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An Introduction to Applied Statistical Thermodynamics(incomplete)
An Introduction to Applied Statistical Thermodynamics(incomplete)
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An Introduction to
Applied Statistical Thermodynamics
Radial distribution function for the LennardJones 126 potential at po’=0.93 produced by molecular dynamics simulation using a MATLAB® program that accompanies this book
Stanley I. Sandler
ISBN
978047091354?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.
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Library of Congress CataloginginPublication Data Sandler, Stanley [., 1940An introduction to applied statistical thermodynamics / Stanley [. Sandler.
p. cm. Includes index.
ISBN 97804709 13475 (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 CaliforniaBerkeley, 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 FacultyScholar 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 twosemester undergraduate textbook in the first half of the onesemester graduate course. My second goal is to introduce the firstyear 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 Onehalf 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, molecularlevel 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, PengRobinson, 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 DebyeHtickel 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 wellknown 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 moleculardynamics simulation (albeit only for simple spherical molecules) and userfriendly 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 ShiangTai Lin of National Taiwan University. Zachery Ulissi, a University of Delaware student, converted a FORTRAN Monte Carlo program for the LennardJones fluid made available to us by Professors Daan Frenkel (Cambridge University) and Berend Smit (University of California, Berkeley) into a userfriendly 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 squarewell, hardsphere, and LennardJones fluids, and a molecular
'MATLAB®
is a registered trademark of The MathWorks,
Inc.
Preface
for Instructors
vii
dynamics program for the LennardJones 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 PercusYevick equation for hardspheres. 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 moleculardynamics 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 equationofstate 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 molecularlevel 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, molecularlevel descriplions, computer simulation, the understandings arising from statistical thermodynamics, and the ability to use molecularlevel 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 moleculardynamics simulation, for simple spherical molecules. There are userfriendly 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 LennardJones
126 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 moleculardynamics 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 moleculardynami . 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 hardsphere the for function A program for computing the radial distribution and screen the on appears ouput Graphical in the PercusYevick approximation. HS. PercusYevick 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, PercusYevick the in force for the hardsphere 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
McGrawHill,
Sratistical Mechanics,
L. Hill, Statistical Mechanics,
McGrawHill,
New
New
York,
1938.
1962.
1956.
AddisonWesley, 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 MaxwellBoltzmann 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 IsobaricIsothermal Ensemble 92 Restricted Grand or SemiGrand 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 SimilarSize Molecules from Lattice Theory 168 10.4 The FloryHuggins 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 Xray 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 YvonBornGreen (YBG) Equation 216 The Kirkwood Superposition Approximation
12.3
The OrnsteinZernike Equation
12.4 12.5 12.6
RADIAL 216
219
220
Closures for the OrnsteinZernike Equation 222 The PercusYevick HardSphere 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 MolecularDynamics Simulation
Chapter
CHAPTER
13 Problems
14 PERTURBATION
242 245
253
255
257
THEORY
257 14.1 Perturbation Theory for the SquareWell Potential 14.2 First Order BarkerHenderson Perturbation Theory 262 14.3. SecondOrder 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
DebyeHiickel 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 StatisticalMechanical 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 HydrogenBonding 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 moleculewall 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 moleculemolecule or
moleculewall 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
(1.31)
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’ ergsec 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;
(1.32)
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. 1.51, 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 (1.51)
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. 1.51. Therefore, the probability of
Infinite heat bath of temperature T
Figure 1.51 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,
PrenticeHall, 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,
(1.52)
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
1.52 with respect to E,
derivative of Eq.
ale,
+
ee ee
Em)
oo
Eom)
=
d(E,)
dpapglkn
Ew
d( En
+
7 Ein) Em)
(1,53)
Now
using Eq.
1.52, we also have +
OpaplEn ee dEy
and therefore
dpally)
d
En) Se
ee
En
—
a. a(E,)
,
ial
Pa
Ein
dE,
En
Pov
Ein
1.54
Pat
we have dpaplky,
d(E,
oe

Fa}
ee
=
Em)
dps(F,)
En
d( Ey)
1,55
pa
By a similar argument, one can show that changing F,, while holding £,, constant gives dpapll,
+
Emit
d(En+ En)
—_—_—
=
dpalEn)
PME aE,
P8) 156
E,.)—_—
Now equating the results of Eqs. .55 and .56 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. 1.57 is that the lefthand side is independent of subsystem B, and the righthand 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. 1.31 has the effect of changing all the energy levels in that subsystem, but no effect on subsystem B, That the relationship given in Eq. 1.57 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
( 1.58)
.

