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Table of contents :
An Introduction to Applied Statistical Thermodynamics(incomplete)
An Introduction to Applied Statistical Thermodynamics(incomplete)
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Citation preview

An Introduction to

Applied Statistical Thermodynamics

Radial distribution function for the Lennard-Jones 12-6 potential at po’=0.93 produced by molecular dynamics simulation using a MATLAB® program that accompanies this book

Stanley I. Sandler

ISBN

978-0-470-91354?-5

COTE a www.wiley.com/college/sandler

\\

|

||

9I7804701913475

il

An Introduction to Applied Statistical Thermodynamics

Stanley I. Sandler University of Delaware

WILEY

John Wiley

& Sons, Inc.

VP & Executive Publisher:

Don

Acquisitions Editor:

Jenniter Welter

.

Editorial Assistant: Marketing Manager: Designer: Production Manager:

Alexandra Spicehandler Christopher Ruel Seng Ping Ngieng Janis Soo



Senior Production Editor:

Joyee Poh

Fowley

This book was set in 10.85/12 Times Roman by Laserwords Private Limited and printed and bound by Hamilton Printing. The cover was printed by Hamilton Printing. This book is printed on acid free paper,

Founded in 1807, John Wiley & Sons, Inc. has been a valued source of knowledge and understanding for more than 200 years, helping people around fulfill their aspirations. Our company ts built on a foundation responsibility to the communities we serve and where we live Corporate Citizenship Initiative, a global effort to address the

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Copyright © 2011 John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc. 222 Rosewood Drive, Danvers, MA 01923, website www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201)748-6011, fax (201)748-6008, website http://www. wiley.com/go/permissions,

Evaluation copies are provided to qualifed academics and professionals tor review purposes only. for use in their courses during the next academic year. These copies are licensed and may not be sold or transferred to a third party. Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions and a free of charge return shipping label are avatlable at www.wiley.com/go/returnlabel. Outside of the United States, please contact your local representative.

Library of Congress Cataloging-in-Publication Data Sandler, Stanley [., 1940An introduction to applied statistical thermodynamics / Stanley [. Sandler.

p. cm. Includes index.

ISBN 978-0-470-9 1347-5 (pbk.) |. Statistical thermodynamics. 2. Thermodynamics—Industrial applications. |. Title. TP155.2.T45836 2010 621.402" |—-de22 2010034666 Printed in the United States of America lOORT7EG65432 |]

To Judith,

Catherine, Joel, And Michael

About the Author

Stanley I. Sandler is the H. B. du Pont Chair of Chemical Engineering and Professor of Chemistry and Biochemistry at the University of Delaware. He was department chair from 1982 to 1986 and Interim Dean of the College of Engineering in 1992. He earned the B. Ch. E degree in 1962 from the City College of New York, and the Ph. D. in chemical engineering from the University of Minnesota. He has been a visiting professor at Imperial College of London, the Technical University of Berlin, the University of California-Berkeley, the University of Queensland (Australia), the National University of Singapore and the University of Melbourne (Australia). In addition to this book, Professor Sandler is the author of the textbook “Chemical, Biochemical and Engineering Thermodynamics” 4" edition published by Wiley, over 350 research papers, the author of one and editor of another book on thermodynamic modeling, and the editor of several conference proceedings. He is also the editor of the AIChE Journal. Among his many awards and honors are the Camille

and Henry

Dreyfus Faculty-Scholar Award

(1971);

a Research

Fellowship (1980)

and U. S. Senior Scientist Award (1988) from the Alexander von Humboldt Foundation (Germany); American Society for Engineering Education Lectureship Award (1988); the Professional Progress Award (1984); the Warren

K. Lewis

Award

(1996)

and Founders Award (2004) from the American Institute of Chemical Engineers; the E.

V.

Murphree

Award

(1996)

from

the

American

Chemical

Society;

the

Rossini

Lectureship Award (1997) from the International Union of Pure and Applied Chemistry; and election to the U. S. National Academy of Engineering (1996). He is a fellow of the American Institute of Chemical Engineers and the Institution of Chemical Engineers (Britain), and a Chartered Engineer (Britain).

s r o t c u r t s n I r o f Preface

GOALS

AND

MOTIVATION As the author of a widely used undergraduate textbook on thermodynamics (Chemical, Biochemical and Engineering Thermodynamics,

4th ed., by S. 1. Sandler, John

Wiley & Sons, Inc.) and a teacher of a graduate course in chemical engineering thermodynamics, | am frequently asked what I do teach in the graduate course. Its content has largely been influenced by our departmental policy of not accepting our own undergraduates into our graduate program. Consequently, I have found that the students in the graduate course have very varied backgrounds in thermodynamics. Some have had instruction in thermodynamics that has been axiomatic; others have had very applied courses; some have had only one course; others two; and still others

have had very little thermodynamics as they studied disciplines other than chemical engineering. Therefore, my first goal in the graduate thermodynamics course is to bring everyone to approximately the same level. I do this by covering much of the material in my two-semester undergraduate textbook in the first half of the one-semester graduate course. My second goal is to introduce the first-year graduate students to the fundamental ideas and engineering uses of statistical thermodynamics, the equilibrium part of the statistical mechanics, in the remainder of the semester. It is for this part of the course that | use the material herein, developed and refined over years of teaching the course.

APPROACH One-half of a semester is a short period in which to introduce statistical thermodynamics and some of its engineering and science applications, so compromises need to be made. The result, this book, will undoubtedly disappoint the true practitioners of statistical mechanics in its brevity; but I hope students will find it sufficiently interesting and useful that they may apply the insights and tools they gain to their own research, and perhaps pursue more rigorous courses devoted to the subject. Indeed, with the present emphasis on nano- and biotechnologies, molecular-level descriptions and understandings offered by statistical thermodynamics should be of increasing

interest. I also hope that those who do not wish to pursue research using statistical thermodynamics will have gained an understanding of its utility and how they apply it in their work. What is not presented here is complete theoretical rigor in introducing statistical thermodynamics. My view, with which you may or may not agree, is that in a short introduction to statistical thermodynamics for students unfamiliar with the subject, there is greater value in showing how it can be useful than in dwelling on the fine details of derivations, These are best left to a full course on the subject. Consequently, while terms such as phase space are mentioned in the text, I do not dwell on such concepts or rigorously derive the fundamentals of ensembles from such

a basis.

The

reader

is referred

to the

excellent

(and

much

larger)

textbooks

by Tolman, Hill, McQuarrie, and others listed at the end of these prefaces for such

Instructors

1s This ied. appl more h muc is here oach appr the ead, Inst s. tion enta pres rous rigo perhaps best evidenced by, for example, the analysis of the virial equation of state first discussed in Chapter 7. There is found a development of the second virial coefficient that is similar to most other statistical mechanics books. However, in later chapters, I show how the exact composition dependence of the second virial coefficient derived from statistical thermodynamics has become the basis for mixing rules used with common equations of state such as the van der Waals, Peng-Robinson, and other cubic

equations of state commonly used by chemical engineers and physical chemists. Then, in Chapter 12, the extension to the osmotic virial equation applicable to colloidal and protein solutions is introduced, and I discuss how the osmotic virial coefficient has been used to identify solution conditions for protein crystallization. Another example 1s the derivation of the Debye-Htickel limiting law for electrolyte solutions in Chapter 15. I then show how this has formed the basis for the development of electrolyte solution equations used in engineering. In the final chapter of this book

is an analysis that allows the reader to understand

the statistical thermodynamic

assumptions that underlie the well-known equations of state and activity coefficient models commonly used in engineering and the sciences. Computer simulation has become increasingly Important in research and in obtaining insight into physical processes at the molecular level. This book provides an introduction to the simplest forms of Monte Carlo and molecular-dynamics simulation (albeit only for simple spherical molecules) and user-friendly MATLAB®! programs for doing such simulations, as well as some other calculations. Only equilibrium properties are considered in this book, not dynamic or kinetic properties like the kinetic theory of gases or liquids. Therefore, statistical thermodynamics is the accurate description for the contents here, rather than the more general term of statistical mechanics. In summary, the purpose of this book is to provide a readable introduction to statistical thermodynamics and show its utility—how the results obtained lead to useful

generalizations for practical application. The book also illustrates the difficulties that arise in the statistical thermodynamics liquids.

of dense fluids as seen in the discussion of

EDGMENTS There are many people whose assistance, direct and indirect, have contributed to this book. First and foremost is my wife, who over the years has put up with me busily working away in seclusion in my offices at home and abroad. Next are colleagues who have seen various versions of the manuscript for this book and have offered their comments and corrections, and especially Professor Shiang-Tai Lin of National Taiwan University. Zachery Ulissi, a University of Delaware student, converted a FORTRAN Monte Carlo program for the Lennard-Jones fluid made available to us by Professors Daan Frenkel (Cambridge University) and Berend Smit (University of California, Berkeley) into a user-friendly MATLAB® format. Zach also wrote the MATLAB®-based molecular dynamics for that potential. Professor Lester Woodcock of the University of Manchester, while a visiting professor at the University of Delaware, provided the excellent FORTRAN programs for the Monte Carlo sim-

ulations of the square-well, hard-sphere, and Lennard-Jones fluids, and a molecular

'MATLAB®

is a registered trademark of The MathWorks,

Inc.

Preface

for Instructors

vii

dynamics program for the Lennard-Jones fluid that University of Delaware student Meghan McCabe adapted into the MATLAB® format. Professor Jaceon Chang of the University of Seoul provided help in the development of the MATLAB® program for the solution of the Percus-Yevick equation for hard-spheres. Special thanks must go to my colleagues at the University of Delaware for their support while this book was being written, and at the University of Melbourne (Australia) for providing an office and other assistance for a month each year that allowed me to work undisturbed. I also need to acknowledge the forbearance of the students over the years who have put up with the early versions of the manuscript, which has largely been rewritten and improved based on their very helpful comments. Finally, | want to thank my editor Jennifer Welter and the production staff at John Wiley & Sons, Inc. for their continued help and encouragement.

The following resources accompanying www.wiley.com/college/sandler. For students

this

book

are

available

on

the

website

and instructors:

The collection of MATLAB®

programs

for some

statistical thermodynamic

cal-

culations, including Monte Carlo and molecular-dynamics simulations described

on page 1x.

For instructors only:

Solutions Manual containing complete solutions to the problems in the text. Image Gallery containing illustrations from the test in a format appropriate to include in lecture slides. These last resources are only available to instructors adopting this text for a course. Visit the instruction section of the website www.wiley.com/college/sandler to register for password access to these resources. Stanley I. Sandler September 20]0

‘face for Students

While classical thermodynamics can be used to describe many processes very well, such as phase behavior, chemical reaction equilibria, and interrelating heat and work flows on changes of state, it barely acknowledges the existence of molecules. In that sense, classical thermodynamics

is not complete.

Indeed, to apply classical thermo-

dynamics, we need to know the properties of a substance or a mixture, such as its internal energy, enthalpy, Gibbs energy and heat of formation, and the parameters to be used in equation-of-state or activity coefficient models. However, classical thermodynamics does not provide us with a path to calculate the values of these parameters from knowledge of the molecules in the fluids of interest. Nor does classical thermodynamics provide any information on the underlying basis or assumptions for an equation of state such as that of van der Waals, or the activity coefficient models used by chemists and engineers. Statistical thermodynamics, which starts with a description of individual molecules, can provide such information. For molecules that do not interact, which lead to an understanding of the ideal gas, the road is a straightforward one. However, as you will see in this book, the analysis for molecules that interact (especially in a dense fluid such as a liquid) is very much more complicated, and generally cannot be solved exactly. Nonetheless, we can obtain useful insights by starting with a molecular-level description and statistical thermodynamics. However, given the specific purpose for this book as an addition to a course on classical thermodynamics, only equilibrium properties are considered here, not dynamic or kinetic properties like the kinetic theory of gases or liquids; hence the use here of the term statistical thermodynamics rather than the more general term statistical mechanics. Statistical thermodynamics Is also not complete because some of the parameters we need, such as bond lengths, bond energies, interaction energies, etc., come from an even deeper look at molecules from various forms of spectroscopy and computational

quantum mechanics. Those subjects are beyond the scope of this introductory text, so here we will assume that such information is available when needed. With the present emphasis on nano- and biotechnologies, molecular-level descriplions, computer simulation, the understandings arising from statistical thermodynamics, and the ability to use molecular-level arguments to make useful predictions are all of increasing interest and importance in chemical engineering and physical chemistry. [ hope the presentation here will be sufficiently interesting to students that some may

be encouraged to apply it to their own research, and perhaps even to study the subject further. But I also hope that even those who do not wish to pursue further study of statistical thermodynamics will have gained some appreciation for the subject and its utility, and a better understanding of the limitations and nuances of the generalizations they apply in their work. As computer simulation is of increasing importance in research and in obtaining insight into what is occurring at the molecular level, this book provides an

Preface for Students

introduction to the simplest forms of Monte Carlo and molecular-dynamics simulation, for simple spherical molecules. There are user-friendly MATLAB®” programs for doing such simulations and some other statistical thermodynamic calculations that

can be downloaded from the website for this book www.wiley.com/college/sandler. Stanley I, Sandler

September 2010

“MATLAB®

is a registered trademark of The MathWorks, Inc.

k o o b is th y n a p m o c c a at MATLAB®® programs th 1. LJ_Virial

aer mp te nd co se d an st fir its d an t en ci fi ef co al ri vi nd co se For computing the er ld fo in is e) fil (M m ra og Pr d. ui fl -6 12 s ne Jo dar nn Le ture derivatives for the LJ_ virial.

MC_Squarewell*

d re te en ta Da d. ui fl l el -w re ua sq e th r fo m ra og pr on ti la mu si o rl Ca A Monte 1s ut tp ou l ca ri me nu d an en re sc on s ar pe ap ut tp ou l ca hi ap gr through a GUI, t pu in on | to l ua eq , R, r te me ra pa h dt wi ll we e th g in tt se By t. in a spreadshee er ld fo in is e fil M d. ui fl e er ph -s rd ha e th r fo ns io at ul lc ca es do m ra this prog

MC_sqwell.

MC_LJ*

ta Da d. ui fl -6 12 s ne Jo dar nn Le e th r fo m ra og pr on ti la mu si o rl Ca A Monte ut tp ou l ca ri me nu d an en re sc on s ar pe ap entered through a GUI, graphical output J. _L MC er ld fo in is e fil M t. ee sh ad is in a spre

MD_LJ*

s ne Jo dar nn Le e th r fo m ra og pr on ti la mu si cs mi na dy rla cu le mo al rm he ot An is d an en re sc on s ar pe ap ut tp ou l ca hi ap gr I, GU a h ug ro th d re te en ta Da d. ui fl -6 12 J. _L MD er ld fo in is e fil M t. ee sh numerical output is in a spread

LJ_MD_MC** A program

for the Lennard-Jones

12-6 fluid that does both Monte Carlo and

ta Da . ns io it nd co e at st me sa e th r fo s isothermal molecular-dynamics simulation ut tp ou l ca ri me nu d an en re sc on s ar pe ap ut tp ou l ca hi ap gr I, GU a h ug ro th d re te en . C M _ D M _ ) L er ld fo in is e fil M is in a spreadsheet.

MD_LJ2° d ui fl -6 s 12 e n o J d r a n n e L e th r fo m a r g s o c r p An isothermal molecular-dynami . 2 J L _ D M er ld fo in is le fi M . on ti bu ri st di d ee sp e th es at ul lc ca so al at th PYHS fluid hard-sphere the for function A program for computing the radial distribution and screen the on appears ouput Graphical in the Percus-Yevick approximation. HS. Percus-Yevick folder in is file M numerical output in a spreadsheet. PYHSPMF

mean of potential and function distribution A program for computing the radial

Graphical approximation, Percus-Yevick the in force for the hard-sphere fluid

in is file M spreadsheet. a in ouput output appears on the screen and numerical folder PY HS PMF. . Inc s, rk Wo th Ma e Th of k ar em ad tr ed er st gi re ‘MATLAB® is a ed us g in be is d an ck co od Wo lie Les r so es of Pr of 4This MATLAB® program is based on a FORTRAN program re wa la De of ty si er iv Un the by ne do en be has ® with his permission. The program conversion to MATLAB undergraduate Meghan McCabe. hem m.c lsi /mo p:/ hit e sit Web the on ms gra pro N RA RT FO ‘These programs in MATLAB® are based on the lar ecu Mol g din tan ers Und t, Smi B. and l nke Fre D. by uva.nl/frenkel_smit that accompanies the excellent book ng hei are ms gra pro The 1. 200 , don Lon ss, Pre ic dem Aca ed. Simulation, From Algorithms to Applications, Ind e hav ® AB TL MA to ns sio ver con m gra pro The t. Smi used with the permission of Daan Frenkel and Berend . ssi Uli y her Zac e uat rad erg und re awa Del of y sit ver Uni by been done of ty si er iv Un by d pe lo ve de was id flu s ne Jo dar nn Le the for de co ® AB TL MA cs mi na dy rla cu le mo ©This Delaware undegraduate Zachery Ulissi.

A Partial List of the Many Books on Statistical Thermodynamics and Statistical Mechanics

This text focuses on the applications and practical utility of statistical thermodynamics.

For further study, including detailed derivations of the fundamental concepts statistical mechanics, you may want to consult the following books, among others. As these books have generally been written by chemists, phyicists and mathematicians, these books focus on the science, and do not include the practical applications that are the focus of this book. Statistical Mechanics,

SH2aeP

D. A. McQuarrie, 2000. . Davidson,

University

McGraw-Hill,

Sratistical Mechanics,

L. Hill, Statistical Mechanics,

McGraw-Hill,

New

New

York,

1938.

1962.

1956.

Addison-Wesley, Reading,

1960.

. Kestin

New

Sausalito, CA,

Press, London,

York,

L. Hill, An Introduction to Statistical Thermodynamics,

MA, —

Oxford

C. Tolman, Statistical Mechanics,

Science Books,

University

and

J. R.

York,

1971.

Dorfman,

A Course

in Statistical Mechanics,

Academic

Press,

St

. E. Mayer and M. G. Mayer, Statistical Mechanics, Wiley, New York, 1940. E. Schrédinger, Statistical Thermodynamics, Cambridge University Press,

Cambridge, 1952. L. D. Landau and E. M. Litshitz, Statistical Physics, Pergamon Press, London,

1958. G. 8S. Rushbrooke, Statistical Mechanics, Oxtord University Press, London, 1949. H. Eyring, D. Henderson, B. J. Stover, and E. M. Eyring, Statistical Mechanics and Dynamics,

Wiley,

New

York,

1964.

R.

H. Fowler and E. A. Guggenheim, Statistical Thermodynamics, Cambridge University Press, Cambridge, 1956. E. A. Mason and T. H. Spurling, The Virial Equation of State, Pergamon, New York, 1969. J. S. Rowlinson

and

worths, Oxford, R. A.

Robinson

and

F. L. Swinton,

Liguids and Liquid Mixtures,

3rd ed., Butter-

1982. R.

H.

Stokes,

Electrolyte Solutions,

2nd ed., Academic

Press,

New York, 1965. K. A. Dill and S. Bromberg, Statistical Thermodynamics in Chemistry and Biology, Garland

Science, New

York, 2003.

D. Chandler, /ntreduction to Modern Statistical Mechanics, Oxford University Press, London, 1987. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1989. D, Frenkel and B. Smit, Understanding Molecular Simulation, From Algorithms to Applications, 2nd ed., Academic Press, London, 2001. A. R. Leach, Molecular Modeling: Principles and Applications, 2nd ed., PrenticeHall, 2001.

Contents

INSTRUCTORS

PREFACE

FOR

PREFACE

FOR STUDENTS

CHAPTER

1

INTRODUCTION l.] Le 1.3 1.4 l.5

CHAPTER

2

TO STATISTICAL THERMODYNAMICS

Probabilistic Description — | 2 Macroscopic States and Microscopic States Quantum Mechanical Description of Microstates The Postulates of Statistical Mechanics 5 The Boltzmann Energy Distribution 6

PARTITION

3

FUNCTION

THE

CANONICAL

2.1 fuk

Some Properties of the Canonical Partition Function

9

Relationship of the Canonical Partition Function to Thermodynamic

Properties 11 Canonical Partition Function for a Molecule with Several Independent 2.3 Energy Modes 12 2.4 Canonical Partition Function for a Collection of Noninteracting Identicz | Atoms 13 [5 Chapter 2 Problems

CHAPTER

3

THE IDEAL MONATOMIC GAS a] Canonical Partition Function for the [deal Monatomic Gas’ 16 Aa Identification of B as 1/kT 18 Duss General Relationships of the Canonical! Partition Function to Other Thermodynamic Quantities 19 3.4 The Thermodynamic Properties of the Ideal Monatomic Gas 22 Energy Fluctuations in the Canonical Ensemble 29 aad 3.6 The Gibbs Entropy Equation 33 Sad Translational State Degeneracy 35 Distinguishability, Indistinguishability, and the Gibbs’ Paradox 37 3.8 3.9 A Classical Mechanics—Quantum Mechanics Comparison: The Maxwell-Boltzmann Distribution of Velocities 39 Chapter 3 Problems 42

CHAPTER

4

THE IDEAL DIATOMIC AND POLYATOMIC GASES 4.] The Partition Function for an Ideal Diatomic Gas 44

4.2 4.3

4.la 4.1b

The Translational and Nuclear Partition Functions The Rotational Partition Function 45

4.lc

The Vibrational Partition Function

4.ld

The

Electronic

Partition Function

45

47 45

The Thermodynamic Properties of the Ideal Diatomic Gas The Partition Function for an Ideal Polyatomic Gas 53

49

55 The Thermodynamic Properties of an Ideal Polyatomic Gas 4.4 58 The Heat Capacities of Ideal Gases 4.5 Normal Mode Analysis: The Vibrations of a Linear Triatomic Molecule 4.6 62 Chapter 4 Problems

CHEMICAL

REACTIONS

IN IDEAL GASES

64 The Nonreacting Ideal Gas Mixture 65 Partition Function of a Reacting Ideal Chemical Mixture Three Different Derivations of the Chemical Equilibrium Constant in an Ideal Gas Mixture 6/7 70 Fluctuations in a Chemically Reacting System 5.4 73 The Chemically Reacting Gas Mixture: The General Case 5.5 80 Two Illustrations 5.6 83 Appendix: The Binomial Expansion 85 Chapter 5 Problems 5.1 5.2 5.3.

FUNCTIONS

OTHER

PARTITION

6.1 6.2 6.3. 6.4

Microcanonical Ensemble for a Pure Fluid 8&7 89 Grand Canonical Ensemble for a Pure Fluid Isobaric-Isothermal Ensemble 92 Restricted Grand or Semi-Grand Canonical Ensemble

The The The The

Comments on the Use of Different Ensembles 6.5 Chapter 6 Problems %6

INTERACTING

MOLECULES

93

94

IN A GAS

7.1. 7.2. 7.3. 7.4. 7.5 7.6 7.7 7.8

The Configuration Integral 98 Thermodynamic Properties from the Configuration Integral 100 The Pairwise Additivity Assumption 101 Mayer Cluster Function and Irreducible Integrals 102 The Virial Equation of State 109 Vinal Equation of State for Polyatomic Molecules 114 Thermodynamic Properties from the Virial Equation of State 116 Derivation of Virial Coefficient Formulae from the Grand Canonical Ensemble 118 7.9 Range of Applicability of the Virial Equation 123 Chapter 7 Problems 124

INTERMOLECULAR POTENTIALS AND THE EVALUATION OF THE SECOND VIRIAL COEFFICIENT %.1

Interaction Potentials for Spherical Molecules

8.2

The Second Virial Coefficient in a Mixture: Unlike Atoms 136

8.3

Interaction Potentials for Multiatom, and Colloids 137

8.4

Engineering Applications and Implications of the Virial Equation of State

140

Chapter § Problems

144

125 Interaction Potentials Betwee

Nonspherical

Molecules,

Proteins,

XV

Contents

147

MONATOMIC CRYSTALS 147 The Einstein Model of a Crystal 9.1 150 The Debye Model of a Crystal 9.2 Test of the Einstein and Debye Heat Capacity Models for a Crystal 9.3 159 Sublimation Pressure and Enthalpy of Crystals 9.4 161 A Comment on the Third Law of Thermodynamics 9.5 161 Chapter 9 Problems

) SIMPLE

LATTICE

MODELS

157

163

FLUIDS

FOR

10.1 10.2 10.3.

Introduction 164 165 Development of Equations of State from Lattice Theory Activity Coefficient Models for Similar-Size Molecules from Lattice Theory 168 10.4 The Flory-Huggins and Other Models for Polymer Systems — 172 178 10.5 The Ising Model 184 Chapter 10 Problems

INTERACTING MOLECULES IN A DENSE FLUID. CONFIGURATIONAL DISTRIBUTION FUNCTIONS 11.1 11.2

Reduced Spatial Probability Density Functions 185 Thermodynamic Properties from the Pair Correlation Function

11.3

The

11.4 11.5

11.6

Pair Correlation

Function (Radial

190

Distribution Function) at Low

194 Density Methods of Determination of the Pair Correlation Function at High Density 197 Fluctuations in the Number of Particles and the Compressibility Equation 199

Determination of the Radial Distribution Function of Fluids using Coherent X-ray or Neutron Diffraction

11.7

185

202

Determination of the Radial Distribution Functions of Molecular Liquids

210

Determination of the Coordination Number from the Radial Distribution Function 211 11.9 Determination of the Radial Distribution Function of Colloids and Proteins 213 Chapter || Problems 214 |

11.8

INTEGRAL EQUATION THEORIES DISTRIBUTION FUNCTION

FOR THE

12.1 12.2

The Yvon-Born-Green (YBG) Equation 216 The Kirkwood Superposition Approximation

12.3

The Ornstein-Zernike Equation

12.4 12.5 12.6

RADIAL 216

219

220

Closures for the Ornstein-Zernike Equation 222 The Percus-Yevick Hard-Sphere Equation of State 227 The Radial Distribution Functions and Thermodynamic Properties of Mixtures 228 12.7 The Potential of Mean Force 230 [2.8 Osmotic Pressure and the Potential of Mean Force for Protein and Colloidal Solutions 237 Chapter 12 Problems 239

xvi

Contents

CHAPTER

ON CTI FUN ION BUT TRI DIS IAL RAD THE OF N IO AT IN RM TE DE 13 241 AND FLUID PROPERTIES BY COMPUTER SIMULATION 13.1 13.2.

Introduction to Molecular Level Computer Simulation Thermodynamic Properties from Molecular Simulation

13.3. 13.4

249 Monte Carlo Simulation Molecular-Dynamics Simulation

Chapter

CHAPTER

13 Problems

14 PERTURBATION

242 245

253

255

257

THEORY

257 14.1 Perturbation Theory for the Square-Well Potential 14.2 First Order Barker-Henderson Perturbation Theory 262 14.3. Second-Order Perturbation Theory 265 14.4 Perturbation Theory Using Other Reference Potentials 269 272 l4.5 Engineering Applications of Perturbation Theory Chapter 14 Problems 274

CHAPTER

15 A THEORY OF DILUTE ELECTROLYTE AND IONIZED GASES [5.1

Solutions Containing Ions (and Electrons)

[5.2

Debye-Hiickel Theory

INDEX

276 276

280

15.3. The Mean Ionic Activity Coefficient Chapter 15 Problems 296 CHAPTER

SOLUTIONS

291

16 THE THE

DERIVATION OF THERMODYNAMIC MODELS FROM GENERALIZED VAN DER WAALS PARTITION FUNCTION 297 16.1 The Statistical-Mechanical Background 298 16.2 Application of the Generalized van der Waals Partition Function to Pure Fluids 301 16.3 Equation of State for Mixtures from the Generalized van der Waals Partition Function 310 16.4 Activity Coefficient Models from the Generalized van der Waals Partition Function 318 16.5 Chain Molecules and Polymers 329 16.6 Hydrogen-Bonding and Associating Fluids 332 Chapter 16 Problems 334

aon

Chapter 1

Introduction to Statistical

Thermodynamics INSTRUCTIONAL

OBJECTIVES

FOR CHAPTER

I

The goals of this chapter are for the student to:

¢ Understand e Understand Understand e Understand e Understand

1.1

PROBABILISTIC

the the the the the

probabilistic description used in statistical thermodynamics distinction between macrostates and microstates quantum mechanics description that will be used postulates of statistical thermodynamics derivation of the Boltzmann energy distribution

DESCRIPTION

The goal of statistical thermodynamics 1s to allow one to make predictions about the macroscopic properties of a system, such as its heat capacity, chemical equilibrium constant, equation of state, etc., using information only about the microscopic (or molecular) nature of the system. The methods used take advantage of the fact that the large numbers of molecules in any system of interest allows the use of statistics. An example of what we are trying to do, and some of the difficulties inherent in any molecular description, 1s evident by considering the pressure of a gas on its container. This pressure is the result of collisions of gas molecules with the container walls. The force exerted on the wall by any one collision 1s almost infinitesimal; there are, however, about 10°47 molecule-wall collisions per second for each square centimeter of surface for a gas at standard conditions. The result of so many collisions is a finite force or pressure. The pressure we measure, then, 1s an average over many, many molecular collisions. In fact, the measured pressure is a longtime average (on a molecular scale) of many molecular events. In a similar fashion, other macroscopic properties of a system can be related to longtime averages of the corresponding molecular processes. A direct and deterministic way to proceed with the development of a microscopic theory would be to use a calculational scheme based on following the trajectories (position and velocity) of each molecule

in the system. At each molecule-molecule or

molecule-wall collision, which occurs about every 10~'!' seconds for each molecule, new trajectories would have to be computed. Any macroscopic property could then be computed by calculating the appropriate longtime average of the appropriate microscopic property. For example, the gas pressure could be related to the average over

Introduction to Statistical Thermodynamics time of the force on the container due to molecular collisions. Such calculations have comusing time of s period short for and les molecu of rs numbe limited for done been puters; this technique is discussed in Chapter 13, but is not yet practical for routine engineering calculations. Furthermore, such calculations yield much more information than we actually need or, for that matter, want. One has little need for the location and velocity of each of the 10°* molecules in a liter of gas, with constant updating each

time a collision occurs,

when

one’s

interest is merely

with a small

number

of

average macroscopic properties such as the pressure, temperature, internal energy, heat capacity, Gibbs and Helmholtz energies, etc. In fact, to compute these average properties from the inestimable reams of computer output corresponding to, say, one second in the history of a real gas is an impossible task. Instead, a way to proceed would be to first reduce all the exact information into a suitably compact statistical form. A thermodynamic property of the system could then be computed as an appro-

priate statistical average. be compactly presented or kinetic energies. The from the average kinetic

For example, the information about particle velocities could in terms of the probability distribution for particle velocities temperature of a monatomic gas could then be computed energy.

An alternative approach to the above for the development of a microscopic theory is to start directly with a statistical or probabilistic description. That is, we no longer inquire about the velocity of each molecule, but only about the probability distribution of the velocities of all molecules. This is the procedure we shall follow here. What we are trying to determine are probability distributions and average values of properties when considering all possible states of the molecules consistent with the constraints on the overall

system—for

example

fixed temperature,

volume,

and

number of molecules; or fixed pressure, volume, and number of molecules, etc. In the language of statistical thermodynamics, the collection of possible states consistent with the constraints 1s referred to as the ensemble of states. Special names are given to these ensembles, depending on the constraints. For example, the canonical ensemble refers to all states consistent with fixed temperature,

volume, and number

of molecules. The microcanonical ensemble refers to all states consistent with fixed total energy,

volume,

and

number of molecules,

while the constraints

canonical ensemble are fixed volume, temperature, molar Gibbs energy).

ISCOPIC

STATES

AND

MICROSCOPIC

for the grand

and chemical potential

(partial

STATES

The macroscopic state of a gas can be completely specified by giving the numerical values for a small number of parameters, such as the temperature (or energy), volume, and number of molecules or moles. The classical mechanical description of the microscopic state of the fluid is much more detailed, in that the position vector and velocity vector of each particle would be specified—that is, the microscopic description would be a specification (r,, v). > Va.**+,f,.U,) where r; and v; are the position and velocity vectors of the i“ molecule in some suitable frame of reference. (When considering an “ideal gas,” that is, a gas of noninteracting particles in the absence of external fields such as gravity, the location of the molecules is unimportant, and therefore the description of a microstate of noninteracting molecules need not include position vectors.) In order for the microstate of a gas to be consistent with the observed macroscopic state, the following criteria must be met:

Description of Microstates

1.3 Quantum Mechanical

3

(a) the number of molecules in the microstate must be the same as the number of molecules in the macroscopic state;

(b) all the position vectors, r;, must be constrained to be within the volume V; and (c) the energy of the microscopic state must be equal to the energy of the macroscopic state, Clearly there are a very large number of microstates consistent with any macroscopic state of the system. That is, there are a very large number of position and velocity assignments for the molecules that are consistent with restrictions (a)—(c).

Any counting of microstates following only the prescriptions outlined above would overcount the number of microstates, in that we would be differentiating between microstates that are indistinguishable. For example, consider the two microstates (U),U5,U3°++,v,) and (v5, v),U3---:,v,) for a system of identical particles. The only difference between these two states is that in the second state particle 2 has the velocity that particle | had in the first state, and vice versa. In fact, there are n! ways this set of velocities could be assigned to the m identical particles. The question

then arises as to whether one can really distinguish between these n! different states. If the particles were of macroscopic size, we could identify each particle by, for example, painting a number on it, and therefore be able to distinguish between the different velocity assignments. For microscopic particles, however, the n! states must be considered identical and indistinguishable, and therefore should not be counted as separate microstates. This concept of indistinguishability of identical molecules is in accord with the Heisenberg uncertainty principle—that is, since we do not exactly Know the position and velocity of any molecule at any one time as a result of the amplification of initial uncertainties in solving the laws of mechanics, at any later time we would not know which molecule was which.

‘UM

MECHANICAL

DESCRIPTION

OF MICROSTATES

In the quantum mechanical description of molecular states not all values of the energy are allowed, only certain discreet values. For example, a single molecule or a particle in a cubic box of volume V and side L = V'!/? cannot have any possible energy, as is allowed by classical mechanics, but only values of the energy ¢ given by h?

>

5

4

e(ly. ly, Ll) = —— (I +h +6 ) nV “B ; ai

(1.3-1)

where /,,/,, and /- are integers that can take on the values 0, 1, 2, etc.; m is the

particle mass; and A = 6.62517 x 1077’ erg-sec is Planck’s constant. It is impor-

tant to distinguish between an energy state and an energy level. An energy state is a particular specification of quantum numbers. For a single particle in a box, the three almost equivalent quantum number assignments given below represent three distinguishable states.

l,

l,

I.

2 | |

| 2 l

| 2

Introduction to Statistical Thermodynamics same the or gy, ener the of e valu same the has s state gy ener e thes of each However, energy level

a .

he

anv

“ 249

h

12) — 92 4 424 amv 2 +! +1")

|

*

=

he

o,

(

BmV

og

3h

I? + 1° + 2°) ) = 4mV —— r

Therefore, we refer to this energy level as being threefold degenerate, that is, there are three energy states consistent with this energy level. In what follows, it will be necessary to enumerate the states of a system, for example, in order of increasing energy. This can be done in either of two equivalent ways. The first is to number the energy states, starting with the lowest state. In such a numbering system, there would be some adjacent states that have the same value of the energy; these correspond to microstates of the same energy level. An alternative procedure is to only number the energy levels. In this case each energy value would be distinct, but one would also have to specify the degeneracy of each energy level—that is, the number of states that have this value of the energy, The degeneracy of the j' molecular energy level will be denoted by «;. Our interest will usually be in the energy states of a large assembly of molecules rather than in that of a single molecule. The energy of an assembly of noninteracting molecules is simply the sum of the energies of the individual molecules. To account for the indistinguishability of identical molecules, an energy state of the collection of molecules is specified by giving a set of occupation numbers—that is, a set of numbers m = (1|,2,3,...) where n; is the number of molecules in the i" molecular state. An important feature of this method of state identification ts that the indistinguishability of identical particles has been incorporated into this description in that we have not specified which molecule is in which energy state, but only how many molecules are in each of the energy states. Each occupation number can only take on the integer values 0, 1, 2, 3, etc. However, as we shall see shortly, our interest will be with those systems and macroscopic conditions where the number of energy states available to each molecule greatly exceeds the number of molecules present. Therefore,

the number

of energy

states of an assembly

of molecules

in which

any

particular molecular energy state has higher than single occupancy is very much smaller than the number of energy states in which each occupation number 1s either zero or one. Each distinct energy state of an assembly of molecules will be given by a set of occupation numbers; the i energy state will be specified by the vector n' = (n',, n',....), where ni is the occupation number of the j" single molecule energy state in the i" energy state of the assembly of molecules. Thus, each n' represents a distinct microstate of the system and the energy of this microstate is E; = energy of the i‘ microstate =

>

nie;

(1.3-2)

all molecular energy states

J

where ¢; is the energy of the j" energy state of a single molecule. There will. of course, be many different microstates (sets of occupation numbers) consistent with any specific value of the energy FE. Therefore, as before, we can shift our attention from energy states to energy levels, provided that for each energy level of the macroscopic system we also specify its degeneracy—that is, the number of

1.4 The Postulates of Statistical Mechanics

5

rgy ene an of y rac ene deg the ote den will We el. lev rgy microstates that have this ene

the of y rac ene deg the m fro it sh gui tin dis to ©, by les ecu level of an assembly of mol energy level of a single molecule, w.

ISTULATES

OF

STATISTICAL

MECHANICS

The rigorous development of the principles of statistical mechanics is a very elegant and beyond the scope of this introduction to the subject. The reader is referred to the many excellent textbooks on the subject for such presentations. In contrast, the presentation here will be quite inelegant, but also very simple; it is based upon two postulates. The first postulate: All microstates of the system of volume V that have the same energy and the same number of particles are equally probable.

This postulate is known as the equal a priort probability principle, and is a statement of complete ignorance. However, there is much to be said for this concept. First, it is the most minimalist statement that can be made. Any other assumption of probability assignment would require much more information about the system, information that, in fact, we do not have. (Think about this. How else would you assign probability to states of equal energy?) We can ultimately test this assumption by comparing the results of calculations for system properties based on this assumption with experimental measurements of these quantities. No evidence has been found to contradict the equal a priori assignment of probability. The second postulate: The (long) time average of any mechanical property in a real macroscopic system is equal to the average value of that property over all the microscopic states of the

system, each state weighted with its probability of occurrence, provided that the microscopic states replicate the thermodynamic state and environment of the actual system.

This postulate, which

ts called the ergodic hypothesis, merely sets down

in words a

concept we alluded to earlier, namely that any experimental measurement ts really a long time measurement on a molecular time scale. So long. in fact, with respect to the rate of transition between the microscopic states, that during the time necessary to perform the measurement, the assembly of molecules will have gone through a very large, statistically representative number of microstates. Therefore, we can replace the time average with a statistical average. Taken together, postulates I and II represent a complete framework for the construction of a statistical theory of thermodynamic processes. The first postulate tells us how to choose a probability distribution, and the second postulate establishes that thermodynamic properties computed with this probability distribution will be equivalent

to those

that

we

would

measure.

There

is, however,

an

important

restriction

embodied in this equivalence of statistical and time averages, namely the replication of the thermodynamic state and environment of the real system. In this introductory chapter to statistical thermodynamics we shall be concerned with a system of constant volume and number of particles, but which is free to exchange energy with its surroundings.

Introduction to Statistical Thermodynamics

OLTZMANN

ENERGY

DISTRIBUTION

In this section we establish the way of assigning probabilities to states of different energies for a system of fixed volume and number of particles in contact with a large heat bath.! (Note the equal a prieri probability principle deals only with assigning probabilities to states of equal energy, not of different energies.) Consider the system shown in Fig. 1.5-1, in which the macroscopic subsystems A and B are in contact with an infinite heat bath of constant temperature. Here, the heat bath is considered to be so large that the subsystems A and B are unaffected by the presence of one another. That is, a fluctuation of energy or temperature in system A has no effect on system B, and vice versa. The subsystems A and B are completely unspecified, and need not be identical. Let pa(£,,) be the probability that the subsystem A is in one particular microstate s4 whose energy in E,,. (Note that this is not the probability of finding system A with the energy EF), since there may be many microscopic states consistent with this energy level. By the first postulate, all these states are equally likely, so that the probability of finding subsystem

A in energy

level

£,

is proportional

to Q,4(£,,)pa(F,).

where

Qa(&,,)

is the degeneracy of the level E,,). By the equal a priori probability postulate, this probability can only be a function of the energy level. Similarly, let pp(E,,) be the probability of finding subsystem B in a particular microstate sg, whose energy level is &,,. We now can ask what the probability is of simultaneously finding system A in the state s4 and subsystem B in the state sg. Since A and B are completely independent, this probability 1s (1.5-1)

PACE,) PplEm)

that is, the product of the two separate probabilities. Now consider a composite system formed trom both subsystems A and B. The probability of finding this composite system in a particular microstate sap can, by the equal a priori probability postulate, only be a function to the total energy of the composite

system. This probability will be written as pag(Eapg),

where

Eap

is the

energy of the microstate. Furthermore, all microstates of the composite system with the energy

-ap have the same probability of occurrence. Suppose, Eag

= EE, + En.

One particular microstate (among many) of the system A + B that has this energy is when subsystem A is in state s4 and subsystem B is in state sg. The probability of occurrence of such an event is given by Eq. 1.5-1. Therefore, the probability of

Infinite heat bath of temperature T

Figure 1.5-1 Systems A and B in an infinite heat bath of constant temperature.

I'The simple argument here is based on one that appears in Equilibrium Statistical Mechanics, by E. A. Jackson,

Prentice-Hall, Englewood Cliffs, NJ, 1968, and other books.

1.5 The Boltzmann Energy Distribution

7

the with tem sys the of e tat ros mic r ula tic par r othe any (or 84g e stat occurrence of the energy FE, + Em) Is =

oe

+

Papal,

(1.5-2)

PplEm)

PACE)

abil prob same this has also gy ener total same the with te osta micr r othe any that Note pa(E, + 46)pp(Em — 4) also equals pap(En + Em).

ity of occurrence, for example

We can now inquire how these probabilities would change if we changed the value E, without changing F,,. (Note that as far as the composite system is concerned, this is just one of many ways of changing the total energy.) In principle, the energy of the system is a discrete variable; however, if the energy levels are very closely spaced, we can treat the energy as a continuous variable and write this probability change in terms of derivatives with respect to energy. For the moment, we will assume that the energy levels are closely spaced; we will return to this question later. Taking the

dpapleE, Papleén

a.

JE,

+

+

Een)

FS,

Lom

Em

cl{ En

+

£,, constant, one obtains

holding

1.5-2 with respect to E,

derivative of Eq.

ale,

+

e-e ee

Em)

oo

Eom)

=

d(E,)

dpapglkn

Ew

d( En

+

7 Ein) Em)

(1,5-3)

Now

using Eq.

1.5-2, we also have +

OpaplEn ee dEy

and therefore

dpally)

d

En) Se

ee

En



a. a(E,)

,

ial

Pa

Ein

dE,

En

Pov

Ein

1.5-4

Pat

we have dpaplky,

d(E,

oe

-

Fa}

ee

=

Em)

dps(F,)

En

d( Ey)

1,5-5

pa

By a similar argument, one can show that changing F,, while holding £,, constant gives dpapll,

+

Emit

d(En+ En)

—_—_—

=

dpalEn)

PME aE,

P8) 15-6

E,.)—_—

Now equating the results of Eqs. |.5-5 and |.5-6 we obtain PRCE

Em

l Pal Ein)

wm)

dpal Ey) SS

dE,

dpplEn,) En =i:

PalEn)

d(E,,)

dpr(En) 1 dpa(En) d

E,.

