Main Physical Chemistry

Physical Chemistry

In this third edition, core applications have been added along with more recent developments in the theories of chemical reaction kinetics and molecular quantum mechanics, as well as in the experimental study of extremely rapid chemical reactions. * Fully revised concise edition covering recent developments in the field * Clear and comprehensive text ideal for undergraduate and graduate course study * Encourages readers to apply theory in practical situations
Year: 2008
Edition: 3rd ed
Publisher: Academic Press/Elsevier
Language: english
Pages: 1405
ISBN 10: 0123706173
ISBN 13: 9780080878591
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Physical Chemistry
Third Edition

Physical
Chemistry
Third Edition

Robert G. Mortimer
Professor Emeritus
Rhodes College
Memphis, Tennessee

AMSTERDAM • BOSTON • HEIDELBERG • LONDON
NEW YORK • OXFORD • PARIS • SAN DIEGO
SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
Academic Press is an imprint of Elsevier

Cover Design: Eric DeCicco
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Library of Congress Catalog-in-Publishing Data
Mortimer, Robert G.
Physical chemistry / Robert G. Mortimer. – 3rd ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-12-370617-1 (hardcover : alk. paper)
1. Chemistry, Physical and theoretical. I. Title.
QD453.2.M67 2008
541–dc22
2008007675
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN-13: 978-0-12-370617-1
For information on all Elsevier Academic Press publications
visit our Web site at www.books.elsevier.com

Printed in Canada
08 09 10
9 8 7 6 5 4 3 2 1

To my wife, Ann,
and to my late father, William E. Mortimer,
who was responsible for my taking my first chemistry course

Contents

Periodic Table
Inside front cover
List of Numerical Tables in Appendix A
Inside front cover
Information Tables
Inside back cover
Preface

xv

Acknowledgments

Part 1
Chapter 1

xvii

Thermodynamics and the Macroscopic
Description of Physical Systems
1
The Behavior of Gases and Liquids
3
1.1
Introduction
4
1.2
Systems and States in Physical Chemistry
12
1.3
Real Gases
21
1.4
The Coexistence of Phases and the Critical Point

27

Chapter 2

Work, Heat, and Energy: The First Law of
Thermodynamics
39
2.1
Work and the State of a System
40
2.2
Heat
51
2.3
Internal Energy: The First Law of Thermodynamics
55
2.4
Calculation of Amounts of Heat and Energy Changes
60
2.5
Enthalpy
74
2.6
Calculation of Enthalpy Changes of Processes without Chemical Reactions
81
2.7
Calculation of Enthalpy Changes of a Class of Chemical
Reactions
86
2.8
Calculation of Energy Changes of Chemical Reactions
94

Chapter 3

The Second and Third Laws of Thermodynamics:
Entropy
105
3.1
The Second Law of Thermodynamics and the Carnot Heat
Engine
106
vii

viii

Contents

3.2
3.3
3.4
3.5

The Mathematical Statement of the Second Law:
Entropy
114
The Calculation of Entropy Changes
121
Statistical Entropy
133
The Third Law of Thermodynamics and Absolute
Entropies
139

Chapter 4

The Thermodynamics of Real Systems
151
4.1
Criteria for Spontaneous Processes and for Equilibrium:
The Gibbs and Helmholtz Energies
152
4.2
Fundamental Relations for Closed Simple Systems
158
4.3 Additional Useful Thermodynamic Identities
167
4.4
Gibbs Energy Calculations
175
4.5
Multicomponent Systems
182
4.6
Euler’s Theorem and the Gibbs–Duhem Relation
188

Chapter 5

Phase Equilibrium
199
5.1
The Fundamental Fact of Phase Equilibrium
200
5.2
The Gibbs Phase Rule
202
5.3
Phase Equilibria in One-Component Systems
205
5.4
The Gibbs Energy and Phase Transitions
215
5.5
Surfaces in One-Component Systems
222
5.6
Surfaces in Multicomponent Systems
230

Chapter 6

The Thermodynamics of Solutions
237
6.1
Ideal Solutions
238
6.2
Henry’s Law and Dilute Nonelectrolyte Solutions
6.3 Activity and Activity Coefficients
258
6.4
The Activities of Nonvolatile Solutes
267
6.5
Thermodynamic Functions of Nonideal Solutions
6.6
Phase Diagrams of Nonideal Mixtures
282
6.7
Colligative Properties
292

248

275

Chapter 7

Chemical Equilibrium
303
7.1
Gibbs Energy Changes and the Equilibrium
Constant
304
7.2
Reactions Involving Gases and Pure Solids or Liquids
310
7.3
Chemical Equilibrium in Solutions
315
7.4
Equilibria in Solutions of Strong Electrolytes
328
7.5
Buffer Solutions
331
7.6
The Temperature Dependence of Chemical Equilibrium.
The Principle of Le Châtelier
335
7.7
Chemical Equilibrium and Biological Systems
343

Chapter 8

The Thermodynamics of Electrochemical Systems
351
8.1
The Chemical Potential and the Electric Potential
352
8.2
Electrochemical Cells
354
8.3
Half-Cell Potentials and Cell Potentials
361
8.4
The Determination of Activities and Activity Coefficients
of Electrolytes
371
8.5
Thermodynamic Information from Electrochemistry
374

ix

Contents

Part 2
Chapter 9

Dynamics

381

Gas Kinetic Theory: The Molecular Theory of Dilute Gases at
Equilibrium
383
9.1
Macroscopic and Microscopic States of Macroscopic
Systems
384
9.2 A Model System to Represent a Dilute Gas
386
9.3
The Velocity Probability Distribution
394
9.4
The Distribution of Molecular Speeds
405
9.5
The Pressure of a Dilute Gas
411
9.6
Effusion and Wall Collisions
416
9.7
The Model System with Potential Energy
418
9.8
The Hard-Sphere Gas
422
9.9
The Molecular Structure of Liquids
434

Chapter 10 Transport Processes
441
10.1 The Macroscopic Description of Nonequilibrium
States
442
10.2 Transport Processes
444
10.3 The Gas Kinetic Theory of Transport Processes in HardSphere Gases
460
10.4 Transport Processes in Liquids
467
10.5 Electrical Conduction in Electrolyte Solutions
475
Chapter 11

Chapter 12

Chapter 13

The Rates of Chemical Reactions
485
11.1 The Macroscopic Description of Chemical Reaction
Rates
486
11.2 Forward Reactions with One Reactant
488
11.3 Forward Reactions with More Than One Reactant
11.4 Inclusion of a Reverse Reaction. Chemical
Equilibrium
507
11.5 A Simple Reaction Mechanism: Two Consecutive
Steps
510
11.6 Competing Reactions
513
11.7 The Experimental Study of Fast Reactions
515
Chemical Reaction Mechanisms I: Rate Laws and
Mechanisms
523
12.1 Reaction Mechanisms and Elementary Processes in
Gases
524
12.2 Elementary Processes in Liquid Solutions
527
12.3 The Temperature Dependence of Rate Constants
12.4 Reaction Mechanisms and Rate Laws
540
12.5 Chain Reactions
556

499

533

Chemical Reaction Mechanisms II: Catalysis and Miscellaneous
Topics
565
13.1 Catalysis
566
13.2 Competing Mechanisms and the Principle of Detailed
Balance
583
13.3 Autocatalysis and Oscillatory Chemical Reactions
585
13.4 The Reaction Kinetics of Polymer Formation
589

x

Contents

13.5
13.6

Part 3

Nonequilibrium Electrochemistry
595
Experimental Molecular Study of Chemical Reaction
Mechanisms
608

The Molecular Nature of Matter

617

Chapter 14

Classical Mechanics and the Old Quantum Theory
14.1 Introduction
620
14.2 Classical Mechanics
621
14.3 Classical Waves
629
14.4 The Old Quantum Theory
640

Chapter 15

The Principles of Quantum Mechanics. I. De Broglie Waves and
the Schrödinger Equation
653
15.1 De Broglie Waves
654
15.2 The Schrödinger Equation
657
15.3 The Particle in a Box and the Free Particle
663
15.4 The Quantum Harmonic Oscillator
674

Chapter 16

The Principles of Quantum Mechanics. II. The Postulates of
Quantum Mechanics
683
16.1 The First Two Postulates of Quantum Mechanics
684
16.2 The Third Postulate. Mathematical Operators and Mechanical
Variables
684
16.3 The Operator Corresponding to a Given Variable
688
16.4 Postulate 4 and Expectation Values
696
16.5 The Uncertainty Principle of Heisenberg
711
16.6 Postulate 5. Measurements and the Determination of the
State of a System
717

Chapter 17

The Electronic States of Atoms. I. The Hydrogen Atom
725
17.1 The Hydrogen Atom and the Central Force System
726
17.2 The Relative Schrödinger Equation. Angular
Momentum
729
17.3 The Radial Factor in the Hydrogen Atom Wave Function.
The Energy Levels of the Hydrogen Atom
736
17.4 The Orbitals of the Hydrogen-Like Atom
741
17.5 Expectation Values in the Hydrogen Atom
749
17.6 The Time-Dependent Wave Functions of the HydrogenAtom
17.7 The Intrinsic Angular Momentum of the Electron.
“Spin”
755

Chapter 18

619

The Electronic States ofAtoms. II. The Zero-OrderApproximation
for Multielectron Atoms
763
18.1 The Helium-Like Atom
764
18.2 The Indistinguishability of Electrons and the Pauli Exclusion
Principle
766
18.3 The Ground State of the Helium Atom in Zero Order
768
18.4 Excited States of the Helium Atom
772
18.5 Angular Momentum in the Helium Atom
774

753

xi

Contents

18.6 The Lithium Atom
781
18.7 Atoms with More Than Three Electrons

784

Chapter 19

The Electronic States of Atoms. III. Higher-Order
Approximations
789
19.1 The Variation Method and Its Application to the Helium
Atom
790
19.2 The Self-Consistent Field Method
796
19.3 The Perturbation Method and Its Application to the Ground
State of the Helium Atom
799
19.4 Excited States of the HeliumAtom. Degenerate Perturbation
Theory
803
19.5 The Density Functional Method
805
19.6 Atoms with More Than Two Electrons
806

Chapter 20

The Electronic States of Diatomic Molecules
823
20.1 The Born–Oppenheimer Approximation and the Hydrogen
Molecule Ion
824
20.2 LCAOMOs.Approximate Molecular Orbitals ThatAre Linear
Combinations of Atomic Orbitals
833
20.3 Homonuclear Diatomic Molecules
838
20.4 Heteronuclear Diatomic Molecules
851

Chapter 21

The Electronic Structure of Polyatomic Molecules
867
868
21.1 The BeH2 Molecule and the sp Hybrid Orbitals
871
21.2 The BH3 Molecule and the sp2 Hybrid Orbitals
21.3 The CH4 , NH3 , and H2 O Molecules
and the sp3 Hybrid Orbitals
873
21.4 Molecules with Multiple Bonds
878
21.5 The Valence-Bond Description of Polyatomic Molecules
21.6 Delocalized Bonding
885
21.7 The Free-Electron Molecular Orbital Method
892
21.8 Applications of Symmetry to Molecular Orbitals
894
21.9 Groups of Symmetry Operators
896
21.10 More Advanced Treatments of Molecular Electronic
Structure. Computational Chemistry
904

881

Chapter 22

Translational, Rotational, and Vibrational States of Atoms and
Molecules
915
22.1 The Translational States of Atoms
916
22.2 The Nonelectronic States of Diatomic Molecules
919
22.3 Nuclear Spins and Wave Function Symmetry
930
22.4 The Rotation and Vibration of Polyatomic
Molecules
933
22.5 The Equilibrium Populations of Molecular States
942

Chapter 23

Optical Spectroscopy and Photochemistry
949
23.1 Emission/Absorption Spectroscopy and Energy Levels
23.2 The Spectra of Atoms
959
23.3 Rotational and Vibrational Spectra of Diatomic
Molecules
961
23.4 Electronic Spectra of Diatomic Molecules
972