(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. 1.58 one obtains
Each
constants,
of the integration
C4
(1.59)
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
S°
states noof
pa(Em) = 
slates a
system A
of
system B
Therefore,
; Cy,
=
(1.510)
I
—————__ >
ef
states a
and
Ch
=
——_
En
»
of
states
system A
eP
(1.511) Em
mt of
system B
We define the canonical partition function
O(N, V, 6) for any system to be oS.
O(N, VB) = > eBEAN.V) 
(1.512)
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
1.513
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. 1.513 we have that the probability of occurrence of a particular microstate i with energy F, is e
Pea
Se8E
e
Pea
O(N. VB)
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
(2.11)
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
(212)
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
2.13) 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
(2.14)
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. 2.11. (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 1104
p,
11
110"
3105
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 2.11 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
(2.21)
However,
aQ \
($F)
— NAV
>»
.
Stones
k
.
BE,
E,e BEE
. (2.22)
Chapter 2: The Canonical Partition Function so thal
y
E,e7 PE
slates
(e) yFOB —
(2.23) ~
 + __ 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 singleparticle 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)
(2.31)
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
(2,32)
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)
(2.33)
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  
(2.34)
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 oneatom system is
g =e Pl 4 e Ber 4 oe Bea Now
consider
a collection
of N
such
identical
(2.41)
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
(2.42)
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'
(2.43)
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,
(2.44)
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
(2.45)
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
( 2.46 )
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
(2.47)
where the factor NV! arises from the indistinguishability of N identical atoms. It is important to compare and understand the difference between Eq. 2.34 and Eq. 2.47. In Eq. 2.34, 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. 2.47, 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
(2.48)
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. (2.46 to 2.48) each of the individual atom partition functions depend upon volume and the stillunknown 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 Nlike 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 Nparticle 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. 3.11.
Expressions for the canonical partition function for a system of N identical noninteracting particles are q™
——
(3.11)
N!
and g=
)
e PF
=
)
slates i of a
energy levelsj of a
single molecule
single molecule
wje Pri
(3.12)
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 +=)
(3.13)
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 3.11 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" ergsec. 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
(314)
fs
one of these sums,
Soe Ms
(3.15)
{~=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. 3.15 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
(3.16)
5
The remaining sums in Eq, 3.14 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
(3.18)
5
:
2mm
O(N, V, B) B) = —5 In (= hp)
+NinV
—InWN!
(3.19)
The internal energy of the Nparticle monatomic gas can then be computed from
4 U=— ( ae) OB
Jyy
as
3N 26
(3.110)
\TIFICATION OF B AS 1/kT By the ergodic hypothesis, the internal energy of Eq. 3.110 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
(3.21)
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. 3.12 then becomes q =
>
(3.22)
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. 3.15 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. 2.46 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
(3.23)
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. 3.13). Also, from Eq. 2.23 we
have that din
U= (= =)
(2.23) ta
Replacing # with (k7)~', it is easily shown that
yn)  (: ~ =)
U=kr?
ay
(3.3la)
ye
or
kT? (22) (3.41b)
= ——  — 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
(3.32)
dV i) OV Jr
Now using
dV
ny
(3.33)
d U 7 d ) T T T = — + ( Ud = and rearranging Eqs. 3.32 and 3.33, 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
(3.34
But from the first and second laws of thermodynamics for a closed system, we know that (3.35) dU = TdS — PdV Comparing these two equations, we have
(3.36)
and
dS == kd (no+
U —j=kd{l =) k (no
alnQ T ( aT ),.)
( 3.37 )
On integrating this last equation we get
 s=e(ino rr (*2) aT
) +e
(3.38a)
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
(3.38b)
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
(3.39)
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:
(3.310)
= 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?
(3.311)
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
(3.313)
+
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. 3.39 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. 202203 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. 3.39 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 BornOppenheimer 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
(3.41)
so that the partition function becomes
g=
Yo
elim tet bee )/AT
3.42)
states of the atom
Now using the independence of the energy Eg. 2.34, it is easily shown that
states
as was
discussed
in deriving
G = iransGelectGnuc
(3.43)
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 
(3.44)
The first of these partition functions is a sum over the translational energy states, which has already been evaluated in Eq. 3.17, using the particleinthebox 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
(=)
(3.17)
The single particle translational partition function is frequently written as
V trans
=
(3.45)
where
A
is the de Broglie wavelength—that is, the wavelength equivalent of the momentum of a particle in the waveparticle duality theory of matter.* [Table 3.41 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 3.41 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 107? 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 ** Js
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 = kgm?/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 )
(3.46)
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 reemitted 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
(3.47a)
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
=
(3.47b)
“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
(3.48)
©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
(3.49)
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
(3.410)
(3.411)
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
(3.412)
elect
Using only Eq. 3.47b for the electronic partition function, we have
Cy
= (= 7
oT
3.413
Nk vy
(0.419)
2
and A=kATINO=
KT
In
q™
= —NkT
Ing+kT InN!
(3.414)
Now using Stirling’s approximation (which will be done frequently in this book), InN!t=NinNn
—N=WNInN
—N Ine
(3.415)
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 
3.416 (3.416)
and
_f §S=kmnO+k T (
4 In O
ain
aT
)
2amkT \2 Ved! wim  Yer’ elect,  ——  —_—______————
= Nk] of(
7
)
Ny
( 3.417 )

This last equation is referred to as the SackurTetrode 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 Inol (=)
q N
oetccr,)   = kT In 4] =
§
(3.418)°
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. 3.410 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 (=)
(3.419)
which ts (in the form tamiliar to chemists and engineers)
wT,
P) = wolT,
P
Po) + kT In (=)
where
(3.420)
° 4
2 Ho (T, Ps)
=
—kT
kT
In  (=r) hie
\2 kT Foes Fs
When using Eq. 3.420, 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. 3.419, 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 permolecule basis. For a permole 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
(3.421)
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 permolecule basis. These should be multiplied by Avogadro's number to obtain a value on a permole basis.
ILLUSTRATION 3.41 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 m3V 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, § = UA
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
3.42
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 103 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
102 102 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*)
(3.51)
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
(3.53)
=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
__
T°
fT
— Up
3 V ] ¥ N
_
IN @elect, 1

—ln
=
(4.44)

)
+n
(57 a \@,OpOc rT
2
me
ect1 [email protected] + ") Se e 1— n( iF ~ iFTr  —euoF
(445)
4.4 The Thermodynamic
Properties of an Ideal Polyatomic Gas
57
and
(4.46)
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 (4.49)
yam ke } >”
iy,
= 5 +
(4.47)
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
(=)
(4.410)
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
(4.411)

and
PV
(4.412)
= 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
(4.51)
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 powerseries 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
,
(4.61)
ts , KE , gy er en c ti ne ki e th and
: 3 m5 KE = > (vj + ¥3 + v3)
vj
where
dx; =
dt
(4.62)
.
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,
(4.63)
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
(4.64)
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)
(4.65)
mio = —K (no — 11) + K (3 — 12)
(4.66)
my, mi
= K(y2—
= —K (3 — 42)
(4.67)
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
(4.68)
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. 4.65 to 4.67 can be rewritten as
— K(n2 —m) = 0
(4.69)
—monz + K (nz — m1) — K (73 — m2) = 0
(4.610)
+ K(ns — no) =0
(4611)
—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
(4.612)
that Eq. 4.612
in the equation to be zero. That
K —ma"
—K
0
—K
2K — ma
—~K
()
—K
K — mor
(4.613)
=0
Or
a (K* — wo m)3mK
(4.614)
— w*m*) = 0
This sixth order equation for @ has six solutions ay
=O;
a
=O,@34
K = 2,/—:
and
Fil
ws
=+,/
3K —
(4.615)
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. 4.69, 4.610 and 4.611. Using wm = 0, we obtain
H2— 7, =0
272 —n1 — 43 =0 and
m3 — 2 =0
(4.616)
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;
(4.617)
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
(4.618)
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 threedime 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. 4.22 to 4.27 for the linear diatomic molecule, starting from the singleparticle partition function of Eq. 4.21.
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. 4.18
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 constantvolume heat capacity Cy for H;, HD, and Ds at 150 K, 250 K, and 350 K, assuming that the atomatom 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 carboncarbon 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 constantpressure 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 constantpressure 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
carboncarbon 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 4.11 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
(5.11)
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
(5.12)
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)
(5.13a)
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
(5.13b)
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 Scomponent
= —kT lnQ =
—kT In ieai
mixture are easily evaluated.
=r
(win ln 1) —
i=]
(5.14)
(5.15a)
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;
__—
5.15b
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!
(5.21)
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
(5.22)
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
(5.23)
= 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. 5.21 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
(5.26)
Np = NR + Ne =
The double summation of Eq. 5.25 with restriction Eq. 5.26 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!
(5.27)
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. 5.23 is recovered.
THREE DIFFERENT DERIVATIONS OF THE CHEMICAL EQUILIBRIUM CONSTANT IN AN IDEAL GAS MIXTURE Starting from Eq. 5.27, 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. 5.26. Alternatively, since
A=—kTInOQ
(5.31)
where the partition function Q for any particular choice of Na and Eq. 5.21, 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)
(5.32)
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
(5.33) Therefore, the condition for equilibrium 1s that N
Ga _ 9B
«XB _ 4B
Na
Na
7
Ne
7
(5.34)
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 singleparticle 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
NNa
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;
(3.57)
i=l
However,
to proceed,
it should
be
noted
that
using
the
stoichiometric
coefficient
notation of Eq. 5.52, 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
(5.58)
ON; = vjdX
(5.39)
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’
(5.511)
Likewise, the internal energy of the reacting system is
UT,
Vv.)
=
,fdln kT? (
ag
dy \¥
where again Eg. 5.55a 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 5.511 and 5.512 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 notsoobvious 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. 5.51 so that
Arxnlt = > vu;
and
(aX
Cy = > Cy + (=)
i
i
}
Arxnld
(5,.514)
v
where Arxnu is the internal energy change of reaction on a molecular (not molar) basis for the stoichiometry of Eqs. 5.51 and 5.52. 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
(5515
)
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. 5.52, 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
reassociate 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
5.516
NAMING
(5.916)
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¥)
(3.517)
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) _
(5518) :
(5.519)
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 inthebox model. (Also, we have neglected
7
eee IAT
(5.520)
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 socalled 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) .521
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. 5.511)
PV = >> NjKT =(Na+ Ni + NoJkT = (NX —X +X + X)KT = (NQ + X)KT j (5.522)
SO V=(NA+X)KT/P and combining Eqs. 5.]19 and 5.121
———
x?
AkT
=
¥Y = X/N¥
“Theat. tf
e
yp
—
(5.523)
P)
F(T,
h
P
(Na — X*) Now, defining
(2amekT\?
to be the degree of ionization, we have f=
/
FF
\ —_—=
(5.524)
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).
(5.525)

