Basic Mathematics for Chemists

Transcription

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

2 Contents Preface to the First Edition xii Preface to the Second Edition xiv 1 Equations, Functions and Graphs Introduction Equations and Functions Graphs, Plots and Coordinates Definitions Properties of Straight Line Functions The Use of Graphs to Represent Experimental Data Calibration Plots Using Computers Plotting Programs and Spreadsheets Powers, Indices and Exponents Definitions Properties The Floating Point Priorities Factors The Binomial Theorem and Pascal s Triangle Manipulating Equations The Rule of the Game The Short Cut: Inverse Functions Dealing with Fractions Solving Equations Simultaneous Equations Quadratic Equations Manipulating Units Units? What Units? Dimensional Analysis Quantity Calculus The Slope of a Curve Concluding Remarks 48

3 Problems 48 2 Special Functions Introduction Logarithms Definition Uses of Logarithms Uses of Logarithms in Chemistry the 'p' Scale Logarithms in Thermodynamic Equations Logarithms in Kinetic Equations The Dependence of Rate on Temperature Absorption of Light the Beer-Lambert Law Using Logarithms to Find Relationships between Variables The Exponential Function Definition Properties of the Exponential Function The Exponential Function in Chemistry the Boltzmann Equation Spectroscopy Kinetics Electrochemical Kinetics Gas Kinetics Thermodynamics Statistical Thermodynamics Trigonometric Functions Sine, Cosine and Tangent Definitions Properties of Sine and Cosine Waves The Uses of Sines and Waves in Chemistry Concluding Remarks 85 Problems 85 3 Practical Statistics Introduction Statistics for Data Analysis Summarising Data Mathematical Approach Summarising Data Graphical Approach 90

4 3.2.3 Accuracy and Precision Errors Definitions Presentation of Errors Significant Figure Notation The Line of Best Fit In Conclusion 106 Problems Differential Calculus Differentiation The Slope of the Function y = x The Slope of the Line y = x 2 + Ax + 3. A General Approach Notation The Short Cut 114

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

6 4.4.3 Finding Minima, Maxima and Inflexions 4.5 Differentials 4.6 Partial Differentiation Definitions and Notation Graphical Representation of Functions Involving More than Two Variables Fundamental Theorem of Partial Differentiation 4.7 Concluding Remarks Problems 5 Integral Calculus 5.1 Introduction 5.2 Indefinite Integration Definition Indefinite Integration the Method Examples of Indefinite Integrals 5.3 Definite Integration Definition as the Area under a Curve The Area under the Curve y = x Area under the Curve y = x 2. A General Approach Area under the Curve y x 2 between Different Limits Properties of Definite Integrals 5.4 Methods of Integration Notation Direct Integration Integration by Substitution The Method of Partial Fractions Integration by Parts 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 Solutions Involving Integration: Separable Equations

7 X CONTENTS Solutions Involving Integration: First Order Linear Equations Simplifications: Avoiding Complex Integration Overview of Differential Equations in Kinetics Differential Equations in Thermodynamics Heat Capacities The Gibbs Function Pressure and Temperature Dependence Concluding Remarks 202 Problems Statistics for Theoretical Chemistry Introduction Permutations Calculating Arrangements Configurations and Microstates Molecular Assemblies The Importance of W = JV!/»,!«! The Boltzmann Distribution 218 Problems Complex Numbers, Vectors, Determinants and Matrices Introduction Complex Numbers Simple Algebra of Complex Numbers More about the Complex Conjugate, z* Graphical Representation of Complex Numbers the Argand Diagram The Link between the Exponential and Trigonometric Functions Uses of Complex Numbers in Chemistry Vectors Notation and Representation of Vectors Formal Notation The Scalar Product The Vector Product Vectors in Chemistry Determinants 244

8 8.4.1 Definitions Determinants for the Solution of Simultaneous Equations Representation of the Vector Product Slater Determinants Secular Equations and Secular Determinants Matrices Matrix Terminology and Simple Algebra The Inverse Matrix Symmetry Transformations 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