950

xii

Contents

23.5
23.6
23.7
23.8
Chapter 24

Part 4

Spectra of Polyatomic Molecules
975
Fluorescence, Phosphorescence, and Photochemistry
Raman Spectroscopy
985
Other Types of Spectroscopy
991

979

Magnetic Resonance Spectroscopy
1001
24.1 Magnetic Fields and Magnetic Dipoles
1002
24.2 Electronic and Nuclear Magnetic Dipoles
1006
24.3 Electron Spin Resonance Spectroscopy
1010
24.4 Nuclear Magnetic Resonance Spectroscopy
1014
24.5 Fourier Transform NMR Spectroscopy
1024

The Reconciliation of the Macroscopic and Molecular
Theories of Matter
1037

Chapter 25

Equilibrium Statistical Mechanics I. The Probability
Distribution for Molecular States
1039
25.1 The Quantum Statistical Mechanics of a Simple Model
System
1040
25.2 The Probability Distribution for a Dilute Gas
1047
25.3 The Probability Distribution and the Molecular Partition
Function
1055
25.4 The Calculation of Molecular Partition Functions
1064

Chapter 26

Equilibrium Statistical Mechanics. II. Statistical
Thermodynamics
1081
26.1 The Statistical Thermodynamics of a Dilute Gas
1082
26.2 Working Equations for the Thermodynamic Functions of a
Dilute Gas
1089
26.3 Chemical Equilibrium in Dilute Gases
1101
26.4 The Activated Complex Theory of Bimolecular Chemical
Reaction Rates in Dilute Gases
1106
26.5 Miscellaneous Topics in Statistical
Thermodynamics
1116

Chapter 27

Equilibrium Statistical Mechanics. III. Ensembles
1121
27.1 The Canonical Ensemble
1122
27.2 Thermodynamic Functions in the Canonical
Ensemble
1128
27.3 The Dilute Gas in the Canonical Ensemble
1130
27.4 Classical Statistical Mechanics
1133
27.5 Thermodynamic Functions in the Classical Canonical
Ensemble
1141
27.6 The Classical Statistical Mechanics of Dense Gases and
Liquids
1147

Chapter 28

The Structure of Solids, Liquids, and Polymers
28.1 The Structure of Solids
1154
28.2 Crystal Vibrations
1162
28.3 The Electronic Structure of Crystalline Solids
28.4 Electrical Resistance in Solids
1179

1153

1171

xiii

Contents

28.5 The Structure of Liquids
1184
28.6 Approximate Theories of Transport Processes in
Liquids
1188
28.7 Polymer Conformation
1194
28.8 Polymers in Solution
1198
28.9 Rubber Elasticity
1200
28.10 Nanomaterials
1205
Appendices
1209
A. Tables of Numerical Data
1209
B. Some Useful Mathematics
1235
C. A Short Table of Integrals
1257
D. Some Derivations of Formulas and Methods
1261
E. Classical Mechanics
1267
F. Some Mathematics Used in Quantum Mechanics
1275
G. The Perturbation Method
1283
H. The Hückel Method
1289
I. Matrix Representations of Groups
1293
J. Symbols Used in This Book
1303
K. Answers to Numerical Exercises and Odd-Numbered
Numerical Problems
1309
Index

1351

Preface

This is the third edition of a physical chemistry textbook designed for a two-semester
undergraduate physical chemistry course. The physical chemistry course is often the
first opportunity that a student has to synthesize descriptive, theoretical, and mathematical knowledge about chemistry into a coherent whole. To facilitate this synthesis, the book is constructed about the idea of defining a system, studying the states
in which it might be found, and analyzing the processes by which it can change
its state.
The book is divided into four parts. The first part focuses on the macroscopic
properties of physical systems. It begins with the descriptive study of gases and liquids,
and proceeds to the study of thermodynamics, which is a comprehensive macroscopic
theory of the behavior of material systems. The second part focuses on dynamics,
including gas kinetic theory, transport processes, and chemical reaction kinetics. The
third part presents quantum mechanics and spectroscopy. The fourth part presents the
relationship between molecular and macroscopic properties of systems through the
study of statistical mechanics. This theory is applied to the structure of condensed
phases. The book is designed so that the first three parts can be studied in any order,
while the fourth part is designed to be a capstone in which the other parts are integrated
into a cohesive whole.
In addition to the standard tables of integrals and numerical values of various
properties, the book contains several appendices that expand on discussions in the body
of the text, such as more detailed discussions of perturbation theory, group theory, and
several mathematical topics. Each chapter begins with a statement of the principal facts
and ideas that are presented in the chapter. There is a summary at the end of each chapter to assist in synthesizing the material of each chapter into a coherent whole. There
are also marginal notes throughout the chapters that present biographical information
and some comments. Each chapter contains examples that illustrate various kinds of
calculations, as well as exercises placed within the chapter. Both these exercises and
the problems at the end of each section are designed to provide practice in applying
techniques and insights obtained through study of the chapter.
Answers to all of the numerical exercises and to the odd-numbered numerical
problems are placed in Appendix K. A solutions manual, with complete solutions
to all exercises and all odd-numbered problems, is available from the publisher. An
instructor’s manual with solutions to the even-numbered problems is available on-line
to instructors. The instructor can choose whether to allow students to have access to
the solutions manual, but can assign even-numbered problems when he or she wants
the students to work problems without access to solutions.
xv

xvi

Preface

The author encourages students and instructors to comment on any part of the book;
please send comments and suggestions to the author’s attention.
Robert G. Mortimer
2769 Mercury St.
Bartlett, TN 38134, USA

Acknowledgments

The writing of the first edition of this book was begun during a sabbatical leave from
Rhodes College, and continued during summer grants from the Faculty Development
Committee of Rhodes College. It is a pleasure to acknowledge this support.
It has been my pleasure to have studied with many dedicated and proficient teachers,
and I acknowledge their influence, example, and inspiration. I am also grateful for the
privilege of working with students, whose efforts to understand the workings of the
physical universe make teaching the most desirable of all professions.
I have benefited from the expert advice of many reviewers. These include:
Jonas Goldsmith
Jason D. Hofstein
Daniel Lawson
Jennifer Mihalick
Cynthia M. Woodbridge

Bryn Mawr College
Sienna College
University of Michigan–Dearborn
University of Wisconsin–Oshkosh
Hillsdale College

and the reviewers of the previous editions.All of these reviewers gave sound advice, and
some of them went beyond the call of duty in searching out errors and unclarities and
in suggesting remedies. The errors that remain are my responsibility, not theirs.
I wish to thank the editorial staff of Elsevier/Academic Press for their guidance
and help during a rather long and complicated project, and also wish to thank Erica
Ellison, who was a valuable consultant. I thank my wife, Ann, for her patience, love,
and support during this project.

xvii

1

Thermodynamics and the Macroscopic
Description of Physical Systems

1

The Behavior of Gases and Liquids

PRINCIPAL FACTS AND IDEAS

1. The principal goal of physical chemistry is to understand the properties
and behavior of material systems and to apply this understanding in
useful ways.
2. The state of a system is specified by giving the values of a certain number
of independent variables (state variables).
3. In an equilibrium one-phase fluid system of one substance, three
macroscopic variables such as temperature, volume, and amount of
substance can be independent variables and can be used to specify the
macroscopic equilibrium state of the system. At least one of the variables
used to specify the state of the system must be proportional to the size of
the system (be extensive). Other macroscopic variables are mathematical
functions of the independent variables.
4. The intensive state, which includes only intensive variables (variables
that are independent of the size of the system), is specified by only two
variables in the case of an equilibrium one-phase fluid system of one
substance.
5. Nonideal gases and liquids are described mathematically by various
equations of state.
6. The coexistence of phases can be described mathematically.
7. The liquid–gas coexistence curve terminates at the critical point, beyond
which there is no distinction between liquid and gas phases.
8. The law of corresponding states asserts that in terms of reduced variables,
all substances obey the same equation of state.

3

4

1 The Behavior of Gases and Liquids

1.1

Antoine Laurent Lavoisier, 1743–1794,
was a great French chemist who was
called the “father of modern chemistry”
because of his discovery of the law of
conservation of mass. He was
beheaded during the French Revolution
because of his involvement in his
father-in-law’s firm, which was
employed by the royal government to
collect taxes. It is said that he arranged
with a friend to observe his head to see
how long he could blink his eyes after
his head was severed. He blinked for
15 seconds.
Joseph Proust, 1754–1826, was a
French chemist who was the first to
isolate sugar from grapes.
John Dalton, 1766–1844, was an
English schoolmaster and chemist.
After he became a famous chemist, he
continued to teach at what we would
now call the elementary school level.

Galileo Galilei, 1564–1642, was a great
Italian mathematician and physicist. He
refuted the assertion of Aristotle that a
heavier object should fall faster than a
lighter one and is said to have dropped
two balls of different masses from the
leaning tower of Pisa to demonstrate
that they fell at the same rate. He
supported the hypothesis of Copernicus
that the earth revolves around the sun
and was convicted of heresy in 1633
by the Roman Catholic Church for this
belief. He spent the rest of his life under
house arrest.

Introduction
This book is a textbook for a standard two-semester physical chemistry course at the
undergraduate level. Physical chemistry involves both physics and chemistry. Physics
has been defined as the study of the properties of matter that are shared by all substances, whereas chemistry has been defined as the study of the properties of individual substances. Chemistry grew out of the ancient occult art of alchemy, which
involved among other things the attempted transmutation of cheaper materials into
gold. Chemistry began as a completely experimental science. Substances were named
and studied without reference to their molecular structures. Sulfuric acid was called
“oil of vitriol,” and chemists memorized the fact that when copper was treated with oil
of vitriol a solution of “blue vitriol” (now known as copper(II) sulfate) resulted. In the
late 18th century, Lavoisier established the law of conservation of mass in chemical
reactions, and Proust established the law of definite proportion. In order to explain
these laws, Dalton proposed his atomic theory in 1803, as well as announcing the
law of multiple proportions. With this theory, chemistry could evolve into a molecular
science, with properties of substances tied to their molecular structures.

Systems
We call any object that we wish to study our system. A large system containing many
atoms or molecules is called a macroscopic system, and a system consisting of a single
atom or molecule is called a microscopic system. We consider two principal types of
properties of systems. Macroscopic properties such as temperature and pressure apply
only to a macroscopic system and are properties of the whole system. They can be
observed and studied without reference to the molecular nature of matter. Microscopic
properties such as kinetic energy and momentum are mechanical in nature. They apply
to either macroscopic or microscopic systems.
The study of macroscopic properties involves thermodynamics, which is the major
topic of this volume, along with gas kinetic theory, transport processes, and reaction
kinetics. Quantum mechanics, spectroscopy, and statistical mechanics are molecular
topics and are discussed in Parts 3 and 4 of this textbook.

Mathematics in Physical Chemistry
The study of any physical chemistry topics requires mathematics. Galileo once wrote,
“The book of nature is written in the language of mathematics.” We will use mathematics
in two different ways. First, we will use it to describe the behavior of systems without
explaining the origin of the behavior. Second, we will use it to develop theories that
explain why certain behaviors occur. This chapter is an example of the first usage, and
the next chapter is an example of the second usage.
Much of the mathematical education that physical chemistry students have received
has focused on mathematical theory rather than on practical applications. A student
who was unable to apply an elementary calculus technique once said to the author,
“I know that was in the calculus course, but nobody told me that I would ever have
to use it.” Mathematical theory is not always important in physical chemistry, but you

5

1.1 Introduction

need to be able to apply mathematical methods. There are several books that cover the
application of mathematics to problems in physical chemistry.1
Arithmetic is the principal branch of numerical mathematics. It involves carrying
out operations such as addition, subtraction, multiplication, and division on actual
numbers. Geometry, algebra, and calculus are parts of symbolic mathematics, in which
symbols that represent numerical quantities and operations are manipulated without
doing the numerical operations. Both kinds of mathematics are applied in physical
chemistry.