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
(5,527)
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
(5.528) .
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 numberbased 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 1916% 1.602
1o!2 (1.38044 x 1018 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. 5.523, the quantity of more interest 1s
erp) = MT (20m) AKT
F(T, P) = —
f2mm,kT
{ —>—
1.38044 x 1073 (2) x T(K)
4
\2
a7 Fivelect, Lf AT €
—_
P (bar) x 105 (44°)
Ky (m > )
1.38044 x 1078 x T(K).
= Sf, P (bar)
To calculate the degree of ionization, Y, Eg. 5.524
is used. Finally, the heat capacity
is computed using Eq. 5.528 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
: 1107
0
0
2107
r
5000
3110%
1.510%
2104
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
110
2107
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 twodimensional 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 twodimensional surface is qg,q. If there are M adsorption sites on
a twodimensional
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
oN_
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 Sshaped curve between these two extremes and Is referred to as the Langmuir adsorption isotherm; A 1s the temperaturedependent 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
_NN
A
ati
Nal(N — Na)!
SUES
Na=0
that
A and B
527
We can evaluate this using the general form of the binomial expansion, written here as
(A51)
=(x+y)"
a — n a w — » M=0 or to match Eq. 5.27
N
 __
! N — Na=
Naqn* dp 0
Nal(N
a
(A54)
— Na)!
* According to Wikipedia, the binomial expansion is attributed to the 7thcentury 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
(5.37 )
Nay!
i
The summation of Eq. A54, written in generic form so that the result can also be used later, is
.
N
DN
Mwy
MT"
M.NM M NM
_*
>
N
M
2 MN
M..N—M My NI
Mh"
AS5) “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. A55 into a form of Eq. A51. 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 yNM 6a, MUN — MM)! "
M eM yNM 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 wellordered 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 Nparticle 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
(7.11)
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 (7.12)
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
(7.13)
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 (7.21)
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
;
'3.45)
(5 mk =)
nar
on
Z(N,V,T
SS av
N.T
(7.22)
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
(7.23)
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
(7.24) 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)
(731)
The first three terms on the righthand side give the total potential energy as a sum of three pairwise or twobody (twoatom) 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 threeparticle system we will assume
Wry. 2.73) = u(ry2) + u(ri3) + u(rz3) = > i

u(r)
(7,32)
i
lsi f
> j l=; yun
u(riz)
(7.33)
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
(7.42)
Si fay +... dey
de,
and as rij
co,
u(r)
Using Eq. 7.41 in Eq. 7.35 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 (7.44)
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. 7.43 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
(7.45)
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
(7.47a)
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
(7.47b)
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
(7.47¢)
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
(7.48a)
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
(7.48b)
0)
In Eq. 7.44, the term
J
(7.49a)
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 
(7.49b)
j>i
Thus, so far we have
—_ y
om
MY
A(N,V,T)=V"+V
N{
No
>
Bi
V
}
(7.410)
The next type of product that arises in the expansion of the configuration integral is
/ _ / DDE V
og
voi
DY. Sify dry. dry
(7.411)
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
!
(7.412a) 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. 7.48b, 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™ (*) : .
(7.412b)
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
(7.413)
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
(7414)
¥
which diagrammatically can be represented as
ee
0
This integral is representative of the second class of integrals appearing in the term in Eq. 7.44 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
(7415)
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
(7.416)
>iy>i
and
£=
v"
I+
By
(= 2 ) (4
a2
rm
N°\ =
2.2)
fy —
2
7.417
\¥
(7417)
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)
(7.418a)
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
(7.418b)
(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
(7.419a)
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,
(7.56)
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
(7.57a)
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
(7.57b) 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 (7.58a)
which
contains the terms indicated
in Table 7.51.
Chapter 7: Interacting Molecules in a Gas Table 7.51 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
(7.510) .
Therefore, by extension, we have
ee AIE) a(S) Je + CB) +(E)G)(BBo) 1
{N3 /2 1B .\ )° N = (1+++} too a (Fe)
(7.511)
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

(7.512)
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
7.513)
=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
_"

7514
Comparing this result with the virial equation of state
P = pkT{1+ Bop + B3p* + Bap? +.....} we have
b
By=——, iH
2p
(7.515)
38
By=——,
a!
B=

etc.
(7.516)
in a Gas
I: Interacting Molecules
Alternatively, if we write P=
pkT
then
Bjyi
41+ >~ Bjsip! j=l
(7.517)
j = a8
(7.518)
(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!
(7.519)
For our purpose, the difference between this result and that of Eq. 7.513 is negligible.) We will not attempt to reduce the higherorder 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
(7.520)
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) (7.521)
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,
(7.61)
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
(7.62)
{ 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. 7.61, 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=
(7.63)
j=l
where
I B.
—
and, in particular
e.
i+
i"
[Jf (eer @0! — V)r'drdw dw B,(T) = —2xn — ]['da,do,
(7.64)
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, =
(7.65)  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
7.71 (771)
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 permolecule) 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
(7.73a) “
pi
S(N, V, T) ) = SN, , V T))—— NKT Lj (
dB; +1]
(7..773b)
—7
and
N,_V' SCN, V.T) ——S_ ) ™(“ T ——$__NE
=T
ps dBjy1
>j aT
! ) ¢ 3 7 , 7 (7.73¢)
or
) T , S — ) Sin, T fos MS Lg RK
; B i n d  = ee j
J
af
(7.73d)
'Compare Egs. 7.72a and 7.73c. Can you explain why the righthand 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
/
(7.74a)
A(N.V,T)
(7.74b)
= 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
(7.74¢)
j
Also
kT
= w'O(N,V, 7) +
w(N,V.T)= (sr)
va (7.75)
= O(N, V,T) +kT Sj + = Bi+l For an ideal gas at any pressure P, we usually write
wT,
(7.76a)
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
(7.76b)
J
(7, P) for a real gas is defined by the relation
, (7.77)
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

(7.78)
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
 (7.79a)
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
(7.79b) 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 (7.79¢)
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 rederive 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.
(7.81)
119
Derivation of Virial Coefficient Formulae
7.8
For convenience, we define the absolute activity as 2 = e!/*", so that Eq. 7.82 can be written as
(7.83)
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 =
(7.84)
N=0
We have shown previously (Eq. 6.212) 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 (7.86)
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
(7.87)
TV
and therefore
_— _
ain
_ y
or
Nvy
me=e
+
=
 Had
j=l
yg 2boz~ + 3b3z°
+ 
(7.88b)
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. 7.86. 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. 7.811
(7.811)
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

Zé
?
ypv~)
dr
4
 fy ine = x // (h—e M12/Edy T)dr, If the potential
is spherically symmetric.

