Exam (elaborations) TEST BANK FOR The Chemistry Maths Book 2nd Edition

Exam (elaborations) TEST BANK FOR The Chemistry Maths Book 2nd Edition Numbers, variables, and units 1 1.1 Concepts 1 1.2 Real numbers 3 1.3 Factorization, factors, and factorials 7 1.4 Decimal representation of numbers 9 1.5 Variables 13 1.6 The algebra of real numbers 14 1.7 Complex numbers 19 1.8 Units 19 1.9 Exercises 29 2 Algebraic functions 31 2.1 Concepts 31 2.2 Graphical representation of functions 32 2.3 Factorization and simplification of expressions 34 2.4 Inverse functions 37 2.5 Polynomials 40 2.6 Rational functions 50 2.7 Partial fractions 52 2.8 Solution of simultaneous equations 55 2.9 Exercises 58 3 Transcendental functions 62 3.1 Concepts 62 3.2 Trigonometric functions 63 3.3 Inverse trigonometric functions 72 3.4 Trigonometric relations 73 3.5 Polar coordinates 77 3.6 The exponential function 80 3.7 The logarithmic function 83 3.8 Values of exponential and logarithmic functions 86 3.9 Hyperbolic functions 87 3.10 Exercises 89 4 Differentiation 93 4.1 Concepts 93 4.2 The process of differentiation 94 4.3 Continuity 97 4.4 Limits 98 4.5 Differentiation from first principles 100 4.6 Differentiation by rule 102 4.7 Implicit functions 110 viii Contents 4.8 Logarithmic differentiation 111 4.9 Successive differentiation 113 4.10 Stationary points 114 4.11 Linear and angular motion 118 4.12 The differential 119 4.13 Exercises 122 5 Integration 126 5.1 Concepts 126 5.2 The indefinite integral 127 5.3 The definite integral 132 5.4 The integral calculus 142 5.5 Uses of the integral calculus 147 5.6 Static properties of matter 148 5.7 Dynamics 152 5.8 Pressure–volume work 157 5.9 Exercises 160 6 Methods of integration 163 6.1 Concepts 163 6.2 The use of trigonometric relations 163 6.3 The method of substitution 165 6.4 Integration by parts 173 6.5 Reduction formulas 176 6.6 Rational integrands. The method of partial fractions 179 6.7 Parametric differentiation of integrals 184 6.8 Exercises 187 7 Sequences and series 191 7.1 Concepts 191 7.2 Sequences 191 7.3 Finite series 196 7.4 Infinite series 203 7.5 Tests of convergence 204 7.6 MacLaurin and Taylor series 208 7.7 Approximate values and limits 214 7.8 Operations with power series 219 7.9 Exercises 221 8 Complex numbers 225 8.1 Concepts 225 8.2 Algebra of complex numbers 226 8.3 Graphical representation 228 8.4 Complex functions 235 8.5 Euler’s formula 236 8.6 Periodicity 240 Contents ix 8.7 Evaluation of integrals 244 8.8 Exercises 245 9 Functions of several variables 247 9.1 Concepts 247 9.2 Graphical representation 248 9.3 Partial differentiation 249 9.4 Stationary points 253 9.5 The total differential 258 9.6 Some differential properties 262 9.7 Exact differentials 272 9.8 Line integrals 275 9.9 Multiple integrals 281 9.10 The double integral 283 9.11 Change of variables 285 9.12 Exercises 289 10 Functions in 3 dimensions 294 10.1 Concepts 294 10.2 Spherical polar coordinates 294 10.3 Functions of position 296 10.4 Volume integrals 299 10.5 The Laplacian operator 304 10.6 Other coordinate systems 307 10.7 Exercises 312 11 First-order differential equations 314 11.1 Concepts 314 11.2 Solution of a differential equation 315 11.3 Separable equations 318 11.4 Separable equations in chemical kinetics 322 11.5 First-order linear equations 328 11.6 An example of linear equations in chemical kinetics 330 11.7 Electric circuits 332 11.8 Exercises 334 12 Second-order differential equations. Constant coefficients 337 12.1 Concepts 337 12.2 Homogeneous linear equations 337 12.3 The general solution 340 12.4 Particular solutions 344 12.5 The harmonic oscillator 348 12.6 The particle in a one-dimensional box 352 12.7 The particle in a ring 356 12.8 Inhomogeneous linear equations 359 12.9 Forced oscillations 363 12.10 Exercises 365 x Contents 13 Second-order differential equations. Some special functions 368 13.1 Concepts 368 13.2 The power-series method 369 13.3 The Frobenius method 371 13.4 The Legendre equation 375 13.5 The Hermite equation 381 13.6 The Laguerre equation 384 13.7 Bessel functions 385 13.8 Exercises 389 14 Partial differential equations 391 14.1 Concepts 391 14.2 General solutions 392 14.3 Separation of variables 393 14.4 The particle in a rectangular box 395 14.5 The particle in a circular box 398 14.6 The hydrogen atom 401 14.7 The vibrating string 410 14.8 Exercises 413 15 Orthogonal expansions. Fourier analysis 416 15.1 Concepts 416 15.2 Orthogonal expansions 416 15.3 Two expansions in Legendre polynomials 421 15.4 Fourier series 425 15.5 The vibrating string 432 15.6 Fourier transforms 433 15.7 Exercises 441 16 Vectors 444 16.1 Concepts 444 16.2 Vector algebra 445 16.3 Components of vectors 448 16.4 Scalar differentiation of a vector 453 16.5 The scalar (dot) product 456 16.6 The vector (cross) product 462 16.7 Scalar and vector fields 