one, two, etc., ligand species An− . In this section, it is assumed that the species An− does not undergoes further hydrolytic reactions. The possible complexation equilibriums are: Mm+ + An− ⇋ MA(m−n)+

(8.38)

MA(m−n)+ + An− ⇋ MA(m−2n)+ 2

(8.39)

[ ] (m−n(p−1))+ MAp−1 + An− ⇋ MA(m−np)+ p

(8.40)

…

with the corresponding constants: ] [ MA(m−n)+ kc1 = [ m+ ][ n− ] M A [ ] MA(m−2n)+ 2 ][ ], kc2 = [ MA(m−n)+ An−

(8.41)

(8.42)

… [

kcp

] MA(m−np)+ p ] , =[ (m−n(p−1))+ [ n− ] MAp−1 A

(8.43)

104

8 Acid–Base Equilibria and Complexes at Complex Substrates. Polyacids …

We will be interested in the so called ligand number or the average number of moles of bound ligands, , defined as: = average number of moles of bound ligands /total number of moles of metal species

(8.44)

So that: ] ] [ [ ∑p (m− jn)+ (m− jn)+ j MA j MA j j j=1 j=1 ]= = [ ] ∑ [ , p (m− jn)+ + C M M + j=1 MAj ∑p

(8.45)

where C M is the analytical concentration of the metal cation. Since for j = 0, the first term in the numerator equals to cero. Following a procedure similar to that of polyacids, we get: [ n− ]j A j=1 jkcj A [ n− ]j , ∑p A j=1 kcj A

∑p =

1+

(8.46)

where k cj A is the equilibrium constant corresponding to the formation of the complex, MAi (m−in)+ , k cj A = k c1 k c2 … k cj . So that if all the complex formation constants and the concentration of free An− are known, can be calculated.

8.4.2 Ligands Undergoing Hydrolysis. Generalization of Competitive Binding We will extend the generalization made for non-competitive binding to the case in which the ligand is a polyacid. Since the ligand is bound to both protons and metal ions, it will be convenient to modify our definition of the average number of bound protons, , as: = number of moles of protons bound /total number of moles of A species not bound to M.

(8.47)

The amount of bound protons is then, from Eqs. (8.5) to (8.10): n ∑ [ [ ] [ ] ] j Hj A(n− j) H1 A(n−1)− + 2 H2 A(n−2)− + · · · + n[Hn A] = j=0

(8.48)

References

105

The amount of A not bound to M is clearly: [

n ∑ ] [ ] [ ] [ ] An− + H1 A(n−1)− + H2 A(n−2)− + · · · + [Hn A] = Hj A(n− j)−

(8.49)

j=0

that, in terms of Eqs. 8.5–8.9, can be written as: ⎛ ⎞ n ∑ [ n− ] [ ] j A ⎝1 + kjH H+ ⎠

(8.50)

j=1

Therefore, [ + ]j H j=1 jkj H [ + ]j ∑n H j=1 kj H

∑n =

1+

(8.51)

Then, if the association constants of the acid and the pH of the solution are known, can be calculated. As in the previous case, if the all the complex formation constants and the concentration of free An− are known, can be also calculated. In this case, the analytical concentration of the acid is: CA =

p n ] ∑ [ ] ∑ [ (m−jn)+ Hi A(n−i)− + j MAj i=0

(8.52)

j=0

where the first sum extends over the i proton association equilibria and the second one over the j complexation equilibria. The second term is the concertation of ligands bound to the metal ion. Then, n [ ]i [ n− ] ∑ CA = A kiH H+ + C M

(8.53)

i=0

References 1. Tanford C, Epstein J (1954) J Am Chem Soc 76(8):2163 2. Stevenson FJ (1994) Humus chemistry: genesis, composition, reactions. Wiley, New York

Chapter 9

Acid–Base Titration of Complex Substrates. Binding Constant Distribution

9.1 Introduction In this chapter we will consider examples of the titration of polyacids and polybases. These may be polymers in solution or substances in different states as colloids, solid films on top of electrodes, or other natural and synthetic macromolecules. We will consider some detail synthetic polymers such as polymethacrylic acid (PMA) and polyvinylamine (PVA). The preferred methods for titrating these substances are the potentiometric and the spectrophotometric ones. These are described in detail in the bibliography [1]. From the viewpoint of the ionization behaviour the important parameter to be obtained from the titration curve is ⟨vH ⟩ or ⟨r H ⟩ or some combination of them. Consequently, we will show, as a representation of the titration curve, the ⟨vH ⟩ versus pH curve. If desired, the actual titration curve may be obtained employing some of the equations of the preceding chapter together with the pertinent experimental parameters, that is, V t , C b , V s and C a . Usually in this case, the titration curves of polyacids and polybases depend on the concentration of the supporting electrolyte. For these and other reasons, as discussed in the previous chapters, titrations of complex substrates are carried out at several ionic strengths and then some corrections are performed in order for discounting for the ionic strength effect. We have emphasized that, in the case of complex substances, more often than not, there is a distribution of binding constants and redox potentials. That is, in complex substances there is not a unique value of the corresponding constant but rather a distribution of them as a function of the average number of bound ions. Moreover, in general, there is a coupling among the states of binding, deformation (tension), redox potential (electron transfer) and the extension of screening. This coupling is the responsible of the constant distribution. Also, and for the reasons just pointed out previously, when investigating the acid– base behaviour of macromolecules, other types of measurements are carried out in order to ascertain the effects of coupling. In polyelectrolytes, there are several physicochemical properties that depend on the state of coiling or folding of the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. A. Marmisollé and D. Posadas, Binding Phenomena, Physical Chemistry in Action, https://doi.org/10.1007/978-3-031-39736-3_9

107

108

9 Acid–Base Titration of Complex Substrates. Binding Constant Distribution

macromolecule. The state of coiling usually depends on the degree of ionization of the macromolecule. Thus, to search about this condition viscosity, light scattering or some other measurement is performed together with the determination of the TC. The apparent binding dissociation constant, K ap , is determined from the following equation: 

⟨vH ⟩ pH = pK ap − log 1 − ⟨vH ⟩

 (9.1)

Then the distribution is visualized in a plot of ⟨vH ⟩ versus pK ap . The name of apparent is because usually in the case of macromolecules, this is not a true constant but it usually depends on the state of binding; i.e. it is a function of ⟨vH ⟩. Let us now split pK ap into two contributions: pK int due to the intrinsic effect of the dissociation reaction, that is the effect of the reaction itself at ⟨vH ⟩ = 0 in the absence of any kind of interactions with other dissociated sites, and a term φ(⟨vH ⟩) containing all the other dependences of pK ap on ⟨vH ⟩. pK ap = pK int + φ(⟨vH ⟩)

(9.2)

which in terms of the change of Gibbs free energy for the dissociation implies ΔG ap = ΔG int + RT φ(⟨vH ⟩)

(9.3)

9.2 Titration of Polyacids and Polybases 9.2.1 Titration of Polyacids: Polymethacrylic Acid One of the most studied polyacid is polymethacrylic acid (PMA). The monomeric unit of this acid is shown in Scheme 9.1. We will consider in some detail the classic work of Arnold and Overbeek [2]. These authors prepared polymers of different mean molecular weight and made potentiometric acid–base titrations of the polyacid as well as they carried out viscosity Scheme 9.1 Scheme showing the monomeric unit of polymethacrylic acid

CH3 C

CH2

COOH

n

9.2 Titration of Polyacids and Polybases

109

measurements of solutions at different degrees of dissociation. Details the about preparation, fractionation and purification of the polymer, the determination of the polymer concentration in the solution and the measurements of pH and viscosity are given in the original paper. The mean molecular weight of the polymers comprised the range 40,000–266,000. These authors carried out these experiments in solutions of different ionic strength of KCl ranging from 10−4 M to 1 M. The titration curves for 139,000 g/mol PMA in different KCl concentrations are shown in Fig. 9.1. Employing the relation (9.1), these workers calculate the pK ap values as a function of ⟨vH ⟩, as shown in Fig. 9.2. It is seen that the viscosity of the polyelectrolyte solutions depends on the protonation state of the polymer. This can be rationalized as that proton binding is coupled to 9 8 7

pH

Fig. 9.1 pH versus ⟨vH ⟩ for the titration of PMA acid of Molar Mass = 139,000, polymer concentration 0.0099 N, at different ionic strengths of KCl. (●) 1 M, (◯) 0.1 M, (▼) 0.01 M, (Δ) 0.001 M, (∎) 0.0001 M. Adapted with permission from [2]. Copyright © 1950 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

6 5 4 3 0.0

0.2

0.4

0.6

0.8

1.0

0.6

0.8

1.0

8

7

pKap

Fig. 9.2 pK ap versus ⟨vH ⟩ for the same ionic strength values presented in Fig. 9.1. Data taken with permission from [2]. Copyright © 1950 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

6

5 0.0

0.2

0.4

110

9 Acid–Base Titration of Complex Substrates. Binding Constant Distribution

Fig. 9.3 Specific viscosity, ηsp , versus ⟨vH ⟩ at different ionic strength. (●) 1 M, (◯) 0.1 M, (▼) 0.01 M, (Δ) 0.001 M, (∎) 0.0001 M. Adapted with permission from [2]. Copyright © 1950 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

conformational changes (coiled/extended states) which notably affect the viscosity of the solutions. Furthermore, as conformational changes are due to alterations of the amount of electrostatic charges on the polymer chains leading to repulsive electrostatic interactions, they are markedly dependent on the ionic strength of the solutions. In this regard, the increment of the ionic strength attenuates the conformational transitions due to proton binding changes. As a result, the changes in the specific viscosity on the protonation degree are more notorious at low ionic strength (Fig. 9.3). Thus, higher viscosity changes along the acid–base titration of polyelectrolytes are expected when working in the absence of supporting electrolyte. It is seen in the figure that for not very small ⟨vH ⟩ values the pK ap does not changes very much with it. However, except for the most diluted solutions, pK ap depends strongly on the ionic strength. Furthermore, these authors also measured the specific viscosity, ηsp , of the PMA solutions at different ionic strengths, I (Fig. 9.3).