Es,

l

Pal

dpal

En)

by,

as)

dl Ean

The interesting characteristic of Eq. 1.5-7 is that the left-hand side is independent of subsystem B, and the right-hand side of the equation is independent of subsystem A. Furthermore, as noted earlier, it is possible to make changes in subsystem A independent of any changes in subsystem B, and vice versa. One example would be to change the volume of subsystem A, which, as can be seen from Eq. 1.3-1 has the effect of changing all the energy levels in that subsystem, but no effect on subsystem B, That the relationship given in Eq. 1.5-7 must be maintained for all such changes means that each side of that equation must be independent of both subsystems A and B and can only depend on the properties of the reservoir, here characterized by

Introduction to Statistical Thermodynamics

its temperature. This can be written as

dinpa(E,)

— dinpalEm)

_





ad Ea

d Em

( 1.5-8)

.

|

(where we have introduced the negative sign for convenience, as will be evident later). From the discussion above, 6 cannot depend on the subsystems A and B, but may be some function of the character of the thermal reservoir or heat bath (such as its temperature) with which both subsystems are in contact. Indeed, one expects that changing the temperature of the reservoir would change the temperature of both subsystems, and that this would affect the energy probability distribution. Integrating Eg. 1.5-8 one obtains

Each

constants,

of the integration

C4

(1.5-9)

ppl(Em) = Cpe?’

and

pr(E,) = CyeFP"

are specific to the characteristics

and Cg,

of

their respective subsystems and can be determined from the normalization condition that each subsystem must be in one of its allowed energy states, that 1s,

SY

pa(Ey) = 1

and



states noof

pa(Em) = |

slates a

system A

of

system B

Therefore,

; Cy,

=

(1.5-10)

I

—————_-_ >

ef

states a

and

Ch

=

——_

En

»-

of

states

system A

eP

(1.5-11) Em

mt of

system B

We define the canonical partition function

O(N, V, 6) for any system to be oS.

O(N, VB) = > e-BEAN.V) |

(1.5-12)

states

i

Note that the summation is over all the energy states of the system. With this definition of the canonical partition function, we have that the probability of occurrence of a particular microstate ¢ with energy Ey, is

|

F ;

eo BEa ur

SS

eo BEa SS

I EL

ee

1.5-13

states f

This is the important result of this introductory chapter. In what follows, it will be assumed that f is a positive number, and later we will show this to be true. The implication of 6 > 0 is that a state of higher energy has a lower probability of occurrence than a lower energy state.

Chapter 2

The Canonical Partition Function INSTRUCTIONAL

OBJECTIVES

2

FOR THIS CHAPTER

The goals of this chapter are for the student to: e Understand the derivation of the canonical partition function e Understand the role of degeneracy in the probability distribution function e Understand how thermodynamic properties are computed from the canonical partition function e Understand the difference between the canonical partition function for a single

molecule with several independent energy modes function for a collection of identical molecules

ZA

SOME PROPERTIES FUNCTION

OF

THE

CANONICAL

and the canonical

partition

PARTITION

In this section we consider some of the properties of the canonical partition function and its relation to thermodynamic properties. From Eq. 1.5-13 we have that the probability of occurrence of a particular microstate i with energy F, is e

Pea

Se8E

e

Pea

O(N. V-B)

Sales

J

However, we can also ask what the probability is of finding the system in any microstate such that its energy is E,. This would be the product of the probability that the system is in a particular microstate with energy E,, and the degeneracy of that energy level (that is, the number of states having that energy), since by the equal

'Note that here and elsewhere, the notation that E; is used for the energy of a particular Microscopic state, while {/ is the average internal energy of the system, and it is L’ that appears in the equations of classical thermodynamics.

Partition Function

The Canonical

d hoo eli lik e sam the e hav rgy ene e sam the of tes sta all n tio ump ass ty a priori probabili of occurrence. Therefore, number of states |

)) pilEa)=pi(Eqa) x | sith eneagy E.,

=

a) p(E probability

= pi(Eq) x & (Ey) degeneracy

states j whose energy

of finding the macrosystem in any microstate with the energy

is Ey

(2.1-1)

Ee

Consequently, energy Ey, 18

the

e

((Eg)

microstate

=

= O(N,

BE;

with

7 5

e bee

Pee

ye

,

a particular

of

of occurrence

probability

'

5

(2-1-2)

V, B)

States

/

and the probability of occurrence of the energy level E,, P(E)

e™™ (Ey) = @( Eg) P pi(Eq) = 2Ed O(N, V.B) a!

=

MLea)Pi\£e)

=

lh

is

2.1-3) ein.

Also, for later reference we note that the canonical partition function can be written either in terms of states or of levels, as follows:

O(N. V,B) = Soe PY = So al EeF* states

Since

the probability

exp(— BE),

of occurrence

(2.1-4)

levels

j

of any one

microstate

is proportional

to

a particular state with a lower energy is more probable than a particular

state with a higher energy. In fact, the state with the lowest energy is most probable. However, the degeneracy w(£) of an energy level is an increasing function of the energy level. That is, generally there are more states possible having higher energy than a lower energy. For example, the kinetic energy of a particle in classical

mechanics

1s o (vy +u.+ v?), where mm is the mass and v; 1s the velocity in the

i" coordinate direction. If, for demonstration, the velocities are restricted to be integers, the degeneracy of the energy level m7/2 is 3 (one of the three velocities is | and the other two are 0), while the degeneracy of the level 10°m/2 is very large. Therefore, the probability that the macroscopic system will have an energy level FE is the product of an exponential term that is decreasing with increasing energy and a degeneracy that is increasing with increasing energy, as shown schematically in Fig. 2.1-1. (The probability distribution for collections of molecules is considered in Section 3.5.) Therefore, the most probable energy level (not energy state!) is a balance between these two factors, as Indicated in the last of the figures on the following page.

2.2 Relationship of the Canonical Partition Function to Thermodynamic Properties 1-104

p,

11

1-10"

3-105

a,

0

0



Si)

SQO0)

0

1 (0)

!

0

_!

50)

j

100)

1

Probability of occurrence of a particular state with energy E;

Degeneracy of states with energy &;

0.04

wp,

0.02

0

a)

[OU

i Probability of occurrence of a particular state with energy £;

Figure 2.1-1 Probability Distribution for the Translational Energy States of a Single Molecule,

RELATIONSHIP OF THE CANONICAL TO THERMODYNAMIC PROPERTIES

PARTITION

FUNCTION

The internal energy U in thermodynamics is equal to the average value of the energy of the system £, That is,

slates

U = 3

Ex plex) =

oO.

Where

Q=

\-

SLOTS

states

k

K

e “PEE

(2.2-1)

However,

aQ \

($F)

— NAV



.

Stones

k

.

BE,

E,e BEE

. (2.2-2)

Chapter 2: The Canonical Partition Function so thal

y

E,e7 PE

slates

-(e) y-FOB —

(2.2-3) ~

- + __ O

Juv

So we see that the “sum over states” or partition function can be used, rather than probability p(£), to obtain an average value of a thermodynamic quantity. We will see in Chapter 3 that many thermodynamic properties can be obtained from the canonical partition function and its derivatives. However, before we can do this, we also have to consider some other general properties of this function.

A MOLECULE MODES

CANONICAL PARTITION FUNCTION FOR WITH SEVERAL INDEPENDENT ENERGY

Consider a single molecule with, for simplicity, two completely independent energy modes. For example, translational energy that depends on the mass of the molecules and the center of mass velocity, and rotational energy that depends on the moment of inertia and the rotational velocity of the molecule. For simplicity of illustration, assume only three translational energy states Exist—étrans. |. €trans.2 @Nd yans,3—and that only three rotational states exist—eyor,), Erot,2 ANd Epor,3. All possible energy states

of this highly simplified system are

+

7

Epot.2s Etrans,2

The single-particle canonical denote by g for this system is gq

which

can g=



eB lttrans. 1 +8 rot, 1} |.

be

partition

e

function,

PlEtrans, 1+ Fro

2)

L

Erot,2s

or sum

Etrans,3

and

over

Erot,3

we

e P(E trans, 1+#ro1,3)

el

trans.2t# roe,

oe

PlFtrans,2 + rot.2)

|. e PtP trans.2+&rot.3)

+

eo

trans.3 Fé rot, |) =- ge

PlEtrans,3 FF rot.2)

“fs ge

rewritten

>

states, which

=

Pl

+

1

Erot,|+ €trans.3

1

Erot,3+ Etrans.3

frot.1 + &trans.2

7

Erot.3+ Strans.2

+

7 €rot,2+ Etrans,|

Erot,1» €trans,1

+

Etrans.1

PAE trans. 3 FF ro3)

(2.3-1)

as

[email protected] PF tans, | e Peron, |

+ ge

PFtrans.1 g~ PErot,2

|. e

trans, | e +

Berat.2

- BE rot3

e 7 PE tans,? 9 — PE ron,3

AF rat, |

+

eo

PFtrans.2 go

P®trans.3 o— FE ren, |

+

e

PF trans. 3 o— PF rot, 2 +. ge” PF trans.3 @—PEra,3

+4

e FF irans,2

+

eo

(2,3-2)

or

g

a

e PFtrans, |

(2

es e Pe en.2



(2

Petrans. 1

trans rot

+

PPro. a7

e

+

e FF ra.2

e PFra.3)

Pttrans.2

+

+

e

+ oe

e

PFret.3)

FF wans,3

PFtrans.3)

(7

+

e 7 Pe wans,2

Pera

(@~P Fron.

fs a

+

ge

(e Por! Perat,2

PFca.2

+

+

eo

ge

PFro.3)

PFren.3)

(2.3-3)

13

ms Ato l tica Iden ng acti nter Noni of on ecti Coll a for tion Func n itio Part l nica 2.4 Cano

Here, girans = ETP Ee! + e PE trams.2 4 @ PF trans. ig the canonical partition function for the translational energy, and gpo, = e7 P| + e Pfr? 4+ e~PFr.3 jg the canonical partition function for the rotational energy. While the model used here was a simple one, the important result is that if there are several

energy

independent

modes

&,, &p, &c,...,

then

the canonical

partition

function is the product of the partition functions for the individual modes | G = Gagne --- |

(2.3-4)

for independent energy modes

NONICAL PARTITION FUNCTION FOR A COLLECTION ‘ NONINTERACTING IDENTICAL ATOMS Consider first a single atom with a collection of accessible energy states €), £3, €3,....

The canonical partition function, or sum over states, for this one-atom system is

g =e Pl 4 e Ber 4 oe Bea Now

consider

a collection

of N

such

identical

(2.4-1)

atoms,

and

assume

that

the

atoms

are sufficiently far apart that we can neglect their potential energy of interaction. Consequently, the atoms, for the present, will be considered to have only translational energy. To continue, it would be useful 1f we could specify the energy state of each atom. However, this is prohibited by the Heisenberg uncertainty principle. That is, since the position and velocity of each atom at an instant are imperfectly known (by the uncertainty principle), at some later time we could no longer be sure which atom had which velocity. (The situation would be simpler if we could paint a number on each atom, which ts, of course, not possible.) Therefore, rather than specifying the energy state of each atom, we instead can define a state of the system by specifying the number of atoms in each energy state, but not indicating which atoms are in that state. In particular, we will use the notation that ni is the Occupation number, or the number of atoms in the j'" atomic state of a

single atom in the i" macroscopic state of the collection of atoms.

Therefore, the i" macroscopic state of the collection of atoms is given by the collection of numbers (n'/,, m5, n4,....) that specify the number of atoms having the energy &), £2, €3,... ete. Clearly, with this definition >

n'

= N = the number of atoms in the system

(2.4-2)

states |

of a single atom

since each atom must be in one of the states of the possible atomic states. Also Ay

E‘

= energy

of the i” state of system

=

) shites j ofa single atom

gjn'

(2.4-3)

The

Canonical

Partition Function

With this notation, we will now consider a system with only two atoms. A listing stin indi are s atom the that ing gniz reco em, syst the of tes osta micr ible poss of all as s ber num on pati occu two of sets ible poss all of list a be then guishable, would below:

shown

.). 0,0, 1,0... (1, ), , 1,0,0.... 1,0, (1, 1,0,0,0,...) (2,0,0.0,...)., (0, 1,0. 1, 0... 2), (1,0,0,0,..., 1,...0), (0, 2,0,0, 0,...), (0, 1, 1,0,0,...), (0, 1,0,0,1,...), (0, 1,0,0,0,..., 1,...0), (0,0, 1, 1,0, ...) ete.

The canonical partition function, or sum over states, for this system is then O=e

4 gp 2PO2 4 pg Blerte3)

Sher 4 g~Bleiten) 4 phlei tes} 4 gp Bleitea) 4. + a bler+e4) 4g

Bleates) dee

pe Bleste4)

a oe Plea tes) +---etc,

(2.4-4)

Compare this with the square of the single particle partition function g q°

=

(oF

4 ge

8e2

te

fey

yo

_

a

oe Alze)

4



Jee

-

+2) .

en

9 eo Blei tes) fs

+ e Peer) 4 Je PlE2 TEN) 4... te,

2:

atin 5 9

(2.4-5)

So if we accept a small error in counting of the states with an occupation number of 2, 1.¢., (2, 0, 0, 0, ...) then

| ;

Y = —g* 54

( 2.4-6 )

This assumption is satisfactory if the number of possible states of a single atom is very much larger than the number of atoms and the energy states are closely spaced, so that the probability of two atoms being in the same energy state is very small. In fact, that will always be the case for systems of interest to us. Also, this result can be generalized for the N particle system to obtain for N identical atoms |

, QO

= wit

(2.4-7)

where the factor NV! arises from the indistinguishability of N identical atoms. It is important to compare and understand the difference between Eq. 2.3-4 and Eq. 2.4-7. In Eq. 2.3-4, the energy modes are independent, but distinguishable (that is, we can tell the difference between a translational motion and a rotation). In that case, the total partition function is the product of the partition functions for each of the energy modes. However, in Eq. 2.4-7, the individual atoms are independent of, but indistinguishable from, each other. The total partition function then is the product of the individual atom partition functions, but now divided by the factor N!

as a result of the indistinguishability of the atoms. Extending this argument to a mixture of N; atoms of species 1, N> atoms of species 2, etc., we obtain the following general

O(N,, N32

result for a (nonreacting) mixture

—— |

nonreacting mixture

(2.4-8)

Chapter 2 Problems

15

Since atoms of any one species are indistinguishable from each other, the partition function for each species is qi | N; !; however, since the atoms of different species are distinguishable, the system partition function is the product of the partition functions for each species. Note also that in Eqs. (2.4-6 to 2.4-8) each of the individual atom partition functions depend upon volume and the still-unknown parameter f. This parameter, which is only a function of the temperature bath and not the system, will be evaluated in the next chapter by considering an especially simple system, the ideal monatomic gas.

CHAPTER 2.1

2 PROBLEMS

For a gas of N-like particles

expression for Q would be

Q=,q"/N!

Ni

ga % Ni!

N No!

where gq is the partition function (sum over states) for one particle, and @ is the N-particle partition function. Show that if the gas consisted of NV) particles of

Here, g; and q2 are the single particle partition func-

species |, and N> particles of species 2, the appropriate

tions for species

1 and 2, respectively.

Chapter 3

The Ideal Monatomic INSTRUCTIONAL

OBJECTIVES

FOR

Gas

3

CHAPTER

The goals of this chapter are for the student to:

¢ Understand the generality of the identification of 6 with (kT)~' ¢ Understand the ideal gas partition function for a monatomic gas « Be able to compute the thermodynamic properties of an ideal monatomic gas e Understand energy fluctuations in the canonical ensemble e Understand the Gibbs entropy equation

¢ Understand the origin of the Gibbs paradox and its resolution

3.1

CANONICAL PARTITION MONATOMIC GAS

THE

FOR

FUNCTION

IDEAL

The canonical ensemble that we have so far been considering is shown in Fig. 3.1-1.

Expressions for the canonical partition function for a system of N identical noninteracting particles are q™

——

(3.1-1)

N!

and g=

)

e PF

=

)

slates i of a

energy levelsj of a

single molecule

single molecule

wje Pri

(3.1-2)

The value of the parameter f has yet to be established. From the derivation in Chapter 1, it is clear that 6 is independent of the macroscopic system and is only a function of the

characteristic of the thermal reservoir, which is its temperature. We will now establish the functional relationship between f and 7 by evaluating the partition function for one particularly simple system—a collection of noninteracting particles in a box, that is, an ideal monatomic gas. Most importantly, since 6 depends only on the reservoir, and not the system being considered, once its value is established using one system, it is applicable to all other systems. We will use the ideal monatomic gas as the test system to evaluate # since the properties of the ideal gas are known. The simplest model for an ideal gas is N atoms contained in a cubic box of volume V. The allowed particle energy states are, from quantum mechanics, given by he E(t,

16

j=

ly. Ei

3 yan ls + 5 +=)

(3.1-3)

3.1

for the Ideal

Function

Partition

Canonical

Monatomic

Gas

17

| System in contact with a thermal | reservoir of temperature T with rigid but thermally conductive walls that are impermeable to the atoms

Figure 3.1-1 The Canonical Ensemble. A system with walls that are rigid (no volume change) and thermally conductive with fixed number of particles in contact with a thermal reservoir of temperature 7’. Therefore, N. WV and 7 are fixed,

where /,./,, and /, are the translational quantum numbers (each of which can take only positive integer values) and #4 is Planck’s constant, which has a value of 6.62517 x 1077" erg-sec. With these energy states, the single particle partition function is

|

Bhr(2+ +12)

g=) DE

Baves

tot

k

p i e r ? p i e r e 8mV2 1 | Sle amv

_ i,

Consider

, _ pier

PLS eo BmV | = gyqyg:

iy

for the moment

(31-4)

fs

one of these sums,

Soe Ms

(3.1-5)

{~=0

where A = Bh?/8mV~*/7. _7

10---

grams

and

;

suppose

The mass of a particle or atom m is of the order of the

system

lever

volume

is

|

liter,

then

V-/3

:

is of

:

the

order

of 10° em?, and 5 a

my 2/3

= O07)

which has units of ergs, and the symbol O is used to indicate the order of magnitude.

Therefore, if 8 is not very large—say of the order of 10!'®, so that A is of the order of 107'” or less—the sums in Eq. 3.1-5 can be simply evaluated by replacing them with integrals. To see this note /, less than 10°, the summand that the summand is almost a larger, the relative or fractional

that with A about 107!’, for small values of i., Say changes very little in value between /, and /, + 1. so continuous variable. For large /,. say /, of 10’ and change in going from /, to /,+1 is very small:

ée+1 . | —— =]14+—=1410°’ Ex Ey

- 3: The Ideal Monatomic

Gas

so that the summand again behaves like a continuous variable. Therefore, it is an excellent approximation to write Soe

Bh

/8m yess

-

=

| o—BhP x? /8mv2/

iP

|

dx

omar

=



V2/3

(3.1-6)

5

The remaining sums in Eq, 3.1-4 may be evaluated in precisely the same manner to obtain a

3

2amV 2/3

«=

(/ay-]

2mm

a

\2

a

= (Gas) ¥

oe?

therefore (= Q(N,

MN

_q

V, B) =

and

NI

3N

In

Q

yo 9

_ \ arp =“Wh

V

N

(3.1-8)

5

:

2mm

O(N, V, B) B) = —5 In (= hp)

+NinV

—InWN!

(3.1-9)

The internal energy of the N-particle monatomic gas can then be computed from

4| U=— ( ae) OB

Jyy

as

3N 26

(3.1-10)

\TIFICATION OF B AS 1/kT By the ergodic hypothesis, the internal energy of Eq. 3.1-10 must be equal to that which is measured for an ideal gas. Since the internal energy of an ideal monatomic

gas is known to be equal to 2N&T, it follows that

(3.2-1)

Furthermore, 6 is a universal parameter and only a function of the properties of the reservoir. Since f is only a function of the reservoir, and not the ideal gas system that we have used to determine its value, this identification of B with (kT)~' is always valid regardless of the system being considered. Therefore. Fq. 3.1-2 then becomes q =

>

(3.2-2)

states j of a

single molecule

The term e *i/*"

is referred to as the Boltzmann factor, and the partition function g is the Boltzmann factor weighted sum over the states available to the system.

Before proceeding further, it is worthwhile checking several of the assumptions we made earlier. One approximation was that the summation in Eq. 3.1-5 could be replaced with an integration. In order for this step to be justified, it was necessary

3.3 General Relationships of the Canonical Partition Function

19

that 6 not be too large. The Boltzmann constant is k = 1.38044 x 107'® erg/deg, and T can, in principle, range from O K to infinity. Therefore, it 1s clear that, except at very low temperatures (less than | K), 6 will be less than 10'°, and the replacement of the summation with an integration was valid. Another assumption that was made was that the number of molecular energy states was very much greater than the number of molecules, so that the likelihood of two molecules being in the same energy state was small. This was used in obtaining Eqs. 2.4-6 and 7. One liter of gas at standard temperature and pressure contains

about 10° molecules. As a rough estimate, the quantity he

4

——

~ |)"

mV 2/3

33

ergs

may be taken as the spacing between energy levels; furthermore, each energy level has a very large degeneracy. Therefore, the number of energy states available to any one of the 10°* molecules is much greater than the number of molecules. We return

to this question in Section 3.7. With 6 now being identified as being I/A7, it then follows that the general expression for the canonical partition function is

Ei ewB QIN, V.T) = Soe FT = S* Q( Slates

(3.2-3)

levels

i

j

GENERAL RELATIONSHIPS OF THE CANONICAL PARTITION FUNCTION TO OTHER THERMODYNAMIC QUANTITIES We now have the partition function as a function of 6 or 7, the volume, and the number of particles. (Notice that the partition function is a function of volume because the energy levels are a function of volume as seen in Eq. 3.1-3). Also, from Eq. 2.2-3 we

have that din

U=- (= =)

(2.2-3) ta

Replacing # with (k7)~', it is easily shown that

yn) | (: ~ =)

U=kr?

ay

(3.3-la)

ye

or

kT? (22) (3.4-1b)

= —— | -— Q

Since the partition function and V, one can write

a]

VoM

Q for a fixed number of particles N is a function at T |

ang = (8) af

dT + oN

al

ae) av

dV N.T

Chapter 3: The Ideal Monatomic Gas or a

dln Q

|

]

3 = kT* | ——

e d e r

dT +kT

).

dT

(

— UdT +r (

52

(

fdln =)

Snr

OV

al

(3.3-2)

dV i) OV Jr

Now using

dV

ny

(3.3-3)

d U 7 d ) T T T = — + ( Ud = and rearranging Eqs. 3.3-2 and 3.3-3, we have

(Ino + z)

Td = kd

al — er* (FF) dV

kT

dU

U —

dV TN

[email protected]

y=Kra(mo+ ir) —#r (Se), = kTd | 1

= kTd

|

(mo

r

|—sF

dlnQ

— kT

( aT ),.J

dV d In )

( aV

Jay

dV

(3.3-4

But from the first and second laws of thermodynamics for a closed system, we know that (3.3-5) dU = TdS — PdV Comparing these two equations, we have

(3.3-6)

and

dS == kd (no+

U —j=kd{l =) k (no

alnQ T ( aT ),.)

( 3.3-7 )

On integrating this last equation we get

| s=e(ino rr (*2) aT

) +e

(3.3-8a)

Vo

The constant of integration, C, may be set equal to zero by the third law of thermodynamics, which requires that the entropy of a system go to zero at O K. Therefore

S= king +47 (

a]

~*) aT vy

(3.3-8b)

on ti nc Fu ion tit Par l ca ni no Ca the of s ip sh on 3.3 General Relati

21

ng usi by on cti fun ion tit par the to d ate rel be now can s tie per pro c mi Other thermodyna S and P, U, for ons ati rel the h wit er eth tog cs mi the usual equations of thermodyna obtained above. For example, A(N, V, T) = Helmholtz energy = U — TS =kT° |

(ane ar

(

ae

—kTInQ—kT-

2

That 1s

| =—kT

ne V,T)

InQ(Nn, g

VoN

ay

(

Q

)

ae) 5

(3.3-9)

V,T)=—-kTInQ(N,V,T)

ACN,

and if we have a mixture of particles at different species

8) cue Denna A (Rea [email protected]

aA

dG

ONi J rvnjg

\ON/

ONG I TN ys

TPs:

(3.3-10)

= chemical potential

where the chemical potential is the Gibbs energy per molecule. For future reference, we also note that

H=U+PV =kT°

,falnQ'

es |

+kTV

| Cy

;

falng =2kT | ——

(au ={—])

(sr),

a 2kT Q

( or

OT

Jun

+ kT?

TN

( dT?

Jy

ead ec kT? Or

dT Jy iy

OQ

|

ne)

5

(PO

kT?

(a

Jan?

(3.3-11)

dV

VN

q

|

alnQ

(aQ\* \aT

Jy

and Cp can be obtained from the relation

Cp= cy -T(

4)

Y)

— dP

r

= — |} =cy aT '¥

|

@ aT

gy

}y

-T-—_—. aP OV

(3.3-13)

+

It is important to note that the relations of this section are always valid, independent of the system being considered. That is, even though we used the ideal monatomic gas to make the identification of 6 with (kT)~', that identification is always valid independent of the system being considered; thus, the equations in this section are valid for any system. In this regard, Eq. 3.3-9 provides the following interesting contrast between classical (or macroscopic) thermodynamics and statistical thermodynamics. The Helmholtz energy A = U —T’S as a function of N,V, and T is a fundamental equation of state! in the terminology of Gibbs in that if we have A as

'See, for example pp. 202-203 in Chemical, Biochemical and Engineering Thermodynamics John Wiley & Sons, Inc., 2006.

by S. |. Sandler,

Chapter 3: The Ideal Monatomic Gas

a function of N,V, and 7, all other thermodynamic functions can be obtained from linear combinations of A and its derivatives with respect to N, V, and 7. However,

classical thermodynamics provides no guidance as to how to develop an equation for A as a function of N,V, and T, while statistical thermodynamics through the partition function Q and Eq. 3.3-9 provides the recipe, which Is to enumerate all the energy states of the system and then do a Boltzmann factor weighted summation

of all those states. This is easily accomplished for a system in which the molecules do not interact, as we show in the next section for the ideal (that is, noninteracting) monatomic gas and in the next chapter for the ideal diatomic and polyatomic gases.

THE THERMODYNAMIC MONATOMIC GAS

PROPERTIES

OF

THE

IDEAL

In the previous section the general equations relating the partition function and various thermodynamic functions were presented. Here we want to use these relationships to develop explicit expressions for the thermodynamic properties of an ideal monatomic gas. Before we can do this, however, we must refine our molecular model. So far, an

atom has been considered to be a point mass in a cubic box. In fact, an atom is not merely a point mass, but an entity with a quite complicated electronic and nuclear structure, and there are numerous energy states associated with these internal degrees of freedom. The question that then arises is how these internal energy modes affect the partition function and thermodynamic properties of an ideal gas. This question is answered by several observations and assumptions. The first of these is the Born-Oppenheimer approximation, which states that the translational (trans) energy states are independent of the electronic (elect) and nuclear (nuc) energy

states. The next assumption is that the electronic and nuclear energy states of an atom may also be considered to be independent. Therefore, we can write the energy of an atom as the sum of three completely independent energy modes: E =

trans + €elect + Enuc

(3.4-1)

so that the partition function becomes

g=

Yo

elim tet bee )/AT

3.4-2)

states of the atom

Now using the independence of the energy Eg. 2.3-4, it is easily shown that

states

as was

discussed

in deriving

G = iransGelectGnuc

(3.4-3)

where Ytrans



e

=

+E Lr. i / kT

'

elect

ma

—€elect.i (RE ‘ é

=

a and

Gruc

imc {k r eg —F- ley

=

translational

electronic

nuclear

states 1

states i

states |

(3.4-4)

The first of these partition functions is a sum over the translational energy states, which has already been evaluated in Eq. 3.1-7, using the particle-in-the-box model

3.4 The Thermodynamic Properties of the Ideal Monatomic Gas

23

for these energy levels, and replacing 6 with (1/AT) ——

27m

Girans = (Fr

4

)

\2

2rmkT

v=

A

\2

(=)

(3.1-7)

The single particle translational partition function is frequently written as

V trans

=

(3.4-5)

where

A

is the de Broglie wavelength—that is, the wavelength equivalent of the momentum of a particle in the wave-particle duality theory of matter.* [Table 3.4-1 contains a list of the values of several constants used here and elsewhere in this book, several conversion factors, and a simplified formula for the calculation of the single particle translational partition function.| The electronic partition function cannot be evaluated in such a general manner, since the electronic energy states depend on the electronic structure of the atoms, which is specific to each atomic species.* Therefore, the energy states used in the Table 3.4-1 Constants and Conversion

Factors in MKS

Units

Constants Avogadro's number

Na,

6.022 x 107° molecules/mol

Boltzmann's constant k

1.38044 x 10-7? J/K = 1.38044 x 107! erg/K

Mass of an electron

9.1094 x 1077! kg

Planck’s constant hh

6.6261 x 10 ** J-s

Speed of light (vacuum) ¢ Gas constant = Nay x k

2.9979 x 10° m/s 8.314 x 107° bar m*/(mol K)

Conversion

Factors

1J = |kg-m?/s?

leV = 1.60206 x 107!’ J = 1.60206 x 10-'* erg = 23.0693 kcal/mol of electrons = 96.49 kJ/mol of electrons

| A=

107% cm

Translational partition function t

~ ( 2 a m k T \ trans __ (a ) =A Vv

hi-

5 = (x

= 1.88 x 10°°(MT)7 m4

hi-

4

(MT)*? = 1.88 x 10° MT)? | em

Nay

where wt is the weight of a single atom = M/N,,, M is the molecular weight in grams, T is the temperature (K), V = volume in cm’ or m* and Na, = Avogadro’s number

“The analysis leading to this relation was developed

in de Broghie’s Ph.D. thesis in 1924, and for which he

was awarded the Nobel Prize in Physics in 1929. See Problem

3.13.

“See, for example, the National Institute of Standards and Technology Chemistry WebBook, http:// webbook.nist.gov/chemistry/, for spectroscopic data on electronic, rotational, and vibrational energy levels (the latter two needed in the study of polyatomic molecules). Much of the data are in terms of wavelengths or frequencies of emitted radiation and not explicitly in terms of energy levels. To convert a frequency to energy multiply by Planck's constant; also, (speed of light)/(wave length) gives the frequency,

Gas

The Ideal Monatomic

es. stat rgy ene c oni ctr ele the of es tabl m fro ed ain obt are on ati cul cal on cti fun n itio part . level each of cy’ nera dege the and ls leve gy ener the list lly usua es tabl such Since rather than energy states, the partition function is computed as follows: Gatect

}

es

e —felecta/ *

}

Welect jC

electronic

electronic

SLULeS 1

levelsj

—Falaet | fkKT

e

Molect.2€

elect. 1/87

= Welect. 1€ —

Tr =

1/kT

‘kT

Prt

Wfeleet.2/ KI

Wotect.2e

++

ET Cay toot |

Pelee

—Eelect

Felect.j/

>

me -)

(3.4-6)

where A€etect.2 = felect.2 — €elect.|- Mote that for the noble gases, the ground electronic state degeneracy is 1—that is. eject.) = 1, while for alkali metal atoms it ts 2. The electronic energy levels mentioned above are determined from ultraviolet (UV) spectroscopic measurements. The principle of these measurements is that when an atom in the electronic ground state is subjected to UV radiation it may be excited to a higher electronic energy level. The energy difference between the ground state and the excited electronic state can then be determined by the frequency of the wavelength of the adsorbed radiation or the re-emitted radiation as the atom returns to its ground electronic energy level. In this manner, the energy levels of excited states relative to the ground state can be determined. However, the electronic energy of the ground state cannot be obtained by this technique. Furthermore, unless there are changes in the electronic structure (for example, by forming chemical bonds), there is no need to know the absolute energy content of the ground electronic state. (We will reconsider this question later when chemical reactions are studied in Chapter 5.) Therefore, by convention, we chose the energy level of the ground electronic state of an atom to be 0. With this convention, the electronic partition function is Gelect = Melect.1 + Melect,2€

— APales

kT

fetect2/KT

(3.4-7a)

4.

For most atoms, the energy level of the lowest excited energy level is rather high. For example, this value is 15.76eV or 1521 kJ/mol for argon. Therefore, at room temperature e

Afetecn 2/4 T

=

ei

me

0

Consequently, the degeneracy of the ground state alone is an excellent approximation to the electronic partition function. That is Gelect

=

(3.4-7b)

“elect. 1

The computation of the nuclear partition function is very similar to that of the electronic

partition

function,

except that the nuclear energy

levels are even

much

more widely spaced than the electronic energy levels, and do not change on chemical reaction, For example, Ae,>. the difference between the ground and first excited “The ground state degeneracy of the electronic energy levels of an atom is equal to 2s+1, where » is total electron spin angular momentum, which involves quantum mechanics that will not be considered here. Suffice

it to say that the inert gases have an electronic degeneracy of |, that of the alkali metals is 2, and oxygen is 3. Values for some atoms and molecules are available in M. Chase et al.. JANAF Thermochemical Tables, J. Phys. Chem. Ref. Data 14, Supplement | (1985).

3.4 The Thermodynamic Properties of the Ideal Monatomic Gas

25

nuclear energy states, is much larger than kT unless T is of the order of 10'° K. be can tion func n itio part ear nucl the us, to rest inte of n atio situ any for e, Therefor |

written as

(3.4-8)

©nuc.1



Gnuc

so that the nuclear partition function is replaced with only the ground nuclear state degeneracy. Since the nuclear energy state of an atom is unchanged for any process we consider, including a chemical reaction, the nuclear partition function will appear only as a multiplicative factor in the total partition function, will not affect any measurable thermodynamic property, and will cancel out of most calculations. Therefore, generally we can set the nuclear partition function equal to unity. Consequently, the partition function of a collection of N identical noninteracting atoms in a volume V can be written as

N

3 2amkT pA.

|

Vv



—Agelect 2/ KF ence (elect, 1 + Melect,2€

o=

+

+++ ")@nuc.!

N! V

y

2

ona

pasa

NPeleet.2/KT

+ Welect,2e

Peco

et

(3.4-9)

Using the relations of the previous section, we can now compute the thermodynamic functions for the assembly of noninteracting atoms as follows: adlnQ’ (=)

P=kf

aV

Also

Ui = kT

5

(

Ju,

=

4InO In ar

NkT

(ideal gas law).

V )

4 3 —NkT

=

VON

2

(3.4-10)

(3.4-11)

if excited electronic states can be ignored and the electronic partition function can be written aS Gelect = Welect.1- However,

if the first excited electronic energy

level must

be considered (generally, only at high temperatures) one obtains (see Problem 3.2) 3

U = = NKT +

kT Welect.2€elec xe Felect.2/

N

—eecnatelect 26

2

(3.4-12)

elect

Using only Eq. 3.4-7b for the electronic partition function, we have

Cy

= (= 7

oT

3.4-13

Nk vy

(0.419)

2

and A=-kATINO=

KT

In

q™

= —NkT

Ing+kT InN!

(3.4-14)

Now using Stirling’s approximation (which will be done frequently in this book), InN!t=NinNn

—-N=WNInN

—N Ine

(3.4-15)

Ideal

The

Gas

Monatomic

we obtain (neglecting the nuclear partition function) A =

_ _Ner NkT | | (

2mmkT

In(ge/N)

= —NkT

— N&KT Ine

InN

Ing + NkT

—NkT

3

\2 Vewetect.|

ae) elects - ) 5 |

3.4-16 (3.4-16)

and

_f §S=kmnO+k T (

4 In O

ain

aT

)

2amkT \2 Ved! wim - Yer’ elect, | —— | —_—______————

= Nk] of(

7

)

Ny

( 3.4-17 )

|

This last equation is referred to as the Sackur-Tetrode equation, and was originally derived based on the kinetic theory of gases. Finally

i

din QO ( aN )

— «7 (So

3

2amkT ae

\2 V Nv

= kT In|ol (=)

q N

oetccr,) | | = kT In| 4] =

§

(3.4-18)°

which is the Gibbs energy per molecule g. This expression for the chemical potential is Of some interest. In particular, replacing V/N with AT/P using Eq. 3.4-10 and arbitrarily defining a pressure, Pj, to be the standard state pressure, then adding and subtracting AT In P, from the expression above and rearranging, we obtain 3

2mmkT CT,

P)

=

—kT

\? kT

P

rss

In | (=)

+kT

|

In (=)

(3.4-19)

which ts (in the form tamiliar to chemists and engineers)

wT,

P) = wolT,

P

Po) + kT In (=)

where

(3.4-20)

° 4

2 Ho (T, Ps)

=

—kT

kT

In | (=r) hie

\2 kT Foes Fs

When using Eq. 3.4-20, one must remember that jz) is a function of both temperature and the standard state pressure, and that this equation is only applicable to an ideal gas. For the case of the ideal monatomic gas considered here, from statistical mechanics we have obtained the temperature and pressure dependence of the chemical potential as given by Eq. 3.4-19, Furthermore, we have also obtained an explicit

expression from which it is possible to calculate a numerical value for the standard

“This chemical potential 1s on a per-molecule basis. For a per-mole basis, multiply by Avogadro's number, or equivalently replace the Boltzmann constant with the gas constant.

3.4 The Thermodynamic Properties of the Ideal Monatomic Gas

27

state chemical potential:

2umkT [to =

—KT

i

\? kT

In | (=r)

Fo

(3.4-21)

It is useful to discuss the units of the properties that are calculated using the equations above. The internal energy U will be depend on the units used for the Boltzmann constant and will be the total energy for N molecules. If Avogadro’s number of molecules is used for N, then U will be energy per mole of molecules. A similar comment applies to the entropy, Helmholtz and Gibbs energies, the constant volume and constant pressure heat capacities, and the enthalpy. The chemical potential denoted by yz or g is on a per-molecule basis. These should be multiplied by Avogadro's number to obtain a value on a per-mole basis.

ILLUSTRATION 3.4-1 Compute the thermodynamic properties of 1 mole of argon at 300 K and | bar.

SOLUTION Using the values of the parameters in Table 3.4-] obtain

V = NRT/P

and the equations of this section we

mole x 8.314107 —ba— r -—m"

=1

303( 0 K/Ibar r== 0.0.020255

x

mole - K

{Tbe

=

QnmkT \? | y= ( a) V wetect, | = 1.88 x 10°°(MT)22 m-3V x | he 4

= 1.88 x 10°°(39.945 x 300)°” x 0.025

— 6.146 x 10°

A = —NkTIn Fa

= —RT In Fa = —4.276 x 10* joule/mol

w= —kT In | 7 | = —4.207x10* joule/mol N |

r

S=NkIn

(

20MkT

he

4

)

\? Ver

Wetect |

——__

N

gel?

| = Nk

"OW

= 155.01.

joule

mol - K

U = 3NkT/2 = 3.743 x 10° joule/mol

‘oul Cy =3Nk/2 = 12.4752" mo

H=U+PV

=U + RT = 6.238 x 10° joule/mol

and

‘oul

Cp = Cy + R = 20.792 200

mol -k

_

As a check, § = U-A

a

ae

|

dy

-

= 3.743 x 10° — (—4.276 x 10 ) joule/mol

r

= 155.9) ee

mol. K

300K

mm°2

he Ideal Monatomic Gas

ILLUSTRATION

3.4-2

The following information argon,

is available about the first four electronic excited states of Ce;

Ae, eV

State i= |

0

Il

i=2

11.548

5

i=3

11.633

3

i=4

11.723

|

i=5

11.828

3

Although we cannot compute the absolute probability of occurrence of each of these excited levels (since we do not have information on all the excited states needed to compute the electronic partition function), we can compute the relative probabilities of occurrence. That is, to compute the absolute probability of energy level /, the following equation would be used: (He ge LD

pl

Felect i (AE

a

EET

elect.i)

y

Wee

FelecrjlhT

electronic energy slates

|

We do not have the information needed to evaluate the electronic partition function in the denominator. However, we can compute the relative probability of occurrence of any

two energy levels using Plfelecti? 9(E,

ny

‘}

fel

Welect.i —

gr

Felect,i/KT

ael,ect ejenteeelaecntjj//kP

) P(Eetect,j)

and in particular, the relative probability of occurrence of any energy level compared to the ground state is obtained from P(Eete ct.i} ee P(Eelect.

l )

Oe tect. Oe tect,



—Eplecs, i AT Felecia

e

Peter i €

—€plec afk

= w; hee I

le —O/AF

® r

&€

The probability of occurrence of each of these energy levels relative to the ground state at 1O000, 20000 and 30000 K are given below. Level

A£etect:

@V

Orion

Pelect.i€

F=

i=l j=2 j= i=4 i=5 Degree of ionization at | atm

0 11.548 11.633 11.723 11.828

| 5 3 | 3

—Eebect {ff AT

eal

10000 K

7.520x10-° 4.088 x 107° 1.227 x 1076 3.260 x 10-° = 0.0120

Melectil

—febect

elect.

if AT

TF = 20000 K

6132x107 3.502% 10% 1.108 x 10-3 3.127 x 1077 ~— 0.94]

elec i€

—Felee

{f/AT

ce"!

T = 30000 K

5.729 3.326 1.071 3.082

| x x x x

~ 1.0

10-2 10-2 10°? 107?

3.5 Energy Fluctuations in the Canonical Ensemble

29

Also shown in the table is the degree of ionization at each temperature—that is the an form to als orbit ible poss all of out ed jump has tron elec an h whic for s atom of ion fract argon ion and free electron. The energy of ionization is 15.76eV = 1520.6 kJ/mol, which ing look e, efor Ther here. ed ider cons s state ted exci the of any than er high ably is consider at the results in the table, it may seem surprising that at the higher temperatures argon is either completely or almost completely ionized, even though the relative populations of the excited states are quite low. The explanation has to do with the degeneracy of the ionized state. Though we have only considered the electronic states here, each particle also has a range of translational states. As a result of ionization, there are now two sets of translational states available—those for the ion (which are essentially the same as for the atom, since the masses are almost identical), and also those for the electron, which

are new. Since translational energy states are very closely spaced, the degeneracy of the ionized state, because of all the translational states available to both ion the electron, is enormous. Consequently, even though the likelihood of any one ionized state is small as a result of the large energy in the Boltzmann factor, the degeneracy multiplying this factor (which is the product of the electron and ion degeneracies) is so large that ionization 1s

a likelier state than any of the excited atomic states.

>¥ FLUCTUATIONS

IN THE

CANONICAL

ENSEMBLE

In the calculations we have done so far, we have computed the average value of thermodynamic

properties—tor example, the average energy of a system in contact

with a bath at fixed temperature. Since energy can be continually transferred between the system and the bath, it of interest to estimate the extent of the fluctuations of energy that are probable. To proceed, we will assume that the distribution of possible

energy states around the average value is given by a normal or Gaussian distribution. The Gaussian distribution is €

fir

2(4*)

(3.5-1)

where

ao = standard deviation

which

is a measure

of the breadth of the distribution

jf = mean of the distribution x = dimensionless variable that can take on any value between —oo and +co

This distribution is normalized; that is, the integral overall values of x is unity as shown below: +6

“ox

/

|

o/20

— GO

|

ao/2r

eH)

— od

=| =

+00

+

7) é

— af JE — oxo

ac

— p)

Ads()) oes

The Ideal Monatomic

letting

Now

l vy = A |

+oo

|—oo wm

———e

Gas

— (* 5 ‘)

. we obtain

°

—3(254

*\*/

foc

|

‘i

Ja

|

dx =—=

e”

e”

G

=

fo

A

f(x) d A

‘+

2

dy

2 dy = —-—

Jr

(3.5-3)

=1

2

G(x), represented as G, is obtained from

value of any function

Also, the average

a

fo | ov2n , .