SIRT)
(7.812a)
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
(7,812b)
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;
++:
(7.813)
': 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 centertocenter position vector is necessary to specify the potential energy. This is the case in molecular fluids in which relative orientation vectors or several atomatom 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
(7.814)
—
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
(7.816a)
)
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
(7.816b)
) 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
(7.816c)
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
7.817
Tees
have obtained
(7.817)
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 7.91 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 hydrogenbonding 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 7.91 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
ispecies 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
(7.54)
()
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
HardSphere
FOR
SPHERICAL
MOLECULES
Potential
The simplest potential used to represent molecular interactions is the rigid or hardsphere potential shown in Fig. 8.11 and is given by ur) =
oo y
r 4 (8.14)
where T° ( ) is the gamma
function. (If n < 3, the value of B> is infinite.) Though the
()
0.5

1.5
Figure 8.13 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. 8.12.

§: 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?
($.13b)
is a special case of the Point Centers of Repulsion
potential,
and leads to
an infinite virial coefficient (Problem 8.3). Charged particles must be treated in a different manner, and this is discussed in Chapter
15.
Potentials with Attraction
From experimental volumetric data for nonideal gases, it is possible to obtain numerical values for the second virial coefficient. For many gases, particularly at low temperatures, the second virial coefficient is found to be negative—as has been shown earlier with the experimental data for helium. From Eq. 7.54 it is evident that the virial coefficient can only be negative if the intermolecular potential is negative (that is, attractive) for some intermolecular separations. The experimental observation that at higher temperatures the second virial coefficient becomes positive, as was shown earlier for helium, requires that any potential model must also have a repulsive part. Below are several simple potential models having both attractive and repulsive regions.
SquareWell Potential The simplest analytical attractiverepulsive potential is the squarewell model OO
u(r) =
r=oa
—€
ogo«r
Ryo
(8.15)
Kewl
which is shown in Fig. 8.14. The second virial coefficient for the squarewell potential is a
Row
Bo(T) = 27 /
(1 —O)r? dr +27 /
()
aT
+2
~
f R
=
2ma7%
=
(1 — ef !*" yp? dr
23
9
(1 — lyrdr
AS ia
si
+ SU e*7(R— 3No, ?=
2no°
a
= [1+ (1 —e®/*7)(R3 — 1) , ($.16a)
vir) 
Figure 8.14 The SquareWell 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)
(8.26h)
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 rigidsphere model 270° /3. This behavior of the second virial coefficient of the squarewell potential for R,, = 1.5 1s shown in Fig. 8.15. 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 squarewell model:
Ths
_
e/k
(8.17)
~ 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 8.11 contains the squarewell 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 8.15 (a) The reduced second virial coefficient B* for the squarewell 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 8.11 SquareWell and LennardJones 26 Potential Parameters Determined from Second Virial Coefficient Data Assuming the Molecules are Spheres'
argon benzene CF, CH, CO, krypton npentane neopentane nitrogen xenon
126 Potential
LennardJones
SquareWell 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 LennardJones
Potentials
Perhaps the most widely used potential for correlating experimental data on simple molecules is the Mie potential: c
(8.18)
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 dipoleinduced 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 commonlyused (126) potential for nonpolar molecules shown in Fig 8.16.7
LennardJones
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. 8.37b.
Interaction Potentials for Spherical Molecules
8.1
131
wiryee
LennardJones 126 Potential
Figure 8.16 The LennardJones
126
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
($.110)
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 8.17 (a) Reduced Second Virial Coefficient of the LennardJones (b) high temperature range.
126 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. 8.17 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 rigidsphere potential. However, since the potential does have a finite—rather than infinite—slope, the effective hardsphere diameter decreases
as the average
energy
(or mean
kinetic energy) of the system
increases,
as it does with increasing temperature. Thus, the effective hardsphere diameter, and therefore the value of the second virial coefficient, decreases with increasing temperature at very high temperatures.
Exponential6 (Modified Buckingham) Potential This three parameter (€, y, and ry,) potential is given below.
u(r) =
=
6 :
é
—\—  —  —
6/y
¥
,_!
ex PY?
fy
—_ —
——
Por
r
(8.113)
> ’
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 hardsphere 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 8.11 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 LennardJones 126 potential using the potential parameters for argon in Table 8.11.
SOLUTION From Table 8.11,
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 126 potential. While the results are not in perfect agreement with experiment, as would be expected from an approximate potential such as the LennardJones 126 potential, the results are very good over the temperature range considered,
ILLUSTRATION
8.12
Repeat [Illustration 8.11 using the squarewell potential.
SOLUTION From Table 8.11,
ao = 3.067 A = 3.067 x 108cm 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. 8.16b 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 squarewell 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 LennardJones 126 and squarewell potentials with appropriately fitted parameters are comparable. (The same will not be true
when
we
consider
highdensity
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
£
—
(8.21)
Ba ij(1)
xia
> >
Br mix(4, T) =
j=!
where
Bz 4;(T) = an f
(8.22)
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)
(8.23)
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 LennardJones
126 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 purespecies 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 socalled LorentzBerthelot combining rules. For the distance parameter or core diameter (i.e., 07), the usual combining rule is  j7=
57
+ a)
($.24)
This equation is exact for the hardsphere potential, and approximate for all other potential functions. The following combining rule is commonly used for the unlike energy parameter in a twoparameter potential (such as the LennardJones potential):
€y2 = /€1€2(1 — 12)
(8.25)
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. 8.23, 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 atomatom potential, such as the potentials discussed above, with parameters specific to each type of atomatom interaction. In this model, the interaction between two molecules is the sum of all atomatom interactions. For example, for a diatomic molecule (such as hydrogen chloride) shown in Fig. 8.31, for each centerofmass 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 sitesite 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 8.31 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 sitesite 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. 8.24 and 8.25 to
for the unlike CH;+
OH
interaction. This is the
sitesite 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 sitesite interaction parameters.
An extreme example of the sitesite 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 squarewell and LennardJones 126 parameters were given in Table 8.11 for molecules such as CF,, CHy, nitrogen, carbon dioxide, npentane, and neopentane (2,2dimethy] propane). However, there are several caveats in using these parameters. First, as expected, the use of a singlesite 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 npentane. 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 squarewell and LennardJones 126 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. 8.32 and use the following notation: (0), A, d2 — @)) = sin @) sin @2 cos @y2 — 2 cos dj cos do
(8.33)
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 8.32 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 —
(8.34)
Stockmayer Potential (LennardJones 126 potential + dipole) 12 u(r, 0). 4, do — d)) = 485 (2)
6
Tp
— (—)
—
80) . 42, @2 — 1)
(8.35)
It has been shown? that by a Boltzmannfactor orientation averaging of this potential, one obtains (u(r)) =
[ [u(r Oy, 02, G2 — Gye
MP
P2PI/IET sin A) sin A> dO, dO, db ddr
f { e 4.81 .02,.02O1 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 (8.37a)
= (2)
(u(r)) = e(T) (2) r
r
with
yt
3
(0) =66(1+ Terese)
6

