www.elsolucionario.net Mathematical Methods for Physical and Analytical Chemistry www.elsolucionario.net Mathematical Methods for Physical and Analytical Chemistry David Z Goodson Department of Chemistry & Biochemistry University of Massachusetts Dartmouth WILEY A JOHN WILEY & SONS, INC., PUBLICATION www.elsolucionario.net The text was typeset by the author using LaTex (copyright 1999, 2002-2008, LaTex3 Project) and the figures were created by the author using gnuplot (copyright 1986-1993, 1998, 2004, Thomas Williams and Colin Kelley) Copyright © 2011 by John Wiley & Sons, Inc All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior 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Mappings 1.2 Differentials and Derivatives 1.3 Partial Derivatives 1.4 Integrals 1.5 Critical Points 3 14 Functions: Examples 2.1 Algebraic Functions 2.2 Transcendental Functions 2.2.1 Logarithm and Exponential 2.2.2 Circular Functions 2.2.3 Gamma and Beta Functions 2.3 Functional 19 19 21 21 24 26 31 Coordinate Systems 3.1 Points in Space 3.2 Coordinate Systems for Molecules 3.3 Abstract Coordinates 3.4 Constraints 3.4.1 Degrees of Freedom 3.4.2 Constrained Extrema* 3.5 Differential Operators in Polar Coordinates 33 33 35 37 39 39 40 43 Integration 4.1 Change of Variables in Integrands 4.1.1 Change of Variable: Examples 4.1.2 Jacobian Determinant 4.2 Gaussian Integrals 4.3 Improper Integrals 4.4 Dirac Delta Function 4.5 Line Integrals 47 47 47 49 51 53 56 57 Numerical Methods 5.1 Interpolation 5.2 Numerical Differentiation 5.3 Numerical Integration 5.4 Random Numbers 5.5 Root Finding 5.6 Minimization* 61 61 63 65 70 71 74 "This section treats an advanced topic It can be skipped without loss of continuity VII www.elsolucionario.net viii CONTENTS Complex Numbers 6.1 Complex Arithmetic 6.2 Fundamental Theorem of Algebra 6.3 The Argand Diagram 6.4 Functions of a Complex Variable* 6.5 Branch Cuts* Extrapolation 7.1 Taylor Series 7.2 Partial Sums 7.3 Applications of Taylor Series 7.4 Convergence 7.5 Summation Approximants* Part II 79 79 81 83 87 89 93 93 97 99 102 104 Statistics Estimation 8.1 Error and Estimation 8.2 Probability Distributions 8.2.1 Probability Distribution Functions 8.2.2 The Normal Distribution 8.2.3 The Poisson Distribution 8.2.4 The Binomial Distribution* 8.2.5 The Boltzmann Distribution* 8.3 Outliers 8.4 Robust Estimation 111 Ill 113 113 115 119 120 121 124 126 Analysis of Significance 9.1 Confidence Intervals 9.2 Propagation of Error 9.3 Monte Carlo Simulation of Error 9.4 Significance of Difference 9.5 Distribution Testing* 131 131 136 139 140 144 10 Fitting 10.1 Method of Least Squares 10.1.1 Polynomial Fitting 10.1.2 Weighted Least Squares 10.1.3 Generalizations of the Least-Squares Method* 10.2 Fitting with Error in Both Variables 10.2.1 Uncontrolled Error in ж 10.2.2 Controlled Error in ж 10.3 Nonlinear Fitting 151 151 151 154 155 157 157 160 162 www.elsolucionario.net ix CONTENTS 11 Quality of Fit 11.1 Confidence Intervals for Parameters 11.2 Confidence Band for a Calibration Line 11.3 Outliers and Leverage Points 11.4 Robust Fitting* 11.5 Model Testing 12 E x p e r i m e n t Design 12.1 Risk Assessment 12.2 Randomization 12.3 Multiple Comparisons 12.3.1 ANOVA* 12.3.2 Post-Hoc Tests* 12.4 Optimization* Part III ' 165 165 168 171 173 176 181 181 185 188 189 191 195 Differential Equations 13 Examples of Differential Equations 13.1 Chemical Reaction Rates 13.2 Classical Mechanics 13.2.1 Newtonian Mechanics 13.2.2 Lagrangian