=

——

ye HY g

A)Je

*

A

Finally,

o? = ("4 —x)? = (& — x)? = x? — (x) = variance of the distribution where each overbar indicates the average value, (or fluctuations) of the instantaneous value of For reference, we note that for this distribution, in a State in which x is between yw — 0.6740 = je —o

i=]

oT

Eelec

—:/

Vi +4

In (1

—f

) rf \O,OpOc

(

in

Ve N

suit)

'

+

a



IN

Wetect.

|

‘he Ideal Diatomic and Polyatomic Gases

om: kT

|

»

*

= —In

2

atoms £

———__.

mkT\:

fJ|—

2m

T?

kT

i

h2

E

atoms

—In

Ly

‘lle In—6

4 o p : 77)

|=

oO

|

(soa)

Eelect,

=

a

y

vz

+i

he

) 4

G

»

m:kT

|

elect, |

|

20

—In

Pal Lat

+

=—-

—In

an—6 +

[

270

(1

—e¢é

)

Owl TY)

mkT

>< 44in| | ome Nk h atoms i

Ss

an—6

'

Saal

Oy

\On,©OpOc a

D

a [In

dX

__



fT

— Up

3 V |] ¥ N

_

IN @elect, 1

|

—ln

=

(4.4-4)

|

)

+n

(57 a \@,OpOc rT

2

me

ect1 [email protected] + ")| Se e 1— n( iF ~ iFTr | —euoF

(44-5)

4.4 The Thermodynamic

Properties of an Ideal Polyatomic Gas

57

and

(4.4-6)

PV = NkT The analogous equations for a linear molecule are: a

2

\atoms i

A

Ve

r

N

oO,

a

h?

|

|

NkT

mkT \V

Balan os

(-)

_

—~e

(1

‘= “ys [z+

e >)e m;kT \

7 7

U

2

NkT

Ve N

aan

S

i Pees. vi

15 5

T jn—S

fo.

_In

. val TY) —

(1 —[email protected]

— IN Welect.1

3

atoms i

aL

+

ou'T)]

dD oF

L a,

— IN Metect,) =

«=e Oval

AT

dD 0

27 LT [eeu ) ~ er

“—"

i=|

Cy~¥

3.2 Sey; fo4 =

Nk

2

2

I

rT

5

3n—5

Cy;

3

ec OvilT Gael

a)

(1 — @-Wi/T

= “ye

Q 3

Kf

inh hs In

Ovi /Ter

e~OvalT2 (4.4-9)

yam ke |} >”

iy,

= 5 +

(4.4-7)

Eelect, |

L[—e @vilt

e Ouil T

qe —In (=)

Oi oT

7

v

| oO,

4

mykT

-

2

;

= he aLOms

jl

kT 3n—5

Q., Td

+ In¢l

\-

m,kT

E Tr —

— a)

— IN @&gect.|

fa.

+4

N|

Qn

atoms

:

ff

h2

+



(In (1

—e

will]

V N

oO,

_

1



IN &elect. | =

—In

(=)

(4.4-10)

s e s a G c i m o t a y l o P d an c i m o The Ideal Diat 20

mykT

> atoms

ej

T »~

ee

_

an—5

T

e

"oO,

N

h2

Ovi

|

f

{—e 8valt

—In(l—e Oe! "| 4+ [email protected],1

(4.4-11)

|

and

PV

(4.4-12)

= NkT

as is, le cu le mo e th d te ca li mp co w ho er tt ma no at th is n io at rv se ob g in st re te in One of on ti ua eq s ga l ea id e th s, le cu le mo ng ti ac er nt ni no of ed os mp co is d ui fl long as the state results.

EAT CAPACITIES

OF IDEAL

GASES

t an st on (c ty ci pa ca at he e th s es pr ex to on mm co is it g, in er ne gi en d an y tr In chemis on ti nc fu a as rm fo es ri se rwe po le mp si in s se ga volume or constant pressure) of ideal of temperature. That is,

However,

we

see from

(4.5-1)

Cy =a+bT +cT? +dT° the results of this chapter

and the previous

d an , ty ci pa ca at he s ga l ea id the for s on si es pr are exact ex to s ge ta an dv sa di d an es ag nt va ad are e er Th series form, r te me ra pa le ng si the ly on s ga ic om at example, for the di r Fo e. ov ab on ti ua eq the in rs te me ra pa the four adjustable

one that there

they are not of the power the exact expressions. For ©, is needed, rather than a triatomic molecule, three

l na io at br vi ur fo d an r, ea in nl no is le cu vibrational temperatures are needed if the mole

1s rs te me ra pa of er mb nu s thi As . le cu temperatures are needed for a linear mole is es ri se r we po the d an n, io ns pa ex cs ri se rwe po the in as me sa the y el at im ox pr ap rwe po the e us to r le mp si y ll ra ne ge is it algebraically simpler (no exponentials),

of er mb nu the s, om at e re th an th re mo ng ni ai series expansion. For molecules cont n ca ty ci pa ca at he the of n io at ul lc ca al ic parameters needed for the statistical mechan on as re s thi for is It n. io ns pa ex es ri se rwe po the in rs te me ra pa ur fo the ed greatly exce it y, nc te is ns co r Fo s. le cu le mo e rg la for that the power-series form is simpler to use t ac ex an gh ou th en ev s, le cu le mo all for rm fo es ri se ris then common to use the powe th wi es ri se r we po the g in us by , so Al expression is available for small molecules.

r ro er the of ct fe ef no 1s e er th , ta da al nt me ri parameters determined by fitting expe

the of on ti mp su as c mi na dy mo er th l ca ti is at st the that may have been introduced by m. do ee fr of s ee gr de n io at br vi d an al on ti ta ro the of on complete separati the of ) PR IP (D ch ar se Re es ti er op Pr al ic ys Ph for e ut it st Recently the Design In e ur ss re -p nt ta ns co s ga al ide ed at el rr co ve ha s er ne gi American Institute of Chemical En heat capacity data using

Cp =k tk

sinh(k3/T)

|

2

ky

:

|

2

cosh(ks/T)

e us the for n io at ic if st ju e Th . ta da al nt me ri pe ex to fit where each of the k; have been n. io ct se us io ev pr the s in on ti va ri de the om fr t en id ev is s on ti nc of hyperbolic fu

46

: S I S Y L A N A E D O M NORMAL E L U C E L O M C I TRIATOM In the modes modes tional is. the

59

e l u c e l o M c i m o t a i r T r a e n i L a of s n o i t a r b i V e h T : s i s Normal Mode Analy

VIBRATIONS

THE

A LINEAR

OF

l a n o i t a r b i v ct in st di e ar e er th at th d e s u e v a h e w discussion of this chapter, e s o h t t a h w d e r e d i s n o c t no e v a h e w t bu , s e l u c e l o m of triatomic and larger abr vi of t se a of n o i t a c i f i t n e d i e h T . ed fi ti en id are or how they could be t a h t — e l u c e l o m a of s n o i t a r b i v e l b i s s o p l al e b i r c s e d motions that completely s ea id e th of n o i t a t n e s e r p of y t i c i l p m i s r Fo . re he d e s s normal modes—is discu

involved.

consider

shown

the molecule

of three

consisting

below

identical

the others by vibrations as be considered molecule will

atoms,

a distance simply as here; the be briefly

m o r f d e t a r a p e s is h c a e m u i r b i l i u q e at d an m, s s a m of h eac r fo s i s y l a n a l e d o m l a m r o n a of t p e c n o c e th te ra st lu il hb. To ll wi e l u c e l o m e th of is ax e th g n o l a s n o i t a r b i v ly on , e l b i s pos e th of is ax e th of t ou s n o i t a r b i v of se ca d e t a c i l p m more co mentioned later. d i s n o c be n ca e l u c e l o m c i m o t a i r t e th at th e m u s s a e w , s n o i t In modeling the vibra

K. t n a t s n o c g n i r p s a th wi h c a e s g n i r p s by d e t c e n n o c m s e s s a m ered to consist of This is shown below. Fir it iif bh

Ay

fh

At

As

IS , PE , m e t s y s is th of y g r e The potential en

K

K

Y B = 42 = s O 5 + ? by — 1 — PE = 5 (2

,

(4.6-1)

ts , KE , gy er en c ti ne ki e th and

: 3 m5 KE = > (vj + ¥3 + v3)

vj

where

dx; =

dt

(4.6-2)

.

h eac h ic wh in ed uc od tr in is les iab var of set w ne a ns, sio res exp se the To simplify

= 4; is t tha on, iti pos with respect to its equilibrium

distance variable is measured

t tha ng usi w, No nt. poi s mas i’ the of on iti pos m iu br li ui eq the is xjo e x; — xj, Wher ain obt we b, = X20 — 30 = 19 — x29 ons iti pos in the equilibrium K

=

K

5

— my

n O > + ) ni = (2 5 = PE and

nH,

(4.6-3)

tr |

ia

KE = ~ (97 +93 +775)

is t in po ss ma y an r fo on ti mo of The classical equation ,

mij; = m—

d* nj dt-

|

= Fy =

i(PE

e; e dn

(4.6-4)

e th of ve ti va ri de the as ed in ta ob is h ic wh i, t in po ss ma on e rc fo e th is F; where , us Th t. in po ss ma e th of on ti si po e th to t ec potential energy with resp 1)

(4.6-5)

mio = —K (no — 11) + K (3 — 12)

(4.6-6)

my, mi

= K(y2—

= —K (3 — 42)

(4.6-7)

The Ideal Diatomic and Polyatomic Gases

ion mot ic iod per a for t tha e not we , ion rat vib for es mod Now, to identify the normal the equation of motion should be of the form

me; = 4am" G

(4.6-8)

which has the solution ¢; = C;sin(27v,;t + C2). Consequently, for the expected periodic motion, we should have that mij; = —w*mn;, where for simplicity the substitution w* = 4 v* has been made. With these substitutions, Eqs. 4.6-5 to 4.6-7 can be rewritten as

— K(n2 —m) = 0

(4.6-9)

—monz + K (nz — m1) — K (73 — m2) = 0

(4.6-10)

+ K(ns — no) =0

(46-11)

—man

—mwn;

or, in matrix form

—K 0

Since

|, 92, and 3

0

ny

—K K — mo"

n2| 3

—K

K — mw* 2K

—ma* —K

0 =

|0 0

can take on any arbitrary values, the only way

can be satisfied at all times is for the determinate is, for

(4.6-12)

that Eq. 4.6-12

in the equation to be zero. That

K —ma"

—K

0

—K

2K — ma

—~K

()

—K

K — mor

(4.6-13)

|=0

Or

a (K* — wo m)3mK

(4.6-14)

— w*m*) = 0

This sixth order equation for @ has six solutions ay

=O;

a

=O,@34

K = 2,/—:

and

Fil

ws

=+,/

3K —

(4.6-15)

IF

The pair @34 and @3— refers to the two different phases of the same periodic motion, and cq, and @4— are the two different phases of another periodic motion; so of these only «#3. (which we refer to as @3) and w4, (m4) are unique periodic motions that

we consider further. To understand what the normal mode motions are in the original coordinate system, we now substitute, in turn, each of the values of « into Eas. 4.6-9, 4.6-10 and 4.6-11. Using wm = 0, we obtain

H2— 7, =0

272 —n1 — 43 =0 and

m3 — 2 =0

(4.6-16)

61

4.6 Normal Mode Analysis: The Vibrations of a Linear Triatomic Molecule

ravib a not is 0 = w h wit e mod the s, Thu 73. = 72 = 7; that which has the solution ion mot the is, t Tha le. who a as le ecu mol the of on ati nsl tra e fre a tion. but rather

is

Chan. rar) > > >

where the arrows, being of the same size and direction, indicate that the displacements of all the atoms are of the same magnitude and direction in this mode of motion. Now,

using w =

nm?

we obtain

) — K(yn2—-1=0 —Kn2+ K(j2 — m1) — K (na — 2) = 0 —Kn3+ K(y3— 92) =0 Kn

—Kyn, —K(j2—m)=0;

(4.6-17)

which has the solution 4; = —y3 and 72 = 0. Thus, in this mode, the central atom remains stationary, and atoms | and 3 vibrate with motions of equal magnitude and opposite direction, as indicated below:

ORONO < > This vibrational mode ts referred to as a symmetric stretching mode. As an example, this vibration in the linear molecule CO z has been reported from spectroscopy to be at a wave number of 1314 cm~', which corresponds to a vibrational frequency of @, = wave number x speed of light = 2.9979 x joc

x 1314 cm!

= 3.938 x 109 5 and a vibrational

temperature

Finally, using @ = \/

©,

of 1890

K.

we have

—3Kn, — K(n2 —m) =O

or

nz = —2n,

—3Ki92+ K(qj2-—m)-K(y3—-—92)=9 —3Ky3+ K(—y 723 ) =0

or or

y= -— 7=2 —2n3

(4.6-18)

Therefore, 72 = —2y, = —2n3, so that the end atoms move in the same direction and the center atom moves in the opposite direction with a displacement of twice the magnitude of each of the end atoms. This is shown below:

OnnOnwnvO > & >» This

vibration

«7 = 2335

cm™!

is

referred

to

as

for this vibration,

an

asymmetric

corresponding

stretching

to a vibrational

mode.

In

temperature

CQ>. ©,,

of 3360 K. Therefore, for a linear molecule (atoms restricted to remain along a line) there are three modes of motion; one translational mode (w = 0), a symmetric stretching mode and an asymmetric stretching mode. Of course, the atoms in a real linear molecule are not restricted to move along a line. If we were to perform the same type of

62

Chapter 4: The Ideal Diatomic and Polyatomic Gases

re the t tha d fin d ul wo we , le cu le mo the of on ti mo l na io ns analysis for the three-dime were 9 modes of motion or degrees of freedom for a linear molecule:

3 translational degrees of freedom 2 rotational degrees of freedom 4 vibrational degrees of freedom. The Two two the

two rotational motions are in directions perpendicular to the axis of the molecule. of the vibrational modes are as discussed above. The other two vibrations are symmetric bending modes, one in a plane and the other perpendicular to it, of form

cos

o Ov—9-——Ovv

For carbon dioxide, these two identical vibration modes are w; = 663 cm!

for this

vibration, corresponding to a vibrational temperature ©, of 954 K. In contrast, a nonlinear triatomic molecule also has 9 degrees of freedom, distributed as follows:

but

3 translational degrees of freedom 3 rotational degrees of freedom 3 vibrational degrees of freedom Two of the rotational degrees of freedom are perpendicular to the axis of the molecule, and the third is along the axis. The three vibrational modes—a symmetric stretch, a bending mode, and an asymmetric stretch—are schematically shown below:

For

water,

the

wave

numbers

for these

vibrations

are

3553 em—!,

1592cm7!.

and

3725 cm_!, respectively. Similar analyses can be done atoms

in a molecule

increases,

However,

for larger molecules. the number

of vibrational

modes

as the number of increases,

and

the

algebra becomes more difficult. (In fact, each additional atom adds three additional vibrational degrees of freedom.) Also, very large molecules contain chains that are flexible, and it becomes increasingly difficult to make a clear distinction between rotations, vibrations,

and, in some cases (i.e., polymers),

even hindered

translational

motions.

CHAPTER 4.1

4 PROBLEMS

a. Derive Eqs. 4.2-2 to 4.2-7 for the linear diatomic molecule, starting from the single-particle partition function of Eq. 4.2-1.

b. Also, show that for the linear diatomic molecule

if T > ©, and T > ©,

ET Cy 7 — —+ kT and — > =k N 2 N 2

4.2 Derive an expression for the chemical potential of an ideal diatomic gas. 4.3 Obtain the expressions for A and

gas when

the second

and

UL

for a diatomic

third terms in Eq. 4.1-8

must be included. 4.4 Compute and plot U and Cy for CO as a function of temperature over the temperature range of 100 to 1500 K.

Chapter 4 Problems 4.5 Calculate the entropy, heat capacity Cp, and chemical potential (Gibbs free energy) for nitrogen and hydrogen bromide at 25°C and | bar. The first electronic state is nondegenerate for both gases. 4.6 Calculate Cy in J/(mol K) and jz in J/mol for No at

25°C and | bar pressure. 4.7 Calculate the fraction of CO; molecules in first four vibrational states at 200 K, 800 K, and 3000 K. 4.8 Calculate the constant-volume heat capacity Cy for H;, HD, and Ds at 150 K, 250 K, and 350 K, assuming that the atom-atom separation distance and bond force constant are the same in these species. 4.9 Calculate the constant volume heat capacity Cy (in joule/mol K) for No and Clz over the temperature range from 300 K to 2700 K. How well can these

results be fit with a polynomial in 7? 4.10 In this chapter, the harmonic oscillator approximation for the vibrational energy modes was used: l = ( + 5)

iin

A more realistic model

Av

n=O,1,2,.....

is to include the first term in

an expansion for anharmonicity: Paine = (» + ;) hv — x (: mom

GL

1\°

+ ;)

he

where y is a small constant. What partition function for this case’?

k

=j(7+

10,

4.17

the perfect crystalline state should be 0 at absolute zero. Is the result you obtained consistent with the third law of thermodynamics? Explain. When a normal mode analysis ts done for a poly-

atomic molecule, one type of mode that may be found is a hindered rotation. For example, consider an ethane molecule. One internal motion of the molecule is a rotation around the carbon-carbon bond. However, the potential energy of the molecule ts higher when the hydrogen atoms on two different carbons are aligned than when the hydrogens are in a staggered conformation so there is an energy barrier for this rotation. The form of the interaction energy for this rotation is periodic, and for ethane, with three

is the vibrational

In this chapter, the rigid rotator approximation was used for the rotational energy modes Erolf

4.14 Compute and plot U’ and Cy for methane at | bar and temperatures between 300 and 600 K. 4.15 Determine the constant volume and constant-pressure heat capacities for an ideal diatomic gas in the limit of T — 0. A corollary to the third law of thermodynamics is that the heat capacity of a material in the perfect crystalline state should be O at absolute zero. Is the result you obtained consistent with the third law of thermodynamics? Explain. 4.16 Determine the constant volume and constant-pressure heat capacities for an ideal linear triatomic gas in the limit of 7 — 0. A corollary to the third law of thermodynamics ts that the heat capacity of a material in

hydrogens spaced

feciees

ys =0,1,2,...-..

A more realistic model ts to include the first term in an expansion to account for the fact that due to centrifugal forces, the molecules stretches slightly with

increasing rotational motion. This is accounted

for

by including the first term in an expansion about the rigid rotator model that results in

et = (G+ IO, — EPG +I? J =0,1,2,..0-. 4.12 4.13

&

is a small

constant.

What

is the rotational

partition function for this case? Compute and plot U' and Cy for H»O as a function of temperature between 100 and 3000 K. Compute and plot U and Cy for carbon dioxide at | bar and temperatures between 100 K and 1500 K.

120° apart, is given by

u(P) = gO

— cos(32)]

where «(Q) ts the height of the energy barrier for the internal rotation of the molecule. At high temperatures (AT > w(Q)), the barrier to rotation is small (or

inconsequential) compared to the kinetic energy of the molecule, and the internal motion is essentially a free rotation. However, at low temperatures (AT = u(Q)), the rotational energy barrier is large compared to the Kinetic energy, and the motion around the

carbon-carbon bond behaves as a vibration around the staggered conformation. What is the contribution to the constant volume heat capacity of this energy mode at a. / =0 K, and at b.

where

63

at high temperatures (Le., as T —

oo)?

4.18 One assumption that is sometimes made is that for isotopic species, the bond length is the same in each of these species, as is the vibrational force constant k. Are the rotational and vibrational temperatures of H2, Ds, and HD given in Table 4.1-1 consistent with this assumption?

Chapter 5

Chemical Reactions in Ideal Gases

In the previous chapters we saw how, from some simple assumptions, a whole framework could be developed permitting the calculation of the thermodynamic properties of dilute gases trom the results of spectroscopic measurements. In this section, another use of statistical thermodynamics is developed. First, the concept of chemical equi-

librium in an ideal gas mixture is discussed. Then it is shown how, from the same spectroscopic information used in the previous sections, the chemical equilibrium constant for an ideal gas phase reaction, the degree of ionization in a plasma (partially ionized gas), and the very large reactive contributions to the heat capacity can be calculated.

INSTRUCTIONAL

OBJECTIVES

FOR

CHAPTER

5

The goals for this chapter are for the student to:

e Understand how the canonical partition function for an reacting differs from that for a nonreacting ideal gas system « Be able to compute the chemical equilibrium constant for a mixture ¢ Be able to calculate the equilibrium compositions in a reacting ¢ Be able to compute the thermodynamic properties of a reacting

5,1

THE

NONREACTING

IDEAL

GAS

ideal gas mixture

reacting ideal gas ideal gas mixture ideal gas mixture

MIXTURE

Consider a system of Na molecules of species A and Ng molecules of species B in

a volume V and a temperature 7. The partition function for this system is computed by evaluating the sum

O(N4. Ng. V.T) =

S>

e Ei/kT

(5.1-1)

all states @

of the system

Let n' be the vector of occupation numbers

for the i'" state of the system;

that is,

Ne = (My. Mya. - +++ Mgy+2+ ---), Where ni’,; is the number of molecules of species A in the j'" energy state of a species A molecule in the macroscopic

64

state / of the

5.2 Partition Function of a Reacting Ideal Chemical Mixture

65

system. The quantity Nb; is similarly defined for species B. Each state occupation number vector must satisfy the following two restrictions: ry

= Na

(5.1-2)

8; = Na

>

and

J

j

Since a species A molecule is indistinguishable from other species A molecules, but distinguishable from a species B molecule (and vice versa), using the analysis developed in Chapter 2, it immediately follows that

Q(Na, Ng, V,T) = Cat Na, V.T)OpCNg, V,T)

(5.1-3a)

Also, if the number of molecular energy states is much larger than the number of molecules (which is always the case for the systems we consider), then N

O(Na. Ng. V, T) = Qa Na. V. T)Op(Np, V. T) = fa

7

(5.1-3b)

and, more generally, for a mixture of S noninteracting species

S

qh

O(N), No,....Ng,V,T) = I] a The thermodynamic For example

properties of the S-component

= —kT lnQ =

—kT In ieai

mixture are easily evaluated.

=r

(win ln 1) —

i=]

(5.1-4)

(5.1-5a)

i=]

Here, g; 1s the partition function for species 1, which may be an atom, a diatomic molecule, or a polyatomic molecule. Also, P=kT

ainO

(SF ).,

kT

yd

=



N;

__—

5.1-5b

which is the ideal equation of state. Earlier we saw that departure from ideal gas behavior was not the result of the internal structure of the molecules, and here we see it is not a result of forming a mixture. In a later chapter, we shall see that nonideal gas behavior results from the interactions between

PARTITION CHEMICAL

FUNCTION MIXTURE

OF

A REACTING

molecules.

IDEAL

Above we showed that the partition function for a simple mixture of Na molecules of species A and Ng molecules of species B is

O(Na, Ng. V.T) =

qu’

an”

Nal Np!

(5.2-1)

Chemical Reactions in Ideal Gases s ga re pu a r fo on ti nc fu n io it rt pa e th th that is to be compared wi N

(5.2-2)

O(N, V.T) = mr

tes sta rgy ene the all r ove sum the is on cti fun ion tit par le In all cases, the single partic accessible to a molecule. s cie spe s ute nsm tra t tha ur occ can ism mer iso or on cti rea e ibl ers rev a e Now, suppos A into species B and vice versa, A"

accesible

s e t a t s y g r e n e | i er

N __

q

(5.2-3)

= NI

N!

o=

4

and for a single molecule g=



ees kT

_

ef MAT als

>

-

accessible

energy states

energy states

energy slates

available to species A

available to species B

eo By ART

= qa tT 9B

So that ~

(ga + Gp)

iV

|

N!

where

N = Na+ Np This expression can also be obtained by a less intuitive but somewhat more general procedure. Here we start with the observation that, due to the chemical reaction, the actual number of molecules of species A present at any time is not a Known quantity, but may be 0, 1, 2,..., N, where NX and Np are the number of molecules of species

A and species B initially present, respectively, and N = NX + Ne. Furthermore, for any particular set of values for Na and Mg, Eq. 5.2-1 is valid. Therefore, the partition function for this chemically reacting mixture of A and B, for which all values of Na

67

Three Different Derivations of the Chemical Equilibrium Constant in an Ideal Gas Mixture and Np are allowed subject to N = Na + Ng, is

ou ih

dx 98, Is

NI!

2

(N—2)!2!

we “Ye

-»Na=0NpaE =0

9B,

—2

IK

OB

N—3

aAWG

N—

1 (N-—1)!

0 0

° (N—3)13!

a

|

N

WN!

(5.2.5)

an?

is

with the restriction on the double

Na+

summation

that

(5.2-6)

Np = NR + Ne =

The double summation of Eq. 5.2-5 with restriction Eq. 5.2-6 can be reduced to a single summation by eliminating Ng in terms of N — Na, that is Ma

9a

44p

N—WNa

o= L t Nali — Na)! (Na

iE

N! ee

NW!A Nal(N—Na)!®

ghagh-® = (qa + 9p)”

78

N!

(5.2-7)

The Jast term in the above equation is obtained from the term that precedes it by use of the binomial expansion, as discussed in the Appendix to this chapter, so that Eq. 5.2-3 is recovered.

THREE DIFFERENT DERIVATIONS OF THE CHEMICAL EQUILIBRIUM CONSTANT IN AN IDEAL GAS MIXTURE Starting from Eq. 5.2-7, we will now obtain the chemical equilibrium constant for this reacting system by three different methods. Each of these methods leads to the same result, and the purpose of obtaining the same result by different methods is to illustrate the variety of methods that are used in statistical mechanics. The first method is based upon the interrelationship between statistical and classical thermodynamics. From classical thermodynamics, we know that the criterion for

equilibrium state in a closed system at constant T and V is the state of minimum Helmholtz energy A with respect to all possible variations consistent with the physical situation. Here, this implies that A should be a minimum

with respect to Na (or Np),

subject to the restriction of Eq. 5.2-6. Alternatively, since

A=—kTInOQ

(5.3-1)

where the partition function Q for any particular choice of Na and Eq. 5.2-1, and the equilibrium requirement is that the In Q have a subject to the constraints of constant V, 7, and N = Na + Ng. This can be found by the straightforward approach of eliminating Np in N, to obtain InQ

= Nalnga

— Naln Na + (N — Na) Ingp

Ng is given by maximum value maximum value terms to Na and

— (N — Na) ln(N — Na)

(5.3-2)

in Ideal Gases

Reactions

Chemical

which is an unconstrained function of the single variable Na. Now, setting! din O —— =0= Ing,

—InNa

a

dNa

Na —Na — — —Inggp + In(N — Na) + ——— Na

N—

NS

AT

— InNa — Ingg + In(N — Na) = Inga — In Nag — Ingg + In Np

= Inga

(5.3-3) Therefore, the condition for equilibrium 1s that N

Ga _ 9B

«XB _ 4B

Na

Na

7

Ne

7

(5.3-4)

in

This equation looks very much like an equilibrium constant relationship, It can be made to look more so by replacing particle numbers by number densities or number concentrations, that is

ll

638

) ( n m — ) ( n i = W K M E M Gaf¥) = Nafvy (qp/V)

Kw

Ga

ga

ES

qKe

(Np/V)

=

|

(=

|

.

a

ark

|

=

a

This is an interesting result, since it establishes that the equilibrium constant for a reacting ideal gas mixture can be computed from a ratio of single-particle partition functions. The second method of computing the equilibrium concentrations in the simple reaction being considered is by computing the average value of the number of A molecules using the probability p(Na) of the occurrence of a state in which Na molecules of species A are present, which is N—N

Na

SS SS

N

Nal(N

OQ

i

rt

—————"eg TA

a

(3

dp

(oa

ON!

— Na)!

5.3.6

VA

)

From this we obtain

Wa.A N

N =

N

>

ADC

N

A)

1

|

O

N!

—_—_— =

=

N »

Va=0

Na=0

NaAlNN! «

| 7

ee

_

Na)!

tA

A

dp

Na

_N

1

|

t NNi

y a t , fo (ga + gp)”

.N—WN,

Na

Nal(N

0

Na=

=

Na

Nal(N

A

Nal(N — Na)!

N! Naw

_

a

NEN

Ne

NaN!

st

Na)!

N!

=

Aw r | r e r a we ICN — Na)! @4

et (Wa

24,

Na

é Ip

N-Na

-7 53 ) (

N!

'Note that

Ny,

is an integer variable;

consider it to be a continuous

however,

as we are dealing

with the order of 107°

variable, as the difference between

Ni, and

Na+1

molecules.

we can

is so small. So we can take

the derivative with respect to M4. “Because of the magnitudes of the individual partition functions, it is frequently desirable to compute Ing rather than q directly, and therefore In Ay or In ke. 7

.

z

i‘

.

=

.

Bok


Frot,N>Gvib, No elect, No / VY (4trans,0> Grot,02 Fvib,O> Felect.0% iV) 4

rs

2

_{— | ——Myo Se _

Co SS

Mn, Mo, ¥

l—e

— 2

(

I

Welect,NO

ee ey

)

2

| — ¢@v.no/t

v.05 /T

e!2Do,.no—Bo.n3— 80.02 )/ TF

.

Melect.Qy

2 71 2 1 — ¢— 2745/3000

2.08 x 2 x 2.89 x (2 \ ? 30.017 (1 x 2.45)? 28.02 x 32 a

( | — @ 7 2278/3000

p— 3390/3000 |

) (2%4.43—9.76—5.08) 1.602% 107 !? (1.38044 10!" < 3000)

Zz

1.007 x 4.006 x 11.13 x 0.677 x 0.266 x 8.987 x 10°'' =7.264 x 107! where each term has been calculated separately to show the dominance of the electronic

energy difference in the calculation. Using this value of the equilibrium constant, x = 5.489 x 10~® or 5.489 parts per million (ppm).

The thermodynamic

properties of the reacting system can, based on the analysis in

this section, be computed from the partition function S

ocr. Vv.) =|]

Ne

vm

qf;

(3.5-7)

i=l

However,

to proceed,

it should

be

noted

that

using

the

stoichiometric

coefficient

notation of Eq. 5.5-2, the number of molecules of species / present at any time (N;) is related to the number of molecules of this species initially present, Nj.o, by the relation

N; = Nig + VX

(5.5-8)

ON; = vjdX

(5.3-9)

so that

Chemical

Reactions in Ideal Gases

, on ti ta no is th g in Us ) s. le cu le mo of er mb nu of s it un s ha X y, wa s thi d ne fi de at th e (Not om fr ed ut mp co is em st sy ng ti ac re e th the pressure of P(N. , P(N

AT (

V.V.T)T) =k

Q* din — av

).

— £T

>

>

a

; +N N? In N* — ; * lng (N —j| a j Ing aI,

] 8 ) 2 M ( ) 2 er [mn ( 7

5

s

gi

ax

kT

(>). 2, n( =)

V 2M i+

(

:

ore ref The 5a. 5.5Eq. to due es ish van side d han htrig the on m ter last the re whe

kT

P(N, V,T) = —- 2 N’

(5.5-11)

Likewise, the internal energy of the reacting system is

UT,

Vv.)

=

,fdln kT? (

ag

dy \¥

where again Eg. 5.5-5a has been used, and U; is the internal energy of N* molecules of species i at the temperature and species equilibrium number, just as in an ideal gas mixture. Equations 5.5-11 and 5.5-12 are the result of the fact that this is an

ideal gas mixture. However, remember when

using these equations that each N*

changes with temperature, and therefore the internal energy of the reacting mixture has an additional temperature dependence above that of a nonreacting mixture, due to the change in the extent of the interconversion of some species into others with temperature.

5.5 The Chemically Reacting Gas Mixture: The General Case

77

A not-so-obvious result is obtained by looking at the constant volume heat capacity, Cy, of the reacting gas

fe =

Y

au

S

N

2X

(4)

v

a Ing;

247 >) >

—2kT

$

: aT “)ne ,

N*

rar

kT?

9: (SeaT*In gi

|

=)

)

AE

(ar), = a7

S

,

3

= > (ar )

aT

"|

(=)

dy

fading;

(oNaT “) (

(ax\

ding;

jy

aN?

Ov, +kT

aN?

aT

),

©

= Devi + (sr). a o

me

where w; 1s the internal energy per molecule of species 7. Here, the first term after the last equal the equilibrium contribution to seen using the

sign is the sum of the heat capacities of the pure components at composition; the second term is new and can be a very significant the heat capacity as a result of the chemical reaction. This can be notation of Eq. 5.5-1 so that

Arxnlt = > vu;

and

(aX

Cy = > Cy + (=)

i

i

}

Arxnld

(5,.5-14)

v

where Arxnu is the internal energy change of reaction on a molecular (not molar) basis for the stoichiometry of Eqs. 5.5-1 and 5.5-2. Consequently, the last term in the equation above is the contribution to the constant volume heat capacity that is a result of the internal energy change on reaction during the course of the reaction, For engineering purposes, this equation is more conveniently written using molar heat capacities. By using Avogadro’s number Na, this results in the following expression:

CY = >a niCyi V j af t+

Ox

(—) ar

ax

,

NavAnne = >» niCyi+ (—]) A ran pe Wal aT I

f

where

now

the species

heat capacities

7

AgnU rxn

(55-15

)

2

are on a molar basis, n* are the number

of moles of species / present at equilibrium, x is the molar extent of reaction for the stoichiometry of Eq. 5.5-2, and A;x,U is the molar internal energy change on reaction for this stoichiometry. This reaction contribution to the heat capacity can

Reactions

Chemical

in Ideal Gases

be very large (as we show in an illustration that follows) for a system with a large energy change on reaction, but only over the temperature range where the extent of reaction is changing appreciably with temperature (that is, over the temperature range where (dx /d07T)y

is nonzero.)

Though large due usual. To molecules

we do not consider transport properties here, when the heat capacity is to reaction, the thermal conductivity of the gas is also much larger than see this, consider a dissociation reaction with a large heat release. The gas would then dissociate at the high temperature surface, absorbing heat of

reaction;

migrate

re-associate and release

to the cooler surface,

the heat of reaction

resulting in a large rate of heat transfer and a high effective thermal conductivity. One situation where it is especially important to account for this is in the design of heat shields for spacecraft entering the atmosphere of earth or other planets. Because of frictional heating, high temperatures sufficient to 1onize the gas in the boundary layer of the spacecraft can result. Engineers need to account for this in their design. To illustrate how to account for the change in heat capacity, consider the problem of computing the degree of ionization of a gas of atoms as a function of temperature. The partition function for the mixture of atoms, ions, and electrons is

C=



gi

qi

Ne

2A

i

te

5.5-16

NAMING

(5.9-16)

where the subscripts A, 1, and e designate the atoms, ions, and electrons, respectively.

The number densities of these species are interrelated by the equilibrium relationship

(gi/V(ge/V) ga/V

_ (Ni/V)(N-/V) (Na/V¥)

(3.5-17)

or, using the molecular extent of reaction, X, as the independent variable, we have Na

=N,—X,Ne=X

InQ =(N° _

and

No=X

—X)In (tsa 7) + Xin (2) + Xin (©) +(NS +X)

Oo

dA

and

qi

V Vlge/ V qa/V

(gi/

de

0

Xx? =

V) _

(55-18) :

(5.5-19)

V(Nx — X)

The partition functions for each of the species are “a ()

amrmakT a

fz

3

\2

Ge OA elect, 1;

and

(*)

2rmeklT =

a

3

\2 We elect, |

;

di (7)

2amjkT ~ (=)

\?

ae Wi cteci,1e

Here, we have taken the state of zero energy the electronic energy difference (between the thermore, the partition function for the electron in-the-box model. (Also, we have neglected

7

eee IAT

(5.5-20)

as the atom at rest and have attributed atoms and the ions) to the ions. Furhas been computed from the particlean important interaction term among

5.5 The Chemically Reacting Gas Mixture: The General Case

79

charged species that results in the so-called ionization potential lowering, a concept that is beyond the current discussion.) The electronic degeneracy for the atom is unity, and it is equal to 2 for the ion and electron. Thus a

;

Ne

Kn

—_ (qi/ V)Ge/V)

@alV)

ie

a

|

=

(Ni /VICNe/ V)

) v V . N T = (Na/V)

e Fielect, 1/ AE

(=)

=

x?

5(5.-21) .5-21

WVI(NNPS—LXY))

where the masses of the ion and the atom have been taken to be identical. Now, from the ideal gas law for the reacting system (Eq. 5.5-11)

PV = >> NjKT =(Na+ Ni + NoJkT = (NX —X +X + X)KT = (NQ + X)KT j (5.5-22)

SO V=(NA+X)KT/P and combining Eqs. 5.]-19 and 5.1-21

———

x?

AkT

=

¥Y = X/N¥

“Theat. tf

e

yp



(5.5-23)

P)

F(T,

h-

P

(Na — X*) Now, defining

(2amekT\?

to be the degree of ionization, we have f=

/

F-F

\ —_—=

(5.5-24)

Also (°

In )

—— ay

=

v



In #)

-

ar

3

= —;

fy

(din qi

and

2T

.

of

Vv

a

Si eclect, |

aT

kT

= —— =}

=

3 =

ge

thn



Nj

_

Ne)

+ Ni€i,elect, |

=

tr} uo

so that

kT (Ng + X) + XEj elect.

or U (per mole of argon initially present) = SRC

+ ¥)4+ YE: etect.1

and Cc

V



fa

— ().

3 2

.

3 —

2

—RFi(| 7 (i+

3. 2

Y)+ b+

ay

—AT 7

— (Sr)

\

(ay ar

ay +

elect. | ees

= (s).

(5.5-25)

|

Reactions in Ideal Gases

5: Chemical Also

U+ PV

H (per mole of atoms initally present)

3 —RTU+Y) + Yeicteoi +

+ Y)RT

i

5

=

qt

+ YY) H+

Pi cect. 1

(5,5-27)

and finally OH

5

3

ay

fo=(—] +R" {= P (or). =2Ra0 ; (1 4") + y+ 5 (Sr), ~-RU+Y)4 =

7

RT 9

+ 6

dy

,

teem

i,clect, |

oY aT

=a

(FF)

p

(5.5-28) .

It is interesting to note that, at low temperatures, Cp = 5R/2, while at high temperatures Cp = 5R. Can you explain why this is so?*

ILLUSTRATIONS

The Ionization of Argon One mole of gaseous argon at | atmosphere is to be heated at constant pressure to very high temperatures. Using the equations above: (a) Compute and plot the degree of ionization of argon as a function of temperature from 1000 to 30000 K at 0.01, 0.1, | and 33.6 bar. The ionization energy for the first

ionization of argon

Are Ar’ te is 15.76 electron volts. The mass of an electron is 9.1083 x 1077*g. (b) Compute and plot the constant pressure heat capacity for this plasma over the same temperature and pressure range.

SOLUTION

The number-based equilibrium constant is calculated as follows: iT\

2 Kw

=

4

(=)

3

oe Fielect 1/RE

2 8)3/ 1077 x 3 08 .1 (9 x 7 10 x 88 1. x =4

74/2 97 19-16% 1.602

1o-!2 (1.38044 x 10-18 x 7) LONS 3 Ili:

“The answer ts that before dissociation, only the atoms

After dissociation translational

is complete,

motions,

each

atom

has been

are present; each atom

has three translational

motions.

replaced by an ion and an electron, for a total of six

5.6

Two Illustrations

The values are shown in the figure below. From Eq. 5.5-23, the quantity of more interest 1s

erp) = MT (20m) AKT

F(T, P) = —

f2mm,kT

{| -—>—

1.38044 x 10-73 (2) x T(K)

4

\2

a7 Fivelect, Lf AT €

—_

P (bar) x 105 (44°)

Ky (m > )

1.38044 x 10-78 x T(K).

= Sf, P (bar)

To calculate the degree of ionization, Y, Eg. 5.5-24

is used. Finally, the heat capacity

is computed using Eq. 5.5-28 after computing the degree of ionization (dY/0T)y. The results of all these calculations are shown in the figures below.

50

=

‘0

Be

=)

—450

: 1-107

0

0

2-107

r

5000

31-10%

1.5-10%

2-104

T

Cp/2.5R

Log of the equilibrium constant Kw (left) and the degree of ionization Y (right) as a function of temperature. In the degree of ionization figure the lines are in order of the pressures are 0.01, 0.1, | and 33.6 bar, respectively.

0

ff

0

MA

1-10

2-107

Tr

The derivative (0¥Y/dT)p and of temperature. 0.1, | and 33.6

of degree of ionization Y with respect to temperature (left), the reduced constant pressure heat capacity Cp/2.5R as a function In these figures the lines are in order of the pressures are 0.01, bar, respectively.4

Does the pressure dependence of the results shown in the figures agree with LeChatelier’s principle?

61

‘hemical

Gases

in Ideal

Reactions

The Adsorption of a Gas unto a Solid Gas molecules at a pressure low enough that the gas can be considered ideal are in contact

with a two-dimensional surface at which some of the molecules may be adsorbed. The partition function for a single gas molecule is q,,,, and the partition function for a single molecule adsorbed on the two-dimensional surface is qg,q. If there are M adsorption sites on

a two-dimensional

surface, and

N

identical

molecules

adsorbed,

where

Mf => N

so

that the adsorbed molecules do not interact with each other. the partition function for the adsorbed molecules is

O(N, M,T)= Nic

M! N — Ny! La!

where the factorials arise from the number of ways of distributing N indistinguishable molecules over M distinguishable adsorption sites (since they are fixed on the surface of the graphite.) Use this partition function to develop an expression for the fraction of the M adsorption sites that are occupied as a function of the pressure of an ideal gas, and qgas and gag. SOLUTION

The chemical potential of an ideal gas molecule is gg =

dinQ

kT

=

a

= —kT —(N 1

—k7 (des

_

hi = = 4 ') = —kT In (=)

.

N

‘Vv —kT In (=) N Where as usual surface 1s [Ladaj

= —AT

— =

kT 47



= Geax

— _

eas

‘kT = —kT In (it) P

V. The chemical potential of a gas molecule adsorbed on the

ailnQ (

aN

a

)

IN In daa —NinN+

WN —(M—

N (Ings — InN — wo ti

tinGt

N)In(M— N)—(M—N)|

— 8) +

| — )



AT

in ( #8

At equilibrium, jtyy, = fad. 80 that

daa(M — N)

Haq = —kT In (a) Gau(M

N



N)

_

YoaskT P

or



= Pyas = —kT In (")

M-—N=—WN

PoaskT Gad P

kD

Maw (r4 ee

GadP

)

f GoaskT

_ =)

N

83

Appendix: The Binomial Expansion

is

Therefore, the fractional coverage 6 = N/M

o-N_

P

1

M4

MT)

oh

PhP

ews | PO peers

au

KT

fad

kT at This equation shows that at low pressure A = —

>> P, the extent of adsorption (fracKT

:

'



;

:

.

PP +h

tional coverage) increases linearly with pressure, while at high pressures P >> =

qo e

= 4,

the coverage saturates at complete coverage @ = 1. The fractional coverage is an S-shaped curve between these two extremes and Is referred to as the Langmuir adsorption isotherm; A 1s the temperature-dependent Langmuir parameter.

THE

BINOMIAL

EXPANSION?

The canonical partition function for a binary mixture of molecules undergo the chemical reaction A < B is, as shown in the text a

N

qn Age

_N-N

A

ati

Nal(N — Na)!

SUES

Na=0

that

A and B

52-7

We can evaluate this using the general form of the binomial expansion, written here as

(A5-1)

=(x+y)"

a — n a w — » M=0 or to match Eq. 5.2-7

N

| __

! N — Na=

Naqn* dp 0

Nal(N

a

(A5-4)

— Na)!

* According to Wikipedia, the binomial expansion is attributed to the |7th-century mathematician Blaise Pascal, but was known to other mathematicians including Halayuhda in India in the 10th century, Omar Khayyam in Persia in the

11th century,

and

Yang

Hui in China

in the

13th century,

all of whom

derived

similar results.

Chapter 5: Chemical Reactions in Ideal Gases that arises in the equation Naqngh- Na

,

N 2d

Na= Y~ Nap(Na) YS. Nal(N Aa fe— A APUNA) =~ O a ¥4=0

(5.3-7 )

Nay!

i

The summation of Eq. A5-4, written in generic form so that the result can also be used later, is

.

N

DN

Mwy

MT"

M.N-M M NM

_*

>

N

M

2 MN

M..N-—M My NI

Mh"

AS-5) “e)

>

Here, in the second form of this equation, we have changed the lower limit of the summation from M = 0 to M = |, since the M = 0 term in the sum Is 0 as a result of M in the numerator. The goal is to transform Eq. A5-5 into a form of Eq. A5-1. To proceed further, we define the new variables MM = M—1 and NN = WN —1, Note that with these substitutions, the variable MM goes from MM =0(M = 1) to

M = N; that is, equivalent to 44M = N —1=WNN. N >

Therefore

N sM x yN-M 6a, MUN — MM)! "

M eM yN-M M\(N—M)! ~ M=0

NA

_

»~ | waren x 4A being a a @ ane oe SEsOe ee Se EV TOD 2G, for a site without an molecule and gg =I! adsorbed molecule.