9.2.2 Titration of Polybases. Polyvinylamine (PVA) As an example of polybases, we will consider the titration of PVA, studied by Bloys van Treslong and Staverman [3]. The monomeric unit of this base has the structure shown in Scheme 9.2. In Fig. 9.4 the dependence of pH on ⟨r H ⟩ for different ionic strengths is presented. Here ⟨r H ⟩ is the average number of moles of dissociated protons. Scheme 9.2 Scheme showing the monomeric unit of polyvinylamine

9.2 Titration of Polyacids and Polybases

111

Fig. 9.4 ⟨r H ⟩ dependence of pH for the ionization of solutions 0.59 × 10−2 M of PVA at different ionic strengths of (●) 0 M. (◯) 0.05 M, (∎) 1 M and (□) 0.1 M. Taken from [3] with permission. Copyright © 1974 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The potentiometric titration of polyacids and bases shows that the pH dependence of the degree of ionization is also affected by the ionic strength as a consequence of the influence of the electrostatic potential of the charged groups in the macromolecule. Let us consider a positively charged polymer chain due to protonation of amine groups. There is an electrostatic repulsion due to the presence of positive charges on neighbour sites that hinder further protonation of empty sites. Then, the protonation of these empty sites requires higher proton activity values (lower pH values) than in the case of simple substances. Also, this effect becomes more intense as the protonation degree increases (Fig. 9.5). Let us now consider the case of protonation of initially negatively charged groups (as in the case of polyacrylate). In this case, the presence of neighbour negative charges favours the protonation of empty sites due to the attractive electrostatic interaction. Then, the protonation requires lower proton activities (higher pH values) than Fig. 9.5 Dependence of the mean number of dissociated protons, ⟨r H ⟩, on the pK ap determined from data in Fig. 9.4. Taken from [3] with permission. Copyright © 1974 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

9 Acid–Base Titration of Complex Substrates. Binding Constant Distribution

Fig. 9.6 Titration curves for PVA at different ionic strength. Adapted with permission from [4] Copyright © 1957 Interscience Publishers, Inc., New York

10

8

pH

112

6

4 0.0

0.2

0.4

0.6

0.8

1.0

that needed for single monomeric acids (acrylic acid). This effect is also dependent on the dissociation degree of the neighbour sites. As a result of the interaction between neighbour sites, the apparent pK ap values are dependent on the protonation (or dissociation) degree. As we have previously commented, a distribution of pK ap values is the observed in the course of a titration. Also, as interactions are mainly electrostatic in nature, these effects are greatly affected by the ionic strength. In this regard, narrower pK a distributions are observed at high ionic strengths. Polyelectrolytes such as polyvinylamine [4] and polyethylenimine (PEI) [5, 6] show a different behaviour: even after the addition of a great amount of indifferent electrolyte, the curve of pH versus ⟨r H ⟩ still shows a very steep slope, very far from the titration curve of the monomer (see Figs. 9.6 and 9.9). In Fig. 9.6 it is seen that the dependence of the proton dissociation degree on pH is affected by the ionic strength, and therefore, the addition of inert salt to the solution does affect the titration curves. However, the influence of the ionic strength could be corrected by considering the electrostatic work, W el (Chap. 7), employing the following equation [4]: pK ap = pK int +

Wel 2.303kB T

(9.4)

where W el means the electrostatic work energy, pK ap is estimated by extrapolation to zero ionic strength. Katchalsky et al. (loc. cit.) considered the polymer as a sphere of radius R, the charge distribution due to the ionized groups as a continuous one and that φ(⟨vH ⟩) was due entirely to electrostatic effects (see Eq. 7.14). The K ap value was estimated from an interpolation to zero ionic strength to be between 4.81 and 4.87 and compares very well with the pK = 4.84 value of trimethylenediamine very similar to the polymeric unit.

9.2 Titration of Polyacids and Polybases Fig. 9.7 Corrected TC (●) and TC of the monomer (◯) trimethylenediamine

113

12 11

pH

10 9 8 7 6 5 0.0

0.2

0.4

0.6

0.8

1.0

However, this analysis cannot explain the pK ap distribution, as after correction, practically all the titration curves yield the same pH dependence of the dissociation degree which is still different to that of a similar monomer (Fig. 9.7). In view of the results of Fig. 9.7, Katchalsky attributed these departures to interaction between the adjacent dissociated sites [4]. To this end, they employed the one-dimensional Ising isotherm (see Chap. 6). In this case, an extra term should be added to the binding constant:  pK ap = pK int + log

1 − 2θ α θ −α

2 +

  6l 2ve2 ln 1 + 2 2.303εl κl0

(9.5)

where the second term comes from considering the Ising model (see Chap. 6) and the last term comes from the effect of the ionic strength and the length. Note that in this case is θ ≡ ⟨r H ⟩. In view of the difficulties to evaluate the term (2νe2 /ε l) ln [1 + (6 l/κ l0 2 )] the authors preferred to evaluate it experimentally taking into account that W el = eψ and that ψ can be estimated from electrophoretic measurements (see Chap. 7). In Fig. 9.8 the dependence of K ap = K ' on the dissociation degree is shown. Defining ΔpK as (see Eq. 6.26):  ΔpK = log

1 − 2θ α θ −α

2 (9.6)

They determined pK = 9.40 and ΔpK = 1.2 for the polymer. It is interesting to compare these values with those of charged trimethylenediamine NH2 (CH2 )3 NH2 , that has pK 1 = 10.64 and pK 2 = 8.86 and therefore ΔpK = 1.16, which is similar to that determined for polyvinylamine.

114

9 Acid–Base Titration of Complex Substrates. Binding Constant Distribution

Fig. 9.8 ⟨r H ⟩ versus pK ' as determined from Eq. 9.4

1.0

0.8 0.6 0.4 0.2 0.0 0

2

4

6

8

10

12

14

16

pK´

12 10 8

pH

Fig. 9.9 pH as a function of ⟨r H ⟩ for polyethyleneimine alone (•) and in 0.1 M KCl (°) and 1.0 KC (+). Reproduced from Ref. [5] with permission from the Royal Society of Chemistry

6 4 2 0.0

0.2

0.4

0.6

0.8

1.0

1.2

9.2.2.1

Polyethylenimine (PEI)

As we mentioned above polyethyleneimine shows similar results to polyvinylamine. This was studied by Shepherd and Kitchener [5]. Their results for two different ionic strengths and no salt added are shown in Fig. 9.9 It is seen in Fig. 9.9 that the effect of added salt is similar to that observed for PVA (Fig. 9.4). Shepherd and Kitchener employed Eq. This implies assuming si = s0 . That ⟨ 7.51.  2 ⟩ implies considering s as independent of log 1−2θα r . The estimation of the rest of H θ−α the parameters is detailed in the original work [5]. Thus, they determined pK = 10. By comparison of the experimental pH versus ⟨r H ⟩ curve with that of the Henderson– Hasselbalch (Fig. 9.10) at ⟨r H ⟩ = 0.5, drawn for pK = 10 (see Fig. 9.10), they determined ΔpK = 2.0. Figure 9.10 is compared to the experimental points (open

9.2 Titration of Polyacids and Polybases

115

14 12

pH

10 8 6 4 2 0.0

0.2

0.4

0.6

0.8

1.0

1.2

Fig. 9.10 pH as a function of ⟨r H ⟩ for PEI. (− − − ) calculated with the Henderson–Hasselbalch employing pK = 10. (•) experimental points for PEI, continuous curve calculated with Eq. 7.51. Reproduced from Ref. [5] with permission from the Royal Society of Chemistry

circles) with the calculated ones (continuous curve). It is seen that only in the region 0.3 ⟨r H ⟩ 0.5, a good agreement is obtained.

9.2.2.2

Linear PEI (LPEI)

PEI results above refer to branched PEI. This could be represented as: H(NHCH2 CH2 )n NH2 as opposed to LPEI that can be represented as: (C2 H5 N)n . LPEI was first synthesized in 1972. The distance between neighbours in LPEI is, on the average, smaller that in branched PEI. Moreover, in branched PEI 50% of the titrable groups are primary or tertiary amines and this complicates the interpretation of TC. LPEI gives asymmetrical curves with respect to the point of half total charge in the pH versus ⟨r H ⟩ curve as predicted by the Ising model (Chap. 6) as shown in Fig. 9.11. So Smits, Koper and Mandel [6] extended this model so to include triplet interactions (Chap. 6). They obtained an expression of the titration curve as follows:  pH = pK int + log

1 − ⟨rH ⟩ ⟨rH ⟩

 + log

⟨rH ⟩(1 − 2⟨rH ⟩ + xd ) (⟨rH ⟩ − 2xd + xt )(1 − ⟨rH ⟩)2

(9.7)

And: εd = log

(⟨rH ⟩ − 2xd + xt )2 xd (1 − 2⟨rH ⟩ + xd )(xd − xt )2

(9.8)

(xd − xt )2 (⟨rH ⟩ − 2xd + xt )xt

(9.9)

εt = log

9 Acid–Base Titration of Complex Substrates. Binding Constant Distribution

Fig. 9.11 pH as a function of ⟨r H ⟩ for LPEI. Continuous line, calculated as described in the text. Experimental points (•) 1 M NaCl, (°) 0.5 M, (+) 0.1 M, (▲) no added salt. Data taken with permission from Ref. [6]

10 9 8

pH

116

7 6 5 4 3 0.0

0.2

0.4

0.6

0.8

1.0

where ε stands for the relative excess energies of dimers (subscript d) and trimers (subscript t) and x for the corresponding fractions. To fit the experimental results to the theory they proceeded as follows. For low values of ⟨r H ⟩ they set εt equal to zero and determined εd and pK int by trial and error. Then, they employed the full TC to determine εt . The results of the fit are shown in Fig. 9.11 as continuous lines. The agreement is very good.

References 1. 2. 3. 4. 5. 6.