= 00  —a (7)
and
[2kTey,e*
(8.37b)
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 LennardJones 126 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, atomatom
particles,
is more
problematic.
While,
or sitesite 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 hardsphere model may be sufficient—but with a diameter characteristic of the macromolecule.®
°J. H. Bae and T. M.
Another possibility is to use the squarewell
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 8.33 Typical
SquareWell
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 sitesite interactions are involved, the potential will look very different than for an atomatom interaction. To be specific, for a 3 A atom with the squarewell] 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 atomatom interactions involved and the different separation distances, the well will be deeper and the centertocenter 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. 8.33 in which the separation distance is in A. In Fig. 8.33b the macromolecule diameter is 30 A; the range of the well remains that for atomatom 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 hardsphere than the typical squarewell 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. (8.41)
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 7.91) 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 higherorder terms in the expansion can extend the range of applicability of the virial equation. For example, considering also all closedcycle interactions between three molecules leads to
—— = 1+ B(T)p + Bs(T)p*
(8.42)
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° + ++:
(8.43)
At some point, the series in density 1s truncated, and the higherorder 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)
+.. .. +
(8.44)
Thus, for example, although the equation
e” pt ) (T Bs + ° )p (T By + * )p (T B3 + oT = 1+ Bo(T)p
(8.45)
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?
(8.46)
+ 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")]
(8.47)
'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?
——
— —
a2
E=ait— =
F
— —
— —:C=
— + a16
ays
ay\4
a3
7s
— zityat
7
—
—GCG=
=
ay
ils
cy
ay
—:
ig
—; and
aya
D=a9+ H
=
a\7
wat
—
—; +
ayy
aig
ya t+ —zs
—
(8.48)
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)
(8.49)
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=
Vb
a —=
Vv
(8.410) Pad
RT
where V is the molar volume; other members of this class include the Soave” version
of the RedlichKwong equation!”
pi
**
~V—b
=e
VV +b)
8.411
we
and the PengRobinson equation!! p
RT

r a(y)
¥ VV +b)+b(—b)
Vb
(8.412)
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 socalled 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
($.413)
where the first two terms on the righthand side of the equation are common to the family of cubic equations, and the further terms in the series are specific to each equation. Now, making the following identification with the virial equation of state Cc 6
_.
xix) Boi (1 J=b—
Bo mix(X, F) = Yo
a(t)
RT
(8.414)
i=l j=l This identification
suggests that the term on the right should also be quadratic
in
composition. One way to insure this is with the following mixing rules: c h6c¢ a(r)=
c lu
3
S xjxjaij(T)
‘=
jf=l
and
b=
\
S > xix bij
i=l
j=]
($.415)
This set of equations, originally used by van der Waals, is referred to as the van der Waals onefiuid mixing rules, since the mixture is described by the same equation as the pure fluids—though with compositionaveraged 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. 8.24 and 8.25.)
The mixing rules of Eq. 8.415 are not exact. For example, the single relation of Eq. 8.414 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. 8.414—for example, one could add any composition dependent function f(x) to a and then add f(x)/RT to b, and satisfy Eq. 8.414.'* Also, the composition of the third virial coefficient can be shown to be
c B3 mix(x, T) =
ce ($.416)
Yo xix pre Bs ijx(T)
»
f=1 j=l k=l which is cubic in composition. However, from the van der Waals equation in
c
bB3 mix
{X,
T)
=
my
2
c
 =
7
D
i=l]
?
D
. XX
jp bj;
($.417)
j=
Thus, while the mixing rules of Eq. 8.414 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 onefluid 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 onefluid 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 squarewell potential and combining rules
Use these data to estimate the LennardJones parameters for CHy
00 18 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 LJ 126 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 squarewell 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 LennardJones 126 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 LennardJones parameters and the critical constants were given in Eq. 8.112. Use these expressions to obtain the LennardJones 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 126 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 LennardJones 126 parameters for these substances first by using the critical prop
alternale set of squarewell 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 LennardJones 126 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 LennardJones 26 parameters given in Eqs. 8.112. } The triangularwell 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 termbyterm, 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
triangularwell potential have a maximum function of temperature?
the
as a
can be described by the LennardJones 126 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. 8.13. 8.17 Derive Eqs. 8.37.
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 welldefined 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 threedimensional 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 threedimensional
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
(9.11)
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
(9.12)
Q(N, V,T) = So en BitnvoreT 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
(9.13)
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
(9.14)
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
_
eSn/2T
O(N. V.T) =e
AN
(a7)
(9.15) 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) (9.16)
= 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 (9.17) ,/{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 (9.18)
S(N.V.T)
= 2!I(N,V,T)— , ) ACN, (
V,T 
rr =
3Nk
Gp ¢ 7/? aera
—
In
(
—
—Og/T E
)
(9,19)
and
au
Ory
Cy = &
—S6/T
= 3Nk (=) —____.
ay
NUV
r
(1
es
(9.110)
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
(9,111)
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
(9.113) 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 constantvolume 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/
(9.21)
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 highfrequency modes for the vibrational motion of a single atom, as considered in the Einstein model, to lowfrequency (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
(9.22)
0 With this approximation
ao
InQ(N,V,T)= = + d, In ( li —
EF
aWva/2T
e + 
e(v)

Avf2kT
In (ar)
dv
(9,23)
0
The problem then becomes one of determining the probability distribution g(v). In the Debye model, it is recognized that the lowfrequency (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 threedimensional 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
(10.511)
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
(10.512)
)
10.5 The Ising Model
181
Further dl!
Cy
~
2
x
(sr).
\ar
(ae)
(ex (kT
kT a2 { x a
tg
sor
=4NK7 7)
(
10.513 )
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 onedimensional 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 twodimensional Ising model;'* there is no known solution for the threedimensional Ising model. (As an aside, it is Interesting to note that Onsager was the Gibbs Professor of Theoretical Chemistry at Yale University.) Though the onedimensional 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 nearestneighbor lattice interactions are involved. For example, in this way the Ising model ts related the helixcoil 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 HH interaction energy, €cc is the CC interaction energy, and cy = €yc is the HC or
CH interaction energy. Consider the twomonomer 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)
(10.514)
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)
(10.515)
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 threemonomer 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
(10.516)
From this, expressions can be obtained for the likelihood of finding a threemonomer chain in each of the conformations listed above. A similar analysis can be done, for longer polymer chains (Problem 10.6). The ZimmBragg 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]
(10.517)
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]
(10.518)
10.5 The Ising Model
These are shown
in Fig. 10.51
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?
(
10.519
The degree of helicity is shown in Fig. 10.52.
pp
Figure 10.51 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 10.52 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 liquidliquid 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) RedlichKwong,
and
(c) PengRobinson 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
threemonomer chain D lattice. 10.6 Develop expressions for the partition function and the probability of occurrence of each of the possible conformations of the fourmonomer chain D lattice.
10.7 Develop a simplified ZimmBragg 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. 10.413.
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 highorder 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
(11.11) 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
(11.12)
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. 11.12, 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).
(11.13)
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
14)
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 reducedprobability 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
(11.15)
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 N1 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,
(11.16)
The factors before the integral arises as follows. There are N choices for the first molecule identified by the index 1, N1 remaining choices for the molecule designated by the index 2 (since one of the identical
molecules has already been chosen),
N2
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
(11.17) '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 (11.18a)
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) (11.19a)
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
11.18b)
“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
(11.19b)
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
(11.110) “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 twoparticle 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, pdr,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. I1.110 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
(11.111)
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 twobody 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). Poee Ey) = >) > uly)
is
(11.22)
a
i¥0
ic fan
>
u(Fij)
(11.28)
[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
76]
 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
_