Mathematical Functions
A mathematical function involves two kinds of variables: An independent variable is
one to which we can assign a value. A mathematical function is a rule that delivers the
value of a dependent variable when values are assigned to the independent variable or
variables. A function can be represented by a formula, a graph, a table, a mathematical
series, and so on. Consider the ideal gas law:
PV  nRT

(1.1-1)

In this equation P represents the pressure of the gas, V represents its volume, n represents the amount of substance in moles, T represents the absolute temperature, and
R stands for the ideal gas constant. The ideal gas law does a good but not perfect
job of representing the equilibrium behavior of real gases under ordinary conditions.
It is more nearly obeyed if the pressure of the gas is made smaller. A gas that is at a
sufficiently low pressure that it obeys the ideal gas law to an adequate approximation
is called a dilute gas. An ideal gas is defined to obey this equation for all pressures
and temperatures. An ideal gas does not exist in the real world, and we call it a model
system. A model system is an imaginary system designed to resemble some real system.
A model system is useful only if its behavior mimics that of a real system to a useful
degree and if it can be more easily analyzed than the real system.
We can solve the ideal gas law for V by symbolically dividing by P:
V 

nRT
P

(1.1-2)

The right-hand side of Eq. (1.1-2) is a formula that represents a mathematical function.
The variables T , P, and n are independent variables, and V is the dependent variable.
If you have the numerical values of T , P, and n, you can now carry out the indicated
arithmetic operations to find the value of V . We can also solve Eq. (1.1-1) for P by
symbolically dividing by V :
P

nRT
V

(1.1-3)

We have now reassigned V to be one of the independent variables and P to be the
dependent variable. This illustrates a general fact: If you have an equation containing

1 Robert G. Mortimer, Mathematics for Physical Chemistry, 3rd ed., Academic Press, San Diego, CA,

U.S.A., 2005; James R. Barrante, Applied Mathematics for Physical Chemistry, 3rd ed., Pearson Prentice Hall,
Upper Saddle River, NJ, 2004; Donald A. McQuarrie, Mathematical Methods for Scientists and Engineers,
University Science Books, 2003.

6

1 The Behavior of Gases and Liquids

several variables, you can manipulate the equation symbolically to turn any one of
them into the dependent variable.
The ideal gas law might not be accurate enough for some gases under some conditions. If so, we can find some other function that will give the value of the pressure to
greater accuracy. It is an experimental fact that the pressure of a gas or liquid of one
substance at equilibrium is given by a function that depends on only three independent
variables. We represent such a function by
P  P(T , V , n)

(1.1-4)

A mathematician would write P  f (T , V , n) for the functional relation in Eq. (1.1-4),
using the letter P for the variable and the letter f for the function. Chemists have
too many variables to use two letters for each variable, so we use the same letter for
the variable and the function. A functional relation that relates P, V , T , and n for a
gas or a liquid at equilibrium is called an equation of state and is said to represent
the volumetric behavior of the gas or liquid. We will introduce several equations of
state later in this chapter.

E X A M P L E 1.1
Assume that the volume of a liquid is a linearly decreasing function of P, is a linearly
increasing function of T , and is proportional to n. Write a formula expressing this functional
relationship.

Solution
Let V0 represent the volume at some reference temperature T0 , some reference pressure P0 ,
and some reference amount of substance n0 .
V  V0

n
[1 − k(P − P0 ) + a(T − T0 )]  nVm0 [1 − k(P − P0 ) + a(T − T0 )]
n0

where k and a are constants and where Vm represents the molar volume, equal to V /n, and
Vm0 represents V0 /n0 .

A two-dimensional graph can represent a function of one independent variable.
You plot the value of the independent variable on the horizontal axis and represent
the value of the dependent variable by the height of a curve in the graph. To make a
two-dimensional graph that represents the ideal gas law, we must keep two of the three
independent variables fixed. Figure 1.1a shows a set of graphical curves that represent
the dependence of P on V for an ideal gas for n  1.000 mol and for several fixed
values of T .
A three-dimensional graph can represent a function of two independent variables.
Figure 1.1b shows a perspective view of a graphical surface in three dimensions that
represents the dependence of P on V and T for an ideal gas with a fixed value of n
(1.000 mol). Just as the height of a curve in Figure 1.1a gives the value of P for a
particular value of V , the height of the surface in Figure 1.1b gives the value of P for
a particular value of T and a particular value of V . Such graphs are not very useful for
numerical purposes, but help in visualizing the general behavior of a function of two
independent variables.

7

P

1.1 Introduction

5 3 105
5 3 105
3 3 105

P/Nm22

T 5 1000 K
T 5 500 K

273
3 3 105

T 5 373 K
T 5 273 K

0

1 3 105

373
500

T
1000

0.05

(a)

0.1

0.1
m

0.05
Vm/m3 mol21

V

0

(b)

Figure 1.1 (a) The pressure of an ideal gas as a function of V at constant n and various constant values of T. (b) The pressure of an ideal gas as a function of V and T at
constant n.

A function can also be represented by a table of values. For a function of one
independent variable, a set of values of the independent variable is placed in one column.
The value of the dependent variable corresponding to each value of the independent
variable is placed in another column on the same line. A mathematician would say that
we have a set of ordered pairs of numbers. Prior to the advent of electronic calculators,
such tables were used to represent logarithms and trigonometric functions. Such a
table provides values only for a finite number of values of the independent variable,
but interpolation between these values can be used to obtain other values.

Units of Measurement
The values of most physical variables consist of two parts, a number and a unit of measurement. Various units of measurement exist. For example, the same distance could
be expressed as 1.000 mile, 1609 meters, 1.609 kilometer, 5280 feet, 63360 inches,
1760 yards, 106.7 rods, 8.000 furlongs, and so on. A given mass could be expressed
as 1.000 kilogram, 1000 grams, 2.205 pounds, 0.1575 stone, 195.3 ounces, and so on.
There are sets of units that are consistent with each other. For example, pounds are
used with feet, kilograms are used with meters, and grams are used with centimeters.
Here is an important fact: To carry out any numerical calculation correctly you must
express all variables with consistent units. If any quantities are expressed in inconsistent units, you will almost certainly get the wrong answer. In September 1999, a space
probe optimistically named the “Mars Climate Orbiter” crashed into the surface of
Mars instead of orbiting that planet. The problem turned out to be that some engineers
had used “English” units such as feet and pounds, while physicists working on the same
project had used metric units such as meters and kilograms. Their failure to convert
units correctly caused the loss of a space vehicle that cost many millions of U.S. dollars.
In another instance, when a Canadian airline converted from English units to metric
units, a ground crew that was accustomed to English units incorrectly calculated how
much fuel in kilograms to put into an airliner for a certain flight. The airplane ran out of

8

The newton is named for Sir Isaac
Newton, 1642–1727, the great English
mathematician and physicist who
invented classical mechanics and who
was one of the inventors of calculus.
The pascal is named for Blaise Pascal,
1623–1662, a famous French
philosopher, theologian, and
mathematician.

The joule is named for James Prescott
Joule, 1818–1889, a great English
physicist who pioneered in the
thermodynamic study of work, heat,
and energy in a laboratory that he
constructed in his family’s brewery.

1 The Behavior of Gases and Liquids

fuel before reaching its destination. Fortunately, the pilot was able to glide to a former
military air field and make a “deadstick” landing on an unused runway. Some people
who were having a picnic on the runway were fortunately able to get out of the way in
time. There was even a movie made about this incident.
The official set of units that physicists and chemists use is the International System
of Units, or SI units. The letters SI stand for Systeme Internationale, the French name
for the set of units. In this system there are seven base units. The unit of length is the
meter (m). The unit of mass is the kilogram (kg). The unit of time is the second (s). The
unit of temperature is the kelvin (K). The unit of electric current is the ampere (A).
The unit of luminous intensity is the candela (cd). The unit for the amount of a
substance is the mole (mol). The SI units are called MKS (meter-kilogram-second)
units. Prior to 1961, most chemists and some physicists used cgs (centimeter-gramsecond) units, but we now use SI units to avoid confusion.
In addition to the seven base units, there are a number of derived units. The newton
(N) is the SI unit of force:
1 N  1 kg m s−2

(definition)

(1.1-5)

The pascal (Pa) is the SI unit of pressure (force per unit area):
1 Pa  1 N m−2  1 kg m−1 s−2

(definition)

(1.1-6)

We have enclosed these defining equations in boxes, and will enclose the most important
equations in boxes throughout the rest of the book.
A force exerted through a distance is equivalent to an amount of work, which is a
form of energy. The SI unit of energy is the joule (J):
1 J  1 N m  1 kg m2 s−2

(definition)

(1.1-7)

Multiples and submultiples of SI units are indicated by prefixes, such as “milli” for
1/1000, “centi” for 1/100, “deci” for 1/10, “kilo” for 1000, and so on. These prefixes are
listed inside the cover of this book. We do not use double prefixes such as millikilogram
for the gram or microkilogram for the milligram.
We will also use some non-SI units. The calorie (cal), which was originally defined
as the amount of heat required to raise the temperature of 1 gram of water by 1◦ C, is
now defined by:
1 cal  4.184 J

(exactly, by definition)

(1.1-8)

We will use several non-SI units of pressure; the atmosphere (atm), the torr, and the bar.
1 atm  101325 Pa

(exactly, by definition)

(1.1-9)

760 torr  1 atm

(exactly, by definition)

(1.1-10)

1 bar 100000 Pa

(exactly, by definition)

(1.1-11)

The angstrom (Å, equal to 10−10 m or 10−8 cm) has been a favorite unit of length
among chemists, because it is roughly equal to a typical atomic radius. Picometers are
nearly as convenient, with 100 pm equal to 1 Å. Chemists are also reluctant to abandon
the liter (L), which is the same as 0.001 m3 or 1 dm3 (cubic decimeter).

9

1.1 Introduction

The Mole and Avogadro’s Constant
Lorenzo Romano Amadeo Carlo
Avogadro, 1776–1856, was an Italian
lawyer and professor of natural
philosophy. He was the first to postulate
that equal volumes of gases under the
same conditions contained the same
number of molecules.

The formula unit of a substance is the smallest amount of a substance that retains the
identity of that substance. It can be an atom, a molecule, or an electrically neutral set
of ions. A mole of any substance is an amount with the same number of formula units
as the number of atoms contained in exactly 0.012 kg of the 12 C (carbon-12) isotope.
The atomic mass unit (amu or u) is defined such that one atom of 12 C has a mass of
exactly 12 amu. Therefore the mass of a mole of any substance expressed in grams is
numerically equal to the mass of a formula unit expressed in atomic mass units.
The number of formula units, N, in a sample of any substance is proportional to the
amount of substance measured in moles, denoted by n:
N  NAv n

Josef Loschmidt, 1821–1895, was an
Austrian chemist who made various
contributions, including being the first to
propose using two line segments to
represent a double bond and three line
segments to represent a triple bond.

The proportionality constant NAv is called Avogadro’s constant in some countries and
Loschmidt’s constant in others. It is known from experiment to have the value
NAv  6.02214 × 1023 mol−1

(1.1-13)

The ideal gas equation can be written in terms of the number of molecules, N:
V 

Boltzmann’s constant is named for
Ludwig Boltzmann, 1844–1906, an
Austrian physicist who was one of the
inventors of gas kinetic theory and
statistical mechanics.

(1.1-12)

nNAv kB T
NkB T
nRT


P
P
p

(1.1-14)

The ideal gas constant R is known from experiment to have the value 8.3145 J K −1
mol−1 . In common non-SI units, it is equal to 0.082058 L atm K −1 mol−1 . The constant
kB is called Boltzmann’s constant:
kB 

R
8.3145 J K −1 mol−1

 1.3807 × 10−23 J K−1
NAv
6.02214 × 1023 mol−1

(1.1-15)

Problem Solving Techniques
If you have a home repair or automotive repair to do, the work will go better if you
have the necessary tools at hand when you start the job. The same thing is true for
physical chemistry problems. You should analyze the problem and make sure that you
know what formulas and techniques are needed and make sure that you have them at
hand. Think of your supply of formulas and techniques as your tools, and try to keep
your toolbox organized.
One of the most important things in problem solving is that you must use consistent
units in any numerical calculation. The conversion to consistent units is conveniently
done by the factor-label method, which is a straightforward use of proportionality
factors. It is illustrated in the following example, and you can review this method in
almost any general chemistry textbook.