Albert A, Serjeant EP (1962) Ionization constants of acids and bases. Methuen & Co, London Arnold R, Overbeek JTG (1950) Rec Trav Chim 69:192 Bloys van Treslong CJ, Staverman AJ (1974) Recueil. J R Neth Chem Soc 93(6):171 Katchalsky A, Mazur JP, Spitnik P (1957) J Pol Sci 23:513 Sheperd EJ, Kitchener JA (1956) J Chem Soc 2448 Smits RG, Koper GJM, Mandel M (1993) J Phys Chem 97:5745

Chapter 10

The Acid–Base Behaviour of Polyampholytes. The Case of Colloidal Oxides

10.1 Introduction In previous chapters we have defined ampholytes as those substances that have acid and basic groups in the same molecule. Polyampholytes may be defined as those that have many acid and basic groups in the macromolecule. There may be synthetic and natural polyelectrolytes. Among the synthetic ones we may mention copolymers and as an example the copolymer of PVA and PMA. Also colloidal oxides are polyampholytes. However, the difference with the polyampholyte just mentioned is that in the latter the acid and basic groups come from the different monomeric units, whereas in the colloidal oxides, as explained below, is the same function that may act as acid or basic according to the pH of the external solution. Also natural substances are polyampholytes. Proteins are macromolecules that are formed by many aminoacids. Aminoacids are polyampholytes that usually have an amino and a carboxylic group per monomer like the example of glycine mentioned in Chap. 3, but also many other functional groups. As an example of proteins, we may quote that of the insulin, to be briefly discussed in the next chapter. Also macromolecules present in soils and known as humic substances. These are conglomerates that contain many groups, mainly phenolic and carboxylic [1]. The humic and fulvic acids are not truly pure substances but rather fractions obtained when the soil is treated chemically. On the other hand, many models have been proposed to describe the titration behaviour of these kinds of substances with great success [2]. One of these will be briefly considered below. In this work we propose to consider the colloidal oxides as a polyampholyte. That is, we will consider it as a macromolecule having acid and basic sites. In doing so, we will first derive the binding polynomial for a polyampholyte. Then, we will describe the titration of such substances in terms of the average binding for both the basic and the acid sites. Finally, we will obtain the distribution function of the corresponding constants and interpret it. Since the formal treatment is the same, we will omit the consideration of macromolecular polyampholytes. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. A. Marmisollé and D. Posadas, Binding Phenomena, Physical Chemistry in Action, https://doi.org/10.1007/978-3-031-39736-3_10

117

118

10 The Acid–Base Behaviour of Polyampholytes. The Case of Colloidal …

A colloid is a dispersion of a particulate phase, of typical size between 2 and 200 nm, in a continuous phase (the dispersion medium). There are many combinations of dispersion and continuous phases (liquid in liquid, solid in liquid, solid in gases, etc.). Many important properties of colloids depend on the surface charge on the particulate phase. Since we will be interested in colloidal oxides, we will be devoted to this kind of surfaces. The surface structure of some oxides as magnetite (Fe3 O4 ), hematite (Fe2 O3 ), silica (SiO2 ), titanium dioxide (TiO2 ), alumina (Al2 O3 ), etc., may be represented as shown in Figs. 10.1, 10.2 and 10.3 Surface metal ions are chemically bound to OH groups. When colloidal particles of these oxides are in contact with water, these water molecules adsorb onto the surface groups. For this reason the surface groups should be represented as MOH(H2 O)n . For simplicity we will omit the adsorbed water molecules. In view of the amphoteric characteristics of the metals that form part of these oxides, in acid media, the surface groups are protonated to give positive charged sites as depicted in Fig. 10.2. In basic media, the OH− surface groups ionize in the way shown in Fig. 10.3.

Fig. 10.1 Schematic representation of the surface of a metallic oxide

Fig. 10.2 Schematic representation of the surface of a metallic oxide in acid media

Fig. 10.3 Schematic representation of the surface of a metallic oxide in basic media

10.2 The Binding Polynomial for a Colloidal Oxide

119

These protonation−ionization reactions are the reason for the development of the surface charge. This charge is the responsible of the stability of colloidal oxides in solution. A similar mechanism is the responsible of the surface charge in other colloidal particles such as the silver halides, AgX [3]. In this case the surface charge is due to an excess of halide or silver ions at the surface, according to the concentration of those ions in solution. The ion determining the charge on the surface is called the potential determining ion. In the absence of specific interactions of the charged groups with other ions in the solution, there is a pH of the solution in contact with the colloid at which the net charge is zero. This is the point of zero net charge or isoelectric point of the colloid.

10.2 The Binding Polynomial for a Colloidal Oxide Colloidal oxides can be considered as a particular case of ampholytes inasmuch as their surface groups may behave both as acids and as bases. Colloidal oxides are characterized by having 2n M-OH groups that can protonate or dissociate according to the acidity of the medium (see Figs. 10.1, 10.2 and 10.3). Since the protonation and the dissociation of the surface groups produce charged sites on the colloid surface that modify the binding energy of the neighbouring sites, not all the surface sites are equivalent. Therefore, we can consider colloidal oxides as a polyacid−polybase ampholyte (see the forthcoming chapter). We will consider there are 2n surface sites M–OH in one colloidal particle (the amount of these sites may be estimated by adsorption methods). Then, association and dissociation equilibria that suffers the surface groups of this substance can be described by Eqs. 10.1–10.3 and 10.4–10.6, respectively. M(OH)n + H+  M(OH)n−1 OH+ 2

(10.1)

+ 2+ M(OH)n−1 OH+ 2 + H  M(OH)n−1 (OH2 )

(10.2)

M(OH)(OH2 )(n−1)+ + H+  M(OH2 )nn+ n−1

(10.3)

…

For the protonation reactions on the MOH sites and:

…

M(OH)n  M(OH)n−1 O− + H+

(10.4)

M(OH)n−1 O−  M(OH)n−2 On− + H+

(10.5)

120

10 The Acid–Base Behaviour of Polyampholytes. The Case of Colloidal … (n−1)− + MOn− n + H  M(OH)On−1

(10.6)

for the ionization of the MO− sites. Note that we are considering the protonation of the MOH sites and the ionization of the MOH sites. For the protonated MOH sites we have:   M(OH)n−1 OH+ 2 −1   = K an ka1 =  (10.7) M(OH)n H+   M(OH)n−2 (OH2 )2+ −1  +  = K a(n−1) ka2 =  (10.8) M(OH)n−1 OH+ H 2 …   M(OH2 )n+ n

−1   = K a1 kan =  + H M(OH)(OH2 )(n−1)+ n−1

(10.9)

where the k ai are the corresponding association constants and K a(n−i) the equilibrium constants for the corresponding dissociation reaction. The equilibrium constants for the ionization of the MO− sites are:    M(OH)n−1 O− H+ −1   K b1 = (10.10) = kan M(OH)n    M(OH)n−2 O2− H+ −1   K b2 = (10.11) = ka(n−1) M(OH)n−1 O− … K bn =

   M(OH)n−21 O2− H+ −1   = ka1 M(OH)n−1 O−

(10.12)

We will introduce the definition of the average number of moles of associated protons, ⟨v1 ⟩, onto MOH sites, as: ⟨v1 ⟩ =

total number of moles of associated protons onto MOH sites total number of moles of M(OH)species

(10.13)

and similarly for the average number of moles of dissociated protons to form the MOH sites becomes: ⟨r2 ⟩ =

total number of moles of ionized protons from the MOH sites total number of moles of M(OH)species

where n is the total number of sites.

(10.14)

10.2 The Binding Polynomial for a Colloidal Oxide

121

In order to calculate ⟨v1 ⟩ and ⟨r 2 ⟩, we define the analytical concentration of sites as C a . Then, the mass balance over the surface metal species can be written as:     + ··· Ca = [M(OH)n ] + M(OH)n−1 O− + M(OH)n−2 O2− 2       + + M(OH)O(n−1)− + MOnn + M(OH)n−1 OH2 n−1       + M(OH)n−1 (OH2 )2+ + · · · + M(OH)(OH2 )(n−1)+ + M(OH2 )n+ n n−1 (10.15) The total number of moles of associated protons onto MOH sites is:       2+ M(OH)n−1 OH+ + · · · + n M(OH2 )n+ n 2 + 2 M(OH)n−2 (OH2 )2

(10.16)

and the total number of moles of dissociated protons onto a MO− sites is:     + · · · + [M(OH)n ] + (n − 1) M(OH)O(n−1)− n MOn− n n−1

(10.17)

so that the average number of moles of associated protons becomes       2+ + · · · + n M(OH2 )n+ M(OH)n−1 OH+ n 2 + 2 M(OH)n−2 (OH2 )2 ⟨v1 ⟩ = Ca (10.18) On the other hand, the average number of moles of dissociated protons from MOH sites is:       n MOn− + · · · + M(OH)n + (n − 1) M(OH)O(n−1)− n n−1 ⟨r1 ⟩ = (10.19) Ca replacing in terms of the equilibrium constants, it results: ⟨v1 ⟩ =

   2  n ka1 H+ + 2ka1 ka2 H+ + · · · + nka1 . . . kan H+  +  + n  + −1  + −2  −n 1 + ka1 H + · · · + ka1 . . . kan H + K b1 H + K b1 K b2 H + · · · + K b1 . . . K bn H+

(10.20)  + j H j=0 jkaj H ⟨v1 ⟩ =  j ∑n  −i ∑ 1 + nj=0 kajH H+ + i=1 K bjH H+ ∑n

(10.21)

where k aj H = k a1 … k ai , etc. The denominator in this rational fraction is the binding polynomial (BP) for the metal oxide

122

10 The Acid–Base Behaviour of Polyampholytes. The Case of Colloidal …

BP = 1 +

n ∑

n  j ∑  −i kajH H+ + K bjH H+

j=0

i=1

(10.22)

Similarly, in terms of the corresponding constants and [H+ ], the total number of moles of dissociated protons is:  −1  −2  −n K b1 H+ + 2K b1 K b2 H+ + · · · + n K b1 . . . K bn H+

(10.23)

So that, similarly to Eq. 10.20, the average number of moles of dissociated protons results:  −i i K bjH H+ ⟨r2 ⟩ =  j ∑  −i ∑ 1 + nj=0 K ajH H+ + ni=1 K bjH H+  + −i ∑n H i=1 i K bj H ⟨r2 ⟩ = BP ∑n

i=1

(10.24)

(10.25)

Note that both, the average number of moles of dissociated protons and the average number of moles of associated protons can be written in terms of the binding polynomial. Particularly, in this case, ⟨v1 ⟩ + ⟨r2 ⟩ = 1 −

1 BP

(10.26)

If there were no interactions between the sites, i.e. if each kind of surface sites were energetically equivalent, we should only consider two association processes: the association reaction for the M(OH) sites: M(OH)n + H+ ↔ M(OH)n−1 OH+ 2

(10.27)

and the association reaction for the MO− sites: (n−1)− + MOn− n + H ↔ M(OH)On−1

(10.28)

with the equilibrium constants: ka1

  M(OH)n−1 OH+ 2 −1   = K an =  M(OH)n H+

(10.29)

and   M(OH)O(n−1)− n−1 −1  +  = kn−2 K2 =  MOn− H n

(10.30)

10.2 The Binding Polynomial for a Colloidal Oxide

123

On the other side if we consider that, in Eqs. 10.22 and 10.23, the equilibrium constants of each site are all equal (see Chap. 13); that is j

kaHj =

n!k1 j! (n − j)!

K bjH =

n!K 2 j!(n − j)!