NYNL
kT
193
0
pues
LT
dry 3V00
tat ed
Iy*
ON
...dz*
EN
()
(11.29)
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,
(11.210)
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 
(11.211)
This is the desired relation between the radial distribution function. the twobody 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.  1.18, the pair correlation function is defined as
dradrg
V2 fio. ferment
)
2?) Gv rat PP) = “4 ZN
diy
eee
(11.18a)
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 lowdensity 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)
(7.33)
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. 11.3la and  1].31b for the LennardJones 126 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 11.31 (a) LennardJones 126 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 11.32 The (a) low density and (b) higher density radial distribution for the hardsphere 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. 11.32a. The radial distribution function is more complicated at higher density as shown in Fig. 11.32b.
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 OrnsteinZernike 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, xray 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 atomatom correlation functions. This 1s shown in Fig.
11.41
for HCl,
where
there are three different
intermolecular
correlation
func
tions: the hydrogenhydrogen, chlorinechlorine, and hydrogenchlorine correlation functions. These three correlation functions cannot be obtained from a single xray 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 hydrogenchlorine separation distances shown in this configuration, both contribute to the single hydrogenchlorine 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 PercusYevick, hypernetted chain,
s tion Func tion ribu Dist nal atio igur Conf d. Flui e Dens a in les ecu Mol ing : Interact
Figure 11.41 Atomatom correlation functions for the twocenter 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 pairwiseadditive potential obtained from an approximate theory, and not an analytic expression. A third method of obtaining the radial distribution function is by molecularlevel 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 weightedaveraging 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 moleculardynamics. 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~
)\
(11.52)
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
(11.53) V.f
However, we have already shown that
aa kT
—
Ins ins
so that
6,.212 (6.212) ,
—
—?3
‘—N
=kTVv (
oP
OM"
>)
(11.54) Jy y
From classical thermodynamics, we have
dG =d(uN) = udN + Ndu = VdP — SdT + ptdN
(11.55)
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, , soSO thaat (=4 l)l ‘ =
vr = V
dl
=
\. hog p ie ( )
/
VT ~
VVl
(on
 a on u
V.T
(11.56)
; 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
(11.57)
== (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
(11.58)
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
(11.69)
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
(11.610)
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)
(11.611)
dr
(11.612)
11.610 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 polarspherical 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 (11.613)
+ 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
(11.614) and
so we
obtain
I'(s) = frls) + 2npf%(s) f ler
— 1]
sin(sr)
5
redr + 2xpf%(s) f
sin(sr)
5 rd
(11.615) The interpretation of the terms on the righthand 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 Xray or Neutron Diffraction
Therefore, the final diffraction equation is
f(s)
lis) =
sin(sr
(11,6l6a)
—  ]——— r°dr
+ 2a pf? 0 fu
or, defining the total structure function H(s) i
2
aa
=
= H(s)= ane  ter) 7 qe
(11.616b)
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
(11.617)
Therefore, the radial distribution function g(r)—or, equivalently, the total correlation
function /i(r)—is directly obtainable from an xray 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
(11.618)
0
Now, we have previously shown (Eq. 1 1.57) 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
(11.57)
)
where k is the Boltzmann constant. Consequently, I dp His (x =0) = 0} = ; kT (so)
— ji
i (11.619)
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 11.62,
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. 11.617 is shown in Fig. 11.64.° 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 lowerthanaverage
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 126 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 onefluid model.) 11.4 Draw the three lowdensity radial distribution functions for a binary mixture of LennardJones 126 fluids for which 11.5
€), = 2622 and oj; = o3, assuming the LorentzBerthelot combining rules of Eqs. 8.24 and 8.25 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 OrnsteinZernike integral equation * Understand the origin of the PercusYevick equation for hardspheres
12.1
THE
YVONBORNGREEN 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(rwi.y)
J lar
d e y c e MAT dr ...ddery
oe HUE ay
us
However, from Eq. 11.19a vs
f..
Je
—ufkt
ry...dry
Toft
te
Ss
gO (ry, ry. hy;
(11.19a)
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
(12.15)
where we have neglected the difference between N and N2. 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.
(12.16) This is known as the equation of YvonBornGreen, derived this integrodifferential equation.
each of whom
independently
The YvonBornGreen (YBG) equation relates the unknown twobody correlation function to the even less known threebody correlation function. So to proceed, one needs some information about the threebody 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
(12.17)
as p —> O
(12.18)
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. 12.18 using the Mayer cluster expansion
(see Section
11.3 and Eg.
11.35).
The Kirkwood Superposition Approximation
12.2
APPROXIMATION
SUPERPOSITION
KIRKWOOD
219
12.16 for the twoparticle radial distribu
At higher densities we cannot solve Eq.
tion function gr) .F95 T. ep), because this equation also contains the threeparticle 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) (1221) 3
2
.
a
2
or that the threeparticle distribution is the product of all twoparticle 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 (12.22)
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,
(12.23)

