SI units, physical constants and conversion factors; the Greek alphabet and a summary of useful relations

Transcription

1 APPENDIX SI units, physical constants and conversion factors; the Greek alphabet and a summary of useful relations SI units SI stands for the agreed international system of units, which is based upon the kilogram (kg), metre (m) and second (s), together with the kelvin (K) unit of temperature, the mole (mol) for amount of substance and the ampere (A). The candela (cd) unit ofluminous intensity is also a basic unit. Other units are derived from these basic units. The kelvin is defined as 1/ of the thermodynamic temperature of the triple point of water. This means any subsequent refinements in absolute temperature measurement will alter the size of the degree and so, say, the absolute temperature of the boiling point of water, without changing the triple point. The ice point is 0.01 K below the triple point so that O C is K. The mole is the amount of substance containing as many elementary units as there are atoms in kg of 12c. This allows experimental measurement of the Avogadro constant by, in effect, counting this number of atoms, and of the gas constant from measurements of the properties of gases. The unit offorce is the newton (N), defined as that which produces an acceleration of 1 m s - 2 in a mass of 1 kg. Pressure is then force per unit area in N m - 2, this unit being called the pascal (Pa). Energy is measured in joule (J), which is 1 N m, and power in watt (w) which is 1 J s - 1.

2 186 SI units, physical constants, conversion factors The ampere (A) is that current which produces a force of 2 x 10-7 N m -1 between long straight parallel conductors 1 m apart in vacuum. The volt (V) is then the potential difference such that a current of 1 A dissipates 1 watt, and the coulomb (C) is 1 A s. The faraday constant (F) can then be measured as the number of coulombs of charge carried by one avogadro of electrons. Standard gravity is m s - 2 and a standard atmosphere is the pressure produced by a mercury column 0.76 m high under standard gravity at ODC; this is kpa. The preferred unit is the bar, defined as 100 kpa. Other units that are still