Chapter 7

Interacting Molecules in a Gas In this chapter we introduce the idea that molecules interact, so that the energy of interaction contributes to the total energy of the system and must to be taken into account in the partition function. We do that in this chapter for the case of the disordered molecules in a dilute gas using graph theory to derive the virial equation of state. In the following chapter We compute values for the second virial coefficient for specific molecule interaction models. In succeeding chapters, we will then consider interacting molecules in a crystal, which is a dense but well-ordered medium; and then a liquid, which is a dense but disordered substance.

INSTRUCTIONAL

OBJECTIVES FOR CHAPTER

7

The goals for this chapter are for the student to: « Understand the concept of the configuration integral ¢ Understand the pairwise additivity assumption for the intermolecular potential e« Understand the graph theoretic method used to derive the virial equation of state from the canonical partition function

e Understand the derivation of the virial equation of state from the grand canonical partition function

7.1

THE

CONFIGURATION

INTEGRAL

Consider a gas composed of N identical atoms. The spatial position of each of these N atoms

is specified

by

the collection

of N

vectors

(r),r5,...,£,),

where

rj has

the components (x;, ¥;,z;) and the kinetic energies by the N translational energy quantum number vectors (/),/5,..-,/j,), where 1; has the components (/;;, /)j,/:;). The potential energy of interaction for the N-particle system will be written as W(F).fo....,Py), Where we have assumed that the particles are spherically symmetric, so that the interaction energy of the system is only a function of the particle location.

Note

also that we

have treated the kinetic and internal

(electronic) ener-

gies of an atom as quantized variables; but the potential energy is being treated as a classical variable, since very small changes in position (allowed by the Heisenberg

98

7.1

The Configuration Integral

99

uncertainty principle) produce very small interaction energy changes. so the spacing between the potential energy levels is infinitesimal. We will also assume that the kinetic energy states available to a particle are not affected by its interactions with other particles. In the analysis of this chapter, we will limit our consideration to a system of N identical monatomic particles. At the expense of greater complexity in both the notation and physical description, these restrictions could be removed. For example, polyatomic molecules could be considered if we specify both the position r and orientation of each molecule, as well as tts translational, rotational, and vibrational quantum numbers (or, classically, the translational and rotational velocity vectors), and realize that the potential energy is now a function of both the position and orientation of each of the molecules. Furthermore, the extension to mixtures of species can easily be made. However, our purpose here is to develop, in a quantitative fashion, the fundamental aspects of studying nonideal gases; and the basic ideas of the development will only be obscured by complexities discussed above. Therefore, our development will be restricted to monatomic gases. Each of the atoms has a set of allowable kinetic and internal energy states, which we shall assume is unaffected by the presence of other atoms. Consequently, if E' represents the i" state of the N particle system, then

= Debt Os

[Ma + a] thot -

i

-

i

\e

f

i

j=l

(7.1-1)

where ¢'.. int is the internal energy of the j'” molecule in the i" system state, which

is the electronic energy for a monatomic molecule (or the electronic, rotational, and vibrational energy for a polyatomic molecule), ry represents its position vector, and each / 1s a kinetic energy quantum number. Assuming that the electronic. kinetic, and potential energy states are independent of each other, the partition function for this system can immediately be written as int)

Q(N, V,T) = “in? N!

N

2

AT

(==) h?

3N/2‘

oer

vr ff

eemeeewat dr,...dry (7.1-2)

where f,,[ |dr; = [J [ ldx;dyidz;. The constant C has been introduced as a normalization factor, and to keep the partition function dimensionally consistent. In particular,

from

the discussions of the previous chapters,

if there are no interactions

between the molecules, then

O(N.

ef

V,T)=

fe

(gin) a

Wir peti

(2amkT

(==)

KR E dr,

VX?

..dry

y"

=!

so that

s Ga a in s le cu le Mo g in ct ra te In Chapter 7:

. V = y r d . . . r d kT e w t t e f -J . 9 = ) y 0 . . , ) r However, when u(

Therefore

\

Vy

cv¥ =1orC = V~*, and

oy

QnmkT\? (gin)

a

O(N,

V,T) =

I/

Wi

he

ele

EN

RE dr).. «Oty,

oy

Q2nmkT\2 (Gint)

he

) T , V , N ( Z — _ = ~—___—__ N!

, al gr te in n io at ur ig nf co d le al -c so e th Here Z(N, V, T) is

.... dry

Z(N,V,T) = ee

(7.1-3)

V

Vv

7. e ur at er mp te d an V, me lu vo the N, s which is a function of the number of particle on ce en nd pe de the N, s le cu le mo of er mb nu The number of integrals depends on the nd pe de or ct ve on ti si po ch ea for ts mi li n T is through the integrand, and the integratio on the shape of the volume V.

THERMODYNAMIC INTEGRAL

PROPERTIES

FROM

CONFIGURATION

THE

c mi na dy mo er th the all n, ow kn is al egr int n io Clearly, once the configurat of the system can be evaluated: A=—kT

InQ=

properties

T) V. N, Z( In KT — A In KT 3N + T Nk — N n TI NK + — NAT Ingim (7.2-1)

on ti ta no gth len ve wa e li og Br De r lie ear the ed us ve ha where for simplicity we hi

& = Then, for example

OA

po fe)

OV

NT

;

'3.4-5)

(5 mk =)

nar

on

Z(N,V,T

SS av

N.T

(7.2-2)

mo le cu the le s— th at be tw ee n int era of cti on ene rgy pot ent no ial is the re if Note that for tha t to red uce s fun cti on par tit the ion Va = nd Z the n is, if w(r,.....0y) = 0, the ideal gas, which implies that NAT P= — V

on cti era int of rgy ene no is re the if y onl ed ain Consequently, the ideal gas law is obt d ne ai nt co are te sta of on ati equ the in s ect eff gas al ide non the ; all les tic between the par in the configuration integral Z(N.V.T).

7.3 The Pairwise Additivity Assumption

101

ty nti qua s thi of n tio lua eva the h wit ned cer con be will r pte cha The remainder of this g a pin elo dev in is st ere int r, our ula tic par In . gas se y den tel era mod —a se ca for one theoretical basis for the virial equation of state P

:

—— = 1+ Bi(T)p + BT)p° +> pkT

(7.2-3)

which is a Taylor series expansion in density around the ideal gas result, with B3(T) =

B,(T)

AIRWISE



a(P/pkT

lim (“ee”)

dp

p0

(n—1)! lim (

ADDITIVITY

r

:

(a?(P/pkT

|

#Aa(T) = = lim (aoe)

dp-

2 p—0

a" aa) (P/pkT)

ap"

r

a

(7.2-4) I

ASSUMPTION

In order to evaluate the configuration integral, it is necessary to have an expression for the interaction energy between the molecules (actually, we consider only atoms here).

Thus, before proceeding with the analysis of the configuration integral, a discussion of the general character of the interaction energy of an assembly of molecules is useful. The interaction energy between two molecules is written as u(r), r5) and is a function of the positions of molecule | and 2 at r, and r5, respectively. However. by the assumption of spherical symmetry of the molecules (atoms), the interaction energy will only be a function of rj2, the distance between molecules | and 2, rig2= lr; —ro| = [Cr — x2)? + (vy — 2)? +z — z2)°]?. Even though at this point we may not know the details of the interaction potential, we can specify two boundary conditions. First, since atoms cannot overlap, we can expect that u(rj2) — co as rj2 — O. Second, at infinite separation, we expect that there is no energy of interaction between the molecules, that is u(rj2) — O as ry2 — ox. The interaction energy between three particles can be written as W(F), fo. Fa) = ulri2)

+ (ria) + u(ro3) + (P12, P13, 723)

(73-1)

The first three terms on the right-hand side give the total potential energy as a sum of three pairwise or two-body (two-atom) interaction terms. The last term represents the correction to this pairwise additivity assumption as a result of the distortion of the electron clouds of the atoms due to the presence of the other atoms in close proximity. We will, for the present, neglect this nonpairwise additivity term, since its contribution may be small, except for very dense fluids. Therefore, for the three-particle system we will assume

Wry. 2.73) = u(ry2) + u(ri3) + u(rz3) = > i

-

u(r)

(7,3-2)

i

lsi f

> j l=; yun

u(riz)

(7.3-3)

Chapter 7: Interacting Molecules in a Gas

This is the assumption of pairwise additivity. Then the Boltzmann factor in the interaction energy can be written as _

Ey) LAT

ME

s :

-§E

wry df kT

;

' lic j

@ eri

I |

_ [|

i

{RT

(7.344)

j

l 0,

u(r) >

co

and

fj = —-I

O

and

fj =0

(7.4-2)

Si fay +... dey

de,

and as rij

co,

u(r)

Using Eq. 7.4-1 in Eq. 7.3-5 we obtain

Vv

y

?

if Lstsyem

which can be expanded into a sum of products

Z(N, virn=f...f

yo

1+

>

Y

si

fit DOOD

pha

fe (7.4-4)

where the complicated restrictions on the summations are necessary to insure that each cluster function is not counted more than once since, for example, fj> and fry represent the same interaction. There are now several similar methods that can be used for the evaluation of

the configuration integral. We will use a technique that amounts to the reduction of Eq. 7.4-3 into a collection successively more complicated integrals. The simplest term in the configuration integral to evaluate is the first one, which can be evaluated exactly

ldr,...dry =V™

[| V

V

(7.4-5)

7.4 Mayer Cluster Function and Irreducible Integrals

103

The next integral is

V

Vv

is only

Since the integrand

a function of r,; and rj (really rj), the integration over

all other position coordinates can be performed to get

fide dr; = v2 ff

v0 ff vv

(7.4-7a)

fydridyidesdsjdyjde,

The choice of the origin of the coordinate system is arbitrary. To do the integration above, it is especially convenient to choose the origin to be at the location of particle j. Then we change the variables of integration from (x;, yj, 2), ¥j, Yj.) to (xy, Yj, ZX j. Vj. Zj), where

xj = x; — x;, etc.

the Jacobian of this transformation

It is a simple

task to show

is unity. Also, the cluster integral

that

fj; 1s only a

function of the variables x,;, yj. and z,;;, so that the integral over particle j/ can be done and we have

yw

/

figdxidyjdzidxjdyj;dzj = V"* / = yl

fig xijd ydzig x jy jdzj

|/

fyaxyd yd zi

(7.4-7b)

Next, the variable of integration is changed from the position vector in rectangular coordinates to one in spherical coordinates, that is fii = (Xij, Vij, Zi) => rij = (ri. @, @)

and

dxjydyyd2j => rj sin@dbd¢dr; where rj is the scalar distance between molecule i and molecule /. Furthermore, since u(r;, rj) = u(rij), it follows that fj; is a function of only rj, so that

yr! / /

a

fd xd yd 2;

|/

fi(rg)rj sin 0dOdddr; oo

=

vlan

f

fiytraarjari

(7.4-7¢)

0)

This is as far as we can go with the evaluation of this integral until the functional

form of the interaction potential u(rj) is specified. We refer to this integral as an irreducible integral, which we indicate as f,, that is kL)

i, = ax f fylrapr}ary 0

(7.4-8a)

and denoted graphically as

ee Here, the filled circles represent molecules whose positions are to be integrated over the volume, and the line shows that there is an interaction between these molecules. For later reference, we note that

By

1

=

V

fy

I

f fidrjdr;

=4n

/

fir

ria ri

(7.4-8b)

0)

In Eq. 7.4-4, the term

J

(7.4-9a)

y e d e d n i f d d |vf of

V

.

=

results in N(N — 1)/2 terms containing 8), where N(N — 1)/2 represents the number

of distinct pairs that can be formed from N molecules. Since N > I, we will neglect terms of order unity with respect to N and write this as N*/2, so that N2

J \

[LY sian ..-dty = yn '— By vot 4 -

(7.4-9b)

j>i

Thus, so far we have

—_ y

om

MY

A(N,V,T)=V"+V

N{

No

>

Bi

V

}

(7.4-10)

The next type of product that arises in the expansion of the configuration integral is

/ _ / DDE V

og

voi

DY. Sify dry. dry

(7.4-11)

jy

ff

j>i

i

Here two cases arise: (a) i, j, i’, and j" are all different; and (b) eitheri = i’ or j = j’. A representative example of the first case is the product f\> f34, which results in the

integral

[of fatades V Vv

dey = vf. v

I|

=

: Interacting Molecules in a Gas

yw"

f fafades dry dry de, v

/ / firdry dry | | / / fra dr dr, ) VV

vv

!

(7.4-12a) The first equality results from the fact that the integrand is only a function of the position vectors r), F5, f3, and ry; the last relation arises because the integrand is a product of two factors, the first of which

depends only on r , and r,, while the

7.4 Mayer Cluster Function and Irreducible Integrals

105

second is a function of r, and ry. Then, from Eq. 7.4-8b, we have

[--[ fofadey...dey V Vv = fo. fdrs..dey ff

V

V

VV

ff

fiatwdey drsdes dr

VV

2 = vet Vv

ffi dries | | fis dr,dr, = V"~*p; = v™ (*) : |.

(7.4-12b)

ii

This type of integral is represented by the cluster diagram

ee Therefore, the contribution nonrepeated indices is

e

@

to the partition function of any term of the form

ee ae ¥

is

v* (=)

with

(7.4-13)

V’

It remains to count the number of terms of this form that occur. The number of pairs (i, 7), @', 7) with the restrictions that j’>i',i'>i,j >i is

1 (x 2!

- ) (*



2

2

) _ Nt ~ 8

where the term 2! is included so that the products f)2 f34 and f44 f)2 are not counted as two separate terms.

The next integral to be considered is [of

fefiade

Y

... dry

(74-14)

¥

which diagrammatically can be represented as

ee

0

This integral is representative of the second class of integrals appearing in the term in Eq. 7.4-4 that is quadratic in /f. In integrals of this type, the integrations over all position vectors other than r,, r>. and r, may be done to give VT

Lf ’eF

fisfiadey desde,

¥

Since f)2 is a function of only rj2, and

f\3

is a function of only r)3, the obvious

transformation is to a coordinate system that has as its origin the location of particle 1.

: Interacting Molecules in a Gas the interparticle separation distance rj3, the interparticle separation distance rj; as these two variables are independent. unity, we have

Note that moving pare 2 does not change and that moving particle3 does not change this is important for doing the integrations Since the Jacobian of this transformation is yr

tI

fio fia dr, dro dr,

— y"- ‘fan

| tafindender

f

vv “v2

fafisden dr \3

vv 2

vn f firdris

fiadr,;=V" (=)

Vv

Vv

(74-15)

The number of terms of this form that occur, neglecting terms of order unity with respect to NV (the number of atoms), is

(2

\(F

\)a*

moog JX a Consequently,

if terms

of order unity

op

ge

are neglected

(since

N >

| and

therefore

N* => N*), we have

|-

|

> peri

vs Y A ( ) S ( 1 = y dr .. y. dr Dd. fifi

(7.4-16)

>iy>i

and

£=

v"|

I+

By

(= 2 -) (4

a2

rm

N°\ =

2.2)

fy —

2

7.4-17



(74-17)

While the remaining terms in the series for the partition function may, at first glance, seem obvious, it is nonetheless useful to consider the next term in the series expansion

of the configuration integral. This term will involve all possible integrals involving a

triple product of Mayer functions fjj fy ;’ fi j. There are a number of different types of terms that contribute to this integral. The first is integrals in which none of the indices are repeated, corresponding to the cluster diagram

eee 68 ce

®

Integrals of this type are treated as follows:

[. vo

| fra fra F

dr,...dry

=V" °f. | fir faa foe dry... dr yo¥

= v6 TT fisdr, drs | f ss dr. ar, | f feodrs dre y

—y*?

¥

I

it

fiodr,dr,|

Ya

YY

=V™ (2)

(7.4-18a)

7.4 Mayer Cluster Function and Irreducible Integrals

107

The number of terms of this type 1s

N(N —1)] [(N — 2)(N — 3) ] [(N —4)(N — 5) 2 2 2

_ | (~)

2)

~ 31h

3!

_ 1N®

318

(7.4-18b)

(neglecting terms of order unity which are much

smaller than the number of parti-

cles N.)

The next class of integrals or diagrams repeated, such as f)2,/23 fs6, which

one of the indices is

is that in which

is represented by

ee

@e.e06

Note that each of these types of integrals has been evaluated above, and from that analysis we obtain

3

Bi

(7.4-19a)

first dr, ...dry =V" (4)

[| Vv

I

The number of integrals of this type is

[= _ 2] [= == [* — 3)(N >| integrals. One could anticipate this by noting that this cluster diagram has what is called an articulation point, which is a molecule (or, more correctly in graph theory, a node), indicated here by an asterisk, separating the diagram into two parts—one corresponding to a #; integral and the other a > integral. If one chooses the location of the articulation molecule as the origin of the coordinate system for the initial integration, the two types of diagrams easily separate. The next type of irreducible integral is in fact a collection of integrals representing all possible cyclic interactions between four molecules, of which there are 10 distinct types shown below. These interactions are represented by the irreducible integral

mma

|

ff

| | [3 fsa fo3 fia fi2 + 6 fa fo3 fia fis fi2 + fra fra fos fia fis fir | x dr, dr5dr,dr,

(7.5-6)

LT Xk iW KAN YX The extension to higher order types of interactions is obvious, in principle, though very tedious in practice. With this background, we now return to the problem of developing expressions for the higher virial coefficients. To obtain an expression for the expansion of the configuration integral (up to and including the third virial coefficient), we will retain all terms of order 6; and f> while neglecting integrals of type 63 and higher. If this is done for all integrals up to >

one

>

>|

ewe / Sip fej Sie jrdridr jdrpdr jdrjedr jn

(7.5-7a)

obtains

Bi zav"ti4 (> -\(@ | 2

N?V (BiY 1 (N7Y (BY | NB (2! +—({(—})(—)+-—(—)(2)4+—(-2@ 2

‘I

2

2

Vv

4

+ 3! \

2

3

Vv

t 3! \

v2

(7.5-7b) The next term in the series arises from the integral

y de gn dt ye . de e dy y de y de de de; ne fi re fi DEEDS fe | taser v

v (7.5-8a)

which

contains the terms indicated

in Table 7.5-1.

Chapter 7: Interacting Molecules in a Gas Table 7.5-1 Different Four Bond Diagrams Number of Different Indices

Example

8

Diagram

Contribution

fi2taa fse F718

term, it is easy to show that for N > |

LLLD |. [silver tinge dey...dey

“DOSE 1

ose

V

Vv

(N2 torn

v4

|

73

3

)

.

V

Here the 3 term, which will lead to the fourth virial coefficient, has been neglected, By analogy, we find that, to the same order -

/

_

ma

a

/

Satvy

ti

L (NPY (Biv

“a

Sipser jer

Cy

fijevee je

1

dr

NP)

-alZ)(G)" aE)

ae

dry;

(N2Y

(2!B\

(8%.

CEG)

|

ose

7.5 The Virial Equation of State

113

and

» | L / Fis fre Fir gn Surry Som yn Fm jr ry.dey

al3)(v)¥ +>

=}

]

No

NE) CHE”

ie

;

2!B)

¥

N

(7.5-10) .

Therefore, by extension, we have

ee AIE) a(S) Je + CB) +(E)G)-(BBo) 1

{N3 /2 1B .\ )° N = (1+---++} too a (Fe)

(7.5-11)

which looks like the terms in the double power series expansion of

— Z=¥V

N° Bi

yy" exp - | a

| exp |

N?2!B> -=-yre 31y2 {= ys exp

N*By V

exp

N*By 32

|

(7.5-12)

Now, by induction, we find that the general result is 2p y svertardts

2V

8.

| =

TTex

=v"

32

4 | exp ||

ces

Ay3

3V2

BL

Nt

7.5-13)

=a

rp = VN ex

, y l t n e u q e s n o C

P=—kT

(

din =) —

OV Jr

olnZ

a

(

aV

=

),



pkT

{1 —-

SB, —

GE

_"

|

75-14

Comparing this result with the virial equation of state

P = pkT{1+ Bop + B3p* + Bap? +.....} we have

b

By=—-—, iH

2p

(7.5-15)

38

By=——,

a!

B=

-

etc.

(7.5-16)

in a Gas

I: Interacting Molecules

Alternatively, if we write P=

pkT

then

Bjyi

41+ >~ Bjsip! j=l

(7.5-17)

j = a8

(7.5-18)

(Note: This result for the configuration integral is not quite correct due to our inexact counting—that is, neglecting terms of order unity with respect to W. Had the counting been done in a more rigorous manner, the result would have been

Z=V

|N

— N! exp 2» Wa7- GF

B;j DV!

(7.5-19)

For our purpose, the difference between this result and that of Eq. 7.5-13 is negligible.) We will not attempt to reduce the higher-order virial coefficients to a simple integral form as was done with the second virial coefficient, as this is a difficult, tedious task.

It is useful, however, to notice that each virial coefficient arises from considering cyclic interactions of a specific class, and that the number of molecules in the closed cycle determines the order of the virial coefficient to which that cyclic interaction contributes. That is, the interaction between only two molecules results in the second

virial coefficient, the interactions in a closed cycle of three molecules results in the third virial coefficient, etc.

We have now established that the configuration integral for a nonideal monatomic gas 18 (oxo

Z=

ve

NB,

|

p

:

§

|

.

where

fp

=4n | (ern

5

— \)r? dr

(7.5-20)

0

and that the other irreducible integrals are considerably more complicated. An important observation, however, is that each of the f integrals is a function of temperature. This configuration integral can now be used to evaluate the partition function for a real gas. For a monatomic

gas, the result is

InmkT a

me,

In Q(N,V,T)

EQUATION

OF

STATE

= In

FOR

VN?

Z(N, V.T) (7.5-21)

N!

POLYATOMIC

MOLECULES

The analysis used here for monatomic particles, resulting in the family of f integrals, is also applicable to diatomic and polyatomic molecules. In fact, the only difference is that the cluster functions result in § integrals that are multidimensional over not only the separation distance between the particles, but also their relative orientations.

115

7.6 Virial Equation of State for Polyatomic Molecules

In particular, if we make the assumption that none of the internal energy modes of a molecule is affected by the interaction of the particles, the partition function is a Ft

(= he

InQ(N, V.T) = In

V7)

gn (T)Z(N,

(7.6-1)

N!

For monatomic molecules, gj, 1s just the partition function for the electronic energy states of the atom; for more complicated molecules, gin, contains contributions from the rotational and vibrational energy modes as well. Also, for monatomic molecules, the interaction energy among the particles is only a function of the distance between them, and not their orientation. While polyatomic molecules are not spherically symmetric, the development of the partition function so far presented is still valid, except that interaction energy between to molecules is now a function of their relative orientation as well as their separation, and this must be included in the configuration

integral. Consequently, each of the irreducible ($8) integrals are then integrals over both orientation and separation distance and are more complicated than those presented in the previous section. For example, for diatomic or triatomic molecules the 6, integral would be oo

An iif

(eH

y=

2 )/ KT

_

l)rdrdw,

dws

(7.6-2)

|{ dw, dw,

where each w is the vector describing the orientation of the molecule in space (two angles for a linear molecule

and three angles for a nonlinear molecule).

Using Eq. 7.6-1, it 1s easily shown that the volumetric equation of state for a polyatomic molecule is unchanged from that obtained previously for a monatomic gas,

i = Te

= pkT | 1—- Yee,

=pkT

}1+ > p! Bi+

f=

(7.6-3)

j=l

where

I B.



and, in particular

e.

i+

i"

[Jf (eer @0! — V)r'drdw dw B,(T) = —2xn — ]['da,do,

(7.6-4)

Therefore, the partition function can be written in terms of the virial coefficients as follows: 3N

In O(N, V,T) = >In

2amkT



|

j

) +N Ingin +Inv™ — }° N= Bis —InN j

for both monatomic molecule).

and

polyatomic

molecules

(where

gin, =

(7.6-5) | for a monatomic

Chapter 7: Interacting Molecules in a Gas THERMODYNAMIC

VIRIAL

THE

FROM

PROPERTIES

EQUATION

OF STATE Once the canonical partition function is known, as is the case here for the slightly nonideal gas resulting in virial equation of state, all the thermodynamic properties can be obtained. For example

U(N.V.T) =k?” ) (

7

Jas

aT

4 NkT?——2™" — NkT dT

2

xy dBj p! pn SP

gim din 3. >| — NET? 3 2 4 2fin

,({damnQ

»

aT

j

dT

7.7-1 (77-1)

oF j aT

Or

J AB.

U(N( .V.T) ) — U"S(N. 2 Oe “ae ( V.T) ) = —NET?2 YO OF

U(N, V.T) — USN, V, T) _

pi Li

2a 7.7) (

TT ;

NkT

where the superscript IG is used to indicate an ideal gas property at the same temperature, volume, and number of molecules. It is convenient to define a virial coefficient

on a molar (rather than per-molecule) basis, as @; = NayB;, where Nay is Avogadro’s number. Then, on a molar basis (indicated by an underbar) l dé.

, n ( ? , ) U T n T — ) U — R T = — Y — , n ( , " n T U ( ) ) U — T L a

b 2 7 . 7 ( )

where nm is the molar density. The other thermodynamic properties are

ain

S=king +47 (

=)

2amkT \'

aT Jy wn

+ NkInV — Nk

= NkIngim —kInN! + Nkin{ —— 7

ear pI Birt

- In gin

+ NkT——

SN

NkT

Lid

p! dBj x)

a

(7.7-3a) “

pi

S(N, V, T) ) = SN, , V T))—— NKT Lj (

dB; +1]

(7..77--3b)

—7

and

N,_V' SCN, V.T) ——S_ ) ™(“ T ——$__NE

-=-T

ps dBjy1

>j aT

! ) ¢ 3 7 , 7 (7.7-3¢)

or

) T , S — ) Sin, T fos MS Lg RK

; B i n d - = ee j

J

af

(7.7-3d)

'Compare Egs. 7.7-2a and 7.7-3c. Can you explain why the right-hand sides of these equations are identical?

7.7 Thermodynamic

e at St of on ti ua Eq al ri Vi the om fr es Properti

117

) T . V . N ( S T — ) T . V , N ( U = T) A(N, V,

dB at — u'S(N. Vv.) — NKT? Oe at dB

MAT OE —

_7 | s'S(N.V.T)— way oH j

/

(7.7-4a)

A(N.V,T)

(7.7-4b)

= A'S(N, V, T)+ NAT Jp j Bix j

and Q, In k7 = A m fro ly ect dir ten got be o als can ch whi

A(n, T) — A (n, T) “=” RT

eal

(7.7-4¢)

j

Also

kT

= w'O(N,V, 7) +

w(N,V.T)= (sr)

va (7.7-5)

= O(N, V,T) +kT Sj + = Bi+l For an ideal gas at any pressure P, we usually write

wT,

(7.7-6a)

P) = wi (T, Po) + kT In ;

where Py is a standard state pressure (frequently chosen to be | bar); and once Py is s ition defin these With only. ure erat temp of ion funct a is Pp) T, j'¢( chosen,

P wT, P) = w'O(T, Py) +kT In— +kT Po

The fugacity

B, > (i + Doe! =a

(7.7-6b)

J

(7, P) for a real gas is defined by the relation

, (7.7-7)

P) f(T, p IG w(T, P) = we (T, Po) + RT In —— {)

so that

meat

f= PeH UP!

ae

FT

3

4

= Pexp}2pBo + 5p Bs 4; 3h Bs spore

|

(7.7-8)

However, expressing fugacity as a power series in density is not the most useful relation for the engineer, since it is usually the pressure not the density that is known. Therefore, one first has to solve the volumetric equation of state for density for the given temperature and pressure, and then use this density in the equation above.

Chapter 7: Interacting Molecules in a Gas The enthalpy of the nonideal gas is gotten from H = L/ + PV, so that

AG

|

dB,

1

(N,V. T) = U'O(N, V, T) — NkT? | p— + ~p?

=

1

4dBy

—— + -=p? —+ ---

torn +3? ar * 3° at

Pa?

he

,dB;

+ NKT{1 + pB2 + p*B3 + p°Ba +---] = U'O(N, VT) 4 NKT

+NnkT lp (my — 72) 4? ON a) TP

(B

r=) + ar

V8

| (7.7-9a)

BS) _d 2 + p- Bs —T—— owes

By pd , yr Mey WT KT4 p| Bs — T—) Nk) H™°(N,V+ ,T H(N,V,T)= dT

(7.7-9b) and

H(N,V,T)— H™(N, VT)

|

Ss

d Bs

B, — T—

NkT

p(B:

aT ) 1 PO

*(

“|

Bz

\3~*

F)

—T—

ap)

]-

t (7.7-9¢)

DERIVATION THE GRAND

OF VIRIAL COEFFICIENT CANONICAL ENSEMBLE

FORMULAE

FROM

Considerable effort was devoted to deriving expressions for the virial coefficients starting from the canonical ensemble, We will now re-derive the expressions for the virial coefficients starting from the grand canonical ensemble. The reasons for doing this are several: (a) the derivation is easier, avoids the cumbersome apparatus of cluster integrals, and shows the advantage (in this case) of using the grand canonical ensemble; (b) the derivation provides a simple method of obtaining expressions for all the higher virial coefficients: (c) the derivation does not require pairwise additivity of the potential; and (d) the derivation is equally applicable to classical or quantum fluids, and so is completely general.

We start with the definition for the grand canonical partition function &

(p

V,

T)



e

EIN

VIVRE

(aN

/KT

E.N

which can be written as

N

>

E for fixed NV

N

since

Se E

BIN VIET

= O(N,

V, T) =canonical

ensemble partition function.

(7.8-1)

119

Derivation of Virial Coefficient Formulae

7.8

For convenience, we define the absolute activity as 2 = e!/*", so that Eq. 7.8-2 can be written as

(7.8-3)

O(N, V.T)AN

S(V,7T, w) = E(V.T,A) = >» Nf

This provides an expression for series expansion in the absolute functions of increasing numbers To proceed, we note that Q(1, particle in the volume V

the grand canonical partition function in terms of a activity, with coefficients that are canonical partition of atoms (or molecules). V, T) is the canonical partition function for a single 2amkT

\2

QU, V,7T) = Qi = dim (—=")

V

since the configuration integral for a single particle is Z(1, V, 7) = V. For simplicity

O

l = 7

(2,V,T)

= dim ( 22m dk | a

we will use the notation that z = ate

2amkT

an

2

h2

——

)

af2

A. The next term is

7

|

AZ (2,V,T)

=

z°Z (2,V.T)

2!

Indistinguishability factor a

where Z(2.

V.

T)

=

[fewer

dr

dr.

=

Z

In general Z(M,V,T)=

/ L Lf emer

| «

A

Af



O(M,V,T)=

M!

AT a,

om

(

_

yy, oe Aig

34M

IxmkT .

o

\2

Ae

)

|

2M Z(M, V.T) M

A

£(M,V,T)=



iicicine

M!

Trait

eee

so that

|N

E(u. V.T) = 9 Q(N,V,T)AN =

_

Z(e N,eT,V) z™ Ee

=}!

N=0

——_—

N=0

Zyz

= J =

(7.8-4)

N=0

We have shown previously (Eq. 6.2-12) that PV = kT In S, which can be written as

= PV s+¥) gp mein

5, I. I Y= Ziz+ 5Zoz°+ eZsz+---

where

Next, expanding the logarithm ¥)= In(l n(l+Y¥)

¥

-—

l

-¥Y* 5

«

]

_y?_ +3

as: Bt oe

(7.85)

': Interacting Molecules in a Gas and grouping terms of similar power in z, we obtain

PV



z

=Inb=2Z\7+

5

—(42.-—Z

or

=

.

—(#3—32,;7:4+22))+--

no PY

oO oT

In 2 ot biz =Iné

=V)

L s (7.8-6)

j=l

where

b} =>, = 1, since

Z) =V,Vb2 = 5 (22 — Z)), Vbs = © (Z3 — 32122 + 2Z))

and so on. This gives PV/kT as a function of z = Q)A/V =gqim (20mkT /h?)*” A. However, what we really would like to have is a volumetric equation of state in the form

of PV/kT

as a function of N/V

or p.

Now, note that from the definition of the partition function, the average number of particles in the system can be computed from the grand canonical ensemble as follows:

N=

SY NOQN, V.T)AN —

50 _

W—2

dln

(TT, V,?

dIn

eS)

G(T, Viz

_- (ee)

dA

r.v

dz

(7.8-7)

TV

and therefore

_— _

ain

_ y

or

Nvy

me=e

+

=

| Had

j=l

yg 2boz~ + 3b3z°

+ ---

(7.8-8b)

This is an equation for p as a function of z, while what is needed is z as a function of p. To obtain such an expression, we must invert the series (called a series reversion)—that

is, develop an expression for z as a function of density

that can be

used in Eq. 7.8-6. To do this we write __

| Z=Ail

a|

i

+A 21>xe

—,

+A aL

->

3

+++“= := Ajp Aggie wee af + p ) 3 b 3 — 3 6 8 ( + bj [o — 2b2p*

and expanding the series and equating powers of o, we obtain

oe

|

P

+ 3 9 3 b 4 — ? p o b + ? * p ) 3 b 3 — ; b 8 ( + 2b2p?

Pe

ip

o P + o p ) b 2 ; — b 4 ( Tp =P — B20" + since b; = 1. Comparing Eq. 7.8-11

(7.8-11)

with the virial equation of state

+ * p ) T ( C + ? p ) T ( B + p = T k / P we

have B(T)

=

=

by

=

|

tay

l I \y), -(5-) (22

y-

2



e

yt -Z) = ay

—H(r yal {kT

-htt

dr

|



?

ypv~)

dr

4

| fy ine = x // (h—e M12/Edy T)dr, If the potential

is spherically symmetric.

|

SIRT)

(7.8-12a)

i.e., u(f),f5) = u(ri2), then

2

Vk

B(T) = sy Var | (l—e@ MOET \ re dp =2n | (1 — MOET) 6? dr 0

(7,8-12b)

0

which is precisely the expression we developed earlier in this chapter. In a similar fashion, we obtain

C(T) = (4b; — 2b3) =

Z2(Z2—-V*)

=

V3sy —Z;

++:

(7.8-13)

': Interacting Molecules in a Gas

Note that in this derivation, we never had to specify the form of the configuration integral or the interaction potential, whether or not the potential was pairwise additive, or even whether the system was described by classical or quantum mechanics (though both of these will be important in the evaluation of the configuration integrals). Therefore, this derivation is also applicable to more complicated configuration integrals—tfor example, when the intermolecular potential function is not spherically symmetric, so that more than just a single center-to-center position vector is necessary to specify the potential energy. This is the case in molecular fluids in which relative orientation vectors or several atom-atom distances are needed. There is one part of this derivation that needs justification. We had the exact expression

PV ap = inB =I

+¥)

— ra =H a

where

Aj =

|

— » Onan

(7.8-14)



and then used the series expansion

Ind+Y)=¥Y-— wy? 42y3.., 2

3

For this series to converge rapidly, Y should be less than unity. In fact, Y is much greater than 1; it is of the order of the number of molecules in the system, NV. Therefore, the series expansion would seem to be divergent. However, notice that Y is a sum of terms, each of which is a product of the form Q(N)A*. In particular, the first term

in the series is Q(1I)A = GuansA = Gtrans@hl *?

To obtain a rough order

of magnitude estimate for the terms in this product, consider the calculation of the argon as an ideal gas at 25°C:

dirans = 0.245 x 10°’em™ x 2.24 x 107%em? = 0.548 x 107 and

i = —9500 cal/mol

—9500) = @2x298 = eW

o A = e!!*!

9

So ¥ is a sum of terms, each of which is a product of the very large number O(N)

and a very small number 4. In the limit of A = 0, ¥Y = 0, and In(1 + ¥) = 0. So if we expand In(] + ¥) about A = 0, we have dln(l1+

i o + 0 , | ) ¥ + (1 In = ] ¥) ( + n I

da

However,

as mentioned

td

Eee

Sa

=

ans

above,

ee

2

(f.

i

In(1 + ¥)|,;-9 = 0, and

yo

AW)

FY

,

vets

din(1+ ¥)

ad

nh

4

| a* In 3!

1 @

Y

ty

+.

V=()

Oya’

N=0

=?

NQwh aA=0

(7.8-16a)

)

123

7.9 Range of Applicability of the Virial Equation

da?

||

|

=20,-—Q7

——_—-| Te a

In(1

1 Ener)

_ = "NIN = 1) Qna%*—

omer y

+

¥)

=—)

|

and

(7.8-16b)

) On = 2)(N = 1I)(N)QnanYN (N—2)(N—1)(N .

_ = (>

V—3

N(N — 1l)Owa® ~) x >

NQwa")

ma

a

3

= 310; —3020;4+Q}

NQwr*- )

aps

(7.8-16c)

Afi

So that

Ink +¥)~ QiaA+ =- (202 — 03) 2° see” |

= Q)A+ Q2d7 + Q3d° — = (Q\iA+ Grd? + Q3h?+

on;

— 501021 I

3,43

ot ae= Oia +

4,3

) = 5 (QiA° + Q1Q2a° +--+)

a

- (Qia° f= =)



The

important

y—+y241y3 3

hz

point is that we

expansion about the large term

7.8-17

Tees

have obtained

(7.8-17)

the desired result not by doing

an

Y’, but rather by expanding about the vanishingly

=

small term A.

OF

APPLICABILITY

OF

THE

VIRIAL

EQUATION

In Table 7.9-1 are data for the compressibility factor of argon at 25°C at a collection of different pressures. We see that for argon at 25°C, which is well above its critical temperature of 150.87 K, there is an insignificant error in the compressibility factor when using the virial equation with only the second virial coefficient up to 10 atm; only about a 2 percent error at pressures up to 100 atm; and significant errors at higher pressures. In general, one can expect similar accuracies with other nonassociating gases well above their critical points. However, the error will be larger for gases below their critical points, and significantly greater for a gas in which association occurs by, for example, hydrogen bonding. Thus, there would be significant error when using a truncated virial expansion for a strong hydrogen-bonding fluid such as hydrogen fluoride, and to a lesser—but not negligible—extent for acetic acid, methanol, and water.

124

Chapter 7: Interacting Molecules in a Gas Table 7.9-1 The Compressibility Factor of Argon at 25°C and Predictions Using the Second and Third Virial Coefficients P(atm)

P/ pkT

1+ Bp

+ Brp7

+remainder

Total

|

1 —0,.00064

+(0).00000

+0.00000

0.99936

10

1 —0.00645

+0.00020

—0).00007

0.99365

—0.013%

100

| —0.06754

+(0.02127

—0.00036

0.95337

—2.19%

1000

1 —0.38404

+0).68788

+0,37272

1.67616

—63.25%

error of using only B> 0%

Based on a table in E. A. Mason and T. Spurling, Wirial Equation of State, Pergamon 1969.

CHAPTER

Press, New York,

7 PROBLEMS

7.1 Show that the second virial coefficient for a mixture of species is given Cc

¢

7.4 Obtain expressions for the thermodynamic properties of a binary mixture described by the following equation of state:

Bo mix(x, T) = Y° Y- xj) Boi (T) i=]

where

/=]

P(x,

og Bo j(T)

=

an

f

Pr a

(1

cy )r

dr

[

and w,(r) is the intermolecular potential for a species

72

i-species j interaction, Obtain expressions for the thermodynamic properties of a binary mixture described by the following equation of state: P(x,

.

p.T) =

7

and

pkT(1 + Bo a

Bo mix(x, T) = »

mix(x, T)

pal

Ff

- XjX 7 Bri;

species 1S given

e

ce

i=1

j=!

Ay Xj Xe By spel r) &

-

,

|

——F = 1+ B3P + B3P*+ ByPP + :

Relate the virial coefficients B* to the coefficients

7.7 Obtain expressions for the thermodynamic

8;

properties

of a gas using the virial expansion in pressure of the previous problem.

yy

pet Bs mix (Xx, T)p7|

in the virial expansion considered in this chapter.

Show that the third virial coefficient for a mixture of

Bs mix(x, [) = >

id By mix {Xs

and the mole fraction dependence of the second and third virial coefficients are as given in Problems 7.| and 7.3. 7.5 Obtain expressions for the constant volume heat capacity for a monatomic gas that obeys the virial equation of state, 7.6 Since pressure is more easily measured than density, it is Sometimes more convenient to use a virial expansion in terms of pressure as shown below

pr

i=l j=l Vo

Ps i)}= pkT (1

7.8 Develop the expressions for the U —U'° and Cy — C1S for a fluid described by the virial equation of state wwith only the second virial coefficient.

Chapter S

Intermolecular Potentials and the Evaluation of the Second Virial Coefficient In the previous chapter, we developed expressions for the equation of state and other thermodynamic properties of a nonideal gas in terms of the virial coefficients. However, to use these formulae, we need values for the virial coefficients as a function

of temperature. The second virial coefficient for a monatomic species has been shown to be oo

vie B,(T) = —>fi = Qa i (1 —@ HVAT) 2 dr

(7.5-4)

()

that can

be explicitly

evaluated

once

the form

of the

intermolecular

potential

ts

specified. Here we will consider a number of models for the intermolecular potential and examine the form of the virial coefficient for each of these models.

INSTRUCTIONAL

OBJECTIVES

FOR

CHAPTER

8

The goals for this chapter are for the student to:

e Be able to temperature « Be able to e Be able to e Understand of state

8.1

INTERACTION

compute values of the second virial coefficient as a function of for different interaction potential models compute thermodynamic properties using the virial equation of state compute the second virial coefficients in mixtures the engineering implications and applications of the virial equation

POTENTIALS

Hard-Sphere

FOR

SPHERICAL

MOLECULES

Potential

The simplest potential used to represent molecular interactions is the rigid or hardsphere potential shown in Fig. 8.1-1 and is given by ur) =

oo y

r 4 (8.1-4)

where T° ( ) is the gamma

function. (If n < 3, the value of B> is infinite.) Though the

()

0.5

|

1.5

Figure 8.1-3 The Point Centers of Repulsion Potential.

hh

wir Mea

virial coefficient is now temperature dependent, it is still only positive; it decreases in value with increasing temperature, unlike the behavior of the second virial coefficient shown tn Fig. 8.1-2.

|

§: Intermolecular

Coulomb

Potentials and

the Evaluation

of the Second

Virial Coefficient

Potential

Two point charges, g; and g2, in a vacuum interact via the Coulomb potential u(r)

which

=

iq?

($.1-3b)

is a special case of the Point Centers of Repulsion

potential,

and leads to

an infinite virial coefficient (Problem 8.3). Charged particles must be treated in a different manner, and this is discussed in Chapter

15.

Potentials with Attraction

From experimental volumetric data for non-ideal gases, it is possible to obtain numerical values for the second virial coefficient. For many gases, particularly at low temperatures, the second virial coefficient is found to be negative—as has been shown earlier with the experimental data for helium. From Eq. 7.5-4 it is evident that the virial coefficient can only be negative if the intermolecular potential is negative (that is, attractive) for some intermolecular separations. The experimental observation that at higher temperatures the second virial coefficient becomes positive, as was shown earlier for helium, requires that any potential model must also have a repulsive part. Below are several simple potential models having both attractive and repulsive regions.

Square-Well Potential The simplest analytical attractive-repulsive potential is the square-well model OO

u(r) =

r=oa

—-€

ogo«r


Ryo

(8.1-5)

Kewl

which is shown in Fig. 8.1-4. The second virial coefficient for the square-well potential is a

Row

Bo(T) = 27 /

(1 —O)r? dr +27 /

()

aT

+2

~

f R

=

2ma7%

=

(1 — ef !*" yp? dr

23

9

(1 — lyr-dr

AS ia

si

+ SU -e*7(R— 3No, ?=

2no°

a

= [1+ (1 —e®/*7)(R3 — 1) , ($.1-6a)

vir) |

Figure 8.1-4 The Square-Well Potential.