This is an integrodifferential equation for the twoparticle 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. 12.16) 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. 12.21) about the threeparticle distribution function in order to be able to solve for the twoparticle 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. 12.16, would now give an integrodifferential 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 threeparticle distribution functions) and solve the resulting equation for g')(r).r5,1r3: T. p).
which
would
be an improvement
over Eq.
12.21.
The
resulting expression
for
g(r), ro, 3: T, e)would then be used in Eq. 12.16 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 integrodifferential 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. 12.16 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
lowerorder (i.e., twoparticle) distribution, one should not make the superposition approximation about that distribution function, but rather about some higherorder particle distribution such as the threeparticle, fourparticle, or higherorder 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 twoparticle distribution function will be.
JIRNSTEINZERNIKE
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 lowdensity result of a Boltzmann factor in the interaction energy), and then an indirect effect as a result of the correlations between intervening single molecules (132) and chains of molecules (1342, ]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)
— ]
(12.31)
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 OrnsteinZernike 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
(12,32) N2) 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 (1342, 13452, 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 +++ , +>: (12.33) It is easy to show that this last equation is equivalent to hir, ° F>)
=
cr,
: rs)
+
Pp /
e(r,
‘ ra)h
(rs,
(12.34)
ry) dr,
which is the OrnsteinZernike equation.' The way that one proves the equivalence between Eqs. 12.33 and 12.34 is by repeatedly substituting Eq. 12.33 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,
(12.35)
Repeating this process leads to Eq. 12.34. 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 OrnsteinZernike 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
ORNSTEINZERNIKE
OrnsteinZernike
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
hardsphere molecule or some other distance measure for soft interaction potentials. Another
commonly
used
closure
is the
PercusYevick
(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)
(12.42)
and when used in the OrnsteinZernike 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 (12.43) This
is an
integral
equation
for g(r),
known
as the
PercusYevick
equation
can
be solved (with some difficulty) for the radial distribution function for a specified interaction potential. Table 12.44 lists the results for the radial distribution function of the hardsphere fluid obtained in the PercusYevick 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).
(12.44)
223
12.4 Closures for the OrnsteinZernike 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
_
(12.45)
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 OrnsteinZernike 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
11.517
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 11.211),
(Eq.
and
the
equation
second
of using
is a result
the
same
distribution
function in the compressibility equation (Eq. 11.517). 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 hardsphere 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
(12.52)
This is the CarnahanStarling 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 CarnahanStarling term can be used to represent the hardcore 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 CarnahanStarling equation.
The
first is the radial distribution for hardspheres
this, we start from
Eq.
at contact. To derive
11.211 oo
In , f dur P = pkT — =p?  7 3 dr
a(r)r3dr
(11.211)
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 hardsphere 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
_
pg™(a*)o* 2
hs
=
where o~ indicates a value of cincrementally greater than the hardsphere diameter (and a does not appear since the Boltzmann factor is 0 there). Then using
Qn pkl
_ltnt+n9
3”
(1 —n)
it is simple algebra to show that hs, re
21 = i e e e 13
+
g(a")
(12.53
(Note that e™(a~) = 0.) Other interesting results from the CarnahanStarling 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
(12.54)
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
(12.61)
)
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 (12.62)
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
(12.64)
+ 2a Np f wiryeiryr? dr
NRE
0
This
(12.63)
u(r) = Y> > o xjxjuii(r) ij
mixture
random
onefluid
or
model
and
be
can
used
with any intermolecular potential. Note that if the LennardJones 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

} . '
}
(12.66)
6
ApH FE j (8;
J
In the case of the onefluid 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 LennardJones 126 potential, although not restricted here to that form): u(r)
= EF
(—)
(12.67)
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
(—)
(12.68)
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
(12.69)
 2: Integral Equation Theories for the Radial Distribution Function with
ea
= »
iy
f
(12.610)
i
Equations 2.67 to 12.610 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
(12.71)
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
(12.72)
12.72, 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
(12.73)
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

(12.74)
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
(12.75)
Using this relation in Eq. 11.104 we
dwityts)

1)
pd
7
dryoté' 9
Fit =
d
= kT
—ufkT dr,dr4...dry
2 V [fe
N(N — 1)Z2(N,V,T) (12.76)
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. 12.72. 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
(12.78) 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 twobody potential between two isolated atoms, and in fact does not resemble it. This is evident in Fig. 12.71 that is the potential of mean force (divided by AT) computed from the PercusYevick solution for the hardsphere potential. Note that while for hardspheres
the twobody 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 hardsphere 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. 12.71, 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 hardsphere 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. 12.72. 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. 12.71. 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 hardsphere
Whs(r)
+
Wanlh)
+
Welect (', T,
Pp)
interaction (resulting in an excluded
(12.79)
volume between
the molecules). This effective hardsphere 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 atomatom interactions in
SPW.
Tavares and S. 1. Sandler, AICHE Journal 43. 218
(1997),
The
12.7
Potential of Mean
Force
233
Hardsphere potential of mean force
PY hardsphere radial distribution function
0.5
0
bh
=
_
=
(=
du(rjj)
Interactions N
=——s) Vv aV 4

—
ar;
.
(13.29) .
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
(
(13.210)
>?
13.210 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. 13.21] and the pressure using Eq. 13.210 to determine whether equilibrium has been obtained in the simulation. Also, it is easy to show (see Problem
13.4) that
Eq. 11.21] and Eq. 13.210 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 moleculardynamics 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
(13.211)
(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
(13.212)
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 pseudorandom 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 eU ChRT )

_ pre yer e u
(13.31)
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 13.31 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 13.31 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. 13.31; 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
(13.32) 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
(13.33)
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 longrange correction changes. Another very useful method is the socalled 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 vaporliquid 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, vaporliquid 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 squarewell and LennardJones 126 fluids. (The squarewell program can be used for the hardsphere fluid by setting the Ry, parameter in the squarewell 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 MolecularDynamics 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
CULARDYNAMICS
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
(13.42)
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 predictorcorrector 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 (onequadrillionth 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
(13.43)
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 constanttemperature moleculardynamics 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. 13.43 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
(13.44)
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)
(13.45)
In this equation, Ar is the time step used in the numerical integration of Eq. 13.42. 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 (+ — )
(13.46)
#
—
°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 moleculardynamics simulations result in a true constant temperature system, because the temperature will fluctuate during the simulation. It should be noted that the adiabatic moleculardynamics 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 moleculardynamics 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 moleculardynamics simulation MATLAB®’ program for the LennardJones 126 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 126 fluid, Finally the LJLMD_MC program does a Monte Carlo simulation followed by an isothermal moleculardynamics simulation for the LennardJones 26 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 moleculardynamics 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 moleculardynamics 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 twodimensional 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 moleculardynamics
to compute
the radial distribu
distribution of speeds (Eq. 3.95) at one of the con
tion function, internal energy, and pressure for the
ditions in Problem 13.1 and compare the results to
LennardJones
126
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 moleculardynamics program MD_LJ2 is the
those obtained from the moleculardynamics 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 LennardJones  26 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 126 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 squarewell
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. 11.211 and 13.210 are equivalent. 13.5 For a state point of your choice for the hardsphere 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 PercusYevick solution using the program PercusYevick HS. 13.6 Compare the values of radial distribution function for the hardsphere fluid at contact given by valEg. 12.53 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 squarewell 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 squarewell 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 hardsphere 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
SQUAREWELL
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 hardsphere 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 hardsphere
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«
(14.11) 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 hardsphere potential uns)
for tr
oo O
=
r