E X A M P L E 1.2
Find the pressure in Pa and in atm of 20.00 g of neon gas (assumed to be ideal) at a temperature
of 0.00◦ C and a volume of 22.400 L.

10

1 The Behavior of Gases and Liquids

Solution
The Celsius temperature differs from the absolute temperature by 273.15 K, but the Celsius
degree is the same size as the kelvin.
T  273.15 K + 0.00◦ C  273.15 K
We convert amount of gas to moles and the volume to m3 using the factor-label method:


1 mol
n  (20.00 g)
 0.9911 mol
20.179 g


1 m3
V  (22.400 L)
 0.022400 m3
1000 L
We now carry out the numerical calculation:


−1 mol−1 (273.15 K)
(0.9911
mol)
8.314
J
K
nRT


P 

V
0.022400 m3
 1.005 × 105 J m−3  1.005 × 105 N m−2  1.005 × 105 Pa
You can see how the symbolic formula is used as a template for setting up the numerical
calculation. The unit conversions can also be included in a single calculation:



20.00 g 8.314 J K −1 mol−1 (273.15 K)  1 mol  1000 L 
P 
(22.400 L)
20.179 g
1 m3
 1.005 × 105 J m−3  1.005 × 105 N m2  1.005 × 105 Pa
The pressure can be expressed in atmospheres by another conversion:


1 atm
P  (1.005 × 105 Pa)
 0.9919 atm
101325 Pa

A calculator displayed 100,486.28725 Pa for the pressure in the previous example.
The answer was then rounded to four digits to display only significant digits. In carrying out operations with a calculator, it is advisable to carry insignificant digits in
intermediate steps in order to avoid round-off error and then to round off insignificant
digits in the final answer. You can review significant digits in any elementary chemistry
textbook. The main idea is that if the calculation produces digits that are probably incorrect, they are insignificant digits and should be rounded away. An important rule is that
in a set of multiplications and divisions, the result generally has as many significant
digits as the factor or divisor with the fewest significant digits.
Another important technique in problem solving is to figure out roughly how large
your answer should be and what its units should be. For example, the author had a
student under time pressure in an examination come up with an answer of roughly
1030 cm for a molecular dimension. A moment’s thought should have revealed that this
distance is greater than the size of the known universe and cannot be correct. Many
common mistakes produce an answer that either has the wrong units or is obviously
too large or too small, and you can spot these errors if you look for them. You should
always write the units on every factor or divisor when setting up a numerical calculation
so that you will be more likely to spot an error in units.

11

1.1 Introduction

E X A M P L E 1.3
The speed of sound in dry air at a density of 1.293 g L−1 and a temperature of 0◦ C is
331.45 m s−1 . Convert this speed to miles per hour.

Solution

(331.45 m s−1 )

1 in
0.0254 m



1 ft
12 in



1 mile
5280 ft



3600 s
1 hour



 741.43 miles hour−1

Note that the conversion ratios do not limit the number of significant digits because they are
defined to be exact values.

Exercise 1.1
a. Express the value of the ideal gas constant in cm3 bar K −1 mol−1 . Report only significant
digits.
b. Find the volume of 2.000 mol of helium (assume ideal) at a temperature of 298.15 K and a
pressure of 0.500 atm.
c. Find the pressure of a sample of 2.000 mol of helium (assume ideal) at a volume of 20.00 L
and a temperature of 500.0 K. Express your answer in terms of Pa, bar, atm, and torr.

PROBLEMS
Section 1.1: Introduction
1.1 Express the speed of light in furlongs per fortnight.
A furlong is 1/8 mile, and a fortnight is 14 days.
1.2 In the “cgs” system, lengths are measured in centimeters,
masses are measured in grams, and time is measured in
seconds. The cgs unit of energy is the erg and the cgs unit
of force is the dyne.
a. Find the conversion factor between ergs and joules.
b. Find the conversion factor between dynes and
newtons.
c. Find the acceleration due to gravity at the earth’s
surface in cgs units.
1.3 In one English system of units, lengths are measured in
feet, masses are measured in pounds, abbreviated lb (1 lb =
0.4536 kg), and time is measured in seconds. The absolute
temperature scale is the Rankine scale, such that 1.8◦ R
corresponds to 1◦ C and to 1 K.
a. Find the acceleration due to gravity at the earth’s
surface in English units.

b. If the pound is a unit of mass, then the unit of force is
called the poundal. Calculate the value of the ideal gas
constant in ft poundals (◦ R)−1 mol−1 .
c. In another English system of units, the pound is a unit
of force, equal to the gravitational force at the earth’s
surface, and the unit of mass is the slug. Find the
acceleration due to gravity at the earth’s surface in this
set of units.
1.4 A light-year is the distance traveled by light in one year.
a. Express the light-year in meters and in kilometers.
b. Express the light-year in miles.
c. If the size of the known universe is estimated to be
20 billion light-years (2 × 1010 light-years) estimate
the size of the known universe in miles.
d. If the closest star other than the sun is at a distance of
4 light-years, express this distance in kilometers and in
miles.
e. The mean distance of the earth from the sun is
149,599,000 km. Express this distance in light-years.

12

1 The Behavior of Gases and Liquids

1.5 The parsec is a distance used in astronomy, defined to be a
distance from the sun such that “the heliocentric parallax is
1 second of arc.” This means that the direction of
observation of an object from the sun differs from the
direction of observation from the earth by one second
of arc.
a. Find the value of 1 parsec in kilometers. Do this by
constructing a right triangle with one side equal to
1 parsec and the other side equal to 1.49599 × 108 km,
the distance from the earth to the sun. Make the angle
opposite the short side equal to 1 second of arc.
b. Find the value of 1 parsec in light-years.
c. Express the distance from the earth to the sun in parsec.
1.6 Making rough estimates of quantities is sometimes a useful
skill.
a. Estimate the number of grains of sand on all of the
beaches of all continents on the earth, excluding
islands. Do this by making suitable estimates of:
(1) the average width of a beach; (2) the average depth
of sand on a beach; (3) the length of the coastlines of all
of the continents; (4) the average size of a grain of sand.
b. Express your estimate in terms of moles of grains of
sand, where a mole of grains of sand is 6.02214 × 1023
grains of sand.
1.7 Estimate the number of piano tuners in Chicago (or any
other large city of your choice). Do this by estimating:
(1) the number of houses, apartments, and other buildings
in the city; (2) the fraction of buildings containing a piano;
(3) the average frequency of tuning; (4) how many pianos
a piano tuner can tune in 1 week.
1.8 Estimate the volume of the oceans of the earth in liters. Use
the fact that the oceans cover about 71% of the earth’s area

1.2

and estimate the average depth of the oceans. The greatest
depth of the ocean is about 7 miles, slightly greater than
the altitude of the highest mountain on the earth.
1.9 Find the volume of CO2 gas produced from 100.0 g of
CaCO3 if the CO2 is at a pressure of 746 torr and a
temperature of 301.0 K. Assume the gas to be ideal.
1.10 According to Dalton’s law of partial pressures, the pressure
of a mixture of ideal gases is the sum of the partial
pressures of the gases. The partial pressure of a gas is
defined to be the pressure that would be exerted if that
gas were alone in the volume occupied by the gas
mixture.
a. A sample of oxygen gas is collected over water at 25◦ C
at a total pressure of 748.5 torr, with a partial pressure
of water vapor equal to 23.8 torr. If the volume of the
collected gas is equal to 454 mL, find the mass of the
oxygen. Assume the gas to be ideal.
b. If the oxygen were produced by the decomposition of
KClO3 , find the mass of KClO3 .
1.11 The relative humidity is defined as the ratio of the partial
pressure of water vapor to the pressure of water vapor at
equilibrium with the liquid at the same temperature. The
equilibrium pressure of water vapor at 25◦ C is 23.756 torr.
If the relative humidity is 49%, estimate the amount of
water vapor in moles contained in a room that is 8.0 m by
8.0 m and 3.0 m in height. Calculate the mass of the
water.
1.12 Assume that the atmosphere is at equilibrium at 25◦ C
with a relative humidity of 100% and assume that the
barometric pressure at sea level is 1.00 atm. Estimate the
total rainfall depth that could occur if all of this moisture is
removed from the air above a certain area of the
earth.

Systems and States in Physical Chemistry
Figure 1.2 depicts a typical macroscopic system, a sample of a single gaseous substance
that is contained in a cylinder with a movable piston. The cylinder is immersed in a
constant-temperature bath that can regulate the temperature of the system. The volume
of the system can be adjusted by moving the piston. There is a valve between the
cylinder and a hose that leads to the atmosphere or to a tank of gas. When the valve
is closed so that no matter can pass into or out of the system, the system is called a
closed system. When the valve is open so that matter can be added to or removed from
the system, it is called an open system. If the system were insulated from the rest of
the universe so that no heat could pass into or out of the system, it would be called
an adiabatic system and any process that it undergoes would be called an adiabatic

13

1.2 Systems and States in Physical Chemistry

External force exerted here
Constant-temperature bath
Hose

Valve
Part of
surroundings

Figure 1.2

Piston

System

Cylinder

A Typical Fluid System Contained in a Cylinder with Variable Volume.

process. If the system were completely separated from the rest of the universe so that
no heat, work, or matter could be transferred to or from the system, it would be called
an isolated system.
The portion of the universe that is outside of the system is called the surroundings.
We must specify exactly what parts of the universe are included in the system. In
this case we define the system to consist only of the gas. The cylinder, piston, and
constant-temperature bath are parts of the surroundings.

The State of a System
Specifying the state of a system means describing the condition of the system by giving
the values of a sufficient set of numerical variables. We have already asserted that for
an equilibrium one-phase liquid or gaseous system of one substance, the pressure is
a function of three independent variables. We now assert as an experimental fact that
for any equilibrium one-phase fluid system (gas or liquid system) of one substance,
there are only three macroscopic independent variables, at least one of which must be
proportional to the size of the system. All other equilibrium macroscopic variables are
dependent variables, with values given as functions of the independent variables. We
say that three independent variables specify the equilibrium macroscopic state of a gas
or liquid of one substance. We can generally choose which three independent variables
to use so long as one is proportional to the size of the system. For fluid system of one
substance, we could choose T , V , and n to specify the equilibrium state. We could also
choose T , P, and n, or we could choose T , P, and V .
All other equilibrium macroscopic variables must be dependent variables that are
functions of the variables chosen to specify the state of the system. We call both the independent variables and the dependent variables state functions or state variables. There
are two principal classes of macroscopic variables. Extensive variables are proportional
to the size of the system if P and T are constant, whereas intensive variables are independent of the size of the system if P and T are constant. For example, V , n, and m

14

1 The Behavior of Gases and Liquids

(the mass of the system) are extensive variables, whereas P and T are intensive
variables. The quotient of two extensive variables is an intensive variable. The density ρ is defined as m/V , and the molar volume Vm is defined to equal V /n. These are
intensive variables. One test to determine whether a variable is extensive or intensive
is to imagine combining two identical systems, keeping P and T fixed. Any variable
that has twice the value for the combined system as for one of the original systems
is extensive, and any variable that has the same value is intensive. In later chapters
we will define a number of extensive thermodynamic variables, such as the internal
energy U, the enthalpy H, the entropy S, and the Gibbs energy G.
We are sometimes faced with systems that are not at equilibrium, and the description
of their states is more complicated. However, there are some nonequilibrium states that
we can treat as though they were equilibrium states. For example, if liquid water at
atmospheric pressure is carefully cooled below 0◦ C in a smooth container it can remain
in the liquid form for a relatively long time. The water is said to be in a metastable
state. At ordinary pressures, carbon in the form of diamond is in a metastable state,
because it spontaneously tends to convert to graphite (although very slowly).