(10.31)

j

(10.32)

Then, it can be shown that we get for ⟨v1 ⟩/n and ⟨r 2 ⟩/n:   k1 H+ ⟨v1 ⟩ =      −1 n 1 + k1 H+ + K 2 H+

(10.33)

  +  −1 K2 H ⟨r2 ⟩ =  +    +  −1 n 1 + k1 H + K 2 H

(10.34)

and

The similitude with the Langmuir isotherms for competitive adsorption is more evident defining the fraction of protonated MOH sites as: θ1 =

⟨v1 ⟩ n

(10.35)

⟨r2 ⟩ n

(10.36)

and the fraction of ionized MOH sites as θ2 =

So that these expressions can be written in more familiar forms using Eqs. (10.33) and (10.34):   θ1 = k1 H+ 1 − θ1 − θ2

(10.37)

   −1 θ1 = K 2 H+ 1 − θ1 − θ2

(10.38)

and

In the case that the constants are not equal, we may define a constant that is dependent on θ and write Eq. (10.37) as:   θ1 = k1 (θ ) H+ 1 − θ1 − θ2

(10.39)

124

10 The Acid–Base Behaviour of Polyampholytes. The Case of Colloidal …

and    −1 θ1 = K 2 (θ ) H+ 1 − θ1 − θ2

(10.40)

10.3 Titration of Colloidal Oxides We will consider, as an example, a suspension of colloidal magnetite in a 0.1 M NaNO3 . The aim of the titration of this substance is to determine the amount of acid and basic surface groups. As we will see, only the difference of these quantities can be determined. This difference is related to what is known as the surface charge of the colloid. We will consider that both surface acid and basic groups behave as a polyprotic acid and as a polybasic base. That is, they are characterized by the fractions ⟨v1 ⟩ and ⟨r 2 ⟩, which can be expressed in terms of the binding polynomial as discussed above. Note that both the polyacid and the polybase are in the volume V s of the sample. For the sake of completeness, we will describe briefly the titration procedure to obtain the fraction of bound protons and hydroxyls. To a determined volume of solution that contains a known concentration of supporting electrolyte it is added a weighted amount, m, of the oxide. The specific area of the oxide, A, has been previously measured by a suitable method, for instance the BET isotherm. Together with the oxide, it is added a small volume of a strong acid solution so as to obtain a final pH of the system of about 2. A blank solution is made following the same procedure but without the added sample. Both the blank and the sample solutions are titrated with a strong base (Fig. 10.4). The titration volume of the magnetite is calculated by making the difference between the volumes of the sample and that of the blank at fixed pH values. This 10 9 8

pH

Fig. 10.4 Experimental titration curve for magnetite following the experimental procedure described in the text. Test solution (solid line), blank solution (dashed line). Taken with permission from Ref. [4]

7 6 5 4 3 0.0

0.1

0.2

0.3

0.4 -3

Vb / cm

0.5

0.6

0.7

10.3 Titration of Colloidal Oxides

125

procedure involves the assumption of the additivity of titration curves (Sect. 4.3). In order to interpret the titration curve we must go a little deeper into the description of the acid–base behaviour of this substance. The charge balance relation in this case is:     +  Na + ⟨v1 ⟩Cs + H+ = ⟨r2 ⟩Cs + OH−

(10.41)

which after replacement of the concentrations (see Chap. 4), and solving for the titrant volume, it results: Vt Δ = Cs (⟨v1 ⟩ − ⟨r2 ⟩) − Vs (Cs + Δ)

(10.42)

We are interested in the difference in the number of moles of protons per unit area, Γ H+ and the number of moles of hydroxyls, Γ OH− , per unit area: ΓH+ − ΓOH− =

(⟨v1 ⟩ − ⟨r2 ⟩)m A

(10.43)

This is proportional to the surface density of charge, σ,  σ = n F ΓH+ − ΓOH− = σ+ + σ−

(10.44)

The curve σ versus pH obtained in this way is shown in Fig. 10.5. In the present case the analysis is complicated by the fact that only the difference (⟨v2 ⟩ − ⟨r 1 ⟩) is obtained. In what follows, for convenience, we will employ k 2 = K 2 −1 for the distributed constant of proton association onto the MO− sites. Therefore the natural choice seems to consider the constants k 1 (σ) and k 2 (σ) as a function of the surface charge, σ: Fig. 10.5 Experimental results from the titration of magnetite expressed as charge per unit area, σ, as a function of pH. Taken with permission from Ref. [4]

126

10 The Acid–Base Behaviour of Polyampholytes. The Case of Colloidal …

σ F(⟨v1 ⟩ − ⟨r2 ⟩) = = F(θ1 − θ2 ) n n

(10.45)

Applying Eqs. (10.39) and (10.40):  −1   k1 (σ ) H+ − K 2 (σ ) H+ σ =F  −1   n 1 + k1 (σ ) H+ − K 2 (σ ) H+

(10.46)

The analysis is further complicated because the functional relation between σ and k 1 (σ ) and K 2 (σ ) is not simple. The latter cannot be obtained straightforwardly from σ, n and the pH values.

10.4 Summary of Models on the Oxide Solution Interface In what follows we will first consider some simple electrical representation of interface oxide/solution. In the introduction section we stated that the oxide charge comes from protonation/ionization reactions of the oxide groups at the surface. As a whole the interface must be electrically neutral, therefore, provided the ions in solution are strongly hydrated and stable towards chemical reactions, we may think there is a charge excess at the solution side, equal in magnitude and of opposite sign, lying on a plane at certain distance, x 2 , of the surface. This is called the compact double layer of Helmholtz layer or, sometimes, the Stern layer (Fig. 10.6). The plane going through the centre of the ions lying closest to the surface is called the outer Helmholtz plane. This is a plane passing through the centre of the ions closest to the oxide surface. The electrical potential at this plane is designated as ψ 2 . The dielectric in between the layers is formed by solvent molecules. The capacitance of this layer would be C H = (ψ 0 − ψ 2 )/σ = 4πx 2 /ε, where ε is the dielectric constant of the solvent in this region. Here ψ 0 designates the electrical potential at Fig. 10.6 Scheme of the colloid/solution interface

10.4 Summary of Models on the Oxide Solution Interface

127

the surface. However, as mentioned in Sect. 7.7, in the presence of a charged wall in a solution will make an electric field to set up at the solution side that, in turn, will make the ions to distribute so as to satisfy the Poisson–Boltzmann equation for the potential (Eq. 7.9). However, the present problem has planar symmetry therefore now the spatial coordinate x is the distance perpendicular to the oxide surface. The capacitance contribution of this region is called diffuse layer, C d . By solving the Poisson–Boltzmann equation an expression of the capacitance contribution of this charge distribution, may be found:

 −z i eψ(x) z i kB T A cosh Cd = e 2kB T

(10.47)

where

A=

kB T n 0 ε 2π

1/2 (10.48)

The symbols have the same meaning as in Sect. 7.7. Note that according to the present view, the region of the solution side immediately adjacent to the surface may be considered as formed by two capacitors in series. The total capacitance, C, is, therefore: 1 1 1 = + C CH Cd

(10.49)

Usually ψ 2 is a small fraction of ψ 0 − ψ 2 , depending on the electrolyte concentration (Fig. 10.6). We will consider two extreme cases of ions in solution: those, as we mentioned above, strongly hydrated and unreactive and those weakly hydrated and reactive towards some species at the oxide surface that may somehow join the surface. We will represent the specific interaction with the oxide surface at the alkaline branch of the TC, in the following way: S−O− (at the oxide surface) + C+ (cation in solution)  S−O−C(cation at plane 1)

(10.50)

In electrochemistry this type of interaction is called specific adsorption. We will call the plane defined by the centres of these ions the internal plane, or plane 1, and designate its electrical potential by ψ 1 (see Fig. 10.7). We will follow the procedure proposed by Sposito [1]. For convenience we will define the point of zero charge (pzc) as the pH at which the surface charge at the surface of the colloids zero (Eq. 10.45). Many models have been proposed to interpret the charge-pH curves of colloidal oxides. Among them we may mention: the model of Block and de Bruyn [5]; the model of Levine and Smith) [6]; the constant capacitance model [1]; the triple layer model [7, 8]; the objective model [1]; the hydrous oxide model [9]; and the transition

128

10 The Acid–Base Behaviour of Polyampholytes. The Case of Colloidal …

Fig. 10.7 Modelling of the double layer including the possibility of specific adsorption

layer model of Dignam and Kalia [10]. A comparison of the electrostatic models has been done by Westall and Hohl [11]. In this case, we will consider only the constant capacitance model. As we will see, this model is similar to that employed for proteins and synthetic polyacids and bases. An obvious difference between the oxides and the just mentioned systems is that being solids the oxides should not show a noticeably expansion of the surface during the titration. A disadvantage of oxides is that, as we have said, they are polyampholytes and the acid and basic constants are not so different so as to be possible to separate them at all pH values. However, as we shall see, the constant capacitance model assumes that at pH below the pzc there are no ionized groups and that above the pzc there are no protonated sites.

10.5 Analysis of Some Experimental Results with the Constant Capacitance Model This model assumes that the charge can be written as: σ = Cψ

(10.51)

where the capacitance C is independent of the charge. As we already mentioned the generation of surface charge comes from the protonation and ionization of the surface sites according to: + SOH+ 2  SOH + H

(10.52)

SOH  SO− + H+

(10.53)

10.5 Analysis of Some Experimental Results with the Constant Capacitance …

129

respectively. Note we have written the first equation as a dissociation one. The apparent equilibrium constants for these reactions are: K a1

K a2

  xSOH H+ = xSOH2+   xSO− H+ = xSOH

(10.54)

(10.55)

where the x s are the corresponding molar fractions of the surface species and [H+ ] is the proton concentration in bulk. Assuming the only contribution to the potential is electrostatic, the electrical work to bring an ion of charge zi from the bulk of the solution to the surface is zi eψ s, the value of the electrostatic potential at the surface. Then, according to the Boltzmann distribution the amount of such ions at the surface would be: 

 +  + −eψ (10.56) H s = H exp kB T where zi eψ is the electrical work to transfer the protons from the bulk to the surface. Replacing [H+ ] in Eqs. 10.54 and 10.55, by its value at the surface, the constants K a1 and K a2 become: 

−eψ xSOH  +  (10.57) H exp K a1 = xSOH2+ kB T 

−eψ xSO−  +  (10.58) H exp K a2 = xSOH kB T And defining the “intrinsic” equilibrium, K a int , constants as: int K a1 = int K a2

xSOH  +  H s

(10.59)

xSOH2+ xSO−  +  H s = xSOH

(10.60)

