Basic Mathematics for Chemists

Basic Mathematics for Chemists Second Edition Peter Tebbutt JOHN WILEY & SONS Chichester. New York. Weinheim. Brisbane. Singapore. Toronto 2001

Contents Preface to the First Edition xii Preface to the Second Edition xiv 1 Equations, Functions and Graphs 1 1.1 Introduction 1 1.2 Equations and Functions 1 1.3 Graphs, Plots and Coordinates 3 1.3.1 Definitions 3 1.3.2 Properties of Straight Line Functions 5 1.3.3 The Use of Graphs to Represent Experimental Data 8 1.3.4 Calibration Plots 9 1.3.5 Using Computers Plotting Programs and Spreadsheets 9 1.4 Powers, Indices and Exponents 11 1.4.1 Definitions 11 1.4.2 Properties 11 1.4.3 The Floating Point 14 1.5 Priorities 16 1.6 Factors 17 1.7 The Binomial Theorem and Pascal s Triangle 19 1.8 Manipulating Equations 23 1.8.1 The Rule of the Game 24 1.8.2 The Short Cut: Inverse Functions 25 1.8.3 Dealing with Fractions 27 1.9 Solving Equations 33 1.9.1 Simultaneous Equations 33 1.9.2 Quadratic Equations 36 1.10 Manipulating Units 39 1.10.1 Units? What Units? 39 1.10.2 Dimensional Analysis 40 1.10.3 Quantity Calculus 43 1.11 The Slope of a Curve 46 1.12 Concluding Remarks 48

Problems 48 2 Special Functions 52 2.1 Introduction 52 2.2 Logarithms 52 2.2.1 Definition 52 2.2.2 Uses of Logarithms 53 2.2.3 Uses of Logarithms in Chemistry the 'p' Scale 54 2.2.4 Logarithms in Thermodynamic Equations 57 2.2.5 Logarithms in Kinetic Equations 63 2.2.6 The Dependence of Rate on Temperature 65 2.2.7 Absorption of Light the Beer-Lambert Law 65 2.2.8 Using Logarithms to Find Relationships between Variables 66 2.3 The Exponential Function 67 2.3.1 Definition 67 2.3.2 Properties of the Exponential Function 69 2.3.3 The Exponential Function in Chemistry the Boltzmann Equation 71 2.3.4 Spectroscopy 73 2.3.5 Kinetics 73 2.3.6 Electrochemical Kinetics 73 2.3.7 Gas Kinetics 76 2.3.8 Thermodynamics 77 2.3.9 Statistical Thermodynamics 78 2.4 Trigonometric Functions Sine, Cosine and Tangent 78 2.4.1 Definitions 78 2.4.2 Properties of Sine and Cosine Waves 80 2.4.3 The Uses of Sines and Waves in Chemistry 83 2.5 Concluding Remarks 85 Problems 85 3 Practical Statistics 88 3.1 Introduction 88 3.2 Statistics for Data Analysis 88 3.2.1 Summarising Data Mathematical Approach 88 3.2.2 Summarising Data Graphical Approach 90

3.2.3 Accuracy and Precision 93 3.2.4 Errors Definitions 96 3.2.5 Presentation of Errors 97 3.2.6 Significant Figure Notation 101 3.2.7 The Line of Best Fit 103 3.3 In Conclusion 106 Problems 107 4 Differential Calculus 110 4.1 Differentiation 110 4.1.1 The Slope of the Function y = x 2 110 4.1.2 The Slope of the Line y = x 2 + Ax + 3. A General Approach 112 4.1.3 Notation 114 4.1.4 The Short Cut 114

of Exponential and Trigonometnc Functions 4.3 More Complex Differentiation 4.3.1 Simplifying Complex Formulae 4.3.2 Using Reduction Formulae 4.4 Higher Derivatives 4.4.1 Definitions 4.4.2 Uses of Second Derivatives 4.2 Proofs of Derivatives

4.4.3 Finding Minima, Maxima and Inflexions 4.5 Differentials 4.6 Partial Differentiation 4.6.1 Definitions and Notation 4.6.2 Graphical Representation of Functions Involving More than Two Variables 4.6.3 Fundamental Theorem of Partial Differentiation 4.7 Concluding Remarks Problems 5 Integral Calculus 5.1 Introduction 5.2 Indefinite Integration 5.2.1 Definition 5.2.2 Indefinite Integration the Method 5.2.3 Examples of Indefinite Integrals 5.3 Definite Integration 5.3.1 Definition as the Area under a Curve 5.3.2 The Area under the Curve y = x 2 5.3.3 Area under the Curve y = x 2. A General Approach 5.3.4 Area under the Curve y x 2 between Different Limits 5.3.5 Properties of Definite Integrals 5.4 Methods of Integration 5.4.1 Notation 5.4.2 Direct Integration 5.4.3 Integration by Substitution 5.4.4 The Method of Partial Fractions 5.4.5 Integration by Parts 5.4.6 Numerical Integration 5.5 Differentials, Line Integrals and Thermodynamic Concepts Problems 6 Differential Equations 6.1 Introduction 6.2 Definitions 6.3 Differential Equations in Kinetics 6.3.1 Solutions Involving Integration: Separable Equations

X CONTENTS 6.3.2 Solutions Involving Integration: First Order Linear Equations 184 6.3.3 Simplifications: Avoiding Complex Integration 189 6.3.4 Overview of Differential Equations in Kinetics 192 6.4 Differential Equations in Thermodynamics 192 6.4.1 Heat Capacities 193 6.4.2 The Gibbs Function Pressure and Temperature Dependence 197 6.5 Concluding Remarks 202 Problems 202 7 Statistics for Theoretical Chemistry 205 7.1 Introduction 205 7.2 Permutations Calculating Arrangements 205 7.3 Configurations and Microstates 209 7.4 Molecular Assemblies 211 7.5 The Importance of W = JV!/»,!«! 216 7.6 The Boltzmann Distribution 218 Problems 222 8 Complex Numbers, Vectors, Determinants and Matrices 223 8.1 Introduction 223 8.2 Complex Numbers 223 8.2.1 Simple Algebra of Complex Numbers 224 8.2.2 More about the Complex Conjugate, z* 226 8.2.3 Graphical Representation of Complex Numbers the Argand Diagram 226 8.2.4 The Link between the Exponential and Trigonometric Functions 229 8.2.5 Uses of Complex Numbers in Chemistry 231 8.3 Vectors 231 8.3.1 Notation and Representation of Vectors 232 8.3.2 Formal Notation 237 8.3.3 The Scalar Product 238 8.3.4 The Vector Product 241 8.3.5 Vectors in Chemistry 243 8.4 Determinants 244

8.4.1 Definitions 244 8.4.2 Determinants for the Solution of Simultaneous Equations 245 8.4.3 Representation of the Vector Product 248 8.4.4 Slater Determinants 248 8.4.5 Secular Equations and Secular Determinants 250 8.5 Matrices 254 8.5.1 Matrix Terminology and Simple Algebra 255 8.5.2 The Inverse Matrix 259 8.5.3 Symmetry Transformations 261 8.6 Concluding Remarks 264 Problems 264 j Appendix 1 The Greek Alphabet 267j CONTENTS xi Appendix 2 Solutions and Hints for Selected Problems 268 Index 273