Differential Calculus and State Variables
Because a dependent variable depends on one or more independent variables, a change
in an independent variable produces a corresponding change in the dependent variable.
If f is a differentiable function of a single independent variable x,
f  f (x)

(1.2-1)

then an infinitesimal change in x given by dx (the differential of x) produces a change
in f given by
df 

df
dx
dx

(1.2-2)

where df/dx represents the derivative of f with respect to x and where df represents the
differential of the dependent variable f . The derivative df/dx gives the rate of change
of f with respect to x and is defined by
df
f (x + h) − f (x)
 lim
dx h→0
h

(1.2-3)

if the limit exists. If the derivative exists, the function is said to be differentiable.
There are standard formulas for the derivatives of many functions. For example, if
f  a sin(bx), where a and b represent constants, then
df
 ab cos(x)
dx

(1.2-4)

If a function depends on several independent variables, each independent variable
makes a contribution like that in Eq. (1.2-2). If f is a differentiable function of x, y,
and z, and if infinitesimal changes dx, dy, and dz are imposed, then the differential df
is given by

15

1.2 Systems and States in Physical Chemistry


df 

∂f
∂x




dx +
y,z

∂f
∂y




dy +
x,z

∂f
∂z


dz

(1.2-5)

x,y

where (∂f /∂x) y,z , (∂f /∂y)x,z , and (∂f /∂z)x,y are partial derivatives. If the function is
represented by a formula, a partial derivative with respect to one independent variable
is obtained by the ordinary procedures of differentiation, treating the other independent variables as though they were constants. The independent variables that are held
constant are usually specified by subscripts.
We assume that our macroscopic equilibrium state functions are differentiable except
possibly at isolated points. For an equilibrium gas or liquid system of one phase and
one substance, we can write
 
 
 
∂V
∂V
∂V
dV 
dT +
dP +
dn
(1.2-6)
∂T P,n
∂P T ,n
∂n T ,P
This equation represents the value of an infinitesimal change in volume that is produced
when we impose arbitrary infinitesimal changes dT , dP, and dn on the system, making
sure that the system is at equilibrium after we make the changes. For an ideal gas
dV 

nRT
RT
nR
dT − 2 dP +
dn
P
P
P

(ideal gas)

(1.2-7)

A mathematical identity is an equation that is valid for all values of the variables
contained in the equation. There are several useful identities involving partial derivatives. Some of these are stated in Appendix B. An important identity is the cycle rule,
which involves three variables such that each can be expressed as a differentiable
function of the other two:


∂z
∂x

 
y

∂x
∂y

 
z

∂y
∂z


 −1

(cycle rule)

(1.2-8)

x

If there is a fourth variable, it is held fixed in all three of the partial derivatives.
Some people are surprised by the occurrence of the negative sign in this equation. See
Appendix B for further discussion.

E X A M P L E 1.4
Take z  xy and show that the three partial derivatives conform to the cycle rule.

Solution

∂z
∂x y
 
∂x
∂y z
 
∂y
∂z x
     
∂x
∂y
∂z
∂x y ∂y z ∂z x


 y
z
 − 2
y


1
x

z
z 1
 −1
 −y 2  −
xy
y x

16

1 The Behavior of Gases and Liquids

Exercise 1.2
Take z  ax ln(y/b), where a and b are constants.
a. Find the partial derivatives (∂z/∂x)y , (∂x/∂y)z , and (∂y/∂z)x .
b. Show that the derivatives of part a conform to the cycle rule.

A second partial derivative is a partial derivative of a partial derivative. Both derivatives can be with respect to the same variable:
   
 2 
∂ f
∂ ∂f

(1.2-9)
2
∂x ∂x y
∂x y
y

The following is called a mixed second partial derivative:
   
∂2 f
∂ ∂f

∂y∂x
∂y ∂x y

(1.2-10)

x

The reciprocity relation is named for its
discoverer, Leonhard Euler, 1707–1783,
a great Swiss mathematician who spent
most of his career in St. Petersburg,
Russia, and who is considered by some
to be the father of mathematical
analysis. The base of natural logarithms
is denoted by e after his initial.

Euler’s reciprocity relation states that the two mixed second partial derivatives of a
differentiable function must be equal to each other:
∂2 f
∂2 f

∂y∂x
∂x∂y

(1.2-11)

Exercise 1.3
Show that the three pairs of mixed partial derivatives that can be obtained from the derivatives
in Eq. (1.2-7) conform to Euler’s reciprocity relation.

Processes
Aprocess is an occurrence that changes the state of a system. Every macroscopic process
has a driving force that causes it to proceed. For example, a temperature difference is the
driving force that causes a flow of heat. A larger value of the driving force corresponds
to a larger rate of the process. A zero rate must correspond to zero value of the driving
force. A reversible process is one that can at any time be reversed in direction by an
infinitesimal change in the driving force. The driving force of a reversible process
must be infinitesimal in magnitude since it must reverse its sign with an infinitesimal
change. A reversible process must therefore occur infinitely slowly, and the system has
time to relax to equilibrium at each stage of the process. The system passes through
a sequence of equilibrium states during a reversible process. A reversible process is
sometimes called a quasi-equilibrium process or a quasi-static process. There can be
no truly reversible processes in the real universe, but we can often make calculations
for them and apply the results to real processes, either exactly or approximately.
An approximate version of Eq. (1.2-6) can be written for a finite reversible process
corresponding to increments ∆P, ∆T , and ∆n:
 
 
 
∂V
∂V
∂V
∆V ≈
∆T +
∆P +
∆n
(1.2-12)
∂T P,n
∂P T ,n
∂n T ,P

17

1.2 Systems and States in Physical Chemistry

where ≈ means “is approximately equal to” and where we use the common notation
∆V  V (final) − V (initial)

(1.2-13)

and so on. Calculations made with Eq. (1.2-12) will usually be more nearly correct if the
finite increments ∆T , ∆P, and ∆n are small, and less nearly correct if the increments
are large.

Variables Related to Partial Derivatives
The isothermal compressibility κT is defined by
 
1 ∂V
(definition of the
κT  −
isothermal compressibility)
V ∂P

(1.2-14)

The factor 1/V is included so that the compressibility is an intensive variable. The fact
that T and n are fixed in the differentiation means that measurements of the isothermal compressibility are made on a closed system at constant temperature. It is found
experimentally that the compressibility of any system is positive. That is, every system
decreases its volume when the pressure on it is increased.
The coefficient of thermal expansion is defined by
 
1 ∂V
α
V ∂P P,n

(definition of the coefficient
of thermal expansion)

(1.2-15)

The coefficient of thermal expansion is an intensive quantity and is usually positive.
That is, if the temperature is raised the volume usually increases. There are a few
systems with negative values of the coefficient of thermal expansion. For example,
liquid water has a negative value of α between 0◦ C and 3.98◦ C. In this range of
temperature the volume of a sample of water decreases if the temperature is raised.
Values of the isothermal compressibility for a few pure liquids at several temperatures
and at two different pressures are given in Table A.1 of Appendix A. The values of the
coefficient of thermal expansion for several substances are listed in Table A.2. Each
value applies only to a single temperature and a single pressure, but the dependence on
temperature and pressure is not large for typical liquids, and these values can usually
be used over fairly wide ranges of temperature and pressure.
For a closed system (constant n) Eq. (1.2-12) can be written
∆V ≈ V α∆T − V κT ∆P

(1.2-16)

E X A M P L E 1.5
The isothermal compressibility of liquid water at 298.15 K and 1.000 atm is equal to
4.57 × 10−5 bar−1  4.57 × 10−10 Pa−1 . Find the fractional change in the volume of a
sample of water if its pressure is changed from 1.000 bar to 50.000 bar at a constant temperature of 298.15 K.

Solution
The compressibility is relatively small in magnitude so we can use Eq. (1.2-16):
∆V ≈ −V κT ∆P

(1.2-17)

18

1 The Behavior of Gases and Liquids

The fractional change is
∆V
≈ −κT ∆P  −(4.57 × 10−5 bar−1 )(49.00 bar)  −2.24 × 10−3
V

E X A M P L E 1.6
For liquid water at 298.15 K and 1.000 atm, α  2.07 × 10−4 K −1 . Find the fractional
change in the volume of a sample of water at 1.000 atm if its temperature is changed from
298.15 K to 303.15 K.

Solution
To a good approximation,
∆V ≈ V α∆T
The fractional change in volume is
∆V
≈ α∆T  (2.07 × 10−4 K−1 )(5.000 K)  1.04 × 10−3
V

Exercise 1.4
a. Find expressions for the isothermal compressibility and coefficient of thermal expansion for
an ideal gas.
b. Find the value of the isothermal compressibility in atm−1 , in bar −1 , and in Pa−1 for an ideal
gas at 298.15 K and 1.000 atm. Find the ratio of this value to that of liquid water at the same
temperature and pressure, using the value from Table A.1.
c. Find the value of the coefficient of thermal expansion of an ideal gas at 20◦ C and 1.000 atm.
Find the ratio of this value to that of liquid water at the same temperature and pressure, using
the value from Table A.2.

In addition to the coefficient of thermal expansion there is a quantity called the
coefficient of linear thermal expansion, defined by
 
1 ∂L
αL 
L ∂T P

(definition of the coefficient
of linear thermal expansion)

(1.2-18)

where L is the length of the object. This coefficient is usually used for solids, whereas
the coefficient of thermal expansion in Eq. (1.2-15) is used for gases and liquids.
Unfortunately, the subscript L is sometimes omitted on the symbol for the coefficient
of linear thermal expansion, and the name “coefficient of thermal expansion” is also
sometimes used for it. Because the units of both coefficients are the same (reciprocal
temperature) there is opportunity for confusion between them.
We can show that the linear coefficient of thermal expansion is equal to one-third
of the coefficient of thermal expansion. Subject a cubical object of length L to an
infinitesimal change in temperature, dT . The new length of the object is
 
∂L
L(T + dT )  L(T ) +
dT  L(T )(1 + αLdT )
(1.2-19)
∂T P

19

1.2 Systems and States in Physical Chemistry

The volume of the object is equal to L3 , so


V (T + dT )  L(T )3 1 + αLdT 3


 L(T )3 1 + 3αLdT + 3(αLdT )2 + (αLdT )3

(1.2-20)

Since dT is small, the last two terms are insignificant compared with the term that is
proportional to dT .
V (T + dT )  L(T )3 (1 + 3αLdT )
The volume at temperature T + dT is given by
 
∂V
V (T + dT )  V (T ) +
dT  V (T )(1 + αdT )
∂T

(1.2-21)

(1.2-22)

Comparison of Eq. (1.2-22) with Eq. (1.2-21) shows that
α  3αL

(1.2-23)

This relationship holds for objects that are not necessarily shaped like a cube.

E X A M P L E 1.7
The linear coefficient of expansion of borosilicate glass, such as Pyrex or Kimax , is equal
to 3.2 × 10−6 K −1 . If a volumetric flask contains 2.000000 L at 20.0◦ C, find its volume at
25.0◦ C.

Solution
V (25◦ C)  V (20◦ C)(1 + 3αL(5.0◦ C))
 (2.000000 L)(1 + 3(3.2 × 10−6 )(5.0◦ C))  2.000096 L

Exercise 1.5
Find the volume of the volumetric flask in Example 1.7 at 100.0◦ C.

Moderate changes in temperature and pressure produce fairly small changes in the
volume of a liquid, as in the examples just presented. The volumes of most solids
are even more nearly constant. We therefore recommend the following practice: For
ordinary calculations, assume that liquids and solids have fixed volumes. For more
precise calculations, calculate changes in volume proportional to changes in pressure
or temperature as in Examples 1.5 and 1.6.
Exercise 1.6
The compressibility of acetone at 20◦ C is 12.39 × 10−10 Pa−1 , and its density is 0.7899 g cm−3
at 20◦ C and 1.000 bar.
a. Find the molar volume of acetone at 20◦ C and a pressure of 1.000 bar.
b. Find the molar volume of acetone at 20◦ C and a pressure of 100.0 bar.