8.1

129

Interaction Potentials for Spherical Molecules

) ume vol of s unit e hav both 7/3 270 and B2 that ing (us ty nti qua s les or as a dimension

22"2 = Br) = (1+ eR, 20° 7+

*

— 1)

(8.2-6h)

3 where

T* =&T/e.

The second virial coefficient computed with this expression has

the property that at low temperatures, the exponential term dominates, and the second virial coefficient is negative. However, at high temperatures, the value of the second virial coefficient is positive. At very high temperatures, where the mean kinetic energy of the molecules is of much larger magnitude than the depth of the potential well ¢, attractive forces are unimportant; and in this limit the virial coefficient becomes equal to that computed from the rigid-sphere model 270° /3. This behavior of the second virial coefficient of the square-well potential for R,, = 1.5 1s shown in Fig. 8.1-5. As the temperature increases, and B) goes from a negative to a positive quantity, there is a temperature at which the second virial coefficient 1s zero; this is known as the Boyle temperature, 7g. It is a simple exercise to show that for the square-well model:

Ths

_

e/k

(8.1-7)

~ In(R3, Wi /(R3, — 1))

At temperatures below the Boyle temperature, the attractive part of the potential is clearly important, and the virial coefficient is negative. At temperatures higher than Ty, the repulsive part of the potential dominates, and the second virial coefficient is positive. Table 8.1-1 contains the square-well potential parameters for some simple fluids.

Lt)

2

—10

0

lo!

B*(T*)

B*(T*) ss=

20

10"

10!

lo”

10°

Reduced Temperature 7*

(a)

0

5

10

[5

T*

(b)

Figure 8.1-5 (a) The reduced second virial coefficient B* for the square-well fluid with Roy = 1.5 as a function of reduced temperature

7*; (b) high temperature range.

20

}- Intermolecular

Potentials and the Evaluation of the Second

Virial Coefficient

Table 8.1-1 Square-Well and Lennard-Jones |2-6 Potential Parameters Determined from Second Virial Coefficient Data Assuming the Molecules are Spheres'

argon benzene CF, CH, CO, krypton n-pentane neopentane nitrogen xenon

12-6 Potential

Lennard-Jones

Square-Well Potential

Molecule

Rew,

a (A)

e/k (K)

o (A)

ek (K)

L.7 1.38 1.48 1.60 1.44 1.68 1.36 1.45 1.58 1.64

3.067 4.830 4,103 3.355 3.571 3.278 4.668 5.422 3.277 3.593

93.3 620.4 191.1 142.5 283.6 136.6 612.3 382.6 95.2 198.5

3.504 8.569 4.744 3.783 4.328

L777 242.7 151.5 148.9 198.2

3.827 8.497 7.445 3.745 4.099

164.0 219.5 i 95.2 222d

Mie and Lennard-Jones

Potentials

Perhaps the most widely used potential for correlating experimental data on simple molecules is the Mie potential: c

(8.1-8)

we)

This form of the potential model has the advantage of being a smooth, continuous

function, and presumably

more realistic than the interaction potentials considered

so far. The evaluation of the virial coefficients for this potential requires numerical integration, The m parameter in the potential has been estimated from quantummechanical dispersion energy calculations to be equal to 6 for nonpolar molecules. It is the result of the instantaneous, coupled fluctuations of the distributions of the electrons around each atom resulting in a induced dipole-induced dipole net attraction, referred to as London

dispersion forces. Largely for mathematical convenience, n is

frequently chosen to be equal to 12, resulting in the commonly-used (12-6) potential for non-polar molecules shown in Fig 8.1-6.7

Lennard-Jones

wiry =a4e| (2) = (2)"| It is easily shown

that o is the value of r for which

(8.1.9)

u(r)

is equal

to zero, and «& is

the depth of the potential energy well which occurs at r = 2!/®o.

‘From BD. A. McQuarrie, Statistical Mechanics, HarperCollins, Sherwood and J. M. Prausnitz. J. Chem. Phys. 41, 29 (1964).

New

York,

1976; original

source

is A. E.

“As shown in Section 8.3, this same potential can be used by rotationally averaging the permanent dipole—permanent dipole interactions among polar molecules, but this results in temperature dependent ¢ and 7 parameters Eq. 8.3-7b.

Interaction Potentials for Spherical Molecules

8.1

131

wiryee

Lennard-Jones 12-6 Potential

Figure 8.1-6 The Lennard-Jones

12-6

Potential.

WG

The second virial coefficient is then computed from Ome

2x,

= 3°

exp

(

f

,6 )}) (—)

7 ny 3

= B'(T" B*(T* ) )= |

=

(

—4 /

dr

|

|

— =) } jay

where y = (r/o)°. Defining a dimensionless reduced temperature reduced (dimensionless) second virial coefficient can be written as B(T*)

9

($.1-10)

T* = kT/e,

|

]

_ exp | — ; |(

io"

10!

Tie

lor

o ——

|

fi

=

bat

Reduced Second Viral Coefficient &*

1

“ft

=

a“al ntanm

| Penal =

Reduced Second Virial Coefficient B*

Virial Coefficient

of the Second

}- Intermolecular Potentials and the Evaluation

4

Reduced Temperature T*

8

Reduced Temperature 7*

(a)

(b)

Figure 8.1-7 (a) Reduced Second Virial Coefficient of the Lennard-Jones (b) high temperature range.

12-6 Potential;

seen in experiments. An interesting feature of B*(7") is that—in agreement with measurements—at very high reduced temperatures, the second virial coefficient 1s a decreasing function of temperature, so that there is an intermediate temperature at which

the second

virial coefficient achieves a maximum.

(This

is difficult to see in

Fig. 8.1-7 because of the scale.) That the second virial coefficient decreases with increasing temperature is easily explained. At high temperatures, it is only the repulsive part of the potential that is important in determining the value of the second virial

coefficient, Since the repulsive part of the potential rises quite sharply with decreasing intermolecular separation distance r, this portion of the potential can almost, but not quite, be represented by a rigid-sphere potential. However, since the potential does have a finite—rather than infinite—slope, the effective hard-sphere diameter decreases

as the average

energy

(or mean

kinetic energy) of the system

increases,

as it does with increasing temperature. Thus, the effective hard-sphere diameter, and therefore the value of the second virial coefficient, decreases with increasing temperature at very high temperatures.

Exponential-6 (Modified Buckingham) Potential This three parameter (€, y, and ry,) potential is given below.

u(r) =

=

6 :

é

—\— | — | —

6/y

¥

,_!

ex PY?

fy

—_- —

—|—

Por

r

(8.1-13)

> ’

m

The advantage of this potential is that the three adjustable parameters allow greater flexibility in fitting experimental data. However, the disadvantages of this potential are that its derivatives

are discontinuous

must be evaluated numerically.

at r = r,, and the second

virial coefficient

8.1

Interaction Potentials for Spherical Molecules

133

The Yukawa Potential Another interaction potential that is frequently used, especially now for colloidal and protein solutions. is that of Yukawa: OO

u(r) =

4

O=roa

In this model, o is the hard-sphere diameter, ¢ is the well depth, and b is a parameter that determines the range of the interaction. This potential is very similar to the shielded Coulomb potential for the interaction between charged particles that is mentioned in Section 8.4 and developed in Chapter 15, and is frequently used in place of it.

ILLUSTRATION 8.1-1 Below are experimental second virial coefficient data* for argon as a function of temperature.

to | 1560 — 2

—47.6

|

Compare the experimental data with predictions using the Lennard-Jones 12-6 potential using the potential parameters for argon in Table 8.1-1.

SOLUTION From Table 8.1-1,

ao = 3.504 A = 3.504 x 10°-8cm and e/k = 117.7K. Therefore,

Now using the MATLAB®

‘From

21°

3

Nay = 54.26ce/mol.

program LJ_VIRLAL to compute values of B* we obtain

The Virial Coefficients of Gases by J. H. Dymond and E. B. Smith, Oxford University Press, 1969,

Intermolecular Potentials and the Evaluation

of the Second

847 AT |=280

14

"250

Virial Coefficient

=28.9

00 400 S00)

The results are plotted below. 50

-150

cn

:

LOO

200

|

OW)

ACW)

5)

600)

r

The points are the measured values and the line is the second virial coeffici ent calculated with the LJ 12-6 potential. While the results are not in perfect agreement with experiment, as would be expected from an approximate potential such as the Lennard-Jones 12-6 potential, the results are very good over the temperature range considered,

ILLUSTRATION

8.1-2

Repeat [Illustration 8.1-1 using the square-well potential.

SOLUTION From Table 8.1-1,

ao = 3.067 A = 3.067 x 10-8cm and e/k = 93.5 and Rew = 1.70. Therefore, = ~

= 36.39 cc/mol.

Nay

8.1

Interaction Potentials for Spherical Molecules

135

Now using Eg. 8.1-6b to compute values of B*, we obtain

The results of the calculation are plotted below.

30)

—30)

— 100

=150

10M)

200

300

400

S00

600)

The points are the measured values and the line is the second virial coefficient calculated

with the square-well potential.

What we see from the plots and tables of these two illustrations is that the results for the second virial coefficient of using the Lennard-Jones 12-6 and square-well potentials with appropriately fitted parameters are comparable. (The same will not be true

when

we

consider

high-density

fluids

in Chapters

11

and

12).

A general

implication of this is that second virial coefficient data generally cannot be used to “work backward” and uniquely determine the interaction potential between atoms, since different potentials with fitted parameters may give comparable results.

nt cie ffi Coe ial Vir ond Sec the of n tio lua Eva the and s ial Chapter 8: Intermolecular Potent

E: UR XT MI A IN T EN CI FI EF CO AL RI VI ND CO THE SE INTERACTION POTENTIALS BETWEEN UNLIKE ATOMS In Problem

7.1

it was

shown

that

eg

£



(8.2-1)

Ba ij(1)

xia

> >

Br mix(4, T) =

j=!

where

Bz 4;(T) = an f

(8.2-2)

2 dy (1 — @ MiilRT 0)

Consequently, to use the virial equation in a mixture, we need to compute values of several virial coefficients at the same temperature. For example, in a binary mixture

Bo mix (XT) = x7 Ba, (T) + 2x1x2B2,12(T) + x3 Bo.23(T)

(8.2-3)

since Bo;2 = Bsa2, as a result of uwj2(r) =u 2)(r). To proceed further, we have to know not only the interaction potentials between like molecules (that is, w))(r) and u22(r)), but also the cross or mixed interaction w;3(r). A reasonable assumption is that if the #),(r) and w22(r) interaction potentials are of the same form—for example,

both Lennard-Jones

12-6 potentials though with different parameters—then the same

potential should also be used

for the w;2(r). That

still leaves

unresolved the choice

of the appropriate potential parameters. One way that potential parameter values are determined is by fitting experimental second virial coefficient data. As there are considerable experimental data for pure fluids, the pure-species potential parameters can be determined in this way. However, evaluating the potential parameters in the cross interaction 4 ;2(r) is more problematic for two reasons. First, there are only limited experimental data on mixture second virial coefficients, Bs »j,. Second, even when such data are available, since u)2(r) must be computed after subtracting the contributions of uw \,(r) and u(r), there is likely to be significant error in w)2(r). Consequently, the usual procedure is to

use a set of combining rules that relate the potential parameters of the

|| and 22

interaction potentials to those in the 12 potential. The most common combining rules are the following, the so-called Lorentz-Berthelot combining rules. For the distance parameter or core diameter (i.e., 07), the usual combining rule is | j7=

57

+ a)

($.2-4)

This equation is exact for the hard-sphere potential, and approximate for all other potential functions. The following combining rule is commonly used for the unlike energy parameter in a two-parameter potential (such as the Lennard-Jones potential):

€y2 = /€1€2(1 — 12)

(8.2-5)

The use of the geometric average for energy parameters has an approximate basis from quantum chemistry calculations, but is not exact. The binary interaction parameter kj»,

which is usually adjusted to fit mixture second virial coefficient or other experimental

Interaction Potentials for Multiatom, Nonspherical Molecules, Proteins, and Colloids

137

r mete para gy ener the of re natu te ima rox app the for e sat pen com to t data, is mean combining rule. (Note that these combining rules are very much like the ones used for the parameters in cubic equations of state, as will be discussed in Section 8.4.) two than more with ls ntia pote ion ract inte for ed need are s rule ing bin com al Addition parameters (for example, the modified Buckingham potential). In closing this section, it is important to note that from Eq. 8.2-3, regardless of the interaction potentials used and their simplicity or complexity, the mixture second virial coefficient depends quadratically on composition. That is, the form of the interaction potentials used and the values of their parameters determine the numerical values of the like and unlike second virial coefficients; but theory leads to the exact result of a quadratic composition dependence.

TERACTION POTENTIALS FOR MULTIATOM, OLECULES, PROTEINS, AND COLLOIDS

NONSPHERICAL

The intermolecular potential functions used for nonspherical molecules can be quite complicated because of the number of interaction sites present. One form of intermolecular potential is the complete atomistic approach in which the known geometry of the molecule is used; and each atom on one molecule is assumed to interact with each atom on other molecules using an atom-atom potential, such as the potentials discussed above, with parameters specific to each type of atom-atom interaction. In this model, the interaction between two molecules is the sum of all atom-atom interactions. For example, for a diatomic molecule (such as hydrogen chloride) shown in Fig. 8.3-1, for each center-of-mass separation and relative orientation of the two molecules, we would need the four interatomic distances shown to compute the total interaction energy for that configuration, and this calculation would have to be repeated for each configuration. As the number of atoms in a molecule increases, such calculations become increasingly difficult and further complicated by the fact that the configuration of larger molecules can change due to internal rotations around bonds and bending. A slight generalization of the atomistic approach is the site-site model. In this model, each molecule is considered to have two or more interaction sites in which potential models such as those discussed in Section 8.2 are used represent each interaction; but each site does not necessarily have to be located at the center of each atom. Also, in some cases, additional sites are added that are not associated with any atom—as, for example, to represent point charges. A simplification of these models is the united atom approximation in which a group

of atoms are taken together as a single interaction site. For example, a —CH; group may be taken as a single interaction site. In such united atom models, hydrogen atoms are frequently

lumped

together with a larger atom (carbon

in the example

here) to

make a single interaction site,

Figure 8.3-1 Interatomic Distances Needed to Calculate the Interaction Energy for One Configuration of a Pair of Diatomic Molecules.

- Intermolecular Potentials and the Evaluation

One problem that nonidentical site-site atom approximation, group, not only will

of the Second Virial Coefficient

y, ll ra ne ge t, tha is ls de mo se the of y an of use arises in the ed it un the in if e, pl am ex For . ed er id ns co be st mu ns interactio OH an and p ou gr 3 CH a of t is ns co to ol an th me ed er id ns co we we need the CH3 + CH; and OH + OH interaction potential

parameters, but also the parameters

same problem of unlike when dealing with the reason as discussed in Eqs. 8.2-4 and 8.2-5 to

for the unlike CH;+

OH

interaction. This is the

site-site interaction arises that we considered in Section 8.2 monatomic interaction potentials for mixtures. For the same Section 8.2, it is common to use the combining rules of obtain the unlike site-site interaction parameters.

An extreme example of the site-site model

is to treat a whole multiatom, non-

h , wit le cu le mo a of e tur pic tic lis rea a not e il Wh e. sit gle sin a as le spherical molecu on as re e a id ov pr l can de , mo s rs thi te me ra l pa ia nt te po nt the of me st ju le ad a suitab able description of the second virial coefficient. It is for this reason that square-well and Lennard-Jones 12-6 parameters were given in Table 8.1-1 for molecules such as CF,, CHy, nitrogen, carbon dioxide, n-pentane, and neopentane (2,2-dimethy] propane). However, there are several caveats in using these parameters. First, as expected, the use of a single-site model is more reasonable the closer the molecule is to being spherical. Thus, one expects the representation to be better for the almost spherical neopentane than for the more linear n-pentane. Second, potential parameter

sets can also be regressed from other data—for example, from viscosity or thermal conductivity using relations obtained from the kinetic theory of gases. Both because the molecules discussed here consist of more than a single atom, and because the square-well and Lennard-Jones 12-6 potentials are just simple models of the interactions between molecules, different potential sets are obtained depending on the data used. Another class of interaction potential models used for multiatom molecules is specific geometric shapes such as cylinders, spherocylinders (cylinders with hemispherical caps), ovalate spheroids, etc. We will not consider such models here. Still

another class of potentials used for nonspherical molecules is the sum of a spherical potential with a nonspherical part, usually representing permanent dipoles (or multipoles) in the molecules. As an example, we will represent the relative orientation of two molecules as in Fig. 8.3-2 and use the following notation: (0), A, d2 — @)) = sin @) sin @2 cos @y2 — 2 cos dj cos do

(8.3-3)

where jt here is the permanent dipole moment of the molecule. With this notation, the following are some of the simplified interaction potentials used for multiatomic molecules.

Figure 8.3-2 Angles Describing the Relative Orientation of Linear Molecules and/or Two Dipoles.

139

Molecules, Proteins, and Colloids

action Potentials for Multiatom, Nonspherical

Rigid Sphere Containing a Dipole rig

oo

:

(r, 0), 65,¢@2 —@)) =

r>o

— =; 8(61, 0, b2 — $1)

wie Bs Pe a —

(8.3-4)

Stockmayer Potential (Lennard-Jones 12-6 potential + dipole) 12 u(r, 0). 4, do — d)) = 485 (2)

6

Tp

— (—)



80) . 42, @2 — 1)

(8.3-5)

It has been shown? that by a Boltzmann-factor orientation averaging of this potential, one obtains (u(r)) =

[ [u(r Oy, 02, G2 — Gye

MP

P2-PI/IET sin A) sin A> dO, dO, db ddr

f { e 4.81 .02,.02-O1 WEE cin gy, sin Os dA, dA. dd, ddr o\!2

—é

"ay

E

(

r



5

To"

4

_

_

Qu" J

)

(

r

)

3kTr®

°

"

which can be rewritten as

12

hs (8.3-7a)

= (2)

(u(r)) = e(T) (2) r

r

with

yt

3

(0) =66(1+ Terese)

6

|

= 00 | —a (7)

and

[2kTey,e*

(8.3-7b)

That ts, with this approximation, to compute the value of the second virial coefficient for the Stockmayer potential, the values of the second virial coefficient for the Lennard-Jones 12-6 fluid can be used, though the the parameters to be used—e(T) and o(7')—are

now

functions of temperature.

The choice of intermolecular potentials that can be used for very large macromolecules,

such

as proteins and colloidal

principle, atom-atom

particles,

is more

problematic.

While,

or site-site interactions can be used, there are so many

in

atoms

involved that summing over these in a large molecule is very difficult. In some cases, especially colloids or other macromolecules with no net charge, a simple hard-sphere model may be sufficient—but with a diameter characteristic of the macromolecule.®

°J. H. Bae and T. M.

Another possibility is to use the square-well

Reed

III, Ine. Eng. Chem.

6, 67 (1967).

"PON. Pusey and W. van Megen, Mature 320, 340 (1986).

potential for neutral

ent fici Coef al Viri nd Seco the of on uati Eval the and ls ntia Pote ar ecul rmol Inte 8: Chapter (a)

aA

ar)

4.54

5

E

(b)

30 A

utr)

aL5A

2 5¢

Figure 8.3-3 Typical

Square-Well

Potential

for

(a) an Atom and for (b) a Macromolecule Such As a Protein or Colloid.

macromolecules, though because of the size of the macromolecule and because so many site-site interactions are involved, the potential will look very different than for an atom-atom interaction. To be specific, for a 3 A atom with the square-well] parameter A. = 1.5, the range of the square well is 1.5 A from (3 A to 4.5 A): for

a 30 A, because of the number of atom-atom interactions involved and the different separation distances, the well will be deeper and the center-to-center separation distance between the macromolecules greater; but the range of the square well will still be only about 1.5 A. This is illustrated in Fig. 8.3-3 in which the separation distance is in A. In Fig. 8.3-3b the macromolecule diameter is 30 A; the range of the well remains that for atom-atom interactions of 1.5 A; and, as an example, the well depth is 2.5 times the atom well depth ¢. Thus, the interaction potential looks more like a sticky hard-sphere than the typical square-well potential. There are many other potentials that have been also proposed for both spherical and nonspherical molecules; however, we will not consider them here.

ENGINEERING APPLICATIONS VIRIAL EQUATION OF STATE

AND

IMPLICATIONS

OF THE

There are a number of applications and implications that arise from the virial equation

of state. Some of these will be considered here.

Use of the Virial Equation of State as an Engineering Tool In Chapter 7 we derived the first correction from ideal gas behavior and showed that the result is

— kT

= 14+ BT 21 )p

4. (8.4-1)

In applications, this volumetric equation of state is only useful for a relatively dilute gas or vapor phase. In particular, it should not used for high pressure systems (see Table 7.9-1) or at temperatures and pressures close to where the fluid would condense to a liquid.

141

8.4 Engineering Applications and Implications of the Virial Equation of State

Including higher-order terms in the expansion can extend the range of applicability of the virial equation. For example, considering also all closed-cycle interactions between three molecules leads to

—— = 1+ B(T)p + Bs(T)p*

(8.4-2)

pkT

This will extend the range of the virial equation to somewhat higher densities. However, the third virial coefficient B3(7) is very difficult to evaluate for commonly used intermolecular interaction potentials. What is commonly done in application Is to fit the second and third virial coefficients to experimental data. Of course, if the virial coefficients are to be treated as fitted parameters, one need not stop at the third virial coefficient; in fact, one can include as many coefficients as can be justified by the quality of the experimental data. That is, one can use

p ‘ oer 71 + BUT )0 + B(T)p* + B4(T)p> + Bs(T)p* + Be(T)p°

+ By(T)p° + ++:

(8.4-3)

At some point, the series in density 1s truncated, and the higher-order terms are neglected. There is a way to compensate for the terms that have been neglected. It is based on the idea that the exponential function has a series expansion with an infinite

number of terms, that is Ap er =

| fin aly | + Ap + = (Ap)

| | 1 + —3, Ap) $4 + z a t ae)

+.. .. +-

(8.4-4)

Thus, for example, although the equation

e” pt ) (T Bs + ° )p (T By + * )p (T B3 + oT = 1+ Bo(T)p

(8.4-5)

may not be very accurate, nonetheless it can be considered to have an infinite number of terms. This equation and the ones that immediately follow are referred to as extended virial equations, with the exponential term accounting for the neglected terms in the series. While this last equation is not very useful, there are other volumetric equations of state of the extended virial form that have been used in engineering. These include,

among many others, the equations of Benedict, Webb, and Rubin’ P



DRT

= 1

+(

A

C

B-—-—

RT

+a

aaa) 0+ (

al

ee

b —- —)p + —p

aT)? * Rr?

(8.4-6)

+ yp") exp(—yp")

where here p is the molar density and R is the gas constant, and of Bender® P=

pT(|R+

Bo+Cp*

+ Dp* + Ep* + Fp? +(G+

Hp7) exp(—arp")]

(8.4-7)

'M. Benedict, G. B. Webb, and L. C. Rubin, J. Chem. Phys. 8, 334 (1940) and later papers by the same authors. *E. Bender, 5th Syaposium of Thermophysical Properties, ASME,

New York (1970), p. 227.

Virial Coefficient

of the Second

'- Intermolecular Potentials and the Evaluation with =_

it?

—-—

— —

a|2

E=ait— =

-F

—- —

-— —:C=

— + a16

ays

ay\4

a|3

7s

— zityat

7



—GCG=

=

ay

ils

cy

ay

—:

ig

—; and

aya

D=a9+ H

=

a\7

wat



—; +

ayy

aig

ya t+ —zs



(8.4-8)

The virial equation origin of each of these equations of state is evident. Also, in these equations, the temperature dependent terms are meant to account for the temperature dependence of each of the virial coefficients. For applications in which high accuracy is needed (for example, for the custody transfer of natural gas—essentially methane—or for steam in order to determine the efficiency of steam turbines), equations with many more adjustable parameters are used, and in some cases even different sets of parameters for different temperaturepressure ranges.

Mixing Rules for Simple Equations of State In Problem 7.1, the exact composition dependence of the second virial coefficient for a mixture was found to be fe Ba mix (x, rj=

SoS

xixj Boi (T)

(8.4-9)

i=l j=l where B> ;;(7) is only a function of temperature, and not composition. Consequently, the second virial coefficient (and therefore the virial equation of state truncated at

the second virial coefficient) is quadratic in composition, In addition to the virial equation, other equations of state have been proposed and are used. For example, the van der Waals P=

V-b

a -—=

Vv

(8.4-10) Pad

RT

where V is the molar volume; other members of this class include the Soave” version

of the Redlich-Kwong equation!”

pi

**

~V—b

=e

VV +b)

8.4-11

we

and the Peng-Robinson equation!! p

RT

-

r a(y)

¥ VV +b)+b(—b)

V-b

(8.4-12)

These equations have been developed by fitting experimental data and are not the results of exact theory. (In later chapters we will consider approximations that

"G. Soave, Chem. Eng, Sci. 27, 1197 (1972). QO. Redlich and J. N.S.

DY.

Kwong,

Chem.

Rev,

d4, 233

Peng and D. B. Robinson, JEC Fundam.

(1949),

15, 59 (1976).

e Stat of on ati Equ al Viri the of ons ati lic Imp and ons ati lic App ng 8.4 Engineeri

143

can be used to derive some of these equations.) Therefore, unlike the second virial coefficient, the composition dependence of the a and b parameters are not specified. However, the three equations above (and others of the so-called cubic form, since they can be written as a cubic equation in volume) can be expanded series In virial form | a(l)\ PY — a ae — =|+[6-

Vv

RT

(

RT

in an infinite

($.4-13)

where the first two terms on the right-hand side of the equation are common to the family of cubic equations, and the further terms in the series are specific to each equation. Now, making the following identification with the virial equation of state Cc 6

_.

xix) Boi (1 J=b—-

Bo mix(X, F) = Yo

a(t)

RT

(8.4-14)

i=l j=l This identification

suggests that the term on the right should also be quadratic

in

composition. One way to insure this is with the following mixing rules: c h6c¢ a(r)=

c lu

3

S| xjxjaij(T)

‘=|

jf=l

and

b=

\-

S > xix bij

i=l

j=]

($.4-15)

This set of equations, originally used by van der Waals, is referred to as the van der Waals one-fiuid mixing rules, since the mixture is described by the same equation as the pure fluids—though with composition-averaged parameters. However, these equations are incomplete in that, while the pure component parameters (1.e., a;; and aj;) can be determined from pure fluid properties, the cross terms (i.¢., a@j;) are obtained from additional equations referred to as combining rules. (Compare these with the molecular level combining rules of Eqs. 8.2-4 and 8.2-5.)

The mixing rules of Eq. 8.4-15 are not exact. For example, the single relation of Eq. 8.4-14 has been used to constrain two functions, a and b. In fact, there are an infinite number of other mixing rules that could be developed that also satisfy Eq. 8.4-14—for example, one could add any composition dependent function f(x) to a and then add f(x)/RT to b, and satisfy Eq. 8.4-14.'* Also, the composition of the third virial coefficient can be shown to be

c B3 mix(x, T) =

ce ($.4-16)

Yo xix pre Bs ijx(T)

»

f=1 j=l k=l which is cubic in composition. However, from the van der Waals equation in

c

bB3 mix

{X,

T)

=

my

2

c

| =

7

D

i=l]

?

D

. XX

jp bj;

($.4-17)

j=

Thus, while the mixing rules of Eq. 8.4-14 give the correct composition dependence of the second virial coefficient, they are incorrect for any higher order virial coefficient. 2D. §. H. Wong and §. |. Sandler, AIChE J. 38. 671 (1992).

144

Chapter 8: Intermolecular Potentials and the Evaluation of the Second Virial Coefficient Nonetheless, the van der Waals one-fluid mixing than a century, and are satisfactory when dealing functionality, such as the family of hydrocarbons. useful mixing rules that are used in place of the van

rules have been in use for more with species of similar chemical There are more complicated and der Waals one-fluid mixing rules

for mixtures of dissimilar species.'* CHAPTER

8 PROBLEMS

8.1 The following second virial coefficient data are available for methane and CFy4

with oj; = soi + oj] and ej; = /8i&;;. What

is the exact

composition

of the gas in the

cylinder?

P

Beng

r

Ber,

(K)

(cm-/mol)

(K)

(cm?/mol)

110)

—344 + 10

273.16

200

—107 +2

373.16

400

-15.5241

523.16

S00

—O.5 + |

600)

+8.5 = |

—lIil —43.]

+1.25

673.16

+23.6

8.5 For calibration of a gas chromatograph we need to prepare a gas mixture containing exactly 0.7 mole fraction methane and 0.3 mole fraction tetrafluoromethane at 300 K and 25 bar in a steel cylinder that is initially completely evacuated. Assume this mixture can be described by the virial equation using only the second virial coefficient, and the molecular interactions are described by the square-well potential and combining rules

Use these data to estimate the Lennard-Jones parameters for CHy

00 1-8 0)

a=

and CFy.

$.2 Calculate the configurational contribution to the inter-

r -

CH, mixtures. To calibrate the response of the GC, a carefully prepared mixture of known composition is used. This mixture is prepared by starting with an evacuated steel cylinder and adding CO» until the pressure is exactly 2.500 atm 25°C. Then CH, is added until the pressure reaches exactly 5.000 atm

at 25: C. Assuming that both CQ, and CHy are represented by the L-J 12-6 potential with the parameters given below:

a(A)

e/k (K)

CH,

4.010

143.87

CO;

4416

192.25

and that the following combining rule applies

wisn")

+

and Reyij =

Ryw, jj ).

The potential follows:

parameters

for CH4

and

CF,

are

a(A)

ejk

CH,

3.400

88.8

L.85

CFy

4.103

191.1

1.48

(K)

as

Rew

The following two procedures will be considered for making the mixture of the desired composition at the specified conditions. a. Procedure 1. CH4 will be added isothermally to the initially evacuated cylinder until a pressure P, is obtained. Then CFy will be added

isother-

mally until the pressure of 25 bar is obtained at 300 K. What should the pressure P; be to obtain exactly the desired composition? b. Procedure 2. CF, will be added isothermally to the initially evacuated cylinder until a pressure Py is obtained. Then CHy will be added isother-

mally until the pressure of 25 bar is obtained at 300 K. What should the pressure P; be to obtain exactly the desired composition?

'*See, for example, D. S. H. Wong and S. 1. Sandler, AIChE Journal 38, 671 (1992).

Chapter 8 Problems The following additional data are available:

8.6 Compute the heat capacity of nitrogen at 150 K and 150 bar. At these conditions, nitrogen is described by the virial equation of state truncated at the second virial coefficient. Assume that the interaction potential between nitrogen molecules can be described by the spherically symmetric square-well potential. The following data are available:

ao = 3.2774:

ée/k = 95.2K;

8.7 Repeat the calculation of Problem

Te

Roy = 1.58 8.6 but using an

erties

virial coefficient by comparing values of the second virial coefficients for the Lennard-Jones 12-6 poten-

tial with those for an inverse

12"°-power potential

having the same values of ¢ and o over the temper8.9

ature range of 7* = kT /e of | to 100. The best estimates for the relations

between the Lennard-Jones parameters and the critical constants were given in Eq. 8.1-12. Use these expressions to obtain the Lennard-Jones parameters for CHyg, CFy,

Ar,

and

COs,

and

then

compute

the

second

virial coefficients for these gases over the temperature range of 200 to 800 K.

8.10 Below are virial coefticient data'* for Kr and N>:

LS pas [|-104 —

(cc/mol)

T(K)

from

the

more

difficult

Comment on the degree to which these substances satisfy the corresponding states principle for the second virial coefficient with the LennardJones 12-6 potential based on the data above. The partition function for a moderately dilute gas is

8.11

=

ayy (NB Wyn )” Granseordvibdetectnue N!

2V

where q/...= queen! V and 6 =4n [ f(r)redr with f(r) = ees e

u(r) = de(a/r)'*

second

given above.

What can you say about the sensitivity of the predic-

of Problems 8.6 and 8.7? $8.8 Determine the importance of the attractive and repulsive parts of the intermolecular potential to the second

and

task of fitting the measured vinal coefficient data

Rew = 1.87

tions on the potential parameters based on the results

data,

>

ao = 3.299A:

es

a. Obtain the Lennard-Jones 12-6 parameters for these substances first by using the critical prop-

alternale set of square-well potential parameters that has been proposed for nitrogen:

e/k = 53.7K:

145

Mir

RE

a. Derive the equation of state for this gas. b. For this gas develop expressions for the following in terms of 6 (i) departure of the internal mune gas behavior U(N, VT) —U'

from ideal (N,V, T)

(11) departure of the constant volume heat capacity from ideal gas behavior

Cy(N.V.T) — CIS(N,V,T) (iii) departure of the ay behavior H(N,V,T)—

from ideal ACN, V.T)

gas

c. Interactions in argon can be describes reasonably well by the Lennard-Jones 12-6 potential with the following parameters ¢/k = 119.8K anda = 3.405 A. Using this information, at 150 K and &% bar, compute the

| 250

75.7 | | -16.2 |

500 [81 | 169 a0 [|280

(1) volume of argon

(11) the internal energy, enthalpy and constant volume heat capacity deviations from ideal gas behavior

$8.12 The series of books “Chemical Process Principles” by QO. A. Hougen, K. M. Watson, and R. A. Ragatz (John Wiley & Sons, New

York) contain many correspond-

ing states charts that were of interest to engineers before computers were available. In particular, there

“From The Virial Coefficients of Gases by J. H. Dymond and E. B. Smith, Oxford U niversity Press, 1969.

|

Chapter 8: Intermolecular Potentials and the Evaluation of the Second Vinal Coefficient are charts of the fugacity for gases and the departures of enthalpy, entropy, and internal energy from ideal

gas behavior.

such

Here,

information

8.14 The Sutherland potential is

u(r) =

computed for small segments of these charts using the virial equation

results should

be presented

in graphi-

cal form. For these calculations, use the relationships

between the critical properties and the Lennard-Jones |2-6 parameters given in Eqs. 8.1-12. } The triangular-well potential is Oo

“r)Sy

r@ cr < Ryo

that to obtain

numerical values for the second virial coefficient for this potential one must either expand the exponential and integrate term-by-term, or evaluate the integral numerically.) b. Sutherland suggested a value of m = 3. Show that the second virial coefficient does not confor this value of m. (A

better value

from

function of temperature? r>

Ryo

8.15 Assuming benzene (7¢ = 562.1 K), ethane (Tr = 305.4 K), and the refrigerant R12 (Te = 385.0 K)

; . a. Obtain an expression for the second virial coef-

ficient for this potential. b. Does the second virial

(Note

land potential with m = 6 have a maximum as a

Rwo —o 0

for this potential.

quantum mechanics 1s m = 6.) c. Does the second vinal coefficient for the Suther-

—r

gi"

7

|

verge

Roo

a\m —F (—)

of state.

Compute the fugacity and enthalpy, entropy and internal energy departures from ideal gas behavior as a function of T, = T/Tc and P, = P/Pe at T. = 1.00, 2.00, and 3.00 using the virial equation of state. The

rio

oO

will be

coefficient

for

triangular-well potential have a maximum function of temperature?

the

as a

can be described by the Lennard-Jones 12-6 potential, what are the Boyle temperatures of these fluids?

8.16 Compare the values of the second virial coefficient as a function of reduced temperature for the two squarewell potentials of Fig. 8.1-3. 8.17 Derive Eqs. 8.3-7.

Chapter 9

Monatomic Crystals In this chapter we consider the statistical mechanics of monatomic crystals as the first example of a dense system of interacting molecules. A crystalline solid might seem like a difficult system to consider because of the small separations between atoms and their strong interactions. However, there is an important simplification, since the atoms are in a well-defined periodic structure or lattice and their locations are known. The only motions of the atoms are small vibrations around their equilibrium positions. Thus, the problems of the locations of the atoms and the thermodynamic properties of the crystal are separated. This is simpler than, for example, the statistical mechanics of liquids in which the average spatial arrangement of the molecules is unknown and must be found as part of the thermodynamic properties calculation. Traditionally, since metals form crystalline structures, the thermodynamics of crystals was a field for metallurgists. The increased interest in crystalline materials among chemists and chemical engineers in recent years is a result of the importance of crystallization of electronic materials, proteins, colloids, biomolecules, polymers, and pharmaceuticals (the latter for both purification and drug delivery). In this chapter we only consider the simplest models of crystals, and only atomic crystals. Each atom in a crystal is in close proximity to, and interacting with, other molecules. In particular, a molecule interacts strongly with tts nearest neighbors and, to a lesser extent, with molecules that are increasingly further away. The sum of all these interactions results in a three-dimensional energy landscape in which the lowest energy state of each atom is at its equilibrium lattice site, and each atom vibrates about this lattice point in the three coordinate directions.

INSTRUCTIONAL

OBJECTIVES

FOR

CHAPTER

9

The goals for this chapter are for the student to:

¢ Understand the Einstein model of a crystal e Understand the Debye model of a crystal ¢ Understand the relation of these models to the third law of thermodynamics e Understand the limitation of both these models by comparing the predicted heat capacities with experimental data

a1

THE

EINSTEIN

MODEL

OF

A CRYSTAL

The simple Einstein model of a crystal results from considering one atom in the crystal, and assuming all the other atoms are fixed at their equilibrium lattice sites.

147

1: Monatomic Crystals

Also, we will assume that the interaction energy landscape for this molecule is spherically symmetric, that is, the same in any direction away from its equilibrium lattice site. This would be rigorously correct if the all the other atoms in the crystal were smeared out on the surface of a sphere surrounding the lattice site of the molecule of interest. However, the other atoms are located at specific lattice sites, so the energy landscape is not symmetric. In this model energy landscape, the interaction energy is very large if an atom

gets very close to another atom in the lattice. Therefore, the motion of an atom is not the free translation as in an ideal gas, but rather a small wavelength vibration in a crystal. Consequently, the three translational motions of an atom in a gas are instead three vibrational motions in a crystal. As in a diatomic molecule, these three

vibrational motions will be modeled as harmonic oscillators, and as a result of the spherically symmetric energy landscape assumption, these three vibrational motions are

In the three-dimensional

identical.

Einstein

model,

all vibrations are at a single

frequency denoted by vp, and each has the following energy levels: En =

l 4 ;)

(

Ave

with

where vg ts the classical vibrational frequency given by

f Ve = —,/— |

2r

a=QO,1,2,.....

of the Einstein

where f = force constant =

Vm

(9.1-1)

model

in principle

d*u(r)

dr?

m is the mass of an atom and u(r) is the interaction energy landscape for an atom in the lattice. The partition function of a crystal containing of N atoms in this model is

(9.1-2)

Q(N, V,T) = So en Bitn-voreT states |

where the energy of any state consists of two contributions. The first is the sum of

the interaction energies of each atom with all others, all at their equilibrium lattice sites. This is a fixed number that we will write as E'™. The second contribution is as a result of the vibrational motions of each of the atoms. Therefore aN

OLN,

eo

-



I)

V,

EUNVVKE

_

e EM

e

IKT

vibrational states n of each atom

SLaLes |

=e

EnseT

—E™ JKT

(qvib)

3N

(9.1-3)

where

r

— Gvib

=

eel

Yoo eve

| —

Sve

f2kE

hvg/kT @—fve/ akT

@

ME LAkT

00:

ce MMes kT

n—0

n=0)

A=0

~

=

_ 2)Ly hve ‘Po

oS

e

PE/et

~~ 1)>~On/T = e~ E/T

(9.1-4)

149

The Einstein Model of a Crystal

9.1

l ona ati not For re. atu per tem al ion rat vib in ste Ein the as to ed err ref /k Ave with @¢ = convenience we use E'" = Nu, where w is the interaction energy for each molecule at its equilibrium lattice site. Therefore

_

e-Sn/2T

O(N. V.T) =e

AN

(a7)

(9.1-5) e

Ge /27

A(N, V.T) = —kT In O(N, V. T) = Nu — 3NKT In (ert — — 3N

bed

= Nu + —hvg + 3NKT In (1 — e 8/7) (9.1-6)

= Nuj, + 3NKT In (1 — e 8/7)

where uy = u + 4hv¢ is the zero point energy (per atom) of the monatomic Einstein crystal, and

A

) N,V,T)| =(—(sr ).

OO, /2T

e

A

aA

—) =Nv" = =u—-3kT n" ((;——aur

3 ) /7 8F e — (1 In T 3k + uo = ) 7 / 8 —e~ (1 In T 3k + op Sh + =u (9.1-7) ,/{ol

NV

T

U(N,V,T) =k? (mee) df

NV

Of =

Nu

aod

3N ye

-f

3NKT

eOn/T

1]

0 Nue

=

+

aN hv

eSe/T ——~eaiF (9.1-8)

S(N.V.T)

= 2!I(N,V,T)-— , ) ACN, (

V,T |

rr =

3Nk

Gp ¢ 7/? aera



In

(|



—Og/T E

)

(9,1-9)

and

au

Ory

Cy = &

—S6/T

= 3Nk (=) —____.

ay

NUV

r

(1

es

(9.1-10)

e~On/T)

There are two limits of these equations that it are interesting to look at. The first is the limit of low

temperature

A(N,

(7 —

ePe/T

0) for which gyi, =

V.T

>

O) = Nu +

U(N,V.T

>

0) = Nu

S(N, V.T — 0) = 3Nk

3NAv 5 =

e

= Nut.

a T

(9,1-11)

er 9: Monatomic Crystals and

e ) = (= 3Nk = “Gr)as >0)=({— 0) Cy(N,V,T > dl

Or

=,

Ob!

/T

— 0, §(N, V, T = 0) = Ofor the Einstein crystal in accordance with the third | law ‘of thermodynamics that the entropy of a perfectly ordered crystal is 0 at the absolute zero of temperature. Also, that Cy(N, V, T = 0) Note that since jim



Jim

.)

—e

l nO; N,V,T >> @p)Op) =

NuAT (—(kT) YN Ave

Nu iT ++ 3N —-—

=e

nw f T)ON Af

(5

T N kT + BN n ( — +) P -+ —— | = —— 1 n (—)

T In Q(N, V, T > Oc) = Nu — 3NeT In (5) JE

A(N, V, 7 > Og) = —kT

(9.1-13) A Ww

u — 3kT

w(N, VT

> Op) =

U(N.V,T

> Og) = Nu + 3NKT

r In (=)

r

(=) —In [ 3Nk = Op) > S(N, V.T

Or

and

Note

Cy(T > @g) = 3Nk that

the

high

temperature

Cy(T > Og) = 3Nk, which

limit

is known

of the

constant

volume

heat

capacity

is

as the law of Dulong and Petit. This is to be

compared with the constant-volume heat capacity of an ideal monatomic gas, which is Cy = 3Nk. The difference arises because each atom of an ideal gas has three degrees of translational motion, each of which contributes +k to the heat capacity, while each atom in a monatomic crystal has three degrees of vibrational motion, each of which—when fully excited (high temperature)—contributes k to the heat capacity.

| DEBYE

MODEL

OF

A CRYSTAL

The Einstein model of a monatomic crystal is a primitive one in that it is assumed that the motion of each atom 1s independent of all others. A better model is to allow all

9.2 The Debye Model ofa Crystal

151

the molecules to vibrate simultaneously. For an N atom crystal, there is 3N degrees of freedom. To identify these, we could (in principle, but not in practice) do a normal mode analysis, as discussed in Section 4.6 for polyatomic molecules, and find that there are three translational motions corresponding to movement of the center of mass of the crystal, three rotational motions of the whole crystal, and 3N—6 vibrational modes. The partition function of such a crystal, keeping the macroscopic crystal fixed in space so that there is no translational or rotational motion of the center of mass, 1s hy,

3N—f

af kT I]

O(N,V,T)=

— eo WikT I]

(FF)

dvib.j

i=]

_ _

ge

I]

anna (

*)

i=l

N76

ia

@

IN—6

9 Oy,i/2T 1; —¢e

Oval?

