Table Of ContentMathematics for Physical Chemistry
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Mathematics for Physical
Chemistry
Fourth Edition
Robert G. Mortimer
Professor Emeritus
Rhodes College
Memphis, Tennessee
AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD • PARIS
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Fourth edition 2013
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Library of Congress Cataloging-in-Publication Data
Mortimer, Robert G.
Mathematics for physical chemistry / Robert G. Mortimer, Professor emeritus,
Rhodes College Memphis, Tennessee. — Fourth edition.
pages cm
Includes bibliographical references and index.
ISBN 978-0-12-415809-2 (pbk.)
1. Chemistry, Physical and theoretical—Mathematics. I. Title.
QD455.3.M3M67 2013
510.24'541—dc23
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
2012047249
ISBN: 978-0-12-415809-2
For information on all Academic Press publications
visit our web site at store.elsevier.com
Printed and bound in USA
13 14 15 16 17 10 9 8 7 6 5 4 3 2 1
Dedication
To my wife, Ann.
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Contents
Preface xi 3.2 Coordinate Systems in Two Dimensions 26
3.3 Coordinate Systems in Three Dimensions 27
3.3.1 Cartesian Coordinates 27
3.3.2 Spherical Polar Coordinates 27
3.3.3 Cylindrical Polar Coordinates 28
1 Problem Solving and Numerical 3.4 Imaginary and Complex Numbers 29
Mathematics 3.4.1 Mathematical Operations with Complex
Numbers 29
1.1 Problem Solving 1
3.4.2 The Argand Diagram 29
1.2 Numbers and Measurements 1
3.4.3 The Complex Conjugate 31
1.3 Numerical Mathematical Operations 2
3.4.4 The Magnitude of a Complex Quantity 31
1.3.1 Binary Arithmetic Operations 2
3.4.5 Roots of a Complex Number 32
1.3.2 Additional Numerical Operations 2
3.5 Problem Solving and Symbolic Mathematics 32
1.4 Units of Measurement 3
1.5 The Factor-Label Method 5 4 Vectors and Vector Algebra
1.6 Measurements, Accuracy, and Significant
Digits 5 4.1 Vectors in Two Dimensions 35
1.6.1 Scientific Notation 6 4.1.1 The Sum and Difference of Two Vectors 35
1.6.2 Rounding 6 4.1.2 The Product of a Vector and a Scalar 36
1.6.3 Significant Digits in a Calculated 4.1.3 Unit Vectors 36
Quantity 7 4.1.4 The Scalar Product of Two Vectors 37
4.1.5 The Magnitude of a Vector 38
2 Mathematical Functions 4.2 Vectors in Three Dimensions 38
4.2.1 Unit Vectors in Three Dimensions 38
2.1 Mathematical Functions in Physical 4.2.2 The Magnitude of a Vector 38
Chemistry 11 4.2.3 The Sum and Difference of Two
2.1.1 Functions in Thermodynamics 11 Vectors 39
2.1.2 Functions in Quantum Mechanics 12 4.2.4 The Product of a Scalar and a Vector 39
2.1.3 Function Notation 12 4.2.5 The Scalar Product of Two Vectors 39
2.1.4 Continuity 12 4.2.6 The Vector Product of Two Vectors 39
2.1.5 Graphs of Functions 12 4.3 Physical Examples of Vector Products 40
2.2 Important Families of Functions 15 4.3.1 Magnetic Force 40
2.2.1 Linear Functions 15 4.3.2 Electrostatic Force 41
2.2.2 Quadratic Functions 16 4.3.3 Angular Momentum 41
2.2.3 Cubic Functions 16
2.2.4 Logarithms 16 5 Problem Solving and the Solution of
2.2.5 Exponentials 17 Algebraic Equations
2.2.6 Trigonometric Functions 18
5.1 Algebraic Methods for Solving One Equation
2.2.7 Inverse Trigonometric Functions 21
with One Unknown 43
2.2.8 Hyperbolic Trigonometric Functions 22