and Hamiltonian Mechanics 13.2.3 Angular Momentum 13.3 Differentials in Thermodynamics 13.4 Transport Equations 203 203 205 205 208 211 212 213 14 Solving Differential Equations, I 14.1 Basic Concepts 14.2 The Superposition Principle 14.3 First-Order ODE's 14.4 Higher-Order ODE's 14.5 Partial Differential Equations 217 217 220 222 225 228 15 Solving Differential Equations, II 15.1 Numerical Solution 15.1.1 Basic Algorithms 15.1.2 The Leapfrog Method* 15.1.3 Systems of Differential Equations 15.2 Chemical Reaction Mechanisms 15.3 Approximation Methods 15.3.1 Taylor Series* 15.3.2 Perturbation Theory* 231 231 231 234 235 236 239 239 242 www.elsolucionario.net x Part IV CONTENTS Linear Algebra 16 Vector Spaces 16.1 Cartesian Coordinate Vectors 16.2 Sets 16.3 Groups 16.4 Vector Spaces 16.5 Functions as Vectors 16.6 Hilbert Spaces 16.7 Basis Sets 247 247 248 249 251 252 253 256 17 Spaces of Functions 17.1 Orthogonal Polynomials 17.2 Function Resolution 17.3 Fourier Series 17.4 Spherical Harmonics 261 261 267 270 275 18 Matrices 18.1 Matrix Representation of Operators 18.2 Matrix Algebra 18.3 Matrix Operations 18.4 Pseudoinverse* 18.5 Determinants 18.6 Orthogonal and Unitary Matrices 18.7 Simultaneous Linear Equations 279 279 282 284 286 288 290 292 19 Eigenvalue Equations 19.1 Matrix Eigenvalue Equations 19.2 Matrix Diagonalization 19.3 Differential Eigenvalue Equations 19.4 Hermitian Operators 19.5 The Variational Principle* 297 297 301 305 306 309 20 Schrödinger's Equation 20.1 Quantum Mechanics 20.1.1 Quantum Mechanical Operators 20.1.2 The Wavefunction 20.1.3 The Basic Postulates* 20.2 Atoms and Molecules 20.3 The One-Electron Atom 20.3.1 Orbitals 20.3.2 The Radial Equation* 20.4 Hybrid Orbitals 20.5 Antisymmetry* 20.6 Molecular Orbitals* 313 313 313 316 317 319 321 321 323 325 327 329 www.elsolucionario.net CONTENTS xi 21 Fourier Analysis 21.1 The Fourier Transform 21.2 Spectral Line Shapes* 21.3 Discrete Fourier Transform* 21.4 Signal Processing 21.4.1 Noise Filtering* 21.4.2 Convolution* 333 333 336 339 342 342 345 A Computer Programs A.l Robust Estimators A.2 FREML A.3 Neider-Mead Simplex Optimization 351 351 352 352 В Answers to Selected Exercises 355 С Bibliography 367 Index 373 www.elsolucionario.net ANSWERS FOR CHAPTER 15 363 14.4 (b), (c), and (e) In each of these cases the eigenvalue is 14.5 x(t) = (67 cm) cos(1.4 s" t) 14.6 [A] 60s = [A]0 - ±Δ[Β] = 0.0854 mol/L, 14.7 (a) ξ(ί) = L/ 2akt bkt\ 2^e2akt_behkt', к = 0.0026 s _ where a = cA,o, b = cB,o· 14.8 (a) dy/dx + (1 + 2/)z = 0, dj//(l + y) = - z d x , y(x) = β ( _ χ ) / - (b) y(x) = - 6a: (c) y{x) = (2\nx - 4x + x2 + / ) _ / (d) y{x) = - 1/(2 - x) 14.9 Let x = cu Then the choice с = α ^ will give the Airy differential equation with independent variable u It follows that y{x) = aAi(a~l^3x) + bBi(a~1^3x) where a and ò are constants of integration 14.10 (a) (b) constants of integration Any solution can be written in the form apx + bpy + cpz with arbitrary constants a, b, and с (с) There is no inconsistency This is a partial differential equation The theorem does not apply 14.11 The substitution z{x,y) = u(x)v(y) leads to the ODE's u"(x) = KU(X) and v"(y) — (к—4)v(y), where к is an arbitrary separation constant We find that u(x) oc ε±κ1/2χ and v(y) ос е ± ( к - ) / \ A general solution is z{x,y) = л е « / * + ( « - ) / у + Ве-к1/2х + (к-4)1/1у CeKU2x_(K_4)l/2y + + 14.12 First make the substitution c(x,y,t) dT T(t) dt , W(x,y) De_Kl/2x_(K_4)l/2y^ which gives = W(x,y)T(t), D(