(=| 3N

or

In O(N. V.T) = —-—-+

6

©,

;,/2T

In(on) l-—e Oy 7/

(9.2-1)

For the Einstein crystal, the vibrational frequency of each atom was assumed to be identical to ve, so the evaluation of the partition function was straightforward. Here, however, the independent vibrational modes have to be determined from a normal mode analysis, and involve 3N—6 vibrational frequencies. One expects that the frequencies will range from high-frequency modes for the vibrational motion of a single atom, as considered in the Einstein model, to low-frequency (and therefore large wavelength) modes resulting from the concerted motion of large numbers of (in the limit, all) atoms in the crystal. Instead of attempting to identify the 3N—6 normal vibrational modes for the crystal, the Debye model uses a probability distribution of frequencies g(v) defined such that the number of vibrational modes in the frequency range v to v + dv is g(v)dv. This probability distribution is normalized such that i

[ eoray

=3N

—-6

(9.2-2)

0 With this approximation

ao

InQ(N,V,T)= -= + d, In ( li —

EF

a-Wva/2T

e + |

e(v)

|

Avf2kT

In (ar)

dv

(9,2-3)

0

The problem then becomes one of determining the probability distribution g(v). In the Debye model, it is recognized that the low-frequency (large wavelength) collective motions make large contributions to the partition function. Such collective motions, which correspond to wavelengths of several or many lattice spacings, are rather insensitive to the specific atoms of the crystal. (This is an interesting and subtle idea. To understand the wave motion of a large amount of fluid, for example the ocean, we need only information about its macroscopic properties of viscosity and density and the laws of fluid mechanics to describe its motion. However, if we are interested in the

I: Monatomic Crystals the on on ati orm inf ed ail det d nee we e, mpl exa for ice, in le vibration of a single molecu l era gen the s, Thu .) les ecu mol ng ndi rou sur h wit mass of the atom and its interaction r, ula tic par in s— om at of s ion mot h ngt ele wav ge lar ant ort imp the of ics ist character ls. sta cry all for m for r ila sim a of be l wil l sta cry a in es— nci que fre the distribution of ewav ge lar the of ion but tri dis ncy que fre the that d ume ass is it el mod ye Deb In the or avi beh the for ics han mec id sol in d un fo that to r ila sim is l sta cry a in s ion mot gth len of elastic waves in three-dimensional continuum, which is given by g(v)= a@v~. This is taken to be valid for all frequencies up to the highest frequency (shortest wavelength) vp corresponding to the motion of a single atom with all other atoms fixed at their equilibrium positions. That 1s g(v) =

av

for0 00, ») = —NAT Infe® + e-°] = —NkT In?

and

(10.5-11)

A(N +1, T > 0, x) = —N&AT Infe*/*" +0] = —Nyx Also from

,f{al

U(N +1, x) = kT? (

KAT

== Nx (Sea

=“)

aT

_ g—x/kT

exikT 4 @ KIT

NOV

)

so that

eXIKT _ g—x/kT UN

+ 1.7

00. 2) =~

Lim mp

;

vx XD

(

extkT 4 g—XIKT

ekfkr

=

eX

Eke

UN + 1D — 0.x) = — Lim Nx (Gar)

)=0

= —Nx

(10.5-12)

)

10.5 The Ising Model

181

Further dl!

Cy

~

2

x

(sr).

\ar

(ae)

(ex (kT

kT a2 { x a

tg

sor

=4NK|7 7)

(

10.5-13 )

These results can be analyzed in terms of cooperative and anticooperative behavior. In particular, suppose that the parameter x is positive, so that the t+ is a higher energy

state than the ¢| state. Therefore, the lowest energy state is that of alternating spins, and this is the most likely state of the system at low temperatures. We consider this to be anticooperativity in that a lattice point being in one state makes it more likely that its neighbors will be in the opposite state. However, if y 1s negative in value (that is, attractive), +? is a lower energy state than the ¢ | state, therefore, the lowest energy state 1s that of parallel spins, and is cooperative behavior in that a lattice point being in one state makes it more likely that its neighbors will be in the same state at low temperatures. Note that with either cooperative or anticooperative behavior, the energy of the system at low temperatures is —N yx, even though in one case the state is aligned parallel spins and in the other it is alternating antiparallel spins.

The analysis above is for the simple one-dimensional Ising model. Conceptually, the model is easily extended to two and three dimensions, though these extensions are mathematically much more difficult. In fact, Lars Onsager received the Nobel

Prize in Chemistry in 1968 for his solution of the two-dimensional Ising model;'* there is no known solution for the three-dimensional Ising model. (As an aside, it is Interesting to note that Onsager was the Gibbs Professor of Theoretical Chemistry at Yale University.) Though the one-dimensional Ising model is a great simplification of real systems, it has been used to obtain insights into some real phenomena. The underlying assumption of such models is that only nearest-neighbor lattice interactions are involved. For example, in this way the Ising model ts related the helix-coil conformations in polymers. A simple model for this is that a monomer in a chain can be either in the helix (H) conformation or the coil (C) conformation, and that the interaction energy of each monomer is only the result of interactions with adjacent monomers. We provide a simple generalization of the model here using the notation that ¢y4 is the H-H interaction energy, €cc is the C-C interaction energy, and cy = €yc is the H-C or

C-H interaction energy. Consider the two-monomer chain where we do not know in advance whether the first element in the chain is in the helix or coil conformation. The possible conformations are HH, HC, CH, and CC. Therefore, the partition function is O(2,V.T)= —

e fHH/ kT i ebHC/ kT ge

fHH/kT

1.

Fe

4

7fHe/kT

oe PCH! kT 4

be e tcc /aT

g—tcc/kT

= (e FHH/AT ait. e fuc/ kT) + (2 ene/at

J. e fec/kT)

(10.5-14)

and the probabilities p of finding a two monomer chain in each of the HH, CH, and CC conformations are —fHH/AT

PCH) = ~~DenenclkT aalkT 4 po =8cc/kT e

'*L. Onsager, Phys. Rev. 65, 117 (1944).

[O: Simple Lattice Models for Fluids

P(CO) =

aan T

ebcc (kT

De uclkT 4 @ Foc /AT

Ve FcH fee = saemn/AT a+ Des De *uclkT euclkF 44 peccikT en cel T

and p(CH)

(10.5-15)

where the factor of 2 arises because the CH and HC conformations are identical, differing only in the conformation of the starting monomer. By the same reasoning, the conformations for a three-monomer chain, the possible conformations are HHH, HHC

(and CHH), HCH, HCC

(and CCH), CHC,

and CCC,

and the partition function can be shown to be

O(3, V, T) = (e~SHHIAT 4 gp eHc/AT 2 4 (psu /kT 4 pg eec/kT)2

(10.5-16)

From this, expressions can be obtained for the likelihood of finding a three-monomer chain in each of the conformations listed above. A similar analysis can be done, for longer polymer chains (Problem 10.6). The Zimm-Bragg model for proteins is somewhat more complicated in that it contains more detail. In that model, it is assumed that it is easier to start a polymerization chain from a coiled amino acid sequence than from a helix sequence, and that it 1s easier to add a coiled sequence to a coiled sequence than it is to add a helix sequence. Only a very simplified version of the model will be considered here, in which €cc = €nc = €HH # &cH—that is, there is no energy difference in adding a col to a coil, a coil to a helix, or a helix to a helix; however,

there is a different (and

greater) energy required to add a helix to a coil. Also, in this model it is assumed that it is more difficult to start a chain from a helix than from a coil, and the energy difference 18 €cH — €cc. (Note that this is also required by symmetry, so that the CCH and HCC trimers are equally likely; and the same is true for the CHH and HHC trimers.) The partition function for this system is

QO(3, VT)

= gece + gecu + 9cue + ¢cun + Guce + gucu + dune + gunn = e *cc/ KE Ty 4 eg (cHeccWAT 4 ecu

Fec

kT (4 4 e

cH

1 g—lecu eee WRT 4 g—tecu eco Fcc ys RE

kT

+141]

=e *ec/kTT] 4. 35 + 35 + 57] = 7 78CC/T] 4 Gs 457]

(10.5-17)

where 5 = e‘*cH~*co/KT is the Boltzmann factor of the energy penalty on forming a CH bond rather than CC bond, and also for initiating the chain from a helix rather than a coil. The probability of occurrence of the various conformations are then

e—2ecc/kT

\

p({CCC) = p(CCH) = p(CHC) = p(CHH) = p(HCC) = p(HHC) = p(HHH) iY

~ [1 +65 +52] s

and p(HCH) =

[1 +6s +52]

(10.5-18)

10.5 The Ising Model

These are shown

in Fig. 10.5-1

183

as a function of the parameter s. Note that for

amino acids, s is thought to be of the order of 10-* to 10~+, so that only coiled trimers would be expected with such a large energy penalty. Another property that we can calculate is the expected helicity of the trimer—that is, the average number of helices in the chain from

(H) =0 x p(CCC) + 1 x [p(CCH) + p(CHC) + p(HCC)] + 2 « [p(CHH)

+ p(HCH) + p(HHC)] + 3 x p(HHH) _

Ix(ststs)+2x(s +5)+s +3 7 x58

7

1+ 6s +5

Os + 3s? =

—_

1+6s

+s?

(

10.5-19

The degree of helicity is shown in Fig. 10.5-2.

pp

Figure 10.5-1 Probability of occurrences

25

3

—34

os

NS

2

ee

logis)

—3

I

0.5

=

a]

logis)

|

of various trimers as a function of the energy penalty s: CCC (solid line),

CCH, CHC, CHH, HCC, HHC or HHH

8)

(dotted line) and HCH (dashed line).

Oo

Figure 10.5-2 Average helicity of amino acid trimers as a function of the energy penalty function sv.

184

Chapter 10: Simple Lattice Models for Fluids

10 PROBLEMS

CHAPTER

Draw the excess free energy versus composition diaeram for the regular solution model (simple lattice model) for various values of x /kT. 10,2 For the regular solution model, find the composition of the coexisting phases and draw the phase boundary as a function of ¥ /AT. Develop the equations to be solved for the compositions of the coex-

10.1

phases.

isting

that

Show

if »

has

a

large

very

value, the two phases are relatively insoluble in each other with the compositions of the coexisting phases being

3 The spinodal curve for liquid-liquid equilibrium in a mixture described is the locus of points for which

(se)

0

OX: J rp

as compositions

for which

(3)

2% QO are not

physically possible, as the chemical potential of a substance must increase as ils concentration increases (just as for a pure gas the situation in

which (a5), > Ois unphysical). Develop an expression for the spinodal composition for the regular solution (simple lattice) model. 10.4 Obtain expressions for the third virial coefficients for the (a) van der Waals,

(b) Redlich-Kwong,

and

(c) Peng-Robinson equations of state. 10.5 Develop expressions for the probabilities of occurrence of the HHH, HHC (and CHH), HCH, HCC (and

CCH),

CHC,

and

CCC

conformations

of the

three-monomer chain |-D lattice. 10.6 Develop expressions for the partition function and the probability of occurrence of each of the possible conformations of the four-monomer chain |-D lattice.

10.7 Develop a simplified Zimm-Bragg model for a fourmonomer chain, and compute probabilities of the different possible chains and the average helicity as a function of the energy penalty function s. 10.8 Derive Eq. 10.4-13.

10.9 Show that for p polymer chains (each of ¢ monomer units on a lattice of NV sites), the number of possible

Chapter 1 1

Interacting Molecules in a Dense Fluid. Configurational Distribution Functions In this chapter, we are interested in the statistical mechanical description of a dense fluid, such as a liquid. In many ways, this is the most difficult class of systems to treat. A liquid is a dense fluid, and at high densities, the virial equation is very slow to converge, so that many high-order virial coefficients would be needed. However, it is not possible to evaluate analytically the very complex multidimensional integrals involved, so the virial equation of state cannot be used to describe liquids. In the past, statistical mechanical lattice and cell models have been used to describe liquids, and such models treat a liquid as a crystalline structure using sophisticated refinements of the simple models presented in the previous chapter. Such models are not very accurate, and miss some of the essential features of liquid behavior—such as that the molecules do not remain at fixed positions. However, there is a completely different, more theoretically based method used to describe liquids, the discussion of which

begins in this chapter and continues through to Chapter [4.

INSTRUCTIONAL

OBJECTIVES

FOR

CHAPTER

II

The goals for this chapter are for the student to: e« Understand the concept of a radial distribution function e¢ Understand the relationship between the radial distribution function and thermodynamic properties e Have an introduction to the methods by which the radial distribution function ts obtained

11.1

REDUCED

SPATIAL

PROBABILITY

DENSITY

FUNCTIONS

As background, it is useful to start by considering at the configuration integral for a system of N identical atoms

Z(N.V,T) = fo. few

ar dry...dry

(11.1-1) 185

ons cti Fun ion but tri Dis nal tio ura fig Con d. Flui se Den a in les ecu Mol ng cti era Int |:

Z(N,V,T)

|

any ose cho can we , ents fici coef al viri the of n tio lua eva the in as Note that here, | le ecu mol of on ati loc the e, mpl exa r —fo tem sys e nat rdi coo the of in convenient orig as n tte wri be can gral inte n tio ura fig con the that —so le) ecu (or any other single mol feet

[| fo

Vf.

few

ae

fae

dein

radris

dradess ...driy

(11.1-2)

where, for example, r)5 is the vector between the origin of the coordinate system (here the location of molecule

1) and molecule 2, etc. However,

since we can choose

the origin of the coordinate system only once, further simplification of the integral by choice of the coordinate system is not possible. So instead of trying to directly evaluate the integral in Eq. 11.1-2, we will proceed differently. Consider

for the moment

a collection of N

(but distinguishable)

identical

atoms

or molecules in a volume V at temperature 7. We are now interested in obtaining an expression for the probability of finding molecules in specific locations near each other. However, since position 1s a continuous variable, there are an infinite number of positions, so the probability of finding a molecule at any point is essentially zero. Instead, as is usually the case with the statistics of continuous variables, we will use a probability distribution or probability density function, and consider the probability that a molecule

is located

in a finite,

but differential,

volume

element

dr about

a

specific location r. In fact, we will initially consider a volume element dr that is so small that it can contain at most a single molecule. We start by considering the probability that molecule | is in a small volume element dr, around the location (or position vector) r;. This probability is just dr) /V since,

given no other information, the molecule is equally likely to be any place in the volume; so the probability that the molecule is in a specific small volume element is just equal to the fraction of the total system volume that the volume element occupies. However, since the molecules are indistinguishable, we really should consider the likelihood that any of the N molecules ts in the volume element dr, independent of their identity. That is, Likelihood that any of the NV molecules

is in volume element dr,

N = yan

= pdr).

(11.1-3)

This is being referred to as a likelihood rather than a probability for semantic reasons. By definition, a probability has a value between 0 and unity—as does dr, /V, which approaches 0 if dr) becomes infinitely small, and is unity when the volume element is equal to the system volume. However, the likelihood function goes from 0 to N over this range, so it cannot strictly be considered a probability function; more correctly, it is the probable number of molecules. Now consider the probability that molecule | is ina volume element dr, around rj, and simultaneously that molecule 2 is in volume element drz around rs, molecule 3 is in volume element dr3 around rs, ..., etc., regardless of their translational motions. This is more complicated, since there can be an energy of interaction between the molecules

in the different

volume

elements,

which

would

influence this probability.

In fact, this probability is equal to the Boltzmann factor of the interaction energy in this configuration, normalized by the configuration integral (the sum of probabilities

11.1

Reduced Spatial Probability Density Functions

187

of all configurations), and consequently is given by! e7#lEL. Peed A [-

fe

—Htr

(KT dr)...

|. fo.

dry

a

_

eee

EWEN

dry

Z(N,

Y,

de

__

dry

(ll

1-4)

Tr)

where Z is the N particle configuration integral discussed earlier. (Note that for simplicity in the following equations, we will frequently write u(r,,r5,...F))) as simply w and Z(N,V.7)

as Zy.)

Of greater interest are reduced-probability functions involving fewer numbers of molecules. For example, the probability that molecule | is in the volume element dr, about r;, and that molecule 2 is in the volume element dr> about rz,..., and that molecule # is in a volume

element dr,, about r,, regardless of the locations of

molecules n+1,n+2,..., N 1s

[fear drjdr,...dr wf

_

ee

...dry fs

UAT oyfr. 40 naa dry

(11.1-5)

However, we must again remember that identical molecules are indistinguishable. Therefore, instead of inquiring about the probability that a specific molecule is in a volume element dr, now we can only ask about the likelihood that any one of the

N indistinguishable molecules could be in that volume element. The likelihood that (simultaneously) one of the VW molecules is in the volume element dr; about rj), one of the N-1 remaining molecules is in the volume element drz about rz, ..., and that one of the remaining MN —n-+1 molecules is in the volume element dr, about r,,

regardless of the positions of the other N —n

drydiy dey fe. fC deg dt ges a9 fs etn Z(N,V,T)

PEE = Darel N!

molecules is

drydry-..dty

fe.

fe

dry

(dt nga

dey

dy

Z(N,V,T)

~ (N—n)!

=p"g"(r),..., r,: T, p)drj dr,...dr,

(11.1-6)

The factors before the integral arises as follows. There are N choices for the first molecule identified by the index 1, N-1 remaining choices for the molecule designated by the index 2 (since one of the identical

molecules has already been chosen),

N-2

choices for the third molecule, etc. Also, in the last part of the equation above, we have defined an #-body correlation function as NI

rT.)

= a

V

(z)

n

[.

. fe

dry

de

oe

ary

——$——$——————SS— drs...drnx | ; fe WIdr KT

(11.1-7) 'For cach spatial vector r, the integral is over the total system volume of integration is not shown, except where it is needed for clarity.

V. For simplicity of notation, the range

188

Chapter

| 1: Interacting Molecules in a Dense Fluid. Configurational Distribution Functions In most cases we will be interested that the functions of interest are

g(r}. 93 T, p) = N(N — 1)

yf

a

in small

fewar;

ee

values

of n—that

Py

a

is, # =?

V2 fo. fe

dry

fo

or 3—so

dry...dry

fewer

.. dP y (11.1-8a)

vf...fe

ow

(3),

—y

if

“KT dry... dry

SF. FoF Tp) S

;

few

ar,

id

on

—yil

vi f..fe

/

I

WT dry. dry

—_——

a

gy

Z(N,V,T) (11.1-9a)

or more commonly written as? N(N



vf.

. /

e MED ED E34

pg ry .053 T, p) =

Zz Ny? fo.

femmes

DIET ay,

.

dry

A

2 3 s r

MRT dy,

...ar My

11.1-8b)

“AN

|

and

oe

T -

=

N(N — 1)(N -2 | coo fe Menta eata WAT gy, .« dey z

,

?

ws

fa.

fl ermere

a5

mn

MT

dr...

dy

ZN

(11.1-9b)

The physical interpretation of the first of these correlation functions is as follows. The likelihood that a molecule is located in a volume elem ent dr, about the position vector r; (which for simplicity can be taken to be the ori gin of a coordinate system)

and simultaneously that a second molecule is in the volume element dr> about r> is, from the discussion above:

uy =

aride / a / eT drsdry...dry e—e—e

x N

nn 7

(N — 2}!

N2=

sj ar)dr,

V-

Vv;

fe" -

dr jdr, / vo / eT drsdrs...dry

Zyo

drsdry...dry ZN

——=

| = Per,

Fo

~

0, Pdr drs oO

(11.1-10) “Here again, for simplicity of notation, we have used fy = Z(N,V,T).

11.1

Reduced Spatial Probability Density Functions

As is clear from its definition above, the two-particle or pair correlation 2 (r .:£5: p,T)

189

function

is a function of the position vectors r; and r3, and also a function

of the molecular density e and ei pertie

T. For simplicity of notation, we will

T) p, ro; ), g(r on cti fun The . 75) ,, g(r as ply sim T) e. ro: (r), g'?’ e writ usually is commonly referred to as the radial distribution function. We will use both terms interchangeably. Among the properties of the pair correlation function is thal its value is unity if there are no intermolecular interactions, and also when the distance between the position vectors r; and rz is large on a molecular scale so that each

molecule no longer feels the presence of the other. Note that if there were no interactions among

the molecules—that

is, « = 0—then

Zy = V%, and the integral in the numerator becomes equal to V“~?. In this case, the likelihood of a molecule being in the volume element dr, and simultaneously a second molecule being in the volume element dr2 is just pdr dr, and g(r), r5: 9,1) = 1. This result is only valid if there is no energy of interaction and therefore

no correlation between

the molecules

in the volume elements dr,

and

dr>, and is a simple extension of the discussion at the beginning of this section. That is, p-dr,jdr, =(pdr)* is what is obtained if the molecules are uniformly distributed, so the number of molecules in any volume element is just the average density times the size of the volume element. However, in general, there is a connection between the two volume elements, since the molecules they contain can interact. That is, if the two volume elements are sufficiently close on a molecular scale, the presence of a molecule in one of the volume elements influences the likelihood of a molecule being in the second volume element. At very low density, this correlation is given by the Boltzmann factor of the interaction energy. However, at high density the correlation is more complicated, and the likelihood of both volume elements containing molecules is given by Eq. I1.1-10 above. Next, we consider a somewhat different question: what is the probable number of molecules in volume element dr2, given that there is a molecule in dr|? This is computed as the likelihood of molecules being simultaneously in dr; and dra, divided

by the likelihood of a molecule being in the volume element dr), and is given by

obable number of molecules indr, given that there is a molecule at dr, probable number of molecules simultaneously indr, and dr, probable number of molecules in dr,

NI

fe —

drydrs f

"IKT dridr,...dry

drjde, | ... [«

...dry Ns

_

~

— 2)! (N=2)! NI

dry | .. |

et

ZN

adry...dry

dry fo. few

a (N—1)!

yp ZN

ats ff

NY

be

4

=F? ar

= pg

dradrsdry..-dry

V '(r).foi p. T)dry

]

v2 fo.

NY SEE oy V =

=

fe!

drdry

dey

ZN

(11.1-11)

ons cti Fun ion but tri Dis nal tio ura fig Con id. Flu se Den a in les ecu Chapter 11: Interacting Mol

... j; .dr drj es nat rdi coo tive rela to ... dr, drj m fro es nat rdi since by a change in coo it is easily shown that

a / e “KT drjdradr,...dry

|

V

Fx PAIR

THE

FROM

THERMODYNAMIC PROPERTIES CORRELATION FUNCTION

The importance of the two-body correlation function or radial distribution function becomes evident in the computation of the thermodynamic properties of dense fluids. For example, the average value of the interaction energy (also called the configurational energy) of an assembly of molecules is

“ENUKT dy dry ...dry

fue, Poy sue Pye ie

|. =

fo

fermen

/ ' / Wr), 6...

Pye

ar dry

dey

EitatN VEE dp dr, ...dry (11.21) ZN

Now assuming, as before (and this is a strong pairwise additive—that is, that

assumption)

that the potential

U(r). Poe-e Ey) = >) > uly)

is

(11.2-2)

a

i¥0

ic fan

>

u(Fij)

(11.2-8)

[xf foo

with I

2

a - a”

ry = [Ge — xy +n —¥P +i —2P]" =| Lf2

&=x.V,2

1 wis

_.

=v

*

ao 2

[2

_ fii

(€; — ©)

2s

f*ax*, y*,2*

Also d(rj;)

=

d

(re)

vis] 13

=a

7-6]

| on MOL tN MET gy

atod GON

|

Lf

4

5

rij

so that

|

_ 8 J OZIN.VT)} oY an av

|

|

0

0 | _

= NV

Nv— |

l

fou fe

I =

J+

=

ata

WAFL o TW)

PRT

dxy...dzy

| Ne

vf.

ee Ly) du(ry,...fy)

7p)

fad!

e HE

..dzy

Tet

dv 0

i

and |

dZ(N,V,T)

|.

=

N

Vv

V

i

|

[-

()

dx)

fe

0

...dZy

|

Ia

ef

if

() |

Vv

—wi ke

u(rjjje'”

%

of

dxy...dZy =

11.2

Thermodynamic Properties from the Pair Correlation Function N

VY

ape

|

(N)CN



|

ff east 0 du(ry2)

|

() |

v

= Vv

ad

ef. 2

tra) drjo

drjo

«

dh

#

_

dct

dV

0

|

|

MED

V2

_..

pee eer

=

ip

| OUP i2) AN? uk ay 0

N

_

|

NYN-L

kT

193

0

pues

LT

dry 3V00

tat ed

Iy*

ON

...dz*

EN

()

(11.2-9)

Consequently du(rj>)

/ ]

Z

(22) —

\av

=

(ier)

=p

av

ANT

N(N—l)y

— ——

6kTV

NT



oy)

a

aVv

=PpNT

N?

i

sav

du(rj)>)

TF

6ATV

2

iy

rye

WRT ar

«me Of

TR



Z

- few (“ In =)

oF

Tar, ...dry

\!

dr dr

dr)?

f

~

\

N?

=p-

6kT V3

//

du(rj2) Frm

rjog'

2,

_

ow

|

(rj2)drjdr,

(11.2-10)

Vv so that

P=kT

vine





kT



du(r}2)

p-

_



| 10

>

Ir

/

V

a | ; ee

= pkT — 6

ot FF

dry \F

= pkT —

6

An / 0

ne) dry

opt

9 rd : P = pkT — aa ef ‘ WY iar a, dr a

drs -

(11.2-11)

This is the desired relation between the radial distribution function. the two-body interaction potential and the pressure. It is this relation, referred to as the pressure equation, which will lead to the volumetric equation of state. An alternative relation

between the radial distribution function and the pressure (or volumetric equation of state) will be developed later in this chapter.

194

11.3

Chapter

11: Interacting Molecules in a Dense Fluid. Configurational Distribution Functions

THE PAIR CORRELATION FUNCTION FUNCTION) AT LOW DENSITY

(RADIAL

DISTRIBUTION

From Eg. | 1.1-8, the pair correlation function is defined as

dradrg

V2 fio. ferment

)

2?) Gv rat PP) = “4 ZN

diy

eee

(11.1-8a)

with

Zn = Z(N,V,T) =|...fe

Wry fo tN WET dy drs ...dr y

To begin the evaluation of the configuration integral for a low-density fluid, we will

follow the same procedure used in Chapter 7 for deriving the expression for the second virial coefficient—that is, we will assume the interaction energy is pairwise additive

wrj,--.-Py)= >>

Do

?

ules)

(7.3-3)

l>o—the

interaction potential w(r) is O, and g(r) = |. This corresponds to there being no spatial correlation between the molecules. That is, at large separations, the number of molecules in a volume element far removed from a central molecule is just equal to the average density times the size of the volume element. Finally, since the radial distribution function at low density is equal to the Boltzmann factor of the interaction energy, the ranges of these two functions are the same. (Here, by range we mean the extent of the intermolecular separation distance over which there is a spatial correlation among the molecules, so that the value of the radial distribution function ts different from unity, indicating no correlation and a completely random distribution of molecules.) That the ranges of intermolecular potential and the radial distribution function are the same at low density is illustrated in Figs. 11.3-la and | 1].3-1b for the Lennard-Jones 12-6 potential. In these figures, the energy uw* is u/e and r* is r/a.

2

4

=

"|

0

Z

:

UW ()

4

|

rr

aad

3 lA

ha

-

re

Figure 11.3-1 (a) Lennard-Jones 12-6 potential w* as a function of r* and (b) the low density radial distribution function for this potential as a function of r*. 4

3 25

3 2

| gir)

eur)

2

oS

| | 0.5 () 0)

|

2 rier

3

0

005

115

225 rier

Figure 11.3-2 The (a) low density and (b) higher density radial distribution for the hard-sphere potential as a function of r/o.

335

4

45

y sit Den h Hig at on cti Fun on ati rel Cor Pair the of n tio ina erm Det of s hod 11.4 Met

197

Note that the region around the peak in the radial distribution function corresponds al radi the l, ntia pote ere -sph hard the For . cule mole the of shell tion to the first coordina ere -sph hard the for gy ener ion ract inte the e Sinc ler. simp even is tion func distribution potential is infinite for r< o, and zero for r > o, g(r) is zero for r= o and unity for r = o. This is shown in Fig. 11.3-2a. The radial distribution function is more complicated at higher density as shown in Fig. 11.3-2b.

THODS OF DETERMINATION ‘CTION AT HIGH DENSITY

OF

THE

CORRELATION

PAIR

At high density, the radial distribution function is more complicated, extending several molecular diameters. since in a dense fluid there are correlations over larger intermolecular separation distances. This is a result of there being both a direct correlation,

between

molecules

| and

?, and

a collection

of indirect correlations,

such

as the correlation between the locations of molecules | and 3 combined with the correlation between molecules 3 and 2, and the correlations between molecules | and 3, 3 and 4, and 4 and |, etc. Consequently, the radial distribution function in a dense fluid is of longer range and has several peaks (corresponding to coordination shells) and valleys (resulting from a steric hindrance to molecules just outside each coordination shell from the molecules in that shell). In fact, we will use the idea of direct and indirect correlations in Chapter 12 to derive one of the important statisti-

cal mechanical equations, the Ornstein-Zernike equation, used in the determination of radial distribution functions in dense fluids. Obtaining the radial distribution function in a liquid or dense fluid is very much more complicated than analysis for the dilute gas. There are a number of very different methods that are used. The first is to obtain the radial distribution from laboratory scattering experiments on the fluid of interest. Typically, x-ray or neutron beams are used for the scattering, since the wavelengths of these beams are comparable to molecular dimensions. The intensity of scattered radiation by the fluid is measured as a function of the scattering angle; and by a mathematical analysis (Fourier transformation), the radial distribution is obtained as discussed in Section 11.6. This direct method does not require the assumption of pairwise additivity and is exact for atomic fluids for which there 1s only a single radial distribution function. However, it is more difficult in the study multiatom molecular fluids that are generally of interest, since In such cases there are a number atom-atom correlation functions. This 1s shown in Fig.

11.4-1

for HCl,

where

there are three different

intermolecular

correlation

func-

tions: the hydrogen-hydrogen, chlorine-chlorine, and hydrogen-chlorine correlation functions. These three correlation functions cannot be obtained from a single x-ray or neutron scattering measurement. Note that each correlation function is the result of averages over many pairs of molecules in different configurations. Also. for the case here the two hydrogen-chlorine separation distances shown in this configuration, both contribute to the single hydrogen-chlorine pair correlation function. For molecules with more atoms, the number of different correlation functions increases. A second method of determining the radial distribution function is by use of statistical mechanical theory, and there are several ways to proceed. One method has been to use the assumption of pairwise additivity of the potential and the graph theory of clusters, as was done in the development of expressions for the virial coefficients from the canonical ensemble, to develop integral equations for the radial distribution function. The graph theory development is extremely complicated, so approximations are made leading to models with names such as the Percus-Yevick, hypernetted chain,

s tion Func tion ribu Dist nal atio igur Conf d. Flui e Dens a in les ecu Mol ing |: Interact

Figure 11.4-1 Atom-atom correlation functions for the two-center HCl molecule.

and mean spherical approximations, as examples. Some of these will be discussed in the Chapter 12. Such integral equations are usually solved numerically to obtain the radial distribution function at various temperatures and densities. However, after numerical evaluation, one only has a table of numbers and not analytic expressions for the radial distribution function as a function of separation distance, temperature, and density for the model pairwise-additive potential obtained from an approximate theory, and not an analytic expression. A third method of obtaining the radial distribution function is by molecular-level computer simulation, which is discussed in some detail in Chapter 13. A brief introduction, for the sake of continuity, is given here. In this method, by computer programming, molecules are described by model potentials (usually, but not always, pairwise additive) and are considered to be in a box that exists in the memory of a computer. Many different configurations of the collection of molecules are considered, and averages over these configurations are used to obtain the radial distribution and thermodynamic properties of these model molecules. There are two general simulation methods

in use; they will be described here in the next few sentences, in the

simplest terms, and in more detail in Chapter 13. The first method involves considering many configurations of the system, essentially randomly generated, and using a Boltzmann weighted-averaging process. (The method actually used is slightly different because, for computational efficiency as explained in Chapter 13, a chain of configurations is generated with a likelihood proportional to the Boltzmann factor of the energy, and then a linear average is used to obtain the radial distribution function and the properties.) Such a procedure ts called Monte Carlo simulation, in reference to the random nature of roulette or other games of pure chance. The other general computer simulation method is molecular-dynamics. In this procedure, an initial configuration and distribution of velocities for the molecules is chosen, and then Newtonian mechanics is used to follow the evolution of the system as a function of time (as discussed in Chapter 13). Once the system has equilibrated—as evidenced by fluctuations being random rather than systematic in the various calculated properties, such as energy and pressure—average values of both thermodynamic and dynamic (that is, transport) properties and the radial distribution function can be obtained.

The efficient implementation of these simulation methods is much more intricate and sophisticated than the simple descriptions given above. If implemented properly, and if the simulations are run long enough, one obtains essentially exact property values for a fluid whose molecules interact with the model potential used. This is very valuable for testing theories and obtaining insight into the behavior of fluids. However, it must be remembered that at each state point, one obtains only numerical

11.5 Fluctuations in the Number of Particles and the Compressibility Equation

199

values of the properties, and not an analytical expression such as an equation of state or an explicit equation

However,

function.

distribution

for the radial

one

may

try

to fit the set of numerical values obtained with an equation. One of the important advantages of these simulation methods is that, by clever computational algorithms, they can be used for very complicated molecular fluids, including polymers.

“TUATIONS IN THE NUMBER OF PARTICLES THE COMPRESSIBILITY EQUATION For later reference, it is useful to have information about the fluctuations in the number of particles in a system described by the grand canonical ensemble. We can get this information as follows. The average number of particles in the system Is computed from ; =

N=)

d

Ne NHIRE

~

NE

NP(N,E,V)=

gp EU(NVVIKT

e(v, 7, 2)

kT (Ee ew) E(V,T, 1) ay

iy (e

aT) ay

vp

sl) vr

and 5

N2 =)

WE

r @NHIKT

NOE

> N?P(N,E,V) =

kT

NA

4

_

6

~

5 E;

(NV

. fkT

©, jw)

ecv,

(CRM)

~ E(V.T. a)

aye

we

(2M e(V,T, = (kT) (Sa)

,(ame(V.T, + (kT)? (Sa) dt

Ve

aya~

)\-

(11.5-2)

V.T

Now, assuming that the fluctuations in the number of particles is a result of a Gaussian distribution, to obtain the standard deviation we look at

— N2

— N-

_

ary (4

din

In

S(V. 7, &( a) ie

(11.5-3) V.f

However, we have already shown that

aa kT



Ins ins

so that

6,.2-12 (6.2-12) ,



—?3

‘—-N

=kTVv (

o-P

OM"

>)

(11.5-4) Jy y

From classical thermodynamics, we have

dG =d(uN) = udN + Ndu = VdP — SdT + ptdN

(11.5-5)

s on ti nc Fu on ti bu ri st Di l na io at ur ig nf Co d. ui Fl e ns De a in s le cu le Mo g in ct |: Intera therefore

NNdep il == VdP —— 4 SdéT, , so|SO thaat (=4 l)l ‘ =

vr = V

dl

=

\. hog p ie ( )

/

VT ~

VVl

(on

| a on u

V.T

(11.5-6)

; es ti er op pr e ag er av be to ed er id ns co are es ti er op pr In classical thermodynamics, the les tic par of er mb nu the , cs mi na dy mo er th al tic so when comparing classical and statis in les tic par of er mb nu e ag er av the be to d te re rp te in is cs mi na dy mo er th cal in classi statistical thermodynamics. Also | : = Ndu=VdP—SdT

\

an

(11.5-7)

== (5)

gr

N \ 0p J yr

vr

\aX

N

a Ni/v.t

aP

faoP\

1

oP

_!

so that

\ — =

)

Opt _—

as d ne fi de is « y it il ib ss re mp co al rm he ot is and the PS

|

(=) -— =—-| o\aP/,

(sn) — V \aP

(11.5-8)

Therefore

_

a

_

aN



— NkT px

— NkT ( = A*(s)- A(s)

I(s)

f(sje

ter

SFul |

oy

»

|

=

ae

| f(sje'=om



,

nr= |

Nfs) + £0)

| »

A=

7

Pe,

| f-(sye™

m=!

oem

NOON

A=]

Puen

(11.68)

m=]

Ax

where we have separated the n = m and the n + m terms in the sum, and used that Pml—f al

=

finn

In a liquid, the scattering sites are not fixed, but move

during

the course of a

scattering experiment (which can take minutes or hours, depending on the intensity of the incident and scattered radiation). From the definition of the radial distribution

function, and using the ergodic hypothesis that allows us to replace a time average with an average over states, we have N s-

a=]

ON S-

Ng's hn

a

(|

/

emer

yd)

=

a

/

e*“"o(r)dr



Np

|e

eirydr

m=!)

MSE

(11.6-9)

where

the brackets

() denote

an average

positions of a pair of scattering Therefore

atoms

over time, and

(and therefore

the integrals are over all

is the average over states).

I(s) = Nf7(s) + Noss) f e'**e(r)de

(11.6-10)

The factor NV, the number of scatters in the target region of the liquid that is subjected to the beam, is not known. What is done is to normalize the measured large scattering

ons cti Fun ion but tri Dis nal tio ura fig Con id. Flu se Den a in les ecu |: Interacting Mol gle sin the m fro g rin tte sca the to ion iat rad red tte sca of angle portion of the intensity . way this In e. don be can this why r late n lai exp will We . nce ere erf int atoms without s: low fol as is ) /‘(s , ion iat rad red tte sca of ity ens int a new lim /(s) = NI'(s) = N f(s)

so that

J/'(s) =

I(s)

K— OO

We then rewrite Eq.

N

= f7(s)

(11.6-11)

dr

(11.6-12)

11.6-10 as

I'(s) = f(s) + pPis) f eee)

2 ide + pfs) |e

Next we choose a coordinate system in which the vector s is along thez axis, so that s —r =srcos@, and polar-spherical coordinates are be used for integration over the vector r leading to

M(s) = f7(s) + pf7ts) / eS"

ar) — |r? sind dé dddr

+ pf*(s) / el TF 2 cin AdAdddr

= f7(s) + 2mpf?(s) / el" 84) o(r) — 1]r? sind ddr (11.6-13)

+ 2npf*(s) | el TOS 2 Sind dé dr

To proceed further, we note that by two changes of the variable of integration 7

fe

scp pme

ils

+1

con? sind d? = fe eddy =

(

| kr

—|

bal

isr

ee

—@

ist

/ exp(y) dy = ——_—_——__ = PAF

sin(sr

ae

oP

—IsF

(11.6-14) and

so we

obtain

I'(s) = frls) + 2npf%(s) f ler

— 1]

sin(sr)

5

redr + 2xpf%(s) f

sin(sr)

5 rd

(11.6-15) The interpretation of the terms on the right-hand side of this equation 1s as follows. The first term is the intensity of scattered radiation if there were no interference from adjacent atoms. The second term is the result of the scattering interference between nearby atoms. The last term 1s proportional to the Fourier transform of the Kronecker delta function, and thus only makes a contribution to the intensity of scattered radiation at ¢ = 0—that is, when 6 = 0—which occurs when the beam generator is pointing directly into the detector. Experimentally, it is not possible to measure the intensity of radiation at very small scattering angles (i.e., near s = 0), so this term is neglected.

207

ial Distribution Function of Fluids using Coherent X-ray or Neutron Diffraction

Therefore, the final diffraction equation is

f(s)

lis) =

sin(sr

(11,6-l6a)

— | ]——— r°dr

+ 2a pf? 0 fu

or, defining the total structure function H(s) i

2

aa

=

= H(s)= ane | ter) 7 qe

(11.6-16b)

2 2 dr

“S

0)

Note that in this equation the radial distribution function g(r) appears

within an

integral. However, this integral is of the Fourier type, so that the function obtained from experiment, H(s), is in fact the Fourier transform of what is referred to as the total correlation function A(r) = g(r)



1. Thus

h(r) = g(r) -1= 5

(11.6-17)

Therefore, the radial distribution function g(r)—or, equivalently, the total correlation

function /i(r)—is directly obtainable from an x-ray or neutron scattering experiment. However,

a value

to obtain

distribution

of the radial

function

at each

value

of the

interatomic separation r, information is needed on the structure function H(s) at all values of s to evaluate the integral. Finally,

note

that

the structure

at zero

A (s)

function

angle

(s =O)

cannot

be

measured due to the interference in the detector from unscattered radiation. However, it can be obtained

in another way. To see this, we note that

sin(s

|

8 6SF

soOSr

a

im eS? — tim— E _ MEE s—a0

5

BT =] — |

3!

3!

so that

oo

A(s =0)

= 2p

| Igir) — |r? dr

(11.6-18)

0

Now, we have previously shown (Eq. 1 1.5-7) that by starting with the grand canonical ensemble and using fluctuation theory, the following result was obtained: xo

kT ().

=f = xo | [g(r) — 1 ]redr

(11.5-7)

)

where k is the Boltzmann constant. Consequently, I dp His (x =0) = 0} = ;| kT (so)

— ji

i (11.6-19)

s tion Func tion ribu Dist nal atio igur Conf d. Flui e Dens a in les ecu Mol ing |: Interact

So even though we cannot measure the scattering function at zero angle (s = 0, because of the unscattered radiation), we can nonetheless obtain the value of H(s = 0) from the measured value of the isothermal compressibility. An example of the measured scattering intensity /(s) for liquid argon is shown 11.6-2,

in Fig.

and

the

function

structure

A(s)

derived

from

that data

is shown

g usin data that from ed put com g(r) tion ribu dist l radia the Then -3. 11.6 Fig. in Eq. 11.6-17 is shown in Fig. 11.6-4.° There we see an excluded region in which the center of a second molecule is not present, due to the repulsive forces, followed by a relatively strong peak in the radial distribution function near r = 4 A (0.4nm)

corresponding to the first coordination shell of an atom. Next is a region of lowerthan-average

of centers

density

(g) >) xyx;u(r, x) is an effective mixture potential. Obtain an expression for this effective mixture potential in terms of u;;(r). uj, Ur) and w;;(r). Further, if each of these intermolecular potentials are of the Lennard-

Jones 12-6 form, how are the parameters of u(r) related to those in w;;(r), uj; (r) and wjj(r)?

(Note that this model is also referred to as a one-fluid model.) 11.4 Draw the three low-density radial distribution functions for a binary mixture of Lennard-Jones 12-6 fluids for which 11.5

€), = 2622 and oj; = o3, assuming the Lorentz-Berthelot combining rules of Eqs. 8.2-4 and 8.2-5 apply with Aya = 0. The average of any property 6(N,V,7T.r).ro,.....7,) In the canonical ensemble, represented by O(N. V,T). is computed from

a

m

|

fa.

Vo Ti 6). fo. een Ey) OO

B(N,V.T)=

|.

fou

wanes F

wiley Le

gq”

) y F . V.T. : Fy), Fo

i

drydry...dry

HIE Eden AD

drjdry...dry

| y E e e P y E r d Y . E . W y r d y r d tp @ =

| . fe

aa

(O(N,

Mo

Fas

ioe

eee Ea PM

drjdr5...dry

where the notation (} has been used to indicate that the property in the system of interaction molecules is to be averaged using the configuration integral.

a. Explain why the chemical potential of a species can be computed using HIN, b.

Show

+ 1,V¥.7T)—A(NLV.T)

V, 7) = A(N

that

W(N,

Vf)

Q(N+1,V,T) = ACN +1, V, 7) — ACN, V, 7) = —AT In Q(N.V.T) /

cf

Spacey

MIE

_

/

open

yt

a

©

T EDA F —_ ppt, | E

drjidry....dryiy

Mey 3 nt)

drjdry....dry



[fe c. Finally, show that

a w(N,V,T) = —kT In la (N+ 1)AA g

|(

im

| Eee,

LAS

p}

ate

hoe

£n2|

5

fit,

°°

Chapter 1 2

Integral Equation Theories for the Radial Distribution Function In this chapter we consider the method of determining radial distribution functions based on the use of integral equations. There are two very different starting points for deriving integral equations for the determination of the radial distribution function. We first consider one method (Section 12.1) and then a completely different, complementary method (Section 12.3).