466 16.8 The gradient of a scalar field 467 16.9 Divergence and curl of a vector field 469 16.10 Vector spaces 471 16.11 Exercises 471 17 Determinants 474 17.1 Concepts 474 17.2 Determinants of order 3 476 17.3 The general case 481 Contents xi 17.4 The solution of linear equations 483 17.5 Properties of determinants 488 17.6 Reduction to triangular form 493 17.7 Alternating functions 494 17.8 Exercises 496 18 Matrices and linear transformations 499 18.1 Concepts 499 18.2 Some special matrices 502 18.3 Matrix algebra 505 18.4 The inverse matrix 513 18.5 Linear transformations 516 18.6 Orthogonal matrices and orthogonal transformations 521 18.7 Symmetry operations 524 18.8 Exercises 529 19 The matrix eigenvalue problem 532 19.1 Concepts 532 19.2 The eigenvalue problem 534 19.3 Properties of the eigenvectors 537 19.4 Matrix diagonalization 543 19.5 Quadratic forms 546 19.6 Complex matrices 551 19.7 Exercises 555 20 Numerical methods 558 20.1 Concepts 558 20.2 Errors 558 20.3 Solution of ordinary equations 562 20.4 Interpolation 566 20.5 Numerical integration 573 20.6 Methods in linear algebra 581 20.7 Gauss elimination for the solution of linear equations 581 20.8 Gauss–Jordan elimination for the inverse of a matrix 584 20.9 First-order differential equations 585 20.10 Systems of differential equations 590 20.11 Exercises 592 21 Probability and statistics 595 21.1 Concepts 595 21.2 Descriptive statistics 595 21.3 Frequency and probability 601 21.4 Combinations of probabilities 603 21.5 The binomial distribution 604 21.6 Permutations and combinations 607 21.7 Continuous distributions 613 21.8 The Gaussian distribution 615 xii Contents 21.9 More than one variable 618 21.10 Least squares 619 21.11 Sample statistics 623 21.12 Exercises 624 Appendix. Standard integrals 627 Solutions to exercises 631 Index 653 1 Numbers, variables, and units 1.1 Concepts Chemistry, in common with the other physical sciences, comprises (i) experiment: the observation of physical phenomena and the measurement of physical quantities, and (ii) theory: the interpretation of the results of experiment, the correlation of one set of measurements with other sets of measurements, the discovery and application of rules to rationalize and interpret these correlations. Both experiment and theory involve the manipulation of numbers and of the symbols that are used to represent numbers and physical quantities. Equations containing these symbols provide relations amongst physical quantities. Examples of such equations are 1. the equation of state of the ideal gas pV = nRT (1.1) 2. Bragg’s Law in the theory of crystal structure nλ = 2d sinθ (1.2) 3. the Arrhenius equation for the temperature dependence of rate of reaction (1.3) 4. the Nernst equation for the emf of an electrochemical cell (1.4) When an equation involves physical quantities, the expressions on the two sides of the equal sign1 must be of the same kind as well as the same magnitude. E E RT nF = − Q o ln k Ae E RT = − a / 1 The sign for equality was introduced by Robert Recorde (c. 1510–1558) in his The whetstone of witte (London, 1557); ‘I will sette as I doe often in woorke use, a paire of paralleles, or Gemowe (twin) lines of one lengthe, thus:
, bicause noe.2. thynges can be moare equalle.’ 2 Chapter 1 Numbers, variables, and units EXAMPLE 1.1 The equation of state of the ideal gas, (1.1), can be written as an equation for the volume, in which the physical quantities on the right of the equal sign are the pressure p of the gas, the temperature T, the amount of substance n, and the molar gas constant R = 8.31447 J K−1 mol−1 . We suppose that we have one tenth of a mole of gas, n = 0.1 mol, at temperature T = 298 K and pressure p = 105 Pa. Then The quantities on the right side of the equation have been expressed in terms of SI units (see Section 1.8), and the combination of these units is the SI unit of volume, m3 (see Example 1.17). Example 1.1 demonstrates a number of concepts: (i) Function. Given any particular set of values of the pressure p, temperature T, and amount of substance n, equation (1.1) allows us to calculate the corresponding volume V. The value of V is determined by the values of p, T, and n; we say V is a function of p, T, and n. This statement is usually expressed in mathematics as V = f( p, T, n) and means that, for given values of p, T and n, the value of V is given by the value of a function f( p, T, n). In the present case, the function is f(p, T, n) = nRT2p. A slightly different form, often used in the sciences, is V = V( p, T, n) which means that V is some function of p, T and n, which may or may not be known. Algebraic functions are discussed in Chapter 2. Transcendental functions, including the trigonometric, exponential and logarithmic functions in equations (1.2) to (1.4), are discussed in Chapter 3. =. × − m =  .