5.1.1 Polynomial Equations 43
2.2.9 Significant Digits in Logarithms,
5.1.2 Approximate Solutions to Equations 44
Exponentials, and Trigonometric
5.2 Numerical Solution of Algebraic Equations 47
Functions 22
5.2.1 Graphical Solution of Algebraic
2.3 Generating Approximate Graphs 22
Equations 47
5.2.2 Trial and Error 48
3 Problem Solving and Symbolic
5.2.3 The Method of Bisection 48
Mathematics: Algebra
5.2.4 Solving Equations Numerically with
3.1 The Algebra of Real Scalar Variables 25 Excel 48
vii
viii Contents
5.3 A Brief Introduction to Mathematica 49 7.5 Techniques of Integration 80
5.3.1 Numerical Calculations with 7.5.1 The Method of Substitution 80
Mathematica 49 7.5.2 Integration by Parts 80
5.3.2 Symbolic Algebra with Mathematica 51 7.5.3 The Method of Partial Fractions 81
5.3.3 Solving Equations with Mathematica 52 7.5.4 Integration with Mathematica 83
5.3.4 Graphing with Mathematica 53 7.6 Numerical Integration 83
5.4 Simultaneous Equations: Two Equations with 7.6.1 The Bar-Graph Approximation 83
Two Unknowns 53 7.6.2 The Trapezoidal Approximation 83
5.4.1 The Method of Substitution 53 7.6.3 Simpson’s Rule 84
5.4.2 The Method of Elimination 54 7.6.4 Numerical Integration with
5.4.3 Consistency and Independence in Mathematica 85
Simultaneous Equations 54
5.4.4 Homogeneous Linear Equations 54 8 Differential Calculus with Several
5.4.5 Using Mathematica to Solve Independent Variables
Simultaneous Equations 55
8.1 Functions of Several Independent Variables 89
8.2 Changes in a Function of Several Variables,
6 Differential Calculus
Partial Derivatives 91
6.1 The Tangent Line and the Derivative of a 8.2.1 Differentials 91
Function 59 8.3 Change of Variables 92
6.1.1 The Derivative 60 8.4 Useful Partial Derivative Identities 93
6.1.2 Derivatives of Specific Functions 61 8.4.1 The Variable-Change Identity 93
6.2 Differentials 61 8.4.2 The Reciprocal Identity 94
6.3 Some Useful Derivative Identities 63 8.4.3 The Euler Reciprocity Relation 94
6.3.1 The Derivative of a Constant 63 8.4.4 The Maxwell Relations 94
6.3.2 The Derivative of a Function Times a 8.4.5 The Cycle Rule 95
Constant 63 8.4.6 The Chain Rule 95
6.3.3 The Derivative of a Product of Two 8.5 Thermodynamic Variables Related to Partial
Functions 63 Derivatives 95
6.3.4 The Derivative of the Sum of Two 8.6 Exact and Inexact Differentials 96
Functions 63 8.6.1 Integrating Factors 97
6.3.5 The Derivative of the Difference of 8.7 Maximum and Minimum Values of Functions
Two Functions 63 of Several Variables 98
6.3.6 The Derivative of the Quotient of Two 8.7.1 Constrained Maximum/Minimum
Functions 63 Problems 99
6.3.7 The Derivative of a Function of a 8.7.2 Lagrange’s Method of Undetermined
Function (The Chain Rule) 63 Multipliers 99
6.4 Newton’s Method 64 8.8 Vector Derivative Operators 101
6.5 Higher-Order Derivatives 65 8.8.1 Vector Derivatives in Cartesian
6.5.1 The Curvature of a Function 66 Coordinates 101
6.6 Maximum–Minimum Problems 66 8.8.2 Vector Derivatives in Other Coordinate
6.7 Limiting Values of Functions 67 Systems 103
6.8 L’Hôpital’s Rule 68
9 Integral Calculus with Several
7 Integral Calculus Independent Variables
7.1 The Antiderivative of a Function 73 9.1 Line Integrals 107
7.1.1 Position, Velocity, and Acceleration 73 9.1.1 Line Integrals of Exact Differentials 108
7.2 The Process of Integration 74 9.1.2 Line Integrals of Inexact Differentials 109