INSTRUCTIONAL

OBJECTIVES

FOR

CHAPTER

12

The goals for this chapter are for the student to:

* Understand the basis for integral equation methods distribution function

used to obtain the radial

¢ Understand the basis for the Ornstein-Zernike integral equation * Understand the origin of the Percus-Yevick equation for hard-spheres

12.1

THE

YVON-BORN-GREEN The

method

function, Eq.

we

(YBG)

consider

first

EQUATION starts

from

the

definition

y2 |.

fe



U(r),.-- Ly) =

-+ +4) ri u( + 3) r) u( + 2) ry u(r) = u(

i

+ u(rw-i.y)

J lar

d e y c e MAT dr ...ddery

oe HUE ay

us

However, from Eq. 11.1-9a vs

f..

Je

—ufkt

ry...dry

Toft

te

Ss

gO (ry, ry. hy;

(11.1-9a)

50 that

dg (r).r9: T, p) drs

_

=

me

4

)

5 g(r.

rs

rs: T. p) — ~

"

q

_

=

SO

Era

1, phar,

r5

| du(r}2) x

p [=

.

(3)

pd T. pdrs rar T. Clete SB 8 | Gp Te P)— se] Te— 123T. ar, 8 Cut

EF

(12.1-5)

where we have neglected the difference between N and N-2. This last equation can be rewritten as

[2

) 2 i r c u d _ _ ) e e P 3 Fa poling),

Ory

ars

ar,

g(r), 0.03: T, p) g(r),F 93 T, p)

dr.

(12.1-6) This is known as the equation of Yvon-Born-Green, derived this integro-differential equation.

each of whom

independently

The Yvon-Born-Green (YBG) equation relates the unknown two-body correlation function to the even less known three-body correlation function. So to proceed, one needs some information about the three-body correlation function. However, before we consider this, notice that in the limit of zero density (eo — Q), this equation reduces

to

AT

ding’ (rf;

T,p)

ars

=



du(ri2) , Ors

as p > 0

(12.1-7)

as p —> O

(12.1-8)

which on integration gives us

gr

ryt T, p) = Cem iikT

= eM

VAT

where the constant of integration C has been set to unity by invoking the condition that g(r. , ro

I, p) = 1, as the molecular separation distance becomes

very large

so that w(rj2) = 0. Note that previously we had obtained Eq. 12.1-8 using the Mayer cluster expansion

(see Section

11.3 and Eg.

11.3-5).

The Kirkwood Superposition Approximation

12.2

APPROXIMATION

SUPERPOSITION

KIRKWOOD

219

12.1-6 for the two-particle radial distribu-

At higher densities we cannot solve Eq.

tion function gr) .F95 T. ep), because this equation also contains the three-particle distribution function a™ (ri tas r4; 7, o). The earliest assumption or closure made

was the superposition approximation of Kirkwood that

BO (ry Foy 033 Typ) = 8 (ry. Pos T, OR (Ey. £35 Ts PIB” (lo. Fai T, pe) (122-1) 3

2

.

a

2

or that the three-particle distribution is the product of all two-particle distribution functions. Note that in the limit of zero density, this assumption is consistent with the assumption of pairwise additivity of the potential. This is seen as follows:

IngO(r).ro.r3i T. p =0) = Ing (r,, 69; T. p = 0) + Ing (r,,643 T, p = 0)

+ Ing (ry, 13: T, p = 0) — a u(r).

_

AT

kT

kT

63)

u(fs,

_ u(r), 03)

F>)



or (12.2-2)

U(r). Fo, 3) = Wr), £2) Fur), 3) + Uh, £3) which is the pairwise additivity assumption. Now, using the Kirkwood superposition

approximation

in the

YBG

equation.

we gel

kT

din 2 (r,,

roi 7, p)

du(iry>)

a

eee ee

ary

ar,

ars

du(r)2)

[—~

ar

pag a

{=

ary

du(rys)

du (rsa)

p {|

drs

ars

g(r

ro. rai T, p)

—,

oa

g(r, 753 T, p)

gO (rirot T, pg’ (ry .r3: T. p)g

|

r, pg

(ry,

a

=

(ro, 03: T. p) '

g(r. r9: Tsp)

erry

;

_

rai TT, p)dr,

—_

-

or

pet in gh

23 T, p) Ors



du(r}>)

=

ars



o|

du (ro3) gp

ary

ra; T,

pig

(rs,

a

ra:

T,

pdr,

(12.2-3)

|

This is an integro-differential equation for the two-particle distribution function gl(r),r5;T, p) that can, in principle, be solved (but only with considerable difficulty). In fact, the results obtained from solving this equation for any potential function are not in good agreement with computer simulation results for the same potential.

|2: Integral Equation Theories for the Radial Distribution Function

the with ted star We -3. 12.2 Eq. at ve arri to used e edur proc the ider Cons r cula mole the to ect resp with ve vati deri its took and e) T, 5; ,,1r g'(r definition of separation distance, which resulted in an equation (Eq. 12.1-6) that contained

g(r), 65, F3: T, e). To be able to solve this equation, we then made an assumption

about g@!(r,, r5, 3: T, @)—that is, we made an assumption (Eq. 12.2-1) about the three-particle distribution function in order to be able to solve for the two-particle distribution function, We can then imagine that a way to improve upon this procedure is to develop a hierarchy of such equations. The next level would be to

start with the definition of g'(r,,r5,r3; T, p) and take a derivative with respect to a position vector, which, following the procedure that led to Eq. 12.1-6, would now give an integro-differential equation in containing g“)(r,.r5.r3.ry; 7. p). We could now make an assumption about this quantity (for example, like the Kirkwood superposition approximation, that it was a product of all three-particle distribution functions) and solve the resulting equation for g')(r).r5,1r3: T. p).

which

would

be an improvement

over Eq.

12.2-1.

The

resulting expression

for

g(r), ro, 3: T, e)would then be used in Eq. 12.1-6 to obtain an improved estimate for g(r), ro: T, e). If this was found to be unsatisfactory, one could then go to the next level of deriving an integro-differential equation for g'?)(r,,r, 173,174: T. p) in terms of g(r).f5.%3.f4.ts5:7.), solving the resulting equation for 2 (ry. ro. r3.r4: Tp), using the result in the equation for g(r), 75,14: T. p), solving it, and then using that result in Eq. 12.1-6 to solve for g(r. rs: I, p). However, the method outlined here 1s so tedious that it is rarely used. The key idea of this hierarchy of equations is that if one wants to obtain a good

lower-order (i.e., two-particle) distribution, one should not make the superposition approximation about that distribution function, but rather about some higher-order particle distribution such as the three-particle, four-particle, or higher-order distribution functions. The expectation, then, is that the higher the order of the distribution

function about which the superposition approximation is made, the more accurate the resulting two-particle distribution function will be.

JIRNSTEIN-ZERNIKE

EQUATION

The second integral equation method we consider for determining the radial distribution function has a very different basis, and makes use of a concept similar to that discussed previously in trying to understand the behavior of the radial distribution function. In particular, the idea is that there is a direct correlation between molecules | and 2 (though it is not simply the low-density result of a Boltzmann factor in the interaction energy), and then an indirect effect as a result of the correlations between intervening single molecules (1-3-2) and chains of molecules (1-3-4-2, ]-3----.-2, etc.) between molecules | and 2. We describe this by introducing the following two functions: the total correlation function A(r),r5) = g(r,

and the direct correlation function c(r), r5).

r5)

— ]

(12.3-1)

The expectation is that the range of the direct correlation function c(r,,r5) will be approximately that of the intermolecular potential. The simplest (and least accurate) approximation is that the total correlation and the direct correlation are equal—that is, A(r,,r5) = e(r,,r>). The next level of approximation is that the total correlation is equal to the sum of the direct correlation between

12.3 The Ornstein-Zernike Equation molecules

221

| and 2 and the effect on the total correlation function due to the indirect

correlations from all third molecules, wherever they are located (consequently, one will have to integrate over their position), that is

N A(r).f9) = clr.)

+ Vv / c(r), r3)e(r4. rs) drs

=c(f).f) +p | e(r), Fae(rs, ro) dry where the factor of NV arises from the fact that we have

N (actually

(12,3-2) N-2) choices for

the third molecule, and V is the normalization factor, since the integration is carried

out over the total volume. Continuing to higher levels of accuracy, we can consider additional indirect correlations from increasingly greater numbers of intermediate molecules (1-3-4-2, 1-3-4-5-2, etc.) and obtain

A(r). £5) = c(t). 6) + p fetes r3)c(r3, ho) dr; +e ff e(r), F3)c(r3, F4)e(4, Fo) drydr,

+p” /I/ CUP), F3)C(3, Fye(hy, Fs) Po)e dr3 ( dry r drs s +++ , +>: (12.3-3) It is easy to show that this last equation is equivalent to hir, ° F>)

=

cr,

: rs)

+

Pp /

e(r,

‘ ra)h

(rs,

(12.3-4)

ry) dr,

which is the Ornstein-Zernike equation.' The way that one proves the equivalence between Eqs. 12.3-3 and 12.3-4 is by repeatedly substituting Eq. 12.3-3 into itself. For example,

after one substitution we have

h(r) Py) = clr. Py) + p few, P3)eUr3, Fy) dry

+ p° // c(r),r3)e(r3, rary,

ro)drsdr,

(12.3-5)

Repeating this process leads to Eq. 12.3-4. Much like the discussion of graphs with connecting points and vertices used in deriving the virial equation of state from the canonical ensemble, we can use graphs to represent the various For example,

integrals comprising

the Ornstein-Zernike equation.

the first term c(r,, 175) will be represented by

C+)

the next term f{ c(r,,r3)c(r4, r5)dry is

(}—_@—_)

and ff c(r).rae(rs, tye(ty, roddradry is

O—o—_0—©

—_—



'_. S. Ornstein

———

and F, Zernike,

Proc. Sect. Sci. K. ned. Akad.

Wet.

17, 793 (19144.

Chapter 12: Integral Equation Theories for the Radial Distribution Function

Higher order terms will result in graphs such as

and so on. In each of these cases, an unfilled circle represents the position vector r, Or ry that is not integrated over, and a filled circle represents a position vector in an integration.

CLOSURES The

FOR

THE

ORNSTEIN-ZERNIKE

Ornstein-Zernike

equation

EQUATION

is a single equation in two unknowns, /i(r,, Ff) and

c(r;,f5). Therefore, in order to use this equation to obtain a solution for the radial distribution function, g(r,, r5)—or, equivalently, A(x, , r,)—we need to either specify

the function e(r,, 75) or provide a relationship between f(r), r5) and c(r,.r>). There are Various approximations or closures that have been used. The simplest is the socalled mean spherical approximation: A(r).f2)2

=-—l

C(ry.f5)

or .

e(r).ro) =0

= —u(ry. 5)/kT

In this equation, d is a characteristic

for ry

d diameter—for

example,

the diameter of the

hard-sphere molecule or some other distance measure for soft interaction potentials. Another

commonly

used

closure

is the

Percus-Yevick

(or

PY)

approximation,

obtained as follows. Defining a new function, y(r) = g(rje"/"" results in c(r) = g(r) — yr) = YT ry — yr) = f(r) y(r)

(12.4-2)

and when used in the Ornstein-Zernike equation gives

yiri2) = 1+ o|

F(riady(riadA(roa) dro

or the following equation containing only g(r)

g(rigetOPVET = 1 + p [cent — Dg(risveP!" (g(ro3) — I dry (12.4-3) This

is an

integral

equation

for g(r),

known

as the

Percus-Yevick

equation

can

be solved (with some difficulty) for the radial distribution function for a specified interaction potential. Table 12.4-4 lists the results for the radial distribution function of the hard-sphere fluid obtained in the Percus-Yevick approximation as a function of r*= r/o using th MATLAB) program PYHS available on the website www.wiley.com/college/sandler. A different approximation is the hypernetted chain (HNC) closure e(r) = flr)y(r) + yr)

— 1 — In y(r)

“J. L. Lebowitz and J. K. Percus, Phys. Rev. 144, 251 (1966).

4J. M. J. van Leeuwen. J. Groeneveld. and J. de Boer, Physica 25, 792 (1959).

(12.4-4)

223

12.4 Closures for the Ornstein-Zernike Equation

leading to the HNC equation

u(r3) ) 3 2 r ) ( 3 ) ) g — lJ dry i 2 [ r | r i ( — r n i g I ( — f f = o Iny ry3) u(T

A(rj3) —Ing(ri3z) — T

=p

_

(12.4-5)

h(ro3) dr,

that can also be solved numerically (again, with difficulty) for the radial distribution function for a specified interaction potential. Though the closures of the Ornstein-Zernike equation have been introduced in an apparently arbitrary manner here, they can in fact be derived by use of complicated graph theoric methods* by summing the contributions of graphs such as those shown in Section 12.3, in which one approximates the sum of this infinite collection of graphs by including some of the terms and neglecting others. Such a discussion

is

well beyond the scope of this book. Each of the equations in this section is most commonly solved numerically rather than analytically. It is useful to note that in terms of the notation

compressibility equation of Eq.

in this section, we can rewrite the

11.5-17

a fo ur (*° in fact, the two equations given below:

(=)

_ Lee

pkT } p

2

(1 — 7)?

P

ond

(=) PkT

Je

= EET

-

(1 — ny

iti

where 9» = 2No*/6V =2po°/6. The first equation above with the subscript P arises from using the radial distribution function obtained in the pressure equation 11.2-11),

(Eq.

and

the

equation

second

of using

is a result

the

same

distribution

function in the compressibility equation (Eq. 11.5-17). Note that if the PY approximation were exact, one should obtain the same equation of state regardless of whether the pressure or compressibility equation was used. Therefore, comparison of the results obtained from both equations can be used as a consistency test of the solution. It is interesting to note that the best description of the hard-sphere fluid is obtained from a weighted average of the two results above:

P

1 / _ |

okT

P

3\pkT]

2( P 42 ( p

)

3 \ pkT

Je

| “=< i = _— eae Ee (I — ny

(12.5-2)

This is the Carnahan-Starling equation.’ It is one of the most used results arising from statisical mechanics. One application is that it has been used as the basis for developing equations of state. For example, if a molecule is considered to consist of a central hard core plus other interactions (attractive dispersion forces, electrostatic forces, etc.), the Carnahan-Starling term can be used to represent the hard-core part, and the effects of other terms added as a Taylor series expansion. This is referred to as perturbation theory and is discussed in Chapter 14. Such an analysis has been widely used for systems of scientific and engineering interest, such as large globular molecules, proteins, and colloids.

For later reference, there are other results that follow from the Carnahan-Starling equation.

The

first is the radial distribution for hard-spheres

this, we start from

Eq.

at contact. To derive

11.2-11 oo

In , f dur P = pkT — =p? | 7 3 dr

a(r)r3dr

(11.2-11)

0)

and note that the derivative du(r)/dr is not well behaved at r =o since u(r) goes from infinity at r incrementally less than o, and to O for r incrementally greater than

a. The Boltzmann factor in the intermolecular potential, e~“"/*", is better behaved in that its value goes to 0 at r incrementally

less than a, to | for r incrementally

greater than o, and tts derivative is related to the delta function. Therefore, to evaluate —_



=



°E. Thiele, J. Chem. Phys. 39, 474 (1963), °M. 5S. Wertheim, J. Math. Phys. 5, 643 (1964), 'N. F. Carnahan and K. E. Starling. J. Chen, Phys. 51. 635 (1969).

Chapter 12: Integral Equation Theories for the Radial Distribution Function the derivative of the hard-sphere potential we do the following: =

-wryAT

4

dr

du(r)

or

_

_ |

kT

ET ett Vat

6(x)

ay

A

oer yfkT

_

kT et OUKT 85

~o)

dr

dr where

du(r)

wnat

is the delta function

whose

value

is | at x = 0, and

is 0 elsewhere.

Therefore

P = pkT+

no

2akT ~

()

2UkT

4

= pk +T —— p g(r =at)o? = pkT +

2ekT

_

p-g™(a*)o* 2

hs

=

where o~ indicates a value of cincrementally greater than the hard-sphere diameter (and a does not appear since the Boltzmann factor is 0 there). Then using

Qn pkl

_ltnt+n-9

3”

(1 —n)

it is simple algebra to show that hs, re

21 = i e e e 1-3

+

g(a")

(12.5-3

(Note that e™(a~) = 0.) Other interesting results from the Carnahan-Starling equation of state are that the Helmholtz energy and pressure above that of an ideal gas at the same density are

A(N,V,T)—A'(N,V.3T20 ) NkT ~ (=n? an

4

(P(N, V,T)—

PIS(N,V,T))V

4—n2n? (l—7)3

——— Tr T ————— —'

NkKT

(12.5-4)

THE RADIAL DISTRIBUTION FUNCTIONS AND THERMODYNAMIC PROPERTIES OF MIXTURES While it is difficult to obtain the radial distribution for a pure fluid, the problem is even more difficult for a mixture; the best way to get such information is from computer simulation, as discussed in the next chapter. Before simulation, a collection of approximations were made in integral equation theory. The starting point is that by using the pairwise additivity assumption, the internal energy of a mixture of monatomic molecules can be shown to be (Problem 11.2) U(N,

VT)

=

3 Tene

| +2nNp

Yoyo

_ xix,

/ Wi (rip eip

ris: p, T)r;, ar

(12.6-1)

)

es tur Mix of s tie per Pro c mi na dy mo er Th and ons cti Fun ion but tri Dis | The Radial

229

e sam the e hav ons cti fun ion but tri dis al radi the all that is n tio ump ass ple A very sim dependence on intermolecular separation, that is (12.6-2)

r) g( = -: =) r ( g = r) a( go gu(r) = Furthermore, by writing

12.6-] becomes

Equation

U(N,V,f)

=

35

,

as

to

referred

is

the

(12.6-4)

+ 2a Np f wiryeiryr? dr

NRE

0

This

(12.6-3)

u(r) = Y> > o xjxjuii(r) ij

mixture

random

one-fluid

or

model

and

be

can

used

with any intermolecular potential. Note that if the Lennard-Jones potential 1s used. we

have

{O"-O']a ["EdCY]

ares

i

which

has the solution

6 AXjX j fj Fj;

J

I é=

)

} : i

}

12 AUX jf E77 Fj;

, and ot

| 2

i

=

12 XX jE; Fj;

jf

)

i

|

} . '

}

(12.6-6)

6

ApH FE j (8;

J

In the case of the one-fluid model, the mixture is treated as a pure component with an effective potential (and potential parameters) that changes with changing composition. A somewhat better approximation is to assume that the like molecule and unlike molecule potentials have the same form (for example, the Lennard-Jones 12-6 potential, although not restricted here to that form): u(r)

= EF

(—)

(12.6-7)

OF;

and further assuming a universal form for the radial distribution function in reduced variables r/o

r

r

Fr

O1]

O22

O12

gu | — } = 822 | — ) = 812

=

r

(—)

(12.6-8)

oa

This leads to

3

U(N,V,T) = SNET +2Np YY aixjeyo) | Feet /

i

3 — atke

o. Tyra

cl

+ 22 Npeo* / Firje(rs p, Tr? dr

0

1

(12.6-9)

- |2: Integral Equation Theories for the Radial Distribution Function with

ea

= »

iy

f

(12.6-10)

i

Equations |2.6-7 to 12.6-10 are referred to as the van der Waals I theory.

POTENTIAL

MEAN

OF

FORCE

We have shown that at low density, the radial distribution function is

(12.7-1)

g(ri2, T,p > 0) =e MeVAT

A new function, the potential of mean force, w(rj2, 7, o). is defined by its relationship

to the radial distribution function at all densities, temperatures, and intermolecular separation distances by _

wirya.

g(ri2.7T, p)=e That is, when

bution can be cles in a fluid, and

(12.7-2)

12.7-2, it reproduces the radial distri-

7,e) is substituted in Eq.

w(rj2,

oye kT

function between two particles at the temperature and density of the system. It interpreted as follows. If w(rj2) is the interaction potential between two partia vacuum, w(rj2, T,pe) 1s the effective potential between these two particles in where their interaction is affected by the presence of intermediate molecules indirect

by all

interactions.

That

1s, in addition

to the interaction

between

atom

| and atom 2, atom | Interacts with atom 3 that also interacts with atom 2, etc. As a result of these indirect interactions, the range of the potential of mean force w(ry3, 7,e) is considerably longer than the (direct) interaction potential w(7r)2). Also, unlike

the true intermolecular potential, the potential of mean force depends on temperature and density (and also the concentration of other species if the atoms of interest are In a mixture), To obtain a more formal expression for the potential of mean force, we note that the force between two atoms in a vacuum as they are moved further apart (or closer together)

a,

is

du(ry>)

Fu=

In a fluid, the force between atoms

(12.7-3)

dr >

| and 2 ts affected by the presence of all other

atoms. What we want to compute is the force between atom | at r, and atom 2 at r, In the fluid obtained by averaging over the locations of all the other atoms. That is

iF) = dwr),Ps) {ees Ea —(F,,) = -——t' = -( eee dP 9 —f,..f

dr jy en

fete).

euro

fs, Pe

ry. ow wal

el

ae dr,dry...dry dry ———FSEOoaeoaoNuouToueeeEee—E—————

ES

| a / eT kT —

d

...fe

—ufkT

dr.dry... dry oye

dridr,...dr jy

—l2

fo

fewar,

d = kT

| Inf...

dry,

dr4...dty

f

"dey

dry.

dey

-

(12.7-4)

12.7

Potential

The

2

=

waren)

a

— I)Z(N,V.T)

N(N

drjy\

\V-

dr 44

nor N

Z(N,V,7)

integral

Separately we note that since neither the configuration and V are a function of rj, we have

=(0

ip Gas)

N N.V,T) — 1)Z( N(

231

Force

of Mean

(12.7-5)

Using this relation in Eq. 11.10-4 we

dwityts)

-

1)

pd

7

dryoté' 9

Fit =

d

= kT

—ufkT dr,dr4...dry

2 V [fe

N(N — 1)Z2(N,V,T) (12.7-6)

In g(r). 6s)

£12

On integration this gives us

—wry.fs) =kT Ing(ry.ty) +e where nite

c

is

the

constant

separation—that

of

1s,

or

glry ry) Se MET Ee IET

integration.

when

When

|r, —r5|—

atoms

co—we

|

and

know

2 that

(12.7.7)

are

at

infi-

g(r).r5)=

| so that w(r,,r.) = 0. Consequently, the integration constant ¢ must be zero, and

therefore g(r).r5) =e Wf MAT or g(r) = e WO VET | Which is Eq. 12.7-2. Alternatively, it is easily shown (Problem 12.9) that _

[fe

dwir;p,T)

_

H(Fy fae

a

+ OU), Po..ee F a r ) —N —|*—2 RT

dr,dr,...dry

or

ect

fi

ar,

[.

v

Lf

ere

caw

dy,

dr,

.

.dr

x

Vv

(12.7-8) where —du(r,.6o,...f,)/dr, the force on atom | at vectorial position r, averaged over the locations of all other atoms. An

interesting

interpretation

for the potential

of mean

force

w(rj2,

7T,p)

1s that

since it 1s the integral of force over distance, it is also the work done to bring a pair

of particles in a dense fluid from infinite separation to a separation distance of r. Since this work is done at fixed N,V, and 7, we then have w(r) is the Helmholtz energy change for this process. As already mentioned, because of the indirect interactions (that is, the interactions between atoms | and 2 through all possible intermediate atoms), the potential of mean force, or PMF as tt is frequently referred to, 1s of much longer range than the two-body potential between two isolated atoms, and in fact does not resemble it. This is evident in Fig. 12.7-1 that is the potential of mean force (divided by AT) computed from the Percus-Yevick solution for the hard-sphere potential. Note that while for hard-spheres

the two-body potential is infinite for r/o < 1, and zero for r/o > 1, the potential of mean force for this interaction is also infinite for r/o < 1, but then is nonzero for r/o = 1, and the range over which it is nonzero increases with density. Also, while the hard-sphere potential is either zero or infinity and independent of temperature, the potential of mean force depends on temperature (which is why w(r)/kT is plotted in the figures). Both

the extended range of the potential of mean

force (beyond

that

on cti Fun ion but tri Dis ial Rad the for es ori The on ati Equ 2: Integral the are e enc end dep re atu per tem its and ) ial ent pot dy -bo two true g yin of the underl result of solvation forces—that is, the effect of the two particles of interest being in a fluid rather than isolated in a vacuum.

It is especially interesting to note that in Fig. 12.7-1, there are regions in which the potential of mean force between the atoms is attractive (negative values of the potential of mean force), even though the hard-sphere potential has no attractive region. The simplest way to understand how this occurs is from a kinetic argument. Consider two atoms sufficiently close to each other than no other atoms can get between them, as shown in Fig. 12.7-2. On each collision of the atoms of interest with surrounding atoms in the fluid, there is a force as indicated by the arrows. However, when two atoms are sufficiently close together, there is a region between them shielded from collisions with other atoms (indicated by a lack of arrows directed toward the center of each atom). Consequently, because of this imbalance of possible collisions, there is a net force of each atom in the pair of interest toward the other as a result of the force imbalance—that is, there 1s an apparent attraction. This 1s what is Shown in the PMF in Fig. 12.7-1. At high densities, a similar argument can be made to explain the weaker regions of attraction at large atomic separations resulting from the higher coordination shells. What should be clear from this discussion is that it is not simple to develop an accurate model for the potential of mean force, so that developing the PMF ts not a “cheap” way of obtaining the radial distribution function. Indeed, in the discussion above, the potential of mean force was obtained from known values of the radial distribution and not the reverse. A

problem

of some

interest

is to try

to understand

the

behavior

of polymers,

colloids or proteins in solution. Predicting the radial distribution function of these macromolecules (with so many atoms) from rigorous theory ts not possible, though some progress can be made using computer simulation that is discussed in Chapter 13. Instead, a common procedure for obtaining an approximate radial distribution function for such molecules is to use physical insight to make a model for the potential of mean force with adjustable parameters and then fit the parameters to some available experimental data, for example, osmotic pressure data (as discussed in the following section) or precipitation data. The underlying idea is that since the proteins or colloids are so large compared to the small solvent molecules (and the ions from salt that are generally in such solutions), the solvent can be considered to be a continuum (rather than a collection of individual atoms or molecules), so that the interactions between the proteins or colloids can be described by a potential of mean force in the solution. For example, to model the precipitation of globular proteins in aqueous solution using a polymer to induce precipitation, the following potential of mean force model has been used:° wr, T,

where

Y=

w.(r') is the hard-sphere

Whs(r)

+

Wanlh)

+

Welect (', T,

Pp)

interaction (resulting in an excluded

(12.7-9)

volume between

the molecules). This effective hard-sphere diameter can be obtained from information on the size of protein or colloid. The next term is the weak van der Waals interactions between the molecules in the presence of the solvent, which is usually attractive. This

term could be obtained by summing all the simultaneous atom-atom interactions in

SPW.

Tavares and S. 1. Sandler, AICHE Journal 43. 218

(1997),

The

12.7

Potential of Mean

Force

233

Hard-sphere potential of mean force

PY hard-sphere radial distribution function

0.5

0

bh

=

_

=


(=

du(rjj)

Interactions N

=——-s) Vv aV 4

|



ar;-

.

(13.2-9) .

Now,

using

above

can

that

ri = Via

be written

— Py)

+ iy

+ (fi. — Ptah:

the

x

equation

as

_

NkT

ty

y(

——

du(rij) rij

‘=|

It is Eq.

— Fish

Se

(

(13.2-10)

>?

13.2-10 that is used to calculate the pressure in any configuration of the

molecules in the simulation, and is then averaged over many configurations to obtain the pressure. In this way the pressure can be obtained during the simulation. Indeed, during the course of a simulation, it is common to monitor the interaction energy using Eq. 13.2-1] and the pressure using Eq. 13.2-10 to determine whether equilibrium has been obtained in the simulation. Also, it is easy to show (see Problem

13.4) that

Eq. 11.2-1] and Eq. 13.2-10 are equivalent. As will be discussed below, there is an interesting type of conceptual symmetry that occurs. In Monte Carlo simulation, only the energies need to be computed when considering transitions between configurations, so there is an additional small computational penalty to also compute the forces on the molecules to calculate the pressure. In molecular-dynamics simulations, it is only the forces that need to be computed when considering transitions between configurations, so there is an additional small computational penalty to also compute the configurational energy. In order to compute the entropy from simulation, we would have to use

S(N, V, E) = kInQ(N, V, E) if the

microcanonical

configurations

ensemble

were

available to the system

used,

where

(13.2-11)

(2(N,V.,E)

at fixed volume,

number

is the

number

of molecules,

of

and

13.3

total energy. Alternatively, using the canonical from

SN,

where

¥, Fp) = ke

OW.

Q(N,V.T)=

Monte

ensemble,

¥, ft ar |

dinQ(n,

Carlo Simulation

the entropy

is computed

VT)

aT

NV

(13.2-12)

Soe PitNEAT Slates

249

|

Even for thousands of molecules, the number of different configurations is so large (only one molecule moved a very small distance is a new configuration) that netther Q(N, V, E) nor O(N, V, T) can be computed. Consequently, the entropy (and therefore also the Gibbs and Helmholtz energies) cannot be computed directly in the molecular simulations that are discussed in this chapter. The properties that can be computed are the configurational energy U’, the pressure, and the radial distribution function. All of these can be obtained from averages over a large number of configurations, but do not require the impossible task of considering all possible configurations in order to evaluate the partition functions &2(N,V,&)

‘E CARLO

and

O(N,

V,T).

SIMULATION

In its very simplest form, a Monte Carlo simulation could be done by starting with an empty

(virtual) box, and then using a random

number

generator (or, more precisely,

a pseudo-random number generator, since computers are deterministic not random) to generate a position for each molecule in the box. This would be repeated for each molecule until the required density is obtained. The simulated pressure, con-

figurational energy, radial distribution function, and so forth would then obtained by averaging the results from many normalized Boltzmann factor

individual box fillings, each weighted with the e-U ChRT )

|

_ pre yer e u

(13.3-1)

all conhguralions

The problem with this very simple Monte Carlo approach is that by randomly insert-

ing molecules into a box, it is likely that one or more pairs of molecules will overlap, and the probability of this happening increases rapidly with density. Since even a single pair of overlapping molecules has infinite energy, the Boltzmann factor for such a configuration is zero. Consequently, at moderate and high density, few (if any) randomly generated configurations will contribute to the average. There is also the conceptual problem that to evaluate the denominator in the above equation, one has to sum over all possible states of the system in order to normalize the probabilities—an impossible task.

Therefore, more efficient Monte Carlo simulation methods have been developed. Each of these methods is based on some form of importance sampling. The basic idea is to start with an acceptable configuration (i.c., one that does not have infinite energy, usually obtained by placing molecules on a regular lattice) and, from configuration, develop other configurations in a way that is biased to result in figurations of lower energy. The bias in choosing successive configurations is accounted for in the averaging process. The most common procedure, and the one we will discuss here, is due to Metropolis et al.

this conthen only

3:

Determination

of the Radial

Distribution

Function

The Metropolis* algorithm is based on generating a Markov chain of states. The two characteristics

of Markov

chains are that there are a finite (or countable) set of

states of the system, and that the probability of transition from one state to another depends only on the properties of each state and not on other states—in particular,

not on states that the system

may

previously

have occupied. xn»

is characterized by a set of states (1,2,3,..., n) and an probabilities between each of the states.

The matnx

Markov

chain

of transition

To illustrate the properties of a Markov chain, we consider the following simple example. Suppose there is a system that consists of three states and the probability that the system is in each of these three states is p;, p2, and p3, respectively, which we denote by the vector P = (p), p2, ps3). Clearly p; + p2 + ps = |. Next, a set of probabilities are formulated for the transition from any state to any other state; for example, 7)_.2 1s the transition probability from state | to state 2, 7;_.3 from state | to state 3, etc. The list of all transition probabilities 1s usually presented 1n matrix form. As an example here, we will use the following transition probability matrix. Ti+, Ty., T3..j

Tho Thar Taus

T\.3 0.5 Thiz]}/=/04 0.4 Tawa

0.2 04 0.3

O03 0.2 O03

Notice that the transition matrix is not necessarily symmetric—that is, the probability of going from state | to state 2 does not have to be the same as going from state 2 to state 1. Also note that the sum of the transition probabilities along any row 1s unity, and that each element along the diagonal of the matrix is the probability that

the system remains in its current state. Next, we consider how the probability distribution among

the three states of the

system (P|. P2, ps) changes whenever a transition occurs. The second column

in the

Table 13.3-1 shows how the probability distribution among the three possible states changes on 10 successive transitions if we start from the system in state 1—that is, the initial probability distribution among the states ts (1 00). In column 3 of the table,

the calculation is repeated starting with the different initial probability distribution of (0 1 QO). The last column shows the results of repeating the calculation starting with an initial probability distribution of (0 0 1). The somewhat surprising observation from this table is that the after a sufficient

number of transitions, the probability distribution among the possible states of the system does not depend on the initial state; therefore, it can only depend on the transition matrix between the states. This is an important observation for Monte Carlo simulation, since it indicates that after many transitions, when the probability Table 13.3-1 Change in State Probabilities as a Function of Initial State and Number of Transitions (0.000 1.000 0.000) (0.400 0.400 0.200) (0.440 0.300 0.260)

Transition 4

(1.000 0.000 0,000) (0.500 0.200 0.300) (0.450 0.270 0.280) (O.445 0,282 0.273) (0.445 0.284 0.272)

Transition

(0.444 0.284 0.272)

(0.444 0.284 0.272)

Initial state

Transition

|

Transition

2

Transition 3

10

(0.444 0.286 0.270) (0.444 0.284 0.271)

(0.000 0.000

1.000)

(0.400 0.300 0.300) (0.440 0.290 0.271) (0.444 0.285 0.272) (0.444 0.284 0.272)

(0.444 0.284 0.272)

13.3

Carlo Simulation

Monte

251

distribution between states is no longer changing (which is taken to be the equilibthe of state initial the on depend not does bution distri ility probab this state), rium system, but only on the transition matrix. Also, this result means that in order to evaluate the probability distribution of the states of the system, we do not have to evaluate the summation in the denominator of Eg. 13.3-1; we only need information about the transition matrix. With this background, the Metropolis algorithm of Monte Carlo molecular simulation in the canonical ensemble (fixed N,V, and 7) can now be presented. In the canonical ensemble, the likelihood of occurrence of any state 1s proportional to the Boltzmann factor in its energy. Therefore, the transition probability for a change

from state m to state n is proportional to e~!!"~!")/*"_ The procedure then is to start with the collection of atoms in some arbitrarily chosen state without overlap.

A change in the state is made, usually by moving a randomly chosen atom. This is accomplished by generating a random integer number in the range from | to N, where N is the number of atoms in the system. The atom chosen in this way at location (%», Yo. Zo) is Moved in the x, y, and z directions. The extent of the movement is determined by generating three additional random numbers between —1 and |, which we designate as a@,, ay, and a,. The new possible location of the atom is (Xo + Gy X bm. Vo + Gy X Om. Zo + Oy X bm), Where dm 1s the maximum allowed distance of a proposed move in each coordinate direction. The value of 6,, is usually adjusted during the early stages of a simulation so that approximately half of the proposed moves are accepted according to the acceptance criterion discussed below.

This acceptance ratio that are so small that large that few moves To properly sample

has been found to be a suitable compromise between moves the simulation is very slow to converge and moves that are so are accepted. the system, microscopic reversibility must be satisfied; that

is, the transition probability of generating a state m from state n must be equal to the transition probability of generating state m from state m. This is ensured by the completely random method of generating possible moves discussed above. However, after a possible move is generated, importance sampling is used to decide whether the

move is accepted; if not, the initial state is retained and counted again in the averaging. This is done with importance sampling as follows. Using the configurational energy of the old and new states, we calculate

PO = eo CEn— Em) {kT If # is greater than

| (that is FE, < E,,), the move

(13.3-2) is accepted.

If #

is less than

|

as a result of &, > E,,, then an additional random number RA between 0 and 1 is generated. If R is smaller than , the move is accepted; if not, it is rejected. That Is, the conditions for the acceptance or rejection of a move are

greater than 1, move is accepted less than |, move accepted if >

R

less than |, move rejected if 7 < R

(13.3-3)

The final condition that must be met is that a sufficiently large number of states of the system must be sampled (that is, the simulation must be long enough) to be representative of all the possible states of interest of the system. Such a simulation is said to

be ergodic. The properties of each of the states in the simulation (after discarding the

3: Determination

of the Radial Distribution

Function

n the are um) bri ili equ m fro far are and n tio ura fig con ial init the t lec ref early states that the m fro e enc fer dif ial ent ess the is this that e Not . ing ght wei any t hou wit averaged simple Monte Carlo simulation. in which the states are generated randomly and then ted era gen are tes sta the e, . Her ing ght wei tor fac ann tzm Bol a h wit ed rag s ave tie proper with a Boltzmann factor weighting and then the properties are linearly averaged. The above is a brief description of the simple Monte Carlo NVT simulation for an atomic system. Monte Carlo simulation techniques have evolved much

beyond this

stage, and the reader is referred to books on this subject for details.’ The obvious improvements are to polyatomic systems, in which moves also consist of rotations of the whole molecule as well as around bonds; to different biasing methods to allow the study of chain molecules and polymers; to mixtures in which moves can include molecule identity swaps; and to the use of other ensembles such as the grand canonical ensemble in which W7jx are fixed (which is especially useful for the simulation of adsorption and osmotic equilibrium). Monte Carlo simulation has also been used for the NPT ensemble. The reason this ts of interest is that most experimental measurements are made at fixed temperature and pressure, rather than temperature and

density. Also, excess thermodynamic

properties on mixing are determined at fixed

temperature and pressure. In this case, a possible Monte Carlo move can be either a particle displacement or a volume change of the simulation box. The Markov chain generation acceptance

criteria are different than

in the NV7

ensemble

(and the sim-

ulation is somewhat slower), because all particle locations must be scaled with each volume change and the long-range correction changes. Another very useful method is the so-called Gibbs ensemble simulation.© which involves two simulation boxes (of different densities for pure fluids, and also different compositions for mixtures) and allows for the calculation of vapor-liquid and other phase equilibria. In this simulation, the total number of molecules (to be distributed between the two boxes) and the total volume (to be divided between the two boxes)

are fixed, and temperatures in both simulation boxes are identical, satisfying one of the conditions for phase equilibrium. The simulation then includes three types of moves for a pure fluid and a fourth type of move for mixtures. First are the particle movements within each box to ensure equilibrium within each box. Next are volume changes of both boxes (at fixed total volume) to ensure that the pressure in both simulation boxes will be equal, a second condition for phase equilibrium. The third type of move is the transfer of a molecule from one box to the other to ensure equality of chemical potentials (the third condition for phase equilibrium). For mixtures, an identity swap move (i.e., interchange a species | molecule with a species 2 molecule) is the fourth type of move, and it is needed to ensure the equality

of chemical potentials for all species in both boxes. After equilibration, the common

pressure in both simulation boxes, the different

molecular densities in each simulation box, and (if a mixture) the different composi-

tions in each box are computed. In this way, the vapor pressure of a pure fluid and densities of the coexisting phases can be computed as a function of temperature, In the case of a mixture, vapor-liquid or other phase boundaries can also be computed. The website www.wiley.com/college/sandler contains simple Monte Carlo simulation programs in MATLAB®’ for the square-well and Lennard-Jones 12-6 fluids. (The square-well program can be used for the hard-sphere fluid by setting the Ry, parameter in the square-well fluid equal to | on input.) These programs, MC_sqwell

"A. Z. Panagiotopoulos, Molecular Physics 61, 813 (1987), 'MATLAB®

is a registered

trademark

of The

MathWorks,

Ince.

253

13.4 Molecular-Dynamics Simulation

The readme.txt file provides

®. AB TL MA er ld fo e th in d un fo be n ca and MC_LJ, information on their use.

SIMULATION

CULAR-DYNAMICS

In on ti la mu si o rl Ca e nt Mo om fr t en er ff di ry ve is on ti la mu si cs mi na dy rla Molecu l tia ini r ei th d an x bo on ti la mu si e th in s om that, although the initial placement of at

be arbitrarily chosen, beyond

may

velocities (speed and direction)

is that, for each atom 1 at a z directions (that is Fj... Fj, from its interactions with all forces are

simulation is completely deterministic. The procedure particular position, the forces acting on it in the x, y, and g in lt su re es rc fo e th g in mm su by ed in ta ob e and F; -) ar other atoms. If the interaction is pairwise additive, the

s F; »

=

»

Fiy.x

=——_

jx

fei

N

N

He

Fi y

=

»

Fiz

y

a

=

duPAT (rj) dy

and

j=!

/=l

j#!

i#t

(13.41)

du(rij)

am

Fz= 2) Fie = where

s

j=l

f=1



~

“\—si duE(ry) ay »

that point the

j=!

=

j#i

j#t

Fj;,, is the force on atom / in the x direction as a result of atom / that is at

r ila sim the is re The ms. ato er oth all r ove is sum the and a separation distance rj, h eac for ion mot of s law 's ton New s. ion ect dir e nat rdi coo r othe the for on ati ret erp int atom i in each of the coordinate directions are 4 d*x;

it

and

dt-

MN

'

)

= Fir =

Fij,x =——

d*z;

_

m—=F;-= dt-

du(rij)

)

j=l j#i

j=l i#i N

j=l f#i

=

d? yj

a,

adit-

dx

N

Fej

Al

N

N

. du(rjj)

——dz . j=l

SH

hy =

)

:

j=l it

Fij.y

=

)

j=! ji

du(rij)

:



dy

(13.4-2)

I#i

The procedure, then, is to numerically integrate the equations of motion over such the in ge chan ant ific sign a not is e ther that Ar s step time or s small time interval velocity of any molecule during this time interval. Also, since many time steps are ium libr equi near a to state al initi the from ve evol to em syst the for t (firs ed need state and then continuing for many more time steps to compute average properties), l smal e, rwis othe ; used be to have es edur proc ion grat inte l rica nume rate accu very numerical errors will accumulate during the simulation. Numerical methods such as finite difference and the more accurate predictor-corrector and other integration

methods have been used. Choosing a time step for the integrations is a balance between steps that are too small, so that the simulation to obtain accurate equilibrium averages will take too long; and steps that are too large and lead to errors due to numerical integration—and because of instabilities that arise if. as a result of an interaction (or collision), an atom undergoes a large velocity change or even velocity reversal within the time step, Time steps of the order of fentoseconds (one-quadrillionth of a second) are typical.

.

3: Determination

Function

Distribution

of the Radial

are we that 1s ve abo bed cri des as hod met ics nam -dy lar ecu mol One problem of the y sit den and re atu per tem ied cif spe a at s tie per pro of ues val the in d ste ere int y all usu bed cri des far so hod ics met nam -dy lar ecu mol the le , whi le) emb ens cal oni can the (in the n mea we , rgy ene l tota by e, Her . rgy ene l tota nt sta con of is, at —th tic aba adi is

ed fix is rgy ene l tota the ce Sin . rgy ene on) cti era int (or ial ent pot and c eti kin of sum on cti era int ir the ct era int les tic par as on, ati cul cal ics nam -dy lar ecu mol in the simple energy may increase at the expense of their kinetic energy, or vice versa. Also, since the temperature of the system Is 5

N

|

a Nk rT = si

)

U:

4

(13.4-3)

j=]

the problem that arises in what has so far been described is that since there is a constant interchange between kinetic and interaction energies, the temperature of the system (which only depends on kinetic energy) is not fixed, but varies during the course of the simulation. There are several ways that the adiabatic simulation method described above can be modified to be a constant-temperature molecular-dynamics simulation, all of which involve a continual change in the kinetic energy of the atoms. The simplest method is that during the simulation, the temperature 7 of the atoms is computed using Eq. 13.4-3 and compared with a desired set temperature 7,. Then a new velocity of each atom 7 in each coordinate direction / (denoted by ve) is scaled to obtain a new

velocity (denoted

by Ue)

as follows:

— new

old

u;

= UG

[Ts

7

(13.4-4)

A less abrupt way of changing the velocities is by use of a virtual thermostat to mimic the way energy is interchanged between a real system and thermostatic bath.