Note that the K a int values should be independent both of the ionic strength and θ. The foregoing equations may be written as:

int pH = pK a1 + log

xSOH xSOH2+

 −

eψ 2.303kB T

for σ > 0

(10.61)

130

10 The Acid–Base Behaviour of Polyampholytes. The Case of Colloidal …

and

int pH = pK a2 + log

xSO− xSOH

 −

eψ 2.303kB T

for σ < 0

(10.62)

The reader would recognize that this model is associated with the picture of the double layer shown in Fig. 10.8, in which there is consideration to neither the diffuse layer nor the presence of specific adsorption. Therefore ψ is ψ 0 , the potential at the oxide surface, and C the capacitance of the entire double layer. Below we briefly describe the so-called triple layer model (TLM), which is a straightforward extension of the constant capacitance model to include specific adsorption (see Ref. [2]). Fig. 10.8 Representation of Eqs. 10.66 (a) and 10.67 (b) for the system magnetite in KNO3 at different concentrations: (●) 0.001 M, (◯) 0.01 M, (∎) 0.1 M. Reproduced with permission from Ref. [4]

10.5 Analysis of Some Experimental Results with the Constant Capacitance …

131

One of the basic assumptions of the model is that at pH < pzc, x SO− is zero and that at pH > pzc, x SOH2+ is zero. Therefore, as in the case of ampholytes, there are two basic equations: one for the acidic branch and other for the basic one. The molar fractions can be expressed in terms of the quantities we have used through the book, ⟨v1 ⟩ and ⟨r 2 ⟩ or still better in terms of the surface charge as defined above. In order to calculate the fraction of sites it is necessary to know the number of sites per unit area, N s . Reggazzoni has employed 6 sites nm−2 [4]. With this number the total maximum charge σ max = F N s /N max results σ max = 96.13 μC cm−2 . Taking into account that: xSOH + xSOH2+ = 1

(10.63)

σ+ = xSOH2+ = 1 − α σmax

(10.64)

σ− = xSO− = 1 − β σmax

(10.65)

and that:

And similarly for xSO− :

And replacing ψ from Eq. (10.51) and rearranging, Eqs. (10.61) and (10.62) result: 

eσ+ α int + pK a1 = pK + = pH + log σ >0 (10.66) 1−α 2.303kB T C+ and int pK a2



eσ− β + = pK − = pH + log σ a) =

z 1 z 2 e2 eκa exp(−κr ) 4π ε(1 + κa)r

(12.12)

where z1 and z2 are the magnitude of the charges, e is the charge of the electron. Also, ε = ε0 εR is the dielectric permittivity, ε0 = 8.8510−12 C2 N−1 m−2 the permittivity in vacuum and ε R the relative dielectric constant of the medium. Here, K is the inverse of the Debye length, which depends on the ionic strength. Then, for the particular case of the redox units, assuming the centres are charged spheres, the energy of the coulombic interactions between the different centre pairs may be expressed as: coul εOO =

(z Ox e)2 eκaOx exp(−κr ) 4π ε(1 + κaOx )r

(12.13)

εcoul RR =

(z R e)2 eκa R exp(−κr ) 4π ε(1 + κa R )r

(12.14)

12.4 Electron Binding to Polyaniline

159

and εcoul OR =

(z O x z R e)2 eκa O R exp(−κr ) 4π ε(1 + κa O R )r

(12.15)

where aOx = 2r Ox , aR = 2r R , aOR = r Ox + r R , being r j the effective radii of the redox centre. Assuming the same mean separation between all the redox centres, r m , and that r Ox ≈ r R ≈ am /2. Then: Δεmcoul =

(z O x − z R )2 F 2 eκam exp(−κrm ) 4π εN AV (1 + κam )rm

(12.16)

By introducing the proton binding isotherms (Eq. 12.9) for O and R into the expressions for the segments charges given by Eqs. (12.10) and (12.11), and employing the dissociation constants, K a,i : −1  −1  − 1 + 10 pH− pK a,R (z Ox − z R ) = 1 + 10pH− pK a,Ox

(12.17)

coul Then, assuming that Δεm ≈ Δεm , it results to be

 −1  −1 2 Δεm = C0 1 + 10pH− pK a,Ox − 1 + 10 pH− pK a,R

(12.18)

where C0 becomes: C0 =

F 2 eκam exp(−κrm ) 4π εN Av (1 + κam )rm

(12.19)

Note that C 0 becomes a constant for a given experimental condition (temperature, ionic strength). The parameter Δεm can be obtained from the slope of a representation of E ap as a function of (1 − 2θ n ). Figure 12.7 shows these plots for the case of Pani at several pH values. Then, the pH dependence of Δεm is shown in Fig. 12.8. This pH dependence can be interpreted as follows. According to the values of pK aLE and pK aE at pH < 0 all the amine groups in the LE form and all the imine groups are protonated; then, there is no change in the ionic interactions when reduced units are converted into oxidized ones. However, at pH > 1.0 all the amine groups will be deprotonated in the reduced units, whereas all the imine groups still remain protonated in the oxidized units. This situation leads to the maximum of the interaction energy change when reduced units are converted into oxidized ones. Only at pH above 3.0 the imino groups are partially deprotonated. In view of Eq. 12.14, for low values of pH, Δεm tends to zero (all segments have the same charge and the difference (zOx − zR ) is zero and, therefore, Δεm = 0) and this parameter is maximum when

160

12 Electron Titrations of Electrochemically Active Macromolecules

Fig. 12.7 Dependence of E ap on 1 − 2θ for the data in Fig. 12.6. Adapted with permission from [16]. Copyright © 2013 Elsevier B.V

Fig. 12.8 Dependence of Δεm on pH. The line is the result of the fit to Eq. 12.16. Adapted with permission from [16]. Copyright © 2013 Elsevier B.V

Table 12.1 Results of the fitting of data in Fig. 12.8 to Eq. (12.18) [16]

C0 /kJ mol−1

pK a,Ox

pK a,R

11.8 ± 0.6

5.5 ± 1.5

0.52 ± 0.07

the highest is the difference of protonation degree of each type of segment (at about pH = pK E − pK LE ~ 2.5 At pH beyond 3.0, ΔεKm starts decreasing eventually tending to zero again as the pH further increases). By fitting data in Fig. 12.8 to Eq. 12.16, the parameters C 0 , pK a,Ox and pK a,R can be obtained. The resulting values of both pKs (Table 12.1) are in agreement with many of the reported in the literature.

References

161

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Tanford C (1961) The physical chemistry of macromolecules. Wiley, New York Tanford C, Buzzell JG, Rands DG, Swanson SA (1955) J Am Chem Soc 77:6421 Cai CX, Ju XH, Chen HY (1995) Electrochim Acta 40:1109 Bowden EF, Dautartas MF, Evans JF (1987) J Electroanal Chem Interfacial Electrochem 219:49 Rees DC (1985) Natl Acad Sci USA 82:3082 Mauk AG, Moore GR (1997) J Biol Inorg Chem 2:119 Valleé BL, Williams RJP (1958) Proc Nat Acad Sci USA 59:498 Pascher T, Chesik JP, Winkler JR, Gray HB (1996) Science 271:1558 Marmisollé WA, Posadas D, Florit MI (2008) J Phys Chem B 112:10800 Ju H, Gong Y, Zhu H (2001) Anal Sci 17:59 Qian W, Wang YH, Wang WH, Yao P, Zhuang JH, Xie Y, Huang ZX (2002) J Electroanal Chem 535:85 Andrade EM, Molina FV, Florit MI, Posadas D (2000) Electrochem Solid St Letts 3:504 Ybarra G, Moina C, Molina FV, Florit MI, Posadas D (2005) Electrochim Acta 50:1505 Antonini E, Wyman J, Brunori M, Taylor JF, Rossi Fanelli A, Caputo A (1964) J Biol Chem 239:907 Brunoni M, Wyman J, Antonini E, Rossi−Fanelli A (1964) J Biol Chem 240:3317 Marmisollé WA, Florit MI, Posadas D (2013) J Electroanal Chem 707:43 Bard AJ, Faulkner LF (1980) Electrochemical methods. Wiley, New York Marmisollé WA, Florit MI, Posadas D (2014) J Electroanal Chem 734:10 Hill TL (1960) An introduction to statistical thermodynamics, Addison−Wesley Pub Co

Chapter 13

Appendices

13.1 Appendix 1. Macroconstants and Microconstants 13.1.1 Macroconstants and Microconstants We have referred before to the fact that two sites may or may not have different binding constants (it is also said they have different affinities). This could happen because the sites have intrinsically different affinities like proton binding to a, say, p–aminobenzoic anion. It may happen that in principle the two sites have the same affinity like in the oxalate anion. However as soon as the first proton binds this anion, the affinity of the second site changes and this is the reason why the first protonation constant is different from the second. We refer to this type of constants as microconstants (also called Site Constants) as opposed to the stoichiometric constants or macroconstants [1] (see also [2–4]). We may represent this situation by considering a molecule that has two independent sites, that is the occupancy of one of them does not affect the affinity of the other. This leads to the conclusion that the affinity of the sites is invariant. Let us consider a molecule with two basic sites, R1 and R2. The first protonation reaction may be written as either: R1 − R22− + H+ ⇋ HR1 − R2−

(13.1)

R1 − R22− + H+ ⇋ R1 − R2H−

(13.2)

or as:

We will define now the site constants or microconstants, k 11 and k 21 as those corresponding to the binding to sites R1 and R2, respectively (see Scheme 13.1) that is:

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. A. Marmisollé and D. Posadas, Binding Phenomena, Physical Chemistry in Action, https://doi.org/10.1007/978-3-031-39736-3_13

163

164

13 Appendices

Scheme 13.1 Scheme showing the different paths corresponding to the protonation of a compound of the type R1 − R22− , following the nomenclature proposed in Ref. [5]

k11

] [ HR1 − R2− ][ ] =[ R1 − R22− H+

(13.3)

and ] [ R1 − RH2− ][ ] k21 = [ R1 − R22− H+ ] [ (R1 − R2)H− ][ + ] , K1 = [ R1 − R22− H

(13.4)

(13.5)

where (R1R2)H− represents either HR1R2− or R1R2H− , that is the initial species protonated once. The second protonation step will depend now on if the presence of a proton in the molecule changes the affinity of the remaining free site or not. In the latter case of invariant affinity, we have: HR1 − R2− + H+ ⇋ HR1 − R2H

(13.6)