7.2.1 The Definite Integral as an Area 76 9.1.3 Line Integrals with Three Integration
7.2.2 Facts about Integrals 76 Variables 109
7.2.3 Derivatives of Definite Integrals 78 9.1.4 Line Integrals in Thermodynamics 110
7.3 Tables of Indefinite Integrals 78 9.2 Multiple Integrals 111
7.4 Improper Integrals 79 9.2.1 Double Integrals 111
Contents ix
9.2.2 The Double Integral Representing 12.4 Differential Equations with Separable
a Volume 112 Variables 149
9.2.3 Triple Integrals 113 12.5 Exact Differential Equations 149
9.2.4 Changing Variables in Multiple Integrals 113 12.6 Solution of Inexact Differential Equations
Using Integrating Factors 150
10 Mathematical Series 12.7 Partial Differential Equations 151
12.7.1 Waves in a String 151
10.1 Constant Series 119
12.7.2 Solution by Separation of Variables 151
10.1.1 Some Convergent Constant Series 120
12.7.3 The Schrödinger Equation 154
10.1.2 The Geometric Series 120
12.8 Solution of Differential Equations Using
10.1.3 The Harmonic Series 121
Laplace Transforms 154
10.1.4 Tests for Convergence 121
12.9 Numerical Solution of Differential
10.2 Power Series 122
Equations 155
10.2.1 Maclaurin Series 122
12.9.1 Euler’s Method 155
10.2.2 Taylor Series 123
12.9.2 The Runge–Kutta Method 156
10.2.3 The Convergence of Power Series 124
12.9.3 Solution of Differential Equations
10.2.4 Power Series in Physical
with Mathematica 156
Chemistry 125
10.3 Mathematical Operations on Series 126 13 Operators, Matrices, and Group
10.4 Power Series with More Than One Theory
Independent Variable 126
13.1 Mathematical Operators 161
11 Functional Series and Integral 13.1.1 Eigenfunctions and Eigenvalues 162
13.1.2 Operator Algebra 162
Transforms
13.1.3 Operators in Quantum
11.1 Fourier Series 129 Mechanics 164
11.1.1 Finding the Coefficients of a 13.2 Symmetry Operators 165
Fourier Series—Orthogonality 129 13.3 The Operation of Symmetry Operators
11.1.2 Fourier Series with Complex on Functions 167
Exponential Basis Functions 132 13.4 Matrix Algebra 169
11.2 Other Functional Series with Orthogonal 13.4.1 The Equality of Two Matrices 169
Basis Sets 132 13.4.2 The Sum of Two Matrices 169
11.2.1 Hilbert Space 132 13.4.3 The Product of a Scalar and a
11.2.2 Determining the Expansion Matrix 169
Coefficients 133 13.4.4 The Product of Two Matrices 169
11.3 Integral Transforms 134 13.4.5 The Identity Matrix 170
11.3.1 Fourier Transforms (Fourier 13.4.6 The Inverse of a Matrix 171
Integrals) 134 13.4.7 Matrix Terminology 172
11.3.2 Laplace Transforms 136 13.5 Determinants 172
13.6 Matrix Algebra with Mathematica 174
12 Differential Equations 13.7 An Elementary Introduction to Group
Theory 175
12.1 Differential Equations and Newton’s
13.8 Symmetry Operators and Matrix
Laws of Motion 139
Representations 177
12.2 Homogeneous Linear Differential
Equations with Constant Coefficients 141 14 The Solution of Simultaneous
12.2.1 The Harmonic Oscillator 141
Algebraic Equations with More
12.2.2 The Damped Harmonic
than Two Unknowns
Oscillator—A Nonconservative
System 144 14.1 Cramer’s Rule 183
12.3 Inhomogeneous Linear Differential 14.2 Linear Dependence and Inconsistency 185
Equations: The Forced Harmonic 14.3 Solution by Matrix lnversion 185
Oscillator 147 14.4 Gauss–Jordan Elimination 186
12.3.1 Variation of Parameters Method 147 14.5 Linear Homogeneous Equations 186