Starting from the idea that the rate of temperature change of the system (by heat input trom a thermostat) should be proportional to the difference between the system temperature 7 and the bath temperature 7;,, we have

dT —

dt

=

AT



At

=

TI), -T)

(13.4-5)

In this equation, Ar is the time step used in the numerical integration of Eq. 13.4-2. and t is a parameter coupling the system and the bath. (For a physical system, t would be the ratio of the total heat transfer coefficient (product of heat transfer coefficient and area) to the total heat capacity of the system (product of mass and constant volume heat capacity)). If t Is small in value, the coupling is weak and the system temperature changes slowly. If t 1s large, there is a tight coupling and the system temperature changes rapidly. With this model,* the velocity scaling is ————_—_———

T | wy ue a iy +TAt (+ — )

(13.4-6)

#



°H. J.C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. Di Nola, and J. R. Haak. J. Chem. Phys. 81, S684 (1984).

Chapter

13 Problems

255

The value of t is adjusted to give good simulation results, which will also depend on the time step Ar used in the integration. Empirically, a value of tAr ~ 0.0025 has been found to give reasonable results. Other, more sophisticated thermostats for molecular dynamics are also used. However, none of these isothermal ensemble molecular-dynamics simulations result in a true constant temperature system, because the temperature will fluctuate during the simulation. It should be noted that the adiabatic molecular-dynamics number of atoms, volume,

simulation is at a fixed

and energy (NVE_) and therefore corresponds

to a simula-

tion in the microcanonical ensemble. The isothermal molecular-dynamics simulation is at a fixed number of atoms, volume, and temperature (VV7) and corresponds to a

simulation in the canonical ensemble (though, since there are temperature fluctuations, in principle it is not a true canonical ensemble simulation). The website www.wiley.com/college/sandler contains a simple molecular-dynamics simulation MATLAB®’ program for the Lennard-Jones 12-6 fluid. These programs, MD_LJ and MD_LJ2, can be found in the MATLAB® programs folder. The readme.txt file provides information on its use. There is also the program MD_LJ2 that does an isothermal molecular dynamics for the LJ 12-6 fluid, Finally the LJLMD_MC program does a Monte Carlo simulation followed by an isothermal molecular-dynamics simulation for the Lennard-Jones |2-6 fluid at the same state conditions. This is useful for comparing the properties computed from both types of simulations. Note that exactly the same results should be obtained for an infinite number of moves in a Monte Carlo

simulation and an infinite number of time steps in a molecular-dynamics simulation. This is the ergodic hypothesis of Chapter |. However, because of the stochastic nature of simulations (the random numbers generated in a Monte Carlo simulation and initial assignments of positions and velocities in a molecular-dynamics simulation), the results obtained from the two simulation techniques will not be identical for finite simulations, though they will converge as the simulation lengths increase.

An excellent two-dimensional illustration of adiabatic and isothermal moleculardynamics simulations, prepared by Professor David Kofke and Dr. Andrew Schultz, can be downloaded from http://www.etomica.org/wiki/LennardJones:Simulator and run on your personal computer. In this simulation, the user can set the state conditions and see the fluctuations in the pressure, temperature, potential and kinetic energies,

and the radial distribution as a function of interatomic separation as the system evolves to equilibrium from an initial lattice configuration. The user can adjust the temperature and atom number density in the isothermal simulation, and the energy and number density in the adiabatic simulation. (To choose the energy in the adiabatic simulation, first choose the isothermal simulation, set a reduced temperature that will fix the initial energy in the system, and then choose “adiabatic simulation” and run.)

CHAPTER 13.1

13 PROBLEMS

Use the MATLAB®

program

MD_LJ2

isothermal molecular-dynamics

to compute

the radial distribu-

distribution of speeds (Eq. 3.9-5) at one of the con-

tion function, internal energy, and pressure for the

ditions in Problem 13.1 and compare the results to

Lennard-Jones

12-6

potential

at one

of following

conditions:

a. T* =kT/e = 1.0 and p* = po* = 04 b. T7 =0.9 and p* =0.3 13.2

velocity distribution function. Compute the Maxwell

c. T* = 1.5 and p* =0.45 One of the results of the MATLAB® isothermal molecular-dynamics program MD_LJ2 is the

those obtained from the molecular-dynamics program at one of the conditions in that problem. 13.3 From having done molecular simulations at a collection of state points (either by yourself our sharing simulation results with colleagues) for the Lennard-Jones | 2-6 potential. comment on how the

256

on ti nc Fu on ti bu ri st Di al di Ra the of n io at in rm te De Chapter 13: maximum

(average over possible states) and molecular dynam-

in the first peak of the radial distribution

ics (average

function changes with T* and p*.

constant

by choosing

a fixed value

of (R3, — le, determine how the radial distribution function, pressure, and compressibility change over R,, — € space using the MATLAB® program MC_sqwell. 13.10 A test of the ergodic hypothesis (Section 1.4) would be to study a system using Monte Carlo simulation

time

interval) and

see

if

states (MC) and short time intervals (MD). Use the

MATLAB®

program LJ_MC_MD

for the Lennard-

Jones 12-6 fluid, and compare the results for the radial distribution function and the thermodynamic properties for each of the following cases. How does the extent of agreement change with the length of the simulation? Use one of the following conditions: a. T* =kT/e = 1.0 and p* = po? =08 b. T* = 0.9 and p* = 0.2

program Monte Carlo program the MATLAB® MC_sqwell (setting Ry, = 1). At a fixed packing fraction, calculate and plot the radial distribution function for the square-well

approximately

a long

the two results agree, at least to within the statistical fluctuations that arise from the simulations being for a small number of atoms (compared to Avogadro's number) and over a relatively small numbers of

13.4 Show that Eqs. 11.2-11 and 13.2-10 are equivalent. 13.5 For a state point of your choice for the hard-sphere fluid, compare the results for the radial distribution function obtained from the Monte Carlo program MC_sqwell (setting Roy = 1) with those obtained from the Percus-Yevick solution using the program Percus-Yevick HS. 13.6 Compare the values of radial distribution function for the hard-sphere fluid at contact given by valEg. 12.5-3 at several densities with the ues obtained from Monte Carlo simulation using

fluid using the MATLAB® Monte Carlo program MC_sqwell at a collection of reduced temperatures kT /e for the well width R,, = 1.5. Comment on the temperature dependence of the radial distribution function, 13.8 At a fixed packing fraction, calculate and plot the radial distribution function for the square-well fluid using the MATLAB® Monte Carlo program MC _sqwell at a fixed value of reduced temperature kT /e for varying well widths. Comment on the well width dependence of the radial distribution program. 13.9 One model for globular proteins is as a square-well fluid with a very deep well ¢ and a very narrow well width Ayy. Keeping the intensity of the interaction

over

ce. T* =0.5 and p* = 0.5 da. J* = 2.0 and p* = 0:75 e. T* = 0.2 and p* = 0.75

f. 7* = 0.2 and p* = 1.0 13.11

From having done molecular simulations at a collec-

tion of state points (either by yourself our sharing simulation results with colleagues) for the squarewell potential, comment on how the range of the radial distribution function (that is, the number of easily visible peaks and valleys) changes with 7* and p*, molecularisothermal MATLAB® the 13.12 Since dynamics program MD_LJ2 does a simulation at constant temperature, the kinetic energy of the system is unchanged in the simulation and depends only on the temperature 7* as shown in the graph produced in the simulation. However, as also shown

in the graph, the average total energy (sum of the kinetic and potential energies) depends on both T* and p*. Comment on the behavior of the total energy on T* and p*, and why the total energy is negative at some densilies.

Chapter 14

Perturbation Theory As should be evident from the previous chapters, considerable effort is involved in obtaining the values of the thermodynamic properties and the radial distribution function for a chosen interaction potential at a single temperature and density, regard-

less of whether an integral equation method or computer simulation is used. Also, either of these methods only results in numerical values of these properties at the chosen temperature and density; they do not provide an analytical equation for use in calculations with other interaction potentials or at other state points. One method of extending the usefulness of the thermodynamic properties and radial distributions functions that have been obtained for one interaction potential (which we call the reference potential) for use with a different potential is by using perturbation theory, wherein one does a Taylor series expansion of a thermodynamic property or the radial distribution function in the difference between the new potential of interest and the reference potential. An introduction to this method is the subject of this chapter.

INSTRUCTIONAL

OBJECTIVES

FOR

CHAPTER

14

The goals for this chapter are for the student to: e« Understand perturbation theory using the hard-sphere as the reference potential « Understand perturbation theory for other reference fluids « See how perturbation theory is used to develop thermodynamic models of use in chemistry and chemical engineering

14.1)

PERTURBATION POTENTIAL As one can

THEORY see from

FOR

THE

the previous

SQUARE-WELL

sections, the calculation

of the thermodynamic

properties and radial distribution function for a liquid is a difficult task. One case where we do know the thermodynamic properties with reasonable accuracy is the hard-sphere fluid. (The thermodynamic properties on other fluids are generally known only from simulation and from fitting general simulation results with complicated polynomials in reduced temperature and reduced density.) Therefore, an obvious

question

is, can we use the information

for the hard-sphere

fluid to estimate

the

257

r 4: Perturbation Theory

is h whic ry,' theo tion urba pert in done is what is This ds? flui r othe of es erti prop based on a Taylor series expansion of a property about the values of that property for some other potential model or in some other state. As a reminder, a Taylor series expansion of some property ©(x, A), whose value is a function of the variables x and A, can be expanded as a function of A as follows:

ws

7

+

dA

(

3!

A—(),4

das

2!

AZO 4

. ri

fa@(x, A)

. i

d@(x, A)

sh

1

(as aa?

A—OLe«

(14.1-1) What is generally done in statistical mechanical perturbation theory is to split the interaction potential in a reference part (A = 0 part) and a perturbation part and then do a Taylor series expansion in 4. To be explicit, consider the hard-sphere potential uns)

for tr

oo O

=

r

|

penn

+a >, 5.

47

dr,...dry

Cy

ryet

afo-fe

|

lUtnglri

5

Da

~ | Z(N,V.T:A)

A=0

~

kT

|

Z(N.V.T:a

dr,



wpeatrane

[LY

|/ar

2. Whe lrg. * = sree

b

...dry =O

ae

(14.2-2)

However

[-

[CV

ths (pray

>

b

z

Mpert (rij) | Jer

i j>t

vntrine

dr,

dry

Z(N,V.T:A) A=()

-|>3 S

J

I

CD

dr,...dry

"ame

wmmtrvr Z(N,

wyslri | Jer

V,T;

I =

{j)

| [er

- bs DS uns _— }” N-p NI n _ w

perp

op

5

bas > any

foe ct

re'

this last equation,

we

see where,

in obtaining

to be evaluated

Js

Lotro

at A = 0, and

“J. A. Barker and D. Henderson. J. Chem.

that since

(14,2-3)

r | fer

dr,...dry, have

recognized

all the molecules

Phys. 47, 2856

...dry

dr,

(1967).

that the derivative are

identical,

there

is are

14.2 First Order Barker-Henderson Perturbation Theory

263

N(N — 1)/2 identical contributions to the integral. So we need only to evaluate the integral for one pair of molecules and multiply the result by this factor. Now,

remembering that the radial distribution function is defined as

eo

[>

—HIF,F.

fe

fe

fe

..dry

ERT ay,

EN

—u(r).o.--.0y )fkT dr

_.dryay

which for the pairwise additive, hard-sphere potential is* ye

fe

hs

Gj? |/
, Ups(ri )/ AT

dr,...dry,

_

N N(a

|

Te

2 Mh try) /kT

, r ; d 2 ) ) t . 1 2 r r 1 p p e ( M Z ( Mp

ay,

> fae, [rentriada eri

| N(N—1l)2x

INMN-l1)f

p)dryy = ir

f |

=

dr.

ov

2tpN (r; pyr? f=

trent

— =

rm

fo

Upen(MiD8p. (ria; P) Ary»

- p)r? adr

(14.2-5)

0

“There is an important

point with

regard to the temperature

dependence of the radial distribution function

for the hard-sphere fluid, In general, the two-particle or radial distribution function g!*!(r: 7, o) is a function of temperature and density, as well as radial separation rjy. However, while the radial distribution function for the hard-sphere fluid is dependent of density, it is independent of temperature. The reason for this is, as

discussed earlier, that it is obtained from integrals involving the Boltzmann

factors in the interaction energy

eT

er

| where u(r) = 90

and e "AT

— OQ for r < o, and a(r) = OQ and

l/4! —

| ¢ > @, So there is

no temperature dependence in the integrals defining the radial distribution function for the hard-sphere fluid, This is to be compared with, for example, the case of the square-well fluid of Eg. 14.1-2, for which where u(r) = 00 and eV? — 0 for r < 0, w(r) = —e, and eV ET & pt /*T for gp er © Rot and u(r) = 0 and e "VAT — | for r> R,ya. This temperature dependence of the Boltzmann factor results in a lemperaturedependent radial distribution function. Consequently, the square-well potential—and, in fact, any potential that has a region in which the interaction is finite in value (rather than infinite or zero) over some range of intermolecular separations—will have a radial distribution function that is dependent on temperature as well as density.

Chapter 14: Perturbation Theory Here, we have replaced the variable of integration r;}2 simply with r and, as usual, ignored the difference between N and NV — I. Therefore, to first order in the perturbation expansion, we have A(N,V,T

>A)

kT = or, for A =

A(N, V,7T; A =0) FT

2m N +A iT

4 | Hooter

p)r? dr

1

A(N,V,T;A=1)

= A(N,V,T;4=0)+ 27pN

axe

p)r? dr

(14.2-6)

and specifically for the perturbation potential given by Eq. (14.1-3) Koya

Agw(N, V,T) = Ans(N. V,T)

— 21pNe / g(r: pyr? dr

Or

(14.2-7a)

k

Aw(N,V,T) — Ans(N.V,T) WET =

NET

e

np Te

f

i:

a

>

(r: p)r*dr

(14.2-7b)

a7

In these last equations, we have used that A(N, that A(N, V,T:A =O) = Ap. (N,V, T).

Now

V, 7:4

= 1) = Agw(N,

V,T),

and

using that

aA paki

| a

p* | — — | —

(a).

N (5

14,2-8

T

we obtain the following equation of state to first-order term in the perturbation expansion around the hard-sphere fluid po

7,

Rat?

d

Px (0. T) = Prs(p, T) — —— | 20 Nope | g(r, pyr? dr N ap , a Rawat

= Py(p, T) — 20p7e / g(r, p)r? dr — 2nep” (F

T Rew

da 5Dp / g(r, . ‘

p)r*dr T

(14.2-9)

14.3

Second-Order Perturbation Theory

265

irox app d iel n-f mea as to ed err ref mes eti som are far so ed ain The expressions obt mations. The name arises from the fact that, to the order so far considered, the

radial distribution function is unchanged from that in the reference potential (here the hard-sphere fluid), and the perturbation contribution is obtained as an average of on) cti fun ion but tri dis al radi the is, t (tha ure uct str the r ove ial ent pot ion bat tur per the

of the reference fluid. Though the results above are explicit for the square-well fluid, they are easily generalized to any other potential that can be represented as the sum of a hardsphere potential w,;(r) plus a perturbation potential upen(r) of any form. The results

In this case are A(N,

VT)

NkT

_ Ans(N VT “a ,

ar

fr

| & “(r; plitpen rr” dr

and

(14.2-10)

P(p,T) oo

= Pry(p.T) + 2mp* [2.0

pritpentryr? ar

Row

+ 2rep"

d =; / dp

(14.2-11)

g(r, P)ltpent (Fr? dr

dG

7

Finally, the obvious extension to perturbation theory would be to the radial distribution. However, this becomes quite complicated involving higher-order distribution functions (which can be simplified somewhat by using the Kirkwood superposition approximation, Section 12.2). See Problem 14.9.

ND-ORDER

PERTURBATION

THEORY

To improve the accuracy of perturbation expansion, one should consider higher order terms in the series. The next term in the series comes from the following equation:

v="



LA

N)7e

L4/

(ear aga) o=¥

Nap’ °° “ap 0=%

jen

ars

an


® = ——,/— Inf 14+



yz nd + he) > 3 Pin(1 +7) RT afvT NayCe — AY SIN vith g@ = On V—b VVtb” 4 3 vio ==

leads to P =

Soave-Redlich-Kwong Ver= = VV -6): MN. =

RT

a(y) — ——— Viv—_ eb

Co(T)In(1 2(7) In(l + Bp) which leads to P = V—-b

with a

given above

Peng-Robinson N

VW = V ~—b: Ne = Cx(T)p which

leads to P =

with a(7)

atan |§ ——$—_—___ (-exe =) / 2N bp — b? p+

RT

a(T)

V-b

ViV+6)4+A(V -—b)

aD

= CT)

(or C3(7)) empirically fit to vapor pressure data

4 Note: C = = oR,

—1) and b= ae

Caren, o—————_

2Nb—Vh@

W2NbV

— bt

16.2 Application of the Generalized van der Waals Partition Function

307

ng eri ine eng d use ly mon com all ’t don why is, ask may der rea the One question that equations of state use the more accurate Carnahan-Starling free-volume expression rather than the less accurate van der Waals expression? There are several reasons for this. First, if the excluded volume parameter > (or #) is treated as an adjustable

parameter, the calculated free volume can be made to be closer to that obtained from computer simulation, at least over some ranges of density. Second, and perhaps more importantly, equations of state commonly used in engineering are applied to nonspherical molecules, and analogues of the Carnahan-Starling free-volume expression are not available for every molecular shape that occurs in chemical processing. Furthermore, even if such expressions were available, their use would require that the forms of the equations of state for molecules of different geometric shapes be ditferent. Since engineers deal almost exclusively with mixtures, not pure components, it then would not be clear how to formulate an equation of state for a mixture. (It should be remembered that the way mixtures are treated now is to use the same equation of state for the mixture and all of the pure components, and obtain the parameters for the mixture from a set of mixing and combining rules.) A final reason that the simple van der Waals free-volume term is used is that calculations involving simple cubic equations of state are computationally very quick. At first glance, this may not seem important given the speed of computers. However, in the analysis and design of a chemical process using computer simulation software (for example Aspen),° or in petroleum reservoir simulation, thermodynamic calculations frequently take up to 90) percent of the computer time, and may be used hundreds of thousands or even millions of times in the iterations while the simulation is converging. Therefore, it is advantageous to have a simple, reasonably accurate equation of state rather than one that is more complicated but only slightly more accurate.

Molecular-level computer simulation, as described in Chapter 13, can be used to test the coordination number models discussed above. A comparison of computer simulation results with some of the models are shown in Fig. 16.2-3 for the square-well fluid for A, = 1.5 and for three values of the reduced well depth (or, equivalently, at different reduced temperatures for the same well depth). Also shown as the solid

line is the result of the following relatively simple model? Nin Ve! c

where

V, = No? //2

2A r

2 —————— V+ Vo (e2@/24F — fj

is the close-packed

volume

of hard-spheres

16.2-17 | (

and

N,,

1s the

coordination number at close-packing (18 for Ry, = 1.5). This equation was derived from a simple argument that the likelihood of the occurrence of two neighboring

molecules is proportional to e*/**", Another test of this model is shown in Fig. 16.2-4. Based on the accurate expression for the free volume (Eq. 16.2-3) and for the coordination number model of Eq. 16.1-17, the following equation of state is obtained for the square-well fluid from the generalized van der Waals partition function (Problem 16.13) PV

RT

l+nt

y+

(l—7)°

‘Aspen Plus® process simulator, Aspen Technology,

Nn Volet!7* — 1)

V+

Vo(et/4T — 1)

Inc, Burlington, MA,

(10-218)

Models

6: The Derivation of Thermodynamic

COORDINATION

NUMBER, NV,

éAT = 0.25

Figure 16.2-3 Coordination number for the square-well fluid (Ry, = 1.5) for various values of the reduced inverse temperature

EAT =0.75

e/kT. The points are the result of Monte Carlo simulation,’ the dotted line is the van der Waals

og “1

16.2-7a), the dashed

line ts

the Redlich-Kwong result, the dash-dot line ts the Peng-Robinson result, and the solid line results from Eq. 16.1-17. Reprinted with permission from S$. I. Sandler, Chem. Eng. Educ., Winter 1990, page 12 et. seq.



0.2 0.4 0.6 0.8 REDUCED DENSITY. po*

oS

10

COORDINATION

NUMBER, WN.

0.0

1

model (Eq.

Figure 16.2-4 Coordination number for the square-well

fluid (A,

=

1.5) for various

values of the reduced temperature e/k7. The points are the result of Monte Carlo ta

simulation,! and the solid line results from

O00

0.1

O22

O38

O14

05

po

(he

OF

OF

Oo

Eq. 16.1-17. Dotted portion of the e/kT = | is in the two-phase region. Reprinted with permission from S. |. Sandler, Chem. Eng. Educ., Winter

1990, page 12 et. seq.

While this equation, for the reasons described earlier, is not used in engineering applications, it has been shown to be reasonably accurate for the description of the vapor-liquid equilibrium of simple, spherical real fluids by adjusting the values of its parameters, as shown in Fig. 16.2-5. For interaction potential models other than the square-well potential, the general procedure is to keep the same structure as above by using the mean-value theorem of calculus

to obtain

16.2 Application of the Generalized van der Waals Partition Function

309

COMPRESSIBILITY, PV/NKT

COMPRESSIBILITY, PVANAT

10!

io

107!

1!

PRESSURE,

bor!

1-4

lots

lor?

1Q*!

ae

mee

PRESSURE,

BAR

lor

lar?

107!

oe

10!

BAR

(b)

(a)

Figure 16.2-5 The compressibility 7 = PV/RT

along the vapor-liquid equilibrium envelope for (a) argon and

(b) methane; the points are the experimental data and the line results from Eq. 16.1-18. Reprinted with permission from S. |. Sandler, Chem. Eng. Educ., Winter 1990, page 12 et. seq. N2

Um(N, VT) =

N-

7 | Bese

5

N.V,T)dr= SS

| 8c

Rt

N,V,T)dr

R*

_ N-NAN, V,T)(uS) — 2

(16.2-19)

where u(r) is the soft-core part of the interaction potential, (w*) is an appropriately chosen average value of this part of the potential, NV. is an estimate of the coordination number defined by Eq. (16.1-10), and A* is the range of the soft interaction. So a number of assumptions need to be made. One 1s to choose an effective hard-core diameter,

for which

in principle

the equations

of Section

14.4 can

be used.

Next

is

to pick an average value for the soft part of the interaction potential, (w°>), Finally, one also has to make an assumption about the temperature and density dependence

about the coordination number. Alternatively, one can dispense with the idea of the coordination number, and make dependence of the integral [ Bese

estimates directly for the temperature

N,V.T)dr=4n

and density

/ w>(r)e(r; N,V, Tyr? dr Re

Re

A less commonly used equation of state (in engineering) is that of Alder et al.,’ that uses the following expression for the free volume: Vy=

Vexp

4) n — 3 ( n jf — 3

"B. J. Alder. D. A. Young and M. A. Mark, J, Chem. Phys, 56, 3013 (1972).

Chapter 16: The Derivation of Thermodynamic Models

er mb nu on ti na di or co ing low fol the n, tio ula sim er ut mp co and, based on the results of model:

eo N=

255

>) mAmn

ii

\m-l

pa?

(=)

nt

(“)

i

leads to

which

RT ——

-

V

Sl+nt+y7t+r

(=)"

—————

=

(l—ny

Amn

Dy fl

5 (

ney

kT .

16.2-20

)

fi

EQUATION OF STATE FOR MIXTURES FROM THE GENERALIZED VAN DER WAALS PARTITION FUNCTION The straightforward extension of the Generalized van der Waals partition function to mixtures 1S i;

Fint.i

[Vi(N), No,....V.7))"

TI (|

Oavaw(N1, No, .--, V.fy=

Onix

N

Na...

N

VY,

Tr

eee)

x eXp (-SSe

k1

|

(16.3-1)

with

(16.3-2)

URS (Ni, N2,...V.T) = >) > UPON, N2,....V,T) !

i

and Ui (M1, N2 pees

V.%j=

N;N; Se

waite

f

i

Nie Nae cc ces V,T)dr

'



|

= VXI SG

uP (r) gi (3

Ny, Nay...

V.)dr

Re, (16.3-3)

where

R7, is the range of the ¢ — j interaction (the well region for the square-well

,

potential), and

kT —— = No,....V.T) Prix(N,.

|

N

No,...,V.T (Ni, 7) Ur ---.V, No, mux Unx(N1, kT?

dT

(16.3-4)

f=0o

In these equations, x; 1s the mole

fractions of species i, x; = Nif dj N; =N;i/N,

u>(r) = u(r), and g;;(r) = g;i(r) is so defined that N;g;;(r)dr is the number of species ¢ molecules in a volume element dr at a distance r from a central / molecule.

There are two different paths for using the Generalized van der Waals partition function for mixtures. The first is where the interest is in equations of state and their mixing rules, in which the density, temperature, and composition dependence

of the coordination number is important. This is discussed in this section. The second way of proceeding ts the use of the Generalized van der Waals partition function for

311

Equation of State for Mixtures

16.3

ion nat rdi coo ies pec s-s cie spe The . els mod nt cie ffi coe vity acti ain obt liquid mixtures to ow bel ly ial pec (es ids liqu e sinc but to, here e anc ort imp ral cent numbers are also of their critical point) are not very compressible, a simplifying assumption is made that

the density dependence of the coordination number can be neglected. This will be considered in Section 16.4. To develop equations of state for mixtures—and the mixing rules to be used with such equations—the starting point 1s

P(N. V.T) =kT (° In =| av



gn.

d _

kT

NOT

In

eS

OV lyr

No,

eo.

V.

T1(3%5) | N®mix

(

[Wet

a —~kT — aVv Nj.T

22

N2,

(N17.

V,

T)|*

=)

N; Fint.i

+ Nin

———_—-

In

T i (3

Ve(N).

Wi 0)

No,...,

i“)

N@mix(N, N2,..-, ie 2] kT din m V¢( Ve(Ny.1 No, 2 ..., V7 ')

= wer (

av

-w(

OD mnix((ANV), No,..., W009 ’) 2

N.T

aV

NT

(16.3-5)

Here, the subscripts on the derivative indicate that the volume derivative is at fixed numbers of molecules of all species and temperature. It is useful to separate the equation of state into an entropic part P*" that arises from the free-volume part of the Generalized van der Waals partition function, and an energetic part P*"= coming from the mean potential—that P'E(N, VT) with pent

ay

Aw

NAT

(

using

is, as before,

V.T)

P(N,

din 2 SL V;(Ny, rN No,...,¥.T ey = Oa

’)

aV

= PS™(N,

and AS

(16.3-5a)

T

IDmix(Ni, No,...,V.7 P™2(N,V,T) = —N (Se) av

V,T)+

(16.3-5b) Nod

To begin, we again consider the simplest case of the square-well potential, so that res

of

(Ny.

No,

eee,

VT)

Ny Ni

_

[ Ri

=

—&i;

N Ni; ”et

;

siren

Treat

V,

l)dr

*

(16.3-6a)

where £;; = €;; 1s the well depth of an j-j interaction and N;; is the average number of ¢ molecules around a central j molecule—that is, the coordination number of / molecules around a / molecule, defined by

\6: The Derivation of Thermodynamic Nii CN),

No,

waa

T)

V,

=

Models (16.3-7)

T)dr

V,

ones

N3,

Ni,

Bij (Fs

/

=

Re To proceed, one has to specify either each of the species-species coordination numbers the and re atu per tem of on cti fun a as Ui) es rgi ene nal tio ura fig con the , ctly dire (or, number densities of each species in the mixture. It should be noted that, as before, for interaction potential models that are different than the square-well potential. we can keep the same structure as above using

; N ; N , t i g u f E S = ) P Y ip (Ni, Na, - ++,

Ni. No...

V,T)dr

Ri

Nj Nj (u?,) f sie

= SE

N,, No,...,V¥,T)dr

Ki

N; Nj; (u?,) =—

(16.3-6b)

*

:

where u?.(r) is the soft-core part of the interaction potential, and (u3) is an appropriately chosen average value for the i-j interaction. For the square-well potential, (u?.) = —«;;. In what follows, we will use ¢;;, but the results are easily generalized by replacing it with —{u?.). Table 16.3-1 summarizes the Generalized van der Waals partition function and its relationship to the thermodynamic properties of a mixture. In what follows, we convert this general formalism into equation of state mixing rules by considering specific choices for the free volume and mean potential.

Free Volume in Mixtures For mixtures of hard spheres (or the hard-core part of the square-well simple van der Waals approximation for the free volume is

Vi(N1, No.0... V, T)=VON}, No...

Vi T)—b

ON;

with

fluid), the

b=S >>> xix dij.

i

i

/

For hard spheres, there is an exact combining rule bij

=

|

5

ii

. bj;)

so that

>

b=

i

Y=

xix

bi,

=

S=

xii.

i

i

In this case,

. penton

V,

T)

—s

NET

.. No,.. Ve(N), Oln (eee

VT

av

aln(V

N,.T

— Nb av

NkT N;.T

Y

—WNb

Of course, expressions other than the van der Waals equation could be used for the free-volume term. For example, the Carnahan-Starling expression discussed earlier and extended to mixtures could be used, resulting in greater complexity.

16.3 Equation of State for Mixtures

313

c ami dyn rmo The and on cti Fun n itio Part ls Waa der van ed liz era Gen The -1 Table 16.3 Properties for a Mixture N

inti OCovawl

1,

A mix(i¥),

.

No wees

No..--,

VT)

)|

n

2

I

OAS N;!



ZHE(N,.

Nz.

V1 « Oh wees Limited

Bi

NOa,e

|

T)exp (-

Vv,

00,

.

FS

NO ni (-=*)

ia Vi imix exp

ro

a of Oi, M,..kT? N

Dimix( Ny, Na....,

f=oc

UT (Ni, Nay

R Na. 5 CUOM 0 NAN),

(k7 >> &), the exponential No,

ce neey

Vi~)

No, ......V.7)

xO;

(16.3-11)

terms are close to unity and —_— UR},

: — 7 XjO7 (RF J

a

I)

— 1)

(16.3-12)

which indicates that the ratio of local compositions is essentially equal to the ratio of species volume fractions (this would be exactly true if Rj; = Rj;, which may be the case, aS Roy is frequently chosen to be about 1.5). Finally, at moderate temperatures, the ratio of local compositions is determined by both volume fractions of the species

and the differences in the attractive parts of their interaction potentials. Another interesting observation is that the total coordination number of a species, for example species | in a binary mixture, again using the low-density result, is

Ny, + Nz) =

N\4

a(R

lI

N 4x

VY

3

up . N24 ir — Wet/kh 4 oa R3 — 1)e2!/Ki

]— *7 !/ ®? De — 3, (R , 03 x9 + " l e D — F, (R [x03

(16.3-13)

which indicates that, unless the molecules are of the same size and have the same interaction energies, the total coordination number of a species in a low-density mixture is a function of composition. This may also be true, but to a lesser extent, al liquid densities; this is different from the lattice theory assumption used in Chapter 10 that each lattice site has a fixed coordination number, The simplest model species-species coordination number at other densities is the completely random mixture; in this case, Nj; = “EC, where C’ is a constant and the

315

16.3 Equation of State for Mixtures same for all species. In this case — 5 bij NGNii

No formes ,¥,rj=

UF? (N 1,

l

N;

|

|

.

=

— shy EN

=

=

BH EN GP

(16.3-14) and

1c

;

a Dl DI 5— =— TV Vi en ve Na Prix(N1 i

NN;

and

J

—~Ceii oii —!vy

N;.is Vj. Fis T)== ®;(pliv

7

Using these expressions, we obtain eng

N*C

tii

LL

NM =a

V,

Pen,

:

roy =

C

N*

~y2

»

with

Xj Xj Qj;

»

5 ei

ap

(16.3-15)



if

So then, the equation of state is

P(N, V,T) = P(N, V,T) + P™(N,V,T)

_

naaas

aN?

NkT

alien

V2

~ VNB

which is the van der Waals equation with the van der Waals one-fiuid mixing rules and

B= »- Y > xix) Bij a

a=

» So xix) ai) ~

(16.3-17)

Other choices for the species-species (or local) coordination numbers result in other equations of state, as shown in Table 16.3-2. Of course, choices other than those shown in the table can be made, leading to different equations of state and activity coefficient models. Among those other choices are the model of Wilson® Nij

Nip

_ Ni eg

Nj

0 jj KT

(16.3-18a)

of Whiting and Prausnitz? Nn.



vie

hi

ests

with

Nij

+

N

jj



Nej

=

j

of Hu et al.!"

"Gg. M. Wilson, J. Am.

Chem.

Soc... 86,

127 (1964).

“W.

B. Whiting and J. M.

Prausnitz, Fluid Phase Equilibria, 9

119 (1982),

My

Hu, D. Ludecke and J. M. Prausnitz, Fluid Phase Equilibria, 17, 217 (1984).

Cp

(16.3-18b)

6: The Derivation of Thermodynamic Models s Rule ing Mix e Stat of ons ati Equ the and els Mod ber Num ion nat rdi Coo Table 16.3-2 Local that Result Equation of state mixing rule Local coordination number —

van der Waals (vdW)

Cc

Ni

Y

1|-fluid

vdW

ie it y

1-fluid

z

N; —C, jes kT V

vdW I-fluid

rf N — with Nj; + Nj; = Nej = vei Puy

nonquadratic mixing rule; ai Di AjX jU;Gj; eS

yo FiUi Nee

iV: whe

Nig

NU;

Fm

1

ee

pltty 8p HAT

As above N

with Nij

+

N jij = Nej

Vv Cj

=

Surface area fractions

N;

Nonquadratic mixing rule

Agr

RE: ;

4

(16.3-18c)

and a model based on the simple extension of Eq. 16.1-1 iN, 1 Vo i

aim

Moist

egg f2ke

TV + Voi (e 747 — 1) with an effective coordination number

(16.3-18d)

Voi; = No /v2

with

N,, = 18 (for Roy = 1.5).

Computer simulation'' can be used to test the accuracy of these approximations.

Figure

16.3-1

composition

shows

the deviations

of the local composition

ratios in terms of the quantities @)7 = we

these ratios would

be unity in a completely

ratios from

the bulk

and 4); = nee

Each of

random mixture.

For the square-well

potential at low density it would be 4 3 o:.ik:.

lim @;; = Si

po



]

Ri De;

a (RF, — fj

Egger

(16.3-19)

Another test of the models ts to look at the total coordination number for the squarewell system with €23 = shown in Fig. 16.3-2.

1.2k7 = 2e,,, but varying diameter ratios and densities as

The main conclusion from the simulation results, especially Fig. 16.3-1, is that none of the local composition models are completely accurate; so if one can devise

"S. L Sandler, Chem. Eng. Educ., Spring 1990, p. 80.

16.3

317

Equation of State for Mixtures

Figure 16.3-1 The ratios of the local compositions to the bulk compositions @)> and @5, as a function of reduced density for the square-well fluid with ey

=

L2kr

JE].

=

and Ry, = Re

=

033

1.5 obtained from

Monte Carlo simulation. The arrows indicate the low density results, the points are the simulation results at different bulk mole fraction, The dash-double dotted line is the model of Eq. 16.3-18a small dotted line is the mode! of Eg. 16.3-18b, the dashed line is the model of Eq. 16.3-18c, and the solid lines result from Eq. 16.3-]8d_. Reprinted with permission from S. 1.

(1.8

A

=

O11

—~ ().6

O8=:F1

00

02

04

O04

O85

D6

Of

O48

REDUCED DENSITY, po"

Sandler,

Chem.

Educ..

Eng.

Spring

Total Coordination Number, ',,

1990, page 80 et. seq.

0.00

O25

O50

O75

Xj

Figure

100

O0F25

O§F.50

67S

AY

16.3-2 The total coordination numbers

100

)060.2506—

S50)

OFS

1 AH)

xj

N,, (unfilled points) and N,> (filled

points) as a function of density and mole fraction from simulation,'* The squares, triangles, circles, and diamonds are for reduced densities of p(x\o; + X2055) = 0.1. 0.3, 0.5 and 0.7, respectively, for Ry; = Ry. = 1.5 Reprinted from Fluid Phase Equilibria, Vol. 34, K.-H. Lee and S. I. Sandler, “The Generalized van der Waals

Partition Function. 1V. Local Composition Models for Mixtures of Unequal Size Molecules” by K.-H. Lee and S. I. Sandler, pp.113—147, Copyright 1987, with permission from Elsevier.

improvements, the result may be mixture equations of state (and also activity coeffi-

cient models as will be seen later) with a better theoretical basis and greater accuracy. Nonetheless, the models presently used in engineering are effective, largely as a result of treating the parameters in the model as adjustable, and fitting to experimental data. '*K.-H. Lee and S. I. Sandler, Fluid Phase Equilibria, 34, 113 (1987).

Chapter

16: The Derivation of Thermodynamic

Models

GENERALIZED

THE

ACTIVITY COEFFICIENT MODELS FROM VAN DER WAALS PARTITION FUNCTION

The second way to use the Generalized van der Waals partition function is to develop an expression for the excess Helmholtz energy of mixing, and from that derive activity coefficient models. The main difference from the analysis used for equations of state discussed in Sec. 16.3 is that here, as is typical in formulating activity coefficient models, because of the limited range of liquid densities of interest, the

total coordination number z of a molecule is not considered to be a function of density, though it will be a function of composition and (to a lesser extent) of temperature. The starting point for the analysis in terms of the Generalized van der Waals partition function is to note that for mixing at constant temperature and total volume (so that V(N|, No,...,7)

= >)

V;CN;, T)), the excess Helmholtz energy of mixing

over that of forming an ideal mixture of the same components at the same temperature and total volume is

AG. (Ni, Nas ees Vs T= ACM N2y ee Vs TI=Y> ACN, Vin TY RT Nj I = —Tin|

N,,No,....V.T

ZONA

VD

| | Qi(Mi. Vi. 7)

N;

-«r Yin (X)

7

\%

UN

i

N; \*

Z(N,, No.....V.T= oc) | | (=) = —kT In

i

[] Zi. Vi. T = 00) i

+ Nt

N>,....V,T) — } > Nj ®iANi, Vi, n|

] I * y v y , . . . , o N . ) N ( Vimix

—kT In

\ ; N (*)

[ | very. Vi. 7™ i

j

|

URS(N1, No... V,T) = S0UPN;, Vi, T) mix

>

a

=o

= APY (M1, No,....V)+Apy (M1. N2...-,V,T)

(16.4-1)

The first term on the right-hand side of this last equation is the athermal contribution to the excess Helmholtz energy due to the hard-core molecular size and shape differences between the species in the mixture (since all soft-core forces are unimportant at high temperatures); it is entirely entropic. The second term is the contribution from other than the hard-core interaction, resulting from the attractive and soft-core

16.4 Activity Coefficient Models

repulsive forces. Also, note that

- (echo

No, ..., V, D)

(2a

No, ..., P. D) P.P.N ji

ali

TV Nii

Bem

= KT

319

Iny;(41, %2,..., 7, P)

(16.4-2)

So, starting with the Generalized van der Waals partition function, we can develop activity coefficients models or “reverse engineer” models commonly used to understand the assumptions inherent in such models. In this way, the Generalized van der Waals partition function can provide a theoretically-based platform for improving current activity coefficient models—or developing new ones, Also, once the model assumptions have been identified, they can be tested using computer simulation, As an aside, it is useful to note that at high densities, and especially in liquids, the molecules are so close to each other that they are always subject a background attractive interaction field determined by their proximity to all the surrounding molecules. In this case, the local compositions are determined primarily by the hard-core molecular volumes. It is for this reason that some activity coefficient models separate the contribution to the excess Helmholtz energy into an entropic (sometimes referred to as the configurational) part depending on volume fractions (as in the Flory-Huggins'*

and UNIQUAC'*

models, though the latter also includes surface area fractions), and

an enthalpic or energetic part (sometimes

referred to as the residual

part, though

the term residual is used differently in this chapter) arising for the soft part of the interactions. However, when the attractive forces are very strong and specific to certain orientations, as in hydrogen bonding, different types of models are required, as

discussed in Section 16.6. Free Volume

Term

As in the discussion of equations of state, the free volume (entropic) and energetic

contributions will be considered separately. Therefore, the starting point is

Apy (Ni,

Nay... V) = ASN, No.0.

V) — DAMN, Vi) f

—kT | Nin Vemix + } > Nj ln; — S*N; InVg,; t

® y p N — v V (

[ |_ / ; v x = @; on fracti

measure

v; is some

Where

of the volume

of species 1,

e th to n o i t u b i r t n o c l a n o i t a r u g i f n o c e th r fo n o i s s e r p x e y r o l F e th is a 3 4 . 6 1 . then Eq be n ca s n o i t p m u s s a r he ot , y l e v i t a n r e t l A . 2 1 4 . 0 1 . Eq in d e free energy of mixing us to g n i d a e l , es ri eo th e ic tt la on d e s a b e s o h t , e l p m a x e r fo ; rm te c i p o r t n e e th r fo e d ma

e m u l o v d n a ea ar e c a f r u s h t o b s e d u l c n i at th n o i s s e r p x e > ' n a the Guggenheim-Staverm a

=

)

ob; I

i

, 5, ...V)

A,

sent

ex ent T.V

x; ol

P|

fractions — + N; In

kT

6;

(16.4-3b)

>— Nigi in 7%

, on ti ac fr e m u l o v s it is @; i, le cu le mo of ea ar e ac rf su e th to where g; is proportional and

on ti ac fr ea ar e ac rf su s it is d,

z is the (same)

for all the

number

coordination

species in the mixture. y a a r fo on si es pr ex e th om fr ed in ta ob is t en ci fi ef co ty vi ti ac To see how the e ur xt mi ry na bi a r fo at th so , ¢; will use as an example ¢;; =

we

vi

— — In j N | T k = — n I N AS" (N, No, T, V) = kT Y ea d *i ; oo

ex,ent

|

PF, i

!

= «| int X,VUy + X2V2

|

Ng In

Xypvy + X2v2

Nv» Nv + No In ———_ ln ———

— kT | Ny

Nyvuy +

+ Nove

N,v,

|

|

(

16.4-4

| species of coefficient!® Therefore, to obtain the activity VT No, (dA™"(N,, 1 a

( a = P) T, x2, in y"(x1, =





ip

7

d

~ tip

———_ In N72. + —————_ ln N, N,v,

+ Nove

v

|

Nyv,

N

+ Novo

x X, X;v)

xX) Uy + X29

+ X2v2

x

+1- $1 - 26) xX |

|

=In #1 +1X |

Nov,

X,U

X,U,) + X9V2

=n

+ N2v2

N,v,

+ Navi

Nd

Nyvy

N

a

N,v

+ Nove

N,

Nv, N,v

21

Nu»

1

T.VNig

aN,

TLVN

ON;

kT

|

’)

ae

A X|

,

»

In o + @2 (: — 1) X|

=Inyf"(x1,%2.T, P)

(164-5)

Us=

to d de ad be n ca t en ci fi ef co ty vi ti ac e th to on ti bu ri nt co e um ol -v ee fr or ic op tr en This . xt ne ed er id ns co is at th l ia nt te po the energetic contribution from the mean

im he en gg Gu A. E. by y ntl nde epe ind d pe lo ve de was el mod SThis

(see “Mixtures”, Claredon Press, Oxford,

. 50) (19 163 69, as, s-B Pay . im Ch v. Tra v. (Re n ma er av St J. A. and 1952)

s. ve ti va ri de l na io it os mp co ing tak of y wa er lSee footnote 8 of Chapter 10 for the prop

321

16.4 Activity Coefficient Models

The Mean Potential or co s ie ec sp sie ec sp e th r fo s l e d o m l, ia nt te po n a e m e th r fo To obtain an expression g in ow ll fo e th in ed us en th e ar e es Th d. pe lo ve de be st mu ;, N; , er dination numb equations: Ui?

.

— 5815 NN

¥.f)=

wins

Na,

(M1.

|

_

1

r -UNS(N), No... VT) dT = neh; | 4a | wij AN Ki* os

Tr

|

oe

Ni

i

(16.4-6)

2

kT-

f=oc

=i.

JTS kT

Dix

oN

ric

Vy,

ees

seer

No

(Ni.

kT

;(N;. Vi. 7) =

and

De

2

,

f

|