20

1 The Behavior of Gases and Liquids

PROBLEMS
Section 1.2: Systems and States in Physical Chemistry
1.13 Show that the three partial derivatives obtained from
PV  nRT with n fixed conform to the cycle rule,
Eq. (B-15) of Appendix B.
1.14 For 1.000 mol of an ideal gas at 298.15 K and 1.000 bar,
find the numerical value of each of the three partial
derivatives in the previous problem and show numerically
that they conform to the cycle rule.
1.15 Finish the equation for an ideal gas and evaluate the partial
derivatives for V  22.4 L, T  273.15 K , and
n  1.000 mol.
 
∂P
dV + ?
dP 
∂V T ,n
1.16 Take z  aye x/b , where a and b are constants.
a. Find the partial derivatives (∂z/∂x)y , (∂x/∂y)z , and
(∂y/∂z)x .
b. Show that the derivatives of part a conform to the cycle
rule, Eq. (B-15) of Appendix B.
1.17 a. Find the fractional change in the volume of a sample of
liquid water if its temperature is changed from 20.00◦ C
to 30.00◦ C and its pressure is changed from 1.000 bar
to 26.000 bar.
b. Estimate the percent change in volume of a sample of
benzene if it is heated from 0◦ C to 45◦ C at 1.000 atm.
c. Estimate the percent change in volume of a sample of
benzene if it is pressurized at 55◦ C from 1.000 atm to
50.0 atm.
1.18 a. Estimate the percent change in the volume of a sample
of carbon tetrachloride if it is pressurized from
1.000 atm to 10.000 atm at 25◦ C.
b. Estimate the percent change in the volume of a sample
of carbon tetrachloride if its temperature is changed
from 20◦ C to 40◦ C.
1.19 Find the change in volume of 100.0 cm3 of liquid carbon
tetrachloride if its temperature is changed from 20.00◦ C to
25.00◦ C and its pressure is changed from 1.000 atm to
10.000 atm.
1.20 Let f (u)  sin(au2 ) and u  x2 + y2 , where a is a
constant. Using the chain rule, find (∂f /∂x)y and
(∂f /∂y)x . (See Appendix B.)

1.21 Show that for any system,
α

κT



∂P
∂T


V ,n

1.22 The coefficient of linear expansion of borosilicate glass is
equal to 3.2 ×10−6 K −1 .
a. Calculate the pressure of a sample of helium (assumed
ideal) in a borosilicate glass vessel at 150◦ C if its
pressure at 0◦ C is equal to 1.000 atm. Compare with the
value of the pressure calculated assuming that the
volume of the vessel is constant.
b. Repeat the calculation of part a using the virial equation
of state truncated at the B2 term. The value of B2 for
helium is 11.8 cm3 mol−1 at 0◦ C and 11.0 cm3 mol−1 at
150◦ C.
1.23 Assuming that the coefficient of thermal expansion of
gasoline is roughly equal to that of benzene, estimate the
fraction of your gasoline expense that could be saved by
purchasing gasoline in the morning instead of in the
afternoon, assuming a temperature difference of 5◦ C.
1.24 The volume of a sample of a liquid at constant pressure can
be represented by
Vm (tC )  Vm (0◦ C)(1 + α tC + β tC2 + γ  tC3 )
where α , β , and γ  are constants and tC is the Celsius
temperature.
a. Find an expression for the coefficient of thermal
expansion as a function of tC .
b. Evaluate the coefficient of thermal expansion
of benzene at 20.00◦ C, using α  1.17626 ×
10−3 (◦ C)−1 , β  1.27776 × 10−6 (◦ C)−2 , and
γ   0.80648 × 10−8 (◦ C)−3 . Compare your value with
the value in Table A.2.
1.25 The coefficient of thermal expansion of ethanol equals
1.12 × 10−3 K −1 at 20◦ C and 1.000 atm. The density at
20◦ C is equal to 0.7893 g cm−3 .
a. Find the volume of 1.000 mol of ethanol at 10.00◦ C
and 1.000 atm.
b. Find the volume of 1.000 mol of ethanol at 30.00◦ C
and 1.000 atm.

21

1.3 Real Gases

1.3
The van der Waals equation of state is
named for Johannes Diderik van der
Waals,1837–1923, a Dutch physicist
who received the 1910 Nobel Prize in
physics for his work on equations of
state.

Real Gases
Most gases obey the ideal gas law to a good approximation when near room temperature
and at a moderate pressure. At higher pressures one might need a better description.
Several equations of state have been devised for this purpose. The van der Waals
equation of state is

P+


an2
(V − nb)  nRT
V2

(1.3-1)

The symbols a and b represent constant parameters that have different values for different substances. Table A.3 in Appendix A gives values of van der Waals parameters
for several substances.
We solve the van der Waals equation for P and note that P is actually a function of
only two intensive variables, the temperature T and the molar volume Vm , defined to
equal V /n.
P

nRT
an2
a
RT
− 2 
−
V − nb
Vm − b Vm2
V

(1.3-2)

This dependence illustrates the fact that intensive variables such as pressure cannot
depend on extensive variables and that the intensive state of a gas or liquid of one
substance is specified by only two intensive variables.

E X A M P L E 1.8
Use the van der Waals equation to calculate the pressure of nitrogen gas at 273.15 K and
a molar volume of 22.414 L mol−1 . Compare with the pressure of an ideal gas at the same
temperature and molar volume.

Solution

P


8.134 J K−1 mol−1 (273.15 K)

0.022414 m3 mol−1 − 0.0000391 m3 mol−1

0.1408 Pa m3 mol−1
2
0.022414 m3 mol−1

−

 1.0122 × 105 Pa  0.9990 atm
For the ideal gas

P

RT

Vm


8.134 J K−1 mol−1 (273.15 K)
0.022414 m3 mol−1

 1.0132 × 105 Pa  1.0000 atm

Exercise 1.7
a. Show that in the limit that Vm becomes large, the van der Waals equation becomes identical
to the ideal gas law.
b. Find the pressure of 1.000 mol of nitrogen at a volume of 24.466 L and a temperature of
298.15 K using the van der Waals equation of state. Find the pressure of an ideal gas under
the same conditions.

22

1 The Behavior of Gases and Liquids
c. Find the pressure of 1.000 mol of nitrogen at a volume of 1.000 L and a temperature of
298.15 K using the van der Waals equation of state. Find the pressure of an ideal gas under
the same conditions.

Another common equation of state is the virial equation of state:
B3
B4
B2
PVm
+ 2 + 3 + ···
1+
RT
Vm
Vm
Vm

(1.3-3)

which is a power series in the independent variable 1/Vm . The B coefficients are called
virial coefficients. The first virial coefficient, B1 , is equal to unity. The other virial
coefficients must depend on temperature in order to provide an adequate representation.
Table A.4 gives values of the second virial coefficient for several gases at several
temperatures.
An equation of state that is a power series in P is called the pressure virial equation
of state:
PVm  RT + A2 P + A3 P 2 + A4 P 3 + · · ·

(1.3-4)

The coefficients A2 , A3 , etc., are called pressure virial coefficients and also must depend
on the temperature. It can be shown that A2 and B2 are equal.

E X A M P L E 1.9
Show that A2  B2 .

Solution
We solve Eq. (1.3-3) for P and substituting this expression for each P in Eq. (1.3-4).
P

RT B2
RT B3
RT
+
+
+ ···
2
3
Vm
Vm
Vm

We substitute this expression into the left-hand side of Eq. (1.3-4).
PVm  RT +

RT B2
RT B3
+
+ ···
2
Vm
Vm

We substitute this expression into the second term on the right-hand side of Eq. (1.3-4).
PVm  RT + A2

RT B2
RT B3
RT
+
+
+ ···
2
3
Vm
Vm
Vm

If two power series in the same variable are equal to each other for all values of the variable,
the coefficients of the terms of the same power of the variable must be equal to each other.
We equate the coefficients of the 1/Vm terms and obtain the desired result:
A2  B2

Exercise 1.8
Show that A3 


1 
B3 − B22 .
RT

23

1.3 Real Gases

Table 1.1 displays several additional equations of state, and values of parameters for
several gases are found in Table A.3. The parameters for a given gas do not necessarily
have the same values in different equations even if the same letters are used. The
accuracy of several of the equations of state has been evaluated.2 The Redlich–Kwong
equation of state seemed to perform better than the other two-parameter equations, with
the van der Waals equation coming in second best. The Gibbons–Laughton modification
of the Redlich–Kwong equation (with four parameters) is more accurate than the twoparameter equations.
Table 1.1

Some Equations of State

The letters a and b stand for constant parameters that have different values for different
substances. These parameters do not necessarily have the values for the same substance
in different equations of state.
The Berthelot Equation of State


a
(Vm − b)  RT
P+
TVm2
The Dieterici Equation of State
Pea/Vm RT (Vm − b)  RT
The Redlich–Kwong Equation of State
P

RT
a
−
Vm − b T 1/2 Vm (Vm + b)

The Soave Modification of the Redlich–Kwong Equation of State
P

RT
aα(T )
−
Vm − b Vm (Vm + b)

where α(T )  {1 + m 1 − (T /Tc )1/2 }2 , where m is a constant parameter and where
Tc is the critical temperature. See the article by Soave for values of the parameter m.
The Gibbons–Laughton Modification of the Redlich–Kwong-Soave Equation
of State
The equation is the same as the Soave modification, but α(T ) is given by
α(T )  1 + X(T /Tc ) − 1 + Y (T /Tc )1/2 − 1
where X and Y are constant parameters. See the article by Gibbons and Laughton for
values of these parameters.
Other equations of state can be found in the book by Hirschfelder, Curtiss, and Bird,
including the Beattie–Bridgeman equation, with five parameters, and the Benedict–
Webb–Rubin equation, with eight parameters.

2 J. B. Ott, J. R. Goates, and H. T. Hall, Jr., J. Chem. Educ., 48, 515 (1971); M. W. Kemp, R. E. Thompson,
and D. J. Zigrang, J. Chem. Educ., 52, 802 (1975).

24

1 The Behavior of Gases and Liquids

Graphical Representation of Volumetric Data for Gases
The compression factor, denoted by Z, is sometimes used to describe the behavior of
real gases:
Z

PVm
RT

(1.3-5)

Some authors call Z the compressibility factor. We avoid this name because it might
be confused with the compressibility. The compression factor equals unity for an ideal
gas. Figure 1.3 shows a graph of the compression factor of nitrogen gas as a function
of pressure at several temperatures. At low temperatures, the value of Z is less than
unity for moderate pressures, but rises above unity for larger pressures. At higher
temperatures, the value of Z is greater than unity for all pressures. Attractions between
the molecules tend to reduce the value of Z and repulsions between the molecules tend
to increase the value of Z. Attractions are more important at lower temperatures and
smaller pressures, and repulsions are more important at higher temperatures and higher
pressures. The temperature at which the curve has zero slope at zero pressure is called
the Boyle temperature. This is the temperature at which the gas most nearly approaches
ideality for small pressures.
For a van der Waals gas, the compression factor is given by
Z

PVm
Vm
a
ay
1

−
−

RT
Vm − b RTVm
1 − by RT

(1.3-6)

where we let y  1/Vm . Since a and b are both positive for all gases, the first term on
the right-hand side of Eq. (1.3-6) gives a positive contribution to Z, and the second
term gives a negative contribution. The parameter b describes the effect of repulsive

2.0

Z

200 K
250 K
1900 K
1.0
0.8
150 K

0.6
0.4
0.3
0.2
0.1
5 10 20 50

100

200
P/bar

500

Figure 1.3 The Compression Factor of Nitrogen as a Function of Pressure at Several
Temperatures.

25

1.3 Real Gases

intermolecular forces and the parameter a describes the effect of attractive intermolecular forces. For higher temperatures the second term is relatively unimportant, and the
compression factor will exceed unity for all values of y. For temperatures below the
Boyle temperature the second term becomes relatively more important, and a value of
Z less than unity will occur if y is not too large.

E X A M P L E 1.10
a. Find an expression for the Boyle temperature of a van der Waals gas.
b. Find the value of the Boyle temperature of nitrogen gas as predicted by the van der Waals
equation.