R1 − R2H2− + H+ ⇋ HR1 − R2H

(13.7)

with a site constant k 12 . If, however, the presence of a proton in site 1 affects the affinity of site 2 and vice versa, we will have to differentiate if the second protonation constant comes from the site 1 protonated or otherwise. Thus the second site constants will be k 21 if the site one is protonated coming from the species that had the site 2 already protonated and k 12 in the other case. If the sites have invariant affinities, then k11 = k21

(13.8)

k12 = k22

(13.9)

and

13.1 Appendix 1. Macroconstants and Microconstants

165

The macroconstant for the second protonation step is: [HR1 − R2H] ][ ] . K2 = [ (R1 − R2)H− H+

(13.10)

In the case of invariant affinities, it is possible to find a relation between K 1 , K 2 and k 1 , k 2 . To this end we must first note that, as a consequence of the conservation of mass: [

] [ ] [ ] (R1 − R2)H− = HR1 − R2− + R1 − R2H−

(13.11)

Then we can write ] ] [ ] [ [ (R1 − R2)H− (HR1 − R2)− + (R1 − R2H)− ][ ] = K 1 = [ ][ ] K1 = [ = k11 + k21 . R1 − R22− H+ R1 − R22− H+ (13.12) In addition, ] [ (HR1 − R2)H− [HR1 − R2H] [HR1 − R2H] ][ + ] [ ][ + ] = [ K1 K2 = [ ][ ]2 − 2− R1 − R2 H (HR1 − R2)H H R1 − R22− H+ =

[HR1 − R2− ] [HR1 − R2H] + [HR1 − R2− ][H ] [R1 − R22− ][H+ ]

(13.13)

which leads to: k11 k12 = K 1 K 2

(13.14)

K 1 K 2 = k21 k22

(13.15)

1 1 1 = + . K2 k11 k21

(13.16)

Thus, in principle, if the constants K 1 and K 2 are measured experimentally and it is known from extra thermodynamic information that the receptor has sites with invariant affinities, it is possible to calculate k 1 and k 2 . In the case the affinities change with occupancy it is not possible to determine the constants k 1 , k 2 , k 12 and k 21 . The analysis of this case is given in Klotz, 1997 [1].

166

13 Appendices

13.1.2 The Example of Ciprofloxacin In this example we will show the determination of the microconstants by applying potentiometric and spectrophotometric methods [5]. Ciprofloxacin is an antibiotic. Its formula is shown in Scheme 13.2. We will represent ciprofloxacin by HQ0 . A proton of the carboxylic acid may dissociate (HQ− ) and protonate the amino group (H2 Q+ ) of the piperazinyl group to form a zwitterion (represented as HQ± ). The dissociation scheme is entirely similar to that shown in Scheme A1.1. The macroscopic constants are given by: K1 =

([HQ± ] + [HQ0 ])[H+ ] [H2 Q+ ]

(13.17)

[Q− ][H+ ] ([HQ± ] + [HQ0 ])

(13.18)

K2 =

And the microscopic constants by: k11 =

[HQ0 ][H+ ] [H2 Q+ ]

(13.19)

k12 =

[Q− ][H+ ] [HQ0 ]

(13.20)

k21 =

[HQ± ][H+ ] [H2 Q+ ]

(13.21)

k22 =

[Q− ][H+ ] . [HQ± ]

(13.22)

As the basis of the present method is the spectrometric determination of the fraction of the total carboxylate (COO− ), defined as: αCOO− = or Scheme 13.2 Chemical formula of ciprofloxacin

[Q− ] + [HQ± ] [HQ± ] + [HQ0 ] + [H2 Q+ ] + [Q− ]

(13.23)

13.1 Appendix 1. Macroconstants and Microconstants

αCOO− =

k21 [H+ ]−1 + K 1 K 2 [H+ ]−2 . 1 + K 1 [H+ ]−1 + K 1 K 2 [H+ ]−2

167

(13.24)

From which, knowing α COO− at each pH allows to determine k 21 . The fraction α COO− can be determined from absorbance measurements at different pHs as: αCOO− =

ApH − ACOOH , ACOO− − ACOOH

(13.25)

where ApH is the absorbance at one pH, ACOOH the absorbance at a very acidic pH and ACOO− at a very basic one. The UV spectra at different pH values of ciprofloxacin is shown in Fig. 13.1. In that work the analytical wavelength length chosen was λ = 324 nm. The calculated values of α COO− for different pH values are shown in Fig. 13.2. Fig. 13.1 Spectra of ciprofloxacin at different pH values. Full line, 10.57; long dash, 7.90; short dash, 7.50; dotted, 6.73; dash-dot, 2.96. Taken with permission from [5]. Copyright © 1997, American Chemical Society

Fig. 13.2 Fraction of carboxylic groups for different pH values. Taken with permission from [5]. Copyright © 1997, American Chemical Society

168

13 Appendices

Table 13.1 Values of macro- and microconstants of ciprofloxacin (From [5]) pK 1

pK 2

pk 11

pk 12

pk 21

pk 22

− 6.08

− 8.58

6.58

− 6.24

− 8.02

− 8.41

1.0 0.8

all fractions

Fig. 13.3 Fraction of all the species involved for different pH values. (solid) H2 Q+ , (long dash) HQ− , (dotted) HQ± , (dash−dot) HQ. Taken with permission from [5]. Copyright © 1997, American Chemical Society

0.6 0.4 0.2 0.0 2

4

6

8

10

12

pH

These values allow for determining k 21 . With these values and the macroconstants determined by acid−base titrations the rest of the microconstants can be calculated. Those are assembled in Table 13.1. With the microconstants all the fractions may be calculated. These are shown in Fig. 13.3.

13.2 Appendix 2. Statistical Factors Let us consider the ionization of a polyacid, AHn , that has n acid groups and all the ionization constants equal, that is, equal probability of each proton to ionize or bind. We will employ the nomenclature proposed by Perrin, Dempsey, Serjeant [6]. In this case the ionization constant will be less of the ionization constant of a closely related monoacid by a factor n. This is so because there are n ways of losing the proton but only one in which the proton can be restored. Conversely, the protonation of the fully ionized anion An− may happen in n different ways while its dissociation only in one. Therefore, the relation between the protonation macroconstant, k 1 , and the microconstant, k, will be: k1 = n k

(13.26)

13.2 Appendix 2. Statistical Factors

169

The binding of the second proton may happen in n − 1 ways, but as there will be two protons bound that are indistinguishable, the number of ways must be divided by two. That is: k2 =

(n − 1)k . 2

(13.27)

It is easy to see that the general expression for the i protonation constant is: ki =

(n − i + 1)k . i

(13.28)

The application of relation 13.28 leads to the fact that the observed pK a of this acid will be smaller than the pK a of a closely related monoprotic acid by a factor of log n. Similarly, for the loss of a second proton by a factor of log [(n − 1)/2] and so on. These effects are called statistical factors. Equation 13.28 can be substituted in the general equation for the average number of bound protons to a polyacid: ∑n

+ j H 0 j kaj [H ] = ( ∑n H + j ) . 1 + 1 kaj [H ]

(13.29)

And replacing the k expressions according to (13.28) = (

∑n ∏i + j j 0 j 1 (n − j + 1)k [H ] ). ∑n ∏i 1 + 1 1 (n − j + 1)k j [H+ ]j

(13.30)

The product in this equation can be simplified considering that n! Π1i (n − j + 1) = j (n − i )!i!

(13.31)

so that becomes ∑n

j (n−in!)!i! k j [H+ ]j = . ∑ n! 1 + n1 (n−i)!i! k j [H+ ]j 0

(13.32)

According to the binomial theorem: (1 + k[H+ ])n = 1 +

n ∑ 1

n! k j [H+ ]i . (n − i )!i!

(13.33)

170

13 Appendices

Differentiation with respect to k [H+ ] leads to: k[H+ ]d(1 + k[H+ ])n = nk[H+ ](1 + k[H+ ])n−1 d(k[H+ ]) n ∑ n! k i [H+ ]i =1+ (n − i )!i! 1

(13.34)

so that results: =

k[H+ ]n(1 + k[H+ ])n−1 nk[H+ ] = (1 + k[H+ ])n (1 + k[H+ ])

(13.35)

that can be written as:

= k[H+ ]. n −

(13.36)

That is the Langmuir isotherm.

13.3 Appendix 3. Elements of Statistical Thermodynamics 13.3.1 Introduction The purpose of statistical thermodynamics is to calculate the average thermodynamic properties starting from molecular properties (see, for instance [7–9]). To do so it would be sufficient to calculate the time average of the property during a time tending to infinite, of the system of interest. Instead, Gibbs proposed to consider an enormous amount of replicas of the system (an ensemble) and average over the states. Several types of ensembles may be considered. The most common one is a system that contains a fixed number of particles, N, fixed volume, V and at a fixed temperature, T. This corresponds to a Canonical Ensemble for a closed system that is a system of fixed composition whose average energy is defined but not fixed. Another system often considered in statistical thermodynamics is the Grand Canonical Ensemble which may exchange matter with the surroundings but, at equilibrium, its average composition is also defined although not fixed. It is convenient to introduce a new state function, ln Q(N, V, T ), where Q is the canonical partition function, defined as: Q = ∑ exp(−βE r )

(13.37)

13.3 Appendix 3. Elements of Statistical Thermodynamics

171

where, for convenience, we have defined β = 1/k B T and E r is the energy of the (micro) state r. The sum is extended over all the possible states accessible to the system, compatibles with the external condition of fixed N, V, and T. The grand canonical partition function, , is defined by a double sum: One over the states and another over the number of particles in the state r, N r . = ∑ exp(−β E r )∑ exp(−α Nr )

(13.38)

where α = μ/kT and μ is the chemical potential. Although many problems are more easily solved with the grand canonical partition function and the results are the same, it is usually preferred to work with the canonical partition function. For a system of independent particles, that is when there is no interaction among the particles, the canonical partition function for the free, indistinguishable, particles (like atoms freely moving in a gas) is related to the partition function of a single particle, q, by: Q=

qN . N!