×. ×      ×   − − 105 1 1 mol J K mol K Pa       V nRT p = = . ×. × − − 10 1 1 5 mol J K mol K Pa V nRT p = 1.2 Real numbers 3 (ii) Constant and variable. Equation (1.1) contains two types of quantity: Constant: a quantity whose value is fixed for the present purposes. The quantity R = 8.31447 J K−1 mol−1 is a constant physical quantity.2 A constant number is any particular number; for example, a = 0.1 and π = 3.14159= Variable: a quantity that can have any value of a given set of allowed values. The quantities p, T, and n are the variables of the function f( p, T, n) = nRT2p. Two types of variable can be distinguished. An independent variable is one whose value does not depend on the value of any other variable. When equation (1.1) is written in the form V = nRT2p, it is implied that the independent variables are p, T, and n. The quantity V is then the dependent variable because its value depends on the values of the independent variables. We could have chosen the dependent variable to be T and the independent variables as p, V, and n; that is, T = pV2nR. In practice, the choice of independent variables is often one of mathematical convenience, but it may also be determined by the conditions of an experiment; it is sometimes easier to measure pressure p, temperature T, and amount of substance n, and to calculate V from them. Numbers are discussed in Sections 1.2 to 1.4, and variables in Section 1.5. The algebra of numbers (arithmetic) is discussed in Section 1.6. (iii) A physical quantity is always the product of two quantities, a number and a unit; for example T = 298.15 K or R = 8.31447 J K−1 mol−1 . In applications of mathematics in the sciences, numbers by themselves have no meaning unless the units of the physical quantities are specified. It is important to know what these units are, but the mathematics does not depend on them. Units are discussed in Section 1.8. 1.2 Real numbers The concept of number, and of counting, is learnt very early in life, and nearly every measurement in the physical world involves numbers and counting in some way. The simplest numbers are the natural numbers, the ‘whole numbers’ or signless integers 1, 2, 3,1= It is easily verified that the addition or multiplication of two natural numbers always gives a natural number, whereas subtraction and division may not. For example 5 − 3 = 2, but 5 − 6 is not a natural number. A set of numbers for which the operation of subtraction is always valid is the set of integers, consisting of all positive and negative whole numbers, and zero: - −3 −2 −1 0 +1 +2 +3 - The operations of addition and subtraction of both positive and negative integers are made possible by the rules m + (−n) = m − n m − (−n) = m + n (1.5) 2 The values of the fundamental physical constants are under continual review. For the latest recommended values, see the NIST (National Institute of Standards and Technology) website at 4 Chapter 1 Numbers, variables, and units so that, for example, the subtraction of a negative number is equivalent to the addition of the corresponding positive number. The operation of multiplication is made possible by the rules (−m) × (−n) = +(m × n) (−m) × (+n) = −(m × n) (1.6) Similarly for division. Note that −m = (−1) × m. EXAMPLES 1.2 Addition and multiplication of negative numbers 2 + (−3) = 2 − 3 = −1 2 − (−3) = 2 + 3 = 5 (−2) × (−3) = 2 × 3 = 6 (2) × (−3) = −2 × 3 = −6 (−6) ÷ (−3) = 6 ÷ 3 = 2 6 ÷ (−3) = −6 ÷ 3 = −2 0 Exercises 1–7 In equations (1.5) and (1.6) the letters m and n are symbols used to represent any pair of integers; they are integer variables, whose values belong to the (infinite) set of integers. Division of one integer by another does not always give an integer; for example 6 ÷ 3 = 2, but 6 ÷ 4 is not an integer. A set of numbers for which the operation of division is always valid is the set of rational numbers, consisting of all the numbers m2n = m ÷ n where m and n are integers (m2n, read as ‘m over n’, is the more commonly used notation for ‘m divided by n’). The definition excludes the case n = 0 because division by zero is not defined (see Section 1.6), but integers are included because an integer m can be written as m21. The rules for the combination of rational numbers (and of fractions in general) are (1.7) (1.8) (1.9) where, for example, mq means m × q. EXAMPLES 1.3 Addition of fractions (1) Add and . 1 4 1 2 m n p q m n q p mq np ÷=×= m n p q mp nq × = m n p q mq np nq + = + 1.2 Real numbers 5 The number one half is equal to two quarters and can be added to one quarterto give three quarters: The value of a fraction like 122 is unchanged if the numerator and the denominator are both multiplied by the same number: and the general method of adding fractions is (a) find a common denominator for the fractions to be added, (b) express all the fractions in terms of this common denominator, (c) add