penn
+a >, 5.
47
dr,...dry
Cy
ryet
afofe

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
(14.22)
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 _— }” Np 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
(14,23)
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 BarkerHenderson 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, hardsphere 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
INMNl1)f
p)dryy = ir
f 
=
dr.
ov
2tpN (r; pyr? f=
trent
— =
rm
fo
Upen(MiD8p. (ria; P) Ary»
 p)r? adr
(14.25)
0
“There is an important
point with
regard to the temperature
dependence of the radial distribution function
for the hardsphere fluid, In general, the twoparticle 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 hardsphere 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 hardsphere fluid, This is to be compared with, for example, the case of the squarewell fluid of Eg. 14.12, 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 squarewell 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
(14.26)
and specifically for the perturbation potential given by Eq. (14.13) Koya
Agw(N, V,T) = Ans(N. V,T)
— 21pNe / g(r: pyr? dr
Or
(14.27a)
k
Aw(N,V,T) — Ans(N.V,T) WET =
NET
e
np Te
f
i:
a
>
(r: p)r*dr
(14.27b)
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
14,28
T
we obtain the following equation of state to firstorder term in the perturbation expansion around the hardsphere 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
(14.29)
14.3
SecondOrder Perturbation Theory
265
irox app d iel nf 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 hardsphere 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 squarewell 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
(14.210)
P(p,T) oo
= Pry(p.T) + 2mp* [2.0
pritpentryr? ar
Row
+ 2rep"
d =; / dp
(14.211)
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 higherorder distribution functions (which can be simplified somewhat by using the Kirkwood superposition approximation, Section 12.2). See Problem 14.9.
NDORDER
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 =
SoaveRedlichKwong 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
PengRobinson N
VW = V ~—b: Ne = Cx(T)p which
leads to P =
with a(7)
atan § ——$—_—___ (exe =) / 2N bp — b? p+
RT
a(T)
Vb
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 CarnahanStarling freevolume 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 CarnahanStarling freevolume 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 freevolume 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.
Molecularlevel 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. 16.23 for the squarewell 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 closepacked
volume
of hardspheres
16.217  (
and
N,,
1s the
coordination number at closepacking (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. 16.24. Based on the accurate expression for the free volume (Eq. 16.23) and for the coordination number model of Eq. 16.117, the following equation of state is obtained for the squarewell 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,
(10218)
Models
6: The Derivation of Thermodynamic
COORDINATION
NUMBER, NV,
éAT = 0.25
Figure 16.23 Coordination number for the squarewell 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
16.27a), the dashed
line ts
the RedlichKwong result, the dashdot line ts the PengRobinson result, and the solid line results from Eq. 16.117. 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 16.24 Coordination number for the squarewell
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. 16.117. Dotted portion of the e/kT =  is in the twophase 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 vaporliquid equilibrium of simple, spherical real fluids by adjusting the values of its parameters, as shown in Fig. 16.25. For interaction potential models other than the squarewell potential, the general procedure is to keep the same structure as above by using the meanvalue 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!
14
lots
lor?
1Q*!
ae
mee
PRESSURE,
BAR
lor
lar?
107!
oe
10!
BAR
(b)
(a)
Figure 16.25 The compressibility 7 = PV/RT
along the vaporliquid equilibrium envelope for (a) argon and
(b) methane; the points are the experimental data and the line results from Eq. 16.118. 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*
_ NNAN, V,T)(uS) — 2
(16.219)
where u(r) is the softcore 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. (16.110), and A* is the range of the soft interaction. So a number of assumptions need to be made. One 1s to choose an effective hardcore 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
\ml
pa?
(=)
nt
(“)
i
leads to
which
RT ——

V
Sl+nt+y7t+r
(=)"
—————
=
(l—ny
Amn
Dy fl
5 (
ney
kT .
16.220
)
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

(16.31)
with
(16.32)
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
'
N°

= VXI SG
uP (r) gi (3
Ny, Nay...
V.)dr
Re, (16.33)
where
R7, is the range of the ¢ — j interaction (the well region for the squarewell
,
potential), and
kT —— = No,....V.T) Prix(N,.

N
No,...,V.T (Ni, 7) Ur .V, No, mux Unx(N1, kT?
dT
(16.34)
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 ss 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
(16.35)
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 freevolume 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
(16.35a)
T
IDmix(Ni, No,...,V.7 P™2(N,V,T) = —N (Se) av
V,T)+
(16.35b) Nod
To begin, we again consider the simplest case of the squarewell potential, so that res
of
(Ny.
No,
eee,
VT)
Ny Ni
_
[ Ri
=
—&i;
N Ni; ”et
;
siren
Treat
V,
l)dr
*
(16.36a)
where £;; = €;; 1s the well depth of an jj 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 (16.37)
T)dr
V,
ones
N3,
Ni,
Bij (Fs
/
=
Re To proceed, one has to specify either each of the speciesspecies 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 squarewell 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?,) =—
(16.36b)
*
:
where u?.(r) is the softcore part of the interaction potential, and (u3) is an appropriately chosen average value for the ij interaction. For the squarewell potential, (u?.) = —«;;. In what follows, we will use ¢;;, but the results are easily generalized by replacing it with —{u?.). Table 16.31 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 hardcore part of the squarewell 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 freevolume term. For example, the CarnahanStarling 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;
(16.311)
terms are close to unity and —_— UR},
: — 7 XjO7 (RF J
a
I)
— 1)
(16.312)
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 lowdensity 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
(16.313)
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 lowdensity 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 speciesspecies 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
(16.314) 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
(16.315)
“
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 onefiuid mixing rules and
B= » Y > xix) Bij a
a=
» So xix) ai) ~
(16.317)
Other choices for the speciesspecies (or local) coordination numbers result in other equations of state, as shown in Table 16.32. 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
(16.318a)
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
(16.318b)
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 16.32 Local that Result Equation of state mixing rule Local coordination number —
van der Waals (vdW)
Cc
Ni
Y
1fluid
vdW
ie it y
1fluid
z
N; —C, jes kT V
vdW Ifluid
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
(16.318c)
and a model based on the simple extension of Eq. 16.11 iN, 1 Vo i
aim
Moist
egg f2ke
TV + Voi (e 747 — 1) with an effective coordination number
(16.318d)
Voi; = No /v2
with
N,, = 18 (for Roy = 1.5).
Computer simulation'' can be used to test the accuracy of these approximations.
Figure
16.31
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 squarewell
potential at low density it would be 4 3 o:.ik:.
lim @;; = Si
po
—
]
Ri De;
a (RF, — fj
Egger
(16.319)
Another test of the models ts to look at the total coordination number for the squarewell system with €23 = shown in Fig. 16.32.
1.2k7 = 2e,,, but varying diameter ratios and densities as
The main conclusion from the simulation results, especially Fig. 16.31, 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 16.31 The ratios of the local compositions to the bulk compositions @)> and @5, as a function of reduced density for the squarewell 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 dashdouble dotted line is the model of Eq. 16.318a small dotted line is the mode! of Eg. 16.318b, the dashed line is the model of Eq. 16.318c, 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
16.32 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)
(16.41)
The first term on the righthand side of this last equation is the athermal contribution to the excess Helmholtz energy due to the hardcore molecular size and shape differences between the species in the mixture (since all softcore forces are unimportant at high temperatures); it is entirely entropic. The second term is the contribution from other than the hardcore interaction, resulting from the attractive and softcore
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)
(16.42)
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 theoreticallybased 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 hardcore 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 FloryHuggins'*
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 GuggenheimStaverm a
=
)
ob; I
i
, 5, ...V)
A,
sent
ex ent T.V
x; ol
P
fractions — + N; In
kT
6;
(16.43b)
>— 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,


(
16.44
 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)
(1645)
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, sB 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
(16.46)
2
kT
f=oc
=i.
JTS kT
Dix
oN
ric
Vy,
ees
seer
No
(Ni.
kT
;(N;. Vi. 7) =
and
De
2
,
f