(13.39)

The case of distinguishable particles (like fixed particles on a reticule) is related by: Q = qN

(13.40)

where q, the molecular partition function, is defined as: ) ( q = ∑ exp −βεr

(13.41)

where εr is the energy of the molecules in state r. For non-interacting molecules, the energy of the state r may be separated into its different energy contributions: translational, vibrational, rotational, electronic, nuclear, etc. εr = εr,t + εr,v + εr,rot + εr,el + εr,nuc

(13.42)

q = qt qv qrot qel qnuc

(13.43)

so that

All the different contributions except the translational are called internal contributions to the partition function since they are not dependent on the volume. The probability of finding the system in the state r, Pr , is given by: exp(−β E r ) exp(−β E r ) = Pr = ∑ exp(−β E r ) Q

(13.44)

172

13 Appendices

In what follows we will denote the average thermodynamic quantities by S, E, A in the understanding that we are speaking of the average values. For instance, the average entropy, will be denoted simply by S and so on. The average entropy of the system is related to Pr through: S = −kB

∑

Pr ln(Pr )

(13.45)

Employing the definition of the average of a magnitude we can find, for instance, the average energy as: E=

∑ E r exp(−β E r ) ∂lnQ =− . ∑ exp(−β E r ) ∂β

(13.46)

From 13.45 and 13.46, an alternative definition for S follows: S = kB (ln Q + β E)

(13.47)

Replacing the values of S and E from (13.47) and (13.46) we obtain for the Helmholtz free energy: A = E − T S = −kB T ln Q

(13.48)

The chemical potential can be obtained from: (

∂A μ = kB T ∂N

) .

(13.49)

V,T

A very important conclusion is that all the state functions can be written in terms of ln Q, thus making ln Q a state function itself. Therefore, to solve a macroscopic problem in terms of a microscopic description it is necessary to calculate Q in the canonical description.

13.3.2 The Example of an Ideal Gas In many cases the internal contribution to the partition function is not the more interesting one. Let us consider the following example. Let us consider an atom of mass m, at the temperature T moving in a cube of size L 3 . Its energy in one direction x, in the state r, according to quantum mechanics, is: εrx =

h 2 n 2x , 8π L 2

(13.50)

13.3 Appendix 3. Elements of Statistical Thermodynamics

173

where h is Planck’s constant. The energy in the three directions is: ε = εx + εy + εz

(13.51)

Applying the definition and considering that, in view of the small separation of the energy levels, we can replace the sum by an integral and we can obtain the molecular partition function: ( q=

2π mkB T h2

)3/2 V

(13.52)

since V = L 3 . Q is obtained applying (13.38). The pressure can be obtained from: ( p = kB T

∂ ln Q ∂V

) = T,N

N kB T . V

(13.53)

That is the well-known state equation for ideal gases. The chemical potential results from (13.52), applying (13.48) and (13.49) and replacing V = Nk B T /p, and it results [( ] ) 2π mkB T −3/2 V μ = −kB T ln . (13.54) h2 kB T That should be compared with the familiar: ) ( μ = μ0 + kB T ln p/ p 0

(13.55)

From which it results: [( μ = −kB T ln 0

2π mkB T h2

)−3/2

] kB T .

(13.56)

Note that μ0 depends only on T. With the same arguments we could have shown that for a two-dimensional ideal gas (see Chap. 5 for the nomenclature): π A = N kB T We will employ this equation also in Sect. 13.5.

(13.57)

174

13 Appendices

13.3.3 Subsystems The subject presented in this section serves of theoretical support to most of the material shown in this book. Thus, for instance, it justifies the fact that when considering the acid–base titration of a poly acid, we concentrate only in the binding process without paying attention to what happens in the rest of the system. Below, we will also consider simpler examples. Under certain circumstances it will be convenient to consider small parts of a system. Following Hill [9], we will call these parts a subsystem. These may be atoms, molecules or a macroscopic part of the system. The only condition is that the subsystems must be independent. Under this assumption it may be shown the partition function of the system is the product of the partition functions of each subsystem and that the extensive state functions (A, S, E, etc.) are the sum of the thermodynamic functions of the subsystem (see Hill, Sect. 3.1). For the sake of simplicity let us consider a gaseous mixture containing some class of adsorbent molecules and adsorbate molecules. We are interested only in the adsorption isotherm that is the adsorbent–adsorbate equilibrium. This will be our subsystem, no matter what happens in the rest of the system. Then it may be shown that, if there is one adsorbate per adsorbent particle, it follows the Langmuir isotherm (Hill, Sect. 7.3). The case of solutions is also considered by Hill (Sect. 19.2). This would clear the surprising fact that the even for molecules moving in a liquid the equilibrium between these particles and other species in the solution can be represented by a Langmuir-type isotherm.

13.4 Appendix 4. The Binding Polynomial as the Partition Function of the Bound Species. Ghost−Site Binding Constants In Chap. 8, we have seen that in the case of the protonation of a multibase the expression for the analytical concentration becomes: ] [ ] [ ] [ ] [ CA = [Hn A] + Hn−1 A− + Hn−2 A2− + · · · + HA(n−1)− + An− ∑[ ] H j A(n− j)− . (13.58) = j

And this was written as ] [ ] [ ]2 [ ]n [ CA = An− (ka1 H+ + ka1 ka2 H+ + · · · + ka1 ka2 . . . kan H+ ⎛ ⎞ ∑ [ ]j [ n− ] = A ⎝1 + kajH H+ ⎠. (13.59) j

13.4 Appendix 4. The Binding Polynomial as the Partition Function …

175

It is seen at once that the expression between parenthesis represents all the possible states the multiacid may have. In the same way as it was done for the Langmuir Isotherm, the state of the empty surface (the totally deprotonated species in the present case) is counted as one. Q =1+

∑

[ ]j kajH H+ .

(13.60)

j

The expression in Eq. (13.60) is also called the binding polynomial (BP). Note also as it happens in this type of problems (Hill, [9]) the average number of bound species is found by derivating the logarithm of the partition function with respect to the activity, in this case [H+ ], and multiplying by [H+ ]. [ ]j [ +] jkajH H+ H ∂Q ∂lnQ [ +] = [ +] . = ∑ H[ +] j = Q ∂ H ∂ln H 1 + j kaj H ∑

j

(13.61)

This is Eq. (8.14). Note that the binding polynomial is a polynomial of degree j in [H+ ]. If we could find the roots a1 , a2 , …, aj , of the polynomial, we could write it as: Q=

([

] )([ ] ) ([ ] ) H+ − a1 H+ − a2 . . . H+ − aj

(13.62)

Finding from this expression, it results: ) ) ) ( ( ( ∂ln [H+ ] − a2 ∂ln [H+ ] − a j ∂ln [H+ ] − a1 = + + ··· + . ∂ln[H+ ] ∂ln[H+ ] ∂ln[H+ ]

(13.63)

Performing the derivations, it results: =

[H+ ] [H+ ] [H+ ] + + + ··· + + . [H ] − a1 [H ] − a2 [H ] − aj +

(13.64)

This can be written as: )[ ] [ ] [ ] ( −1/aj H+ (−1/a1 ) H+ (−1/a2 ) H+ )[ ] . ( [ ]+ [ ] + ··· + = 1 + (−1/a1 ) H+ 1 + (−1/a2 ) H+ 1 + −1/a j H+ (13.65) Note that each individual term has the form of a Langmuir isotherm with constants α = (−1/a1 ), β = (−1/a2 ) and so on. That is: [ ] [ ] α H+ β H+ [ [ ] + ··· ] = + 1 + α H+ 1 + β H+

(13.66)

176

13 Appendices

This equation corresponds to a system of n independent sites each one with a different binding constant. Since the constants α, β, and so on, may be complex numbers these are called virtual or ghost-site binding constants.

13.5 Appendix 5. The Gibbs Adsorption Isotherm. Two-Dimensional State Equations This appendix is almost entirely oriented to the field of adsorption. For finding the surface state equation it is convenient to discuss the so-called Gibbs Adsorption Isotherm and in doing so it is unavoidable to consider the presence of an interface.

13.5.1 The Gibbs–Duhem Equation Let us consider the total Gibbs free energy, G, of one phase with n components at constant T and p: G=

∑

n i μi

(13.67)

i

The total differential of G is: dG =

∑

n i dμi +

∑

i

μi dn i

(13.68)

i

But dG =

∑

μi dn i

(13.69)

i

equating the two expressions: 0 = ∑i n i dμi

(13.70)

This is the Gibbs–Duhem equation, which express that not all the chemical potentials can be varied independently.

13.5 Appendix 5. The Gibbs Adsorption Isotherm. Two-Dimensional State …

177

13.5.2 The Gibbs Adsorption Isotherm Let us now consider a system of n components, two phases, α and β, and one interface, σ, at T and p constants. Because the system is at equilibrium: β

μσi = μi = μαi = μ

(13.71)

We must include in G the surface work: γ A, where γ is the surface (or interfacial) tension and A the area of the interface. Then: ∑ ∑ β ∑ G= n σi μi + n i μi + n αi μi + γ A (13.72) i

i

i

Proceeding as in Sect. 13.5.1, we arrive to: −dγ =

∑

Γi dμi at T , p = constants.

(13.73)

i

where we have defined the surface concentration as Γ i = ni σ /A. This is the Gibbs adsorption isotherm. Because the surface tension of a solution is always smaller than that of the pure liquid, γ 0 , and that this is a constant at constant temperature, we may define the two-dimensional pressure, π = γ 0 − γ , so that dπ =

∑

Γi dμ =

∑

i

Γi dμ

(13.74)

i

Note that Eq. (13.74) is a thermodynamic relation between the two-dimensional pressure, the surface concentration and the chemical potential of the species we are considering, or the concentration in the external phase. Suppose we are dealing with an ideal two-dimensional gas for which the adsorption isotherm is: Γi = Γimax kp,

(13.75)

where Γ max is the maximum concentration at the surface. Note that θ = Γ i /Γ imax and we can define the area per molecule as Ai = 1/Γ i . Eliminating p between (13.74) and (13.75) it results: π Ai = kB T .

(13.76)

That is the two-dimensional state equation for an ideal surface gas. In the same way we may have found the state equation for the Langmuir case: π = −kB T Γmax ln(1 − θ ).

(13.77)

178

13 Appendices

Fig. 13.4 Representation of the two-dimensional ideal gas (solid line) and for the Langmuir (dashed line) models

In Fig. 13.4, we show the state equations, represented in the form π /k B TΓ max versus θ, for the two-dimensional ideal gas and for the Langmuir models.

13.6 The General Binding Process and the Binding Driving Force In Chap. 2, Sect. 2.4, we considered a general binding process S∗ + B ↔ SB.