Solution
a. Since y is proportional to P for small values of P, we seek the temperature at which
 


∂Z
b
a
a
0
−
b−
2
∂y y0
RT
RT
(1 − by)
y0
where the subscript y  0 indicates the value of y at which the derivative is evaluated.
The Boyle temperature is
TBoyle  a/Rb
b. For nitrogen,
0.1408 Pa m2 mol−1

  433 K
8.134 J K−1 mol−1 3.913 × 10−5 m3 mol−1

TBoyle  

Exercise 1.9
a. Find an expression for the Boyle temperature of a gas obeying the Dieterici equation of state.
b. Find the value of the Boyle temperature of nitrogen according to the Dieterici equation of
state.
c. Find the expression for the molar volume at which Z  1 for the van der Waals gas for a
given temperature below the Boyle temperature. Hint: Find the nonzero value of y in Eq.
(1.3-6) that makes Z  1.
d. Find the value of the molar volume and the pressure at which Z  1 for nitrogen at 273.15 K,
according to the van der Waals equation.

PROBLEMS
Section 1.3: Real Gases
1.26 For the van der Waals equation of state, obtain formulas
for the partial derivatives (∂P/∂T )V ,n , (∂P/∂V )T ,n , and
(∂P/∂n)T ,V .

1.27 For the virial equation of state,
a. Find the expressions for (∂P/∂V )T ,n and (∂P/∂T )V ,n .
b. Show that (∂2 P/∂V ∂T )n  (∂2 P/∂T ∂V )n .

26

1 The Behavior of Gases and Liquids

1.28 Evaluate each of the partial derivatives in Problem 1.26 for
carbon dioxide at 298.15 K and 10.000 bar.
1.29 a. Derive an expression for the isothermal compressibility of a gas obeying the van der Waals equation of
state. Hint: Use the reciprocal identity, Eq. (B-8).
b. Evaluate the isothermal compressibility of carbon
dioxide gas at a temperature of 298.15 K and a molar
volume of 0.01000 m3 mol−1 . Compare with the value
obtained from the ideal gas law.
1.30 Write the expressions giving the compression factor Z
as a function of temperature and molar volume for the
van der Waals, Dieterici, and Redlich–Kwong equations
of state.
1.31 a. For the van der Waals equation of state at temperatures
below the Boyle temperature, find an expression for a
value of the pressure other than P  0 for which
PVm  RT .

Vm  0.1497 L mol−1 is 1.1336. Find the values of Z
predicted by the van der Waals, Dieterici, and
Redlich–Kwong equations of state for these conditions.
Calculate the percent error for each.
1.38 The parameters for the van der Waals equation of state for
a mixture of gases can be approximated by use of the
mixing rules:
a  a1 x12 + a12 x1 x2 + a2 x22
b  b1 x12 + b12 x1 x2 + b2 x22
where x1 and x2 are the mole fractions of the two
substances and where a1 , b1 , a2 , and b2 are the van der
Waals parameters of the two substances. The quantities a12
and b12 are defined by
a12  (a1 a2 )1/2
and

b. Find the value of this pressure for nitrogen gas at
298.15 K.
1.32 a. By differentiation, find an expression for the isothermal
compressibility of a gas obeying the Dieterici equation
of state.
b. Find the value of the isothermal compressibility of
nitrogen gas at 298.15 K and Vm  24.4 L. Compare
with that of an ideal gas.
1.33 a. By differentiation, find an expression for the coefficient
of thermal expansion of a gas obeying the van der
Waals equation of state.
b. Find the value of the coefficient of thermal expansion
of nitrogen gas at 298.15 K and Vm  24.4 L mol−1 .
1.34 By differentiation, find an expression for the coefficient of
thermal expansion of a gas obeying the Dieterici equation
of state.
1.35 Manipulate the Dieterici equation of state into the virial
form. Use the identity
e−x  1 − x +

x3
xn
x2
−
+ · · · + (−1)n
+ ···
2!
3!
n!

where n!  n(n − 1)(n − 2)(n − 3) . . . (3)(2)(1). Write
expressions for the second, third, and fourth virial
coefficients.
1.36 Write an expression for the isothermal compressibility of a
nonideal gas obeying the Redlich–Kwong equation of state.
1.37 The experimental value of the compression factor
Z  PVm /RT for hydrogen gas at T  273.15 K and


b12 

1/3

b1

1/3

+ b2
3

3

a. Using these mixing rules and the van der Waals
equation of state, find the pressure of a mixture of
0.79 mol of N2 and 0.21 mol of O2 at 298.15 K and at a
mean molar volume (defined as V /ntotal ) of
0.00350 m3 mol−1 . Compare your answer with the
pressure of an ideal gas under the same conditions.
b. Using the van der Waals equation of state, find the
pressure of pure N2 at 298.15 K and at a molar volume
of 0.00350 m3 mol−1 .
c. Using the van der Waals equation of state, find the
pressure of pure O2 at 298.15 K and at a molar volume
of 0.00350 m3 mol−1 .
1.39 Find the value of the isothermal compressibility of carbon
dioxide gas at 298.15 K and a molar volume of
24.4 L mol−1 ,
a. According to the ideal gas law.
b. According to the truncated virial equation of state
B2
PVm
1+
RT
Vm
For carbon dioxide at 298.15 K,
B2  −12.5 × 10−5 m3 mol−1 .
1.40 Considering P to be a function of T , V , and n, obtain the
expression for dP for a gas obeying the van der Waals
equation of state.

27

1.4 The Coexistence of Phases and the Critical Point

The Coexistence of Phases and the
Critical Point
Transitions from a gaseous state to a liquid state or from a liquid state to a solid state,
and so forth, are called phase transitions and the samples of matter in the different
states are called phases. Such transitions can take place abruptly. If a gas is initially
at a temperature slightly above its condensation temperature, a small decrease in the
temperature can produce a liquid phase that coexists with the gas phases, and a further
small decrease in the temperature can cause the system to become a single liquid phase.
This remarkable behavior is an exception to the general rule that in nature small changes
produce small effects and large changes produce large effects.
It is an experimental fact that for any pure substance the pressure at which two phases
can coexist at equilibrium is a smooth function of the temperature. Equivalently, the
temperature is a smooth function of the pressure. Figure 1.4 shows schematic curves
representing these functions for a typical substance. The curves are called coexistence
curves and the figure is called a phase diagram. The three curves shown are the solid–
gas (sublimation) curve at the bottom of the figure, the liquid–gas (vaporization) curve
at the upper right, and the solid–liquid (fusion, melting, or freezing) curve at the upper
left. The three curves meet at a point called the triple point. This point corresponds to

(3-Step process to liquefy gas
without phase transition)

ce c
urve

Expansion
at constant
temperature

Liquid

Critical
point

Compression
at constant
temperature

Vapor (gas)

Li
qu

id

va
p

or
c

oe
x is

te n

Solid–liquid

Solid

coexistenc
e cu

rve

Cooling at
constant pressure

P

1.4

1 atm

Normal freezing temperature
Triple point
Normal boiling temperature
T
Solid–vapor
coexistence curve

Figure 1.4 The Coexistence Curves for a Typical Pure Substance (Schematic).

28

1 The Behavior of Gases and Liquids

the unique value of the pressure and the unique value of the temperature at which all
three phases can coexist.
The equilibrium temperature for coexistence of the liquid and solid at a pressure
equal to 1 atmosphere is called the normal melting temperature or normal freezing
temperature. The equilibrium temperature for coexistence of the liquid and gas phases
at a pressure equal to 1 atmosphere is called the normal boiling temperature. These
temperatures are marked on Figure 1.4. If the triple point is at a higher pressure than
1 atmosphere the substance does not have a normal freezing temperature or a normal
boiling temperature, but has a normal sublimation temperature at which the solid and
gas coexist at a pressure equal to 1 atmosphere. The triple point of carbon dioxide
occurs at a pressure of 5.112 atm and a temperature of 216.55 K (−56.60◦ C) and its
normal sublimation temperature is equal to 194.6 K (−78.5◦ C). Equilibrium liquid
carbon dioxide can be observed only at pressures greater than 5.112 atm. At lower
pressures the solid sublimes directly into the vapor phase.

The Critical Point
There is a remarkable feature that is shown in Figure 1.4. The liquid–vapor coexistence
curve terminates at a point that is called the critical point. The temperature, molar
volume, and pressure at this point are called the critical temperature, denoted by Tc ,
the critical molar volume, denoted by Vmc , and the critical pressure, denoted by Pc .
These three quantities are called critical constants. Table A.5 in the appendix gives
values of the critical constants for several substances. At temperatures higher than the
critical temperature and pressures higher than the critical pressure there is no transition
between liquid and gas phases. It is possible to heat a gas to a temperature higher than
the critical temperature, then to compress it until its density is as large as that of a
liquid, and then to cool it until it is a liquid without ever having passed through a phase
transition. A path representing this kind of process is drawn in Figure 1.4. Fluids at
supercritical temperatures are often referred to as gases, but it is better to refer to them
as supercritical fluids. Some industrial extractions, such as the decaffeination of coffee,
are carried out with supercritical fluids such as carbon dioxide.3 Supercritical carbon
dioxide is also used as a solvent in some HPLC applications.4 Using a chiral stationary
phase, enantiomers can be separated. The liquid–solid coexistence curve apparently
does not terminate at a critical point. Nobody has found such a termination, and it
seems reasonable that the presence of a lattice structure in the solid, which makes it
qualitatively different from the liquid, makes the existence of such a point impossible.
Figure 1.5 schematically shows the pressure of a fluid as a function of molar volume for several fixed temperatures, with one curve for each fixed temperature. These
constant-temperature curves are called isotherms. For temperatures above the critical
temperature there is only one fluid phase, and the isotherms are smooth curves. The
liquid branch is nearly vertical since the liquid is almost incompressible while the gas
branch of the curve is similar to the curve for an ideal gas. For subcritical temperatures,
the isotherm consists of two smooth curves (branches) and a horizontal line segment,
which is called a tie line. A tie line connects the two points representing the molar
volumes of the coexisting liquid and gas phases. As subcritical temperatures closer and
closer to the critical temperature are chosen the tie lines become shorter and shorter

3 Chem. Eng. Sci., 36(11), 1769(1981); Env. Sci. Technol., 20(4), 319 (1986); Chemtech., 21(4), 250
(1991), Anal. Chem., 66(12), 106R (1994).
4A.M. Thayer, Chem. Eng. News, 83, 49 (September 5, 2005).

29

1.4 The Coexistence of Phases and the Critical Point

Pc

Liquid branches

Supercritical isotherms
(no distinction between
liquid and vapor)

Critical isotherm

T 6 > Tc

Tie line
P

T 5 > Tc
T 4 > Tc
Tc
Tie line

T2 <Tc
T1 <Tc

Vapor branch
0

Figure 1.5

Vm,c

Vm

Isotherms for a Typical Pure Substance (Schematic).

until they shrink to zero length at the critical point. No two isotherms can intersect, so
the isotherm that passes through the critical point must have a horizontal tangent line
at the critical point. This point on the critical isotherm is an inflection point, with a zero
value of (∂P/∂Vm )T and a zero value of (∂2 P/∂Vm2 )T .
At the critical point, a fluid exhibits some unusual properties such as strong scattering
of light, infinite heat capacity, and infinite compressibility. If a sample of a pure fluid
is confined in a rigid closed container such that the average molar volume is equal to
that of the critical state and if the temperature is raised through the critical temperature,
the meniscus between the liquid and gas phases becomes diffuse and then disappears
at the critical temperature. Figure 1.6 shows photographs illustrating this behavior in
carbon dioxide.5 The system contains three balls that are slightly different in density,
with densities close to the critical density of carbon dioxide.
Figure 1.7 depicts a perspective view of a three-dimensional graph with a surface
representing the pressure of a fluid as a function of temperature and molar volume. The
isotherms in Figure 1.5 are produced by passing planes of constant temperature through
the surface of this graph. Several isotherms are drawn on the surface in Figure 1.7. The
liquid–gas tie lines are seen in the tongue-shaped region. When the three-dimensional
graph is viewed in a direction perpendicular to the T –P plane each liquid–gas tie line
is seen as a point. The set of all such points makes up the gas–liquid coexistence curve
seen in Figure 1.4.

5 J. V. Sengers and A. L. Sengers, Chem. Eng. News., 46, 54 (June 10, 1968). This figure can be seen on
the Web at http://sfu.ca/chemcai/critical.html, courtesy of Dr. Steven Lower of Simon Fraser University.

30

1 The Behavior of Gases and Liquids

(a)

(b)

(c)

(d)

Figure 1.6 Liquid–Gas Equilibrium near the Critical Point. (a) At a temperature slightly
above the critical temperature. The density of the fluid depends slightly on height, due to
gravity. (b) At the critical temperature, and showing the scattering of light known as critical
opalescence. (c) and (d) At subcritical temperatures, showing a definite meniscus. From
J. V. Sengers and A. L. Sengers, Chem. Eng. News, June 10, 1968, p. 104. Used by permission of the copyright holder.

P

T
Tc isotherm

0
Tie lines
V

m

Figure 1.7 Surface Giving Pressure as a Function of Molar Volume and Temperature
for a Typical Pure Substance in the Liquid–Vapor Region (Schematic).

31

1.4 The Coexistence of Phases and the Critical Point

Because the entire fluid (liquid and gas) surface in Figure 1.7 is connected, a completely successful equation of state should represent the entire surface. The equations
of state that we have discussed yield surfaces that resemble the true surface in the
liquid region as well as in the gas region, although they do not represent the tie lines.
In Chapter 5 we will discuss a technique for constructing the tie lines for a particular equation of state. The modified Redlich–Kwong–Soave equation of Gibbons and
Laughton seems to be fairly accurate in representing both the liquid and the gas, and the
van der Waals equation is often used to give qualitative information. For any equation
of state, we can obtain equations that locate the critical point.

E X A M P L E 1.11
Derive formulas for the critical temperature and critical molar volume for a gas obeying the
van der Waals equation of state.

Solution
We seek the point at which
(∂P/∂Vm )T  0

(1.4-1)

2)  0
(∂2 P/∂Vm
T

(1.4-2)

The first derivative of Eq. (1.3-2) with respect to Vm is


∂P
RT
2a
−
+ 3
∂Vm T
(Vm − b)2
Vm

(1.4-3)

and the second derivative is


∂2 P
2
∂Vm


−
T

2RT
(Vm

− b)3

+

6a
4
Vm

(1.4-4)

Setting the right-hand side of each of these two equations equal to zero gives us two simultaneous algebraic equations, which are solved to give the values of the critical temperature
Tc and the critical molar volume Vmc :
Tc 

8a
,
27Rb

Vmc  3b

(1.4-5)

Exercise 1.10
Solve the simultaneous equations to verify Eq. (1.4-5). One way to proceed is as follows: Obtain
Eq. (I) by setting the right-hand side of Eq. (1.4-3) equal to zero, and Eq. (II) by setting the
right-hand side of Eq. (1.4-4) equal to zero. Solve Eq. (I) for T and substitute this expression
into Eq. (II).

When the values of Tc and Vmc are substituted into the van der Waals equation of
state the value of the critical pressure for a van der Waals gas is obtained:
Pc 

a
27b2

(1.4-6)

32

1 The Behavior of Gases and Liquids

For a van der Waals gas, the compression factor at the critical point is
Zc 

Pc Vmc
3
  0.375
RTc
8

(1.4-7)

Exercise 1.11
Verify Eqs. (1.4-6) and (1.4-7).

Equations (1.4-5) and (1.4-6) can be solved for a and b:
2
Pc 
a  3Vmc

b

27R2 Tc2
9R Vmc Tc

8
64Pc

Vmc
RTc

3
8Pc

(1.4-8)
(1.4-9)

There are two or three formulas for each parameter. Since no substance exactly fits
the equation different values can result from the different formulas. The best values of
a and b are obtained by using Pc and Tc as independent variables. The values of the
parameters for any two-parameter or three-parameter equation of state can be obtained
from critical constants.
Exercise 1.12
a. Show that for the Dieterici equation of state,
Vmc  2b,

Tc 

a
,
4bR

Pc 

a −2
e
4b2

(1.4-10)

b. Show that for the Dieterici equation of state,
Zc  2e−2  0.27067
c. Obtain the formulas giving the Dieterici parameters a and b as functions of Pc and Tc . Find
the values of a and b for nitrogen and compare with the values in Table A.3.

The parameters a and b in the Redlich–Kwong equation of state can be obtained
from the relations

 1/3
5/2
2 − 1 RTc
R2 Tc
 , b
(1.4-11)
a   1/3
3Pc
9 2 − 1 Pc
The value of the compression factor at the critical point according to the Redlich–
Kwong equation of state is 1/3.
Exercise 1.13
Find the values of a and b in the Redlich–Kwong equation of state for nitrogen.

Figure 1.8 shows schematically a more complete view of the three-dimensional
graph of Figure 1.7, including the solid–liquid and solid–gas phase transitions. There

33

1.4 The Coexistence of Phases and the Critical Point

Solid–liquid
tie line

P

Liquid–gas
tie line
Triple point
tie lines
Solid–gas
tie line
0

Tc

V

Tt

T

m

Figure 1.8 Surface Giving Pressure as a Function of Molar Volume and Temperature
Showing All Three Phases (Schematic).

are three sets of tie lines, corresponding to the three curves in Figure 1.4. At the triple
point, all three tie lines come together in a single tie line connecting three phases. As
shown in Figures 1.7 and 1.8, the pressure of a one-phase system of one substance
is a function of only two intensive variables, T and Vm . Any intensive variable in a
one-component fluid system is also a function of two intensive variables. The intensive
state of an equilibrium system is the state of the system so far as only intensive variables
are concerned, and is specified by two independent intensive variables if an equilibrium
system contains a single substance and a single fluid phase. The size of the system is not
specified. For a one-phase fluid (liquid or gas) system of c substances, c + 1 intensive
variables specify the intensive state of the system.

The Law of Corresponding States
The van der Waals equation predicts that the value of the compression factor at the
critical point is equal to 0.375 for all substances. There is even a greater degree of
generality, expressed by an empirical law called the law of corresponding states:6 All
substances obey the same equation of state in terms of reduced variables. The reduced
variables are dimensionless variables defined as follows: The reduced volume is the
ratio of the molar volume to the critical molar volume:
Vm
Vr 
(1.4-12)
Vmc
The reduced pressure is the ratio of the pressure to the critical pressure:
Pr 

P
Pc

6 Hirschfelder, Curtiss, and Bird, op. cit., p. 235 [see Table 2.1].

(1.4-13)

34

1 The Behavior of Gases and Liquids

The reduced temperature is the ratio of the temperature to the critical temperature:
Tr 

T
Tc

(1.4-14)

Using the definitions in Eqs. (1.4-12), (1.4-13), and (1.4-14) and the relations in Eqs.
(1.4-5) and (1.4-6) we obtain for a fluid obeying the van der Waals equation of state:
P

aPr
,
27b2

Vm  3bVr ,

T 

8aTr
27Rb

When these relations are substituted into Eq. (1.3-1), the result is



3
1
8Tr

Vr −
Pr + 2
3
3
Vr

(1.4-15)

Exercise 1.14
Carry out the algebraic steps to obtain Eq. (1.4-15).

In Eq. (1.4-15), the parameters a and b have canceled out. The van der Waals equation
of state thus conforms to the law of corresponding states. The same equation of state
without adjustable parameters applies to every substance that obeys the van der Waals
equation of state if the reduced variables are used instead of P, Vm , and T . The other
two-parameter equations of state also conform to the law of corresponding states.
Figure 1.9 is a graph of the experimentally measured compression factor of a number
of polar and nonpolar fluids as a function of reduced pressure at a number of reduced
Methane
Isopentane
1.0

Ethylene
n-Heptane

Ethane
Nitrogen

Propane
Carbon dioxide

n-Butane
Water

Tr ⫽ 2.00

0.9

1.50

0.8
1.30

Z⫽

PVm
RT

0.7
0.6

1.20

0.5

1.10

0.4
1.00

0.3
0.2
0.1
0

0

0.5

1.0

1.5

2.0

2.5 3.0 3.5 4.0 4.5
Reduced pressure, Pr

5.0

5.5

6.0

6.5 7.0

Figure 1.9 The Compression Factor as a Function of Reduced Pressure and
Reduced Temperature for a Number of Gases. From G.-J. Su, Ind. Eng. Chem., 38, 803
(1946). Used by permission of the copyright holder.

35

1.4 The Coexistence of Phases and the Critical Point

temperatures.7 The agreement of the data for different substances with the law of
corresponding states is generally better than the agreement of the data with any simple
equation of state.
Exercise 1.15
Show that the Dieterici equation of state conforms to the law of corresponding states by expressing
it in terms of the reduced variables.

PROBLEMS
Section 1.4: The Coexistence of Phases and the Critical
Point
1.41 a. Use the van der Waals equation of state in terms of
reduced variables, Eq. (1.4-15), to calculate the
pressure of 1.000 mol of CO2 in a volume of 1.000 L at
100.0◦ C. The critical constants are in Table A.5 in
Appendix A. Since the critical compression factor of
carbon dioxide does not conform to the van der Waals
value, Zc  0.375, you must replace the experimental
th  (0.375)RT /P .
critical molar volume by Vmc
c
c
b. Repeat the calculation using the ordinary form of the
van der Waals equation of state.
1.42 a. Find the formulas for the parameters a and b in the
Soave and Gibbons–Laughton modifications of the
Redlich–Kwong equation of state in terms of the

critical constants. Show that information about the
extra parameters is not needed.
b. Find the values of the parameters a and b for nitrogen.
1.43 The critical temperature of xenon is 289.73 K, and its
critical pressure is 5.840 MPa (5.840 × 106 Pa).
a. Find the values of the van der Waals constants a and b
for xenon.
b. Find the value of the compression factor, Z, for xenon
at a reduced temperature of 1.35 and a reduced pressure
of 1.75.
1.44 a. Evaluate the parameters in the Dieterici equation of
state for argon from critical point data.
b. Find the Boyle temperature of argon according to the
Dieterici equation of state.

Summary of the Chapter
A system is defined as the material object that one is studying at a specific time. The
state of a system is the circumstance in which it is found, expressed by numerical values of a sufficient set of variables. A macroscopic system has two important kinds of
states: the macroscopic state, which concerns only variables pertaining to the system
as a whole, and the microscopic state, which pertains to the mechanical variables of
individual molecules. The equilibrium macroscopic state of a one-phase fluid (liquid
or gas) system of one component is specified by the values of three independent state
variables. All other macroscopic state variables are dependent variables, with values
given by mathematical functions of the independent variables.
The volumetric (P-V-T) behavior of gases under ordinary pressures is described
approximately by the ideal gas law. For higher pressures, several more accurate equations of state were introduced. A calculation practice was introduced: for ordinary
calculations: Gases are treated as though they were ideal. The volumes of solids and
liquids are computed with the compressibility and the coefficient of thermal expansion.
For ordinary calculations they are treated as though they had constant volume.

7 G.-J. Su, Ind. Eng. Chem., 38, 803 (1946).

36

1 The Behavior of Gases and Liquids

When two phases of a single substance are at equilibrium, the pressure is a function
only of the temperature. A phase diagram for a pure substance contains three curves
representing this dependence for the solid–liquid, solid–gas, and liquid–gas equilibria.
These three curves meet at a point called the triple point. The liquid–vapor coexistence
curve terminates at the critical point. Above the critical temperature, no gas–liquid
phase transition occurs and there is only one fluid phase. The law of corresponding
states was introduced, according to which all substances obey the same equation of
state in terms of reduced variables

ADDITIONAL PROBLEMS
1.45 Assume that when Julius Caesar exhaled for the last time
he exhaled 1.0 L of air.
a. Estimate the number of nitrogen molecules that were
exhaled in Julius Caesar’s last breath. The mole fraction
of nitrogen in air is approximately 0.78. (The mole
fraction of a substance is its amount in moles divided
by the total amount of all substances.)
b. Estimate the number of nitrogen molecules out of those
in part a that are now present in the physical chemistry
classroom.