(13.78)

where S* (the substrate) is a species with unbound sites (unoccupied sites), SB means the bound state (occupied sites) and B means the free binding species. The purpose of this section is to analyse in more detail the change of the free energy of process 13.80 and the corresponding affinity. We will analyse this substrate-binding species equilibrium for the case of 1:1 binding. From a chemical equilibrium point of view, at T and p constant, binding proceeds if it leads to a decrease of the Gibbs free energy, ∆G. Then, binding takes place if the final state (SB) has lower free energy than the initial state (S* + B). The driving force is defined as the Affinity, A, of the reaction. In turn, the affinity is defined as: A = −(dG/dξ ) [10], where ξ is the extent of the reaction. Thus if the driving force is positive reaction (13.78) will proceed to the right and vice versa. At a given condition, the binding process could have been produced to a certain extent ξ. The number of moles of free B species is then, n B = n 0B + νB ξ,

(13.79)

13.6 The General Binding Process and the Binding Driving Force

179

where ν B means the stoichiometric number (−1 in the present case). Correspondingly, the number of free binding sites (unoccupied) is n S∗ = n 0S∗ − ξ.

(13.80)

And the number of occupied binding sites is n SB = n 0SB + ξ.

(13.81)

In terms of the binding extent, then (n 0SB + ξ )G m (SB) + (n 0S∗ − ξ )G m (S∗ ) + (n 0B − ξ )G m (B),

(13.82)

where molar Gibbs free energies were written for every species. Equilibrium requires the Gibbs free energy to reach a minimum as a function of the extent of the reaction [11]. That is dG =0 dξ

(13.83)

(dG/dξ ) = G m (SB) − G m (S∗ ) − G m (B) = 0.

(13.84)

which means

The expression −(dG/dξ ) represents a driving force [12] as the process proceeds towards binding when −(dG/dξ ) > 0, it reaches equilibrium when −(dG/dξ ) = 0 and it shifts the unbound state when −(dG/dξ ) < 0. Equivalently, the term ( ) − G m (SB) − G m (S∗ ) − G m (B) = −ΔB G m 0

(13.85)

can be considered a driving force for the binding process to occur. In general, dn i = νi dξ and dG T,p =

∑ i

μi dn i =

∑

μi νi dξ = ΔB Gdξ,

(13.86)

i

where μi refers to the chemical potential of the ith species, which is equivalent to the molar Gibbs free energy for this species [11]. Therefore, in a general form, the driving force (or affinity) can be written as −(dG/dξ ) = −ΔB G = −

∑ i

μi νi .

(13.87)

180

13 Appendices

Using the general binding process (13.80) we can express the average relative number of moles of bound species as =

moles of bound B [SB] = [ ∗] moles of substrate species S + [SB]

(13.88)

whereas the binding species coverage, θ B , can be defined as the fraction of occupied sites; that is θB =

moles of bound B . moles of binding sites

(13.89)

Of course, these expressions do coincide for the case in which each substrate entity has a single binding site. However, in the case of multiple equivalent sites per substrate, the binding process can be described by the following chemical reaction S∗ + nB ↔ SBn

(13.90)

In this case, other binding states could be also possible, as SB1, SB2 , …, SBn-1 . Then, the average relative number of bound B becomes moles of bound B [S B1 ] + 2[S B2 ] + · · · + n[S Bn ] = ∗ moles of substrate species [S ] + [S B1 ] + [S B2 ] + · · · + [S Bn ] ∑n n[S Bn ] , (13.91) = ∑1n 0 [S Bn ]

=

where we have defined [SB0 ] = S∗ . In this case, definition (13.90) leads to moles of bound B [S B1 ] + 2[S B2 ] + · · · + n[S Bn ] = moles of binding sites n([S ∗ ] + [S B1 ] + [S B2 ] + · · · + [S Bn ]) ∑n 1 n[S Bn ] . (13.92) = ∑ n n0 [S Bn ]

θB =

By comparing Eqs. (13.91) and (13.92), θB =

. n

(13.93)

References

181

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Klotz IM (1997) Ligand-receptor energetics. A guide for the perplexed. Wiley, New York Saroff HA (1994) J Chem Ed 71:637 Woodbury CP (2008) Introduction to macromolecular binding equilibria. CRC, Boca Raton Wyman J, Gill SJ (1990) Binding and linkage. University Science Books, Mill Valley Hernández MT, Montero J (1997) Chem Ed 74:1311 Perrin DD, Dempsey B, Serjeant EP (1981) pKa prediction for organic acids and bases. Chapman and Hall, London, p 16 Davidson N (1962) Statistical thermodynamics. McGraw−Hill, New York Dickerson RE (1969) Molecular thermodynamics, W. A. Benjamin, Menlo Park Hill TL (1960) An introduction to statistical thermodynamics. Addison−Wesley Pub Co Muller P (1994) Glossary of terms used in physical organic chemistry. Pure Appl Chem 66:1077–1184 Levine IN (1995) Physical chemistry, 4th edn. Mc Graw-Hill, New York Everett DH (1960) An introduction to the study of chemical thermodynamics. Longmans, Green and Co, London, Sect. 7−7

Index

A Acid base Brönsted, 7 Lewis, 1, 12 Adsorption, 4, 13 heterogeneous, 2, 4 isotherm Bragg-Williams, 72 ideal gas, 3 Ising, 69, 70, 113 Langmuir, 3, 15, 30, 67 mobile, 3 Aminoacids, 117, 142, 143, 147 Aminophenol, 28 Ampholyte, 25, 28–30, 46, 117, 119, 131, 135 Average, 4, 8–10, 12, 24, 26, 28, 30–32, 34, 37, 42–44, 46, 71, 74, 80, 97, 98, 100–102, 104, 107, 110, 115, 117, 120–122, 135, 136, 141, 142, 144, 169, 170, 172, 175, 180

B Binding between protonation and complexation, 25 competitive binding, 2, 4, 12, 15, 23–25, 103, 104 polynomial, 99, 117, 119, 121, 122, 124, 174, 175

C Charge screening, 132

Ciprofloxacin, 166, 167 Complexation, 3, 4, 12–15, 19, 24–26, 44, 103, 105 Constants association, 7–11, 26, 37, 98, 99, 105, 120 dissociation, 9, 10, 37, 40, 43, 100, 102, 108, 155, 159 macroconstants, 163, 165, 168 microconstants, 33, 34, 163, 166, 168 operational, 7, 10, 15 Cooperativity, 2, 70 Coordination, 1, 72, 89, 141, 151 Coupling, 92, 93, 107, 148, 149, 151–154, 157

D Deformation, 79, 88, 90–94, 107, 148, 149 Degree, 3, 4, 8–11, 13–17, 20, 24, 64, 66, 83, 95, 98, 100, 108–113, 144, 148, 149, 154, 156, 158, 160, 175 Dissociation, 3, 4, 7, 9–12, 38, 63, 71, 91, 100, 108, 109, 112, 113, 119, 120, 129, 132, 144, 147, 166, 168 Distribution functions, 55 cumulative and density, 55 Dirac, 57, 58, 60 Gauss, 56, 59, 60, 64 Maxwell-Boltzmann, 56 Uniform, 84, 86 of equilibrium constants, 4, 14, 25, 29, 33, 40, 60, 74, 98, 104, 120–123, 129, 137 Driving force, 16, 19, 20, 178, 179

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. A. Marmisollé and D. Posadas, Binding Phenomena, Physical Chemistry in Action, https://doi.org/10.1007/978-3-031-39736-3

183

184 E Electrochemically Active Macromolecules (EAM), 80, 147, 148, 152 Electromotive force, 17, 47 Electron binding, 15–18, 27, 151, 153, 157 Electron potential, pe, 18, 28 Electrostatic screening, 148 Electrostatic work, 85, 87, 112, 144 End-to-end distance, 79, 95 Energy of adsorption, 60, 63–65, 73

F Formation, 3, 12–14, 30, 43, 45, 70, 78, 79, 88, 104, 105, 137, 156 Fraction volume, 88

G Glycine, 28, 31–34, 52, 53, 117

H Half-reactions, 50 Heterogeneity, 2, 59, 64 Humic substances, 63, 117, 141

I Insulin, 103, 117, 142, 143 Interaction electrostatic, 75, 82–84, 87, 88, 92–94, 110, 111, 148, 158 of charged cylinders, 87 of charged spheres, 84 energy, 68, 89, 156, 157, 159

M Macromolecular deformation, 79, 93, 94 folding, 80, 108 stretched, 80 Mixing energy, 89 entropy, 88 parameter, 89, 90 Model Bragg-Williams or Mean Field, 72, 74 Debye- Huckel, 82, 85 Ising, 4, 68, 71, 74, 113, 115 Langmuir, 15, 60, 178

Index N Node, 13, 65, 66, 68, 88, 89

O O- aminophenol, 28, 29, 46 Os−PVP, 149, 150

P Poisson-Boltzmann, 85, 127 Poly acids, 3, 4, 83, 97, 100–104, 107, 108, 111, 119, 124, 128, 135, 168, 169 aniline, 147, 153 bases, 4, 70, 83, 97, 101, 103, 107, 108, 110, 119, 124, 135 complexants, 103 electrolytes, 4, 77, 80, 82–84, 88, 91, 92, 95, 107, 109, 110, 112, 117, 136–138, 147, 148, 153 methacrylic acid (PMA), 107–110, 117, 135, 136 vinylamine (PVA), 71, 95, 107, 110–114, 117, 135, 136 vynilpyridine (PVP), 97, 136, 150 zwitterions, 135 Polymers flexible, 78–80 gels, 77, 88 swelling, 90 Potential chemical, 66, 67, 69, 73, 89, 91, 171–173, 176, 177, 179 electric, 17 Galvani, 17 isolated electrode, 17 Pressure osmotic, 89–91 two dimensional, 3, 67, 177 Protonation, 3, 8, 9, 11, 12, 18, 20, 23–25, 91, 97, 109–112, 119, 120, 126, 128, 141, 148, 154, 158, 160, 163–165, 168, 169, 174

R Radius of gyration, 80, 81 Redox couple, 16 reaction, 4, 7, 10, 15, 18, 47, 151, 153, 157

Index S Structure globular, 84 Surface charge, 87, 118, 119, 124, 125, 127, 128, 131, 141 heterogeneous, 59 homogeneous, 2, 9 of colloidal oxides, 119, 124

T Theory Flory-Huggins, 88 Thermodynamics statistical, 3, 20, 80, 170 Titrations additivity of, 40, 125

185 curves (TC), 37–43, 45, 46, 50, 53, 60, 70, 72, 83, 97, 99, 107, 109, 112, 113, 115, 124, 135, 136, 140–143 electron, 50, 147, 155 hydrogen chloride, 38 of ampholytes, 46 of colloidal oxides, 40 of magnetite, 5, 124 of mixtures, 41, 43 of proteins, 142 of zwitterions, 52, 135 phosforic acid, 38 redox, 37, 46, 151, 152 with complex formation, 43 Trimethylenediamine, 112, 113

Z Zwitterions, 30, 31, 34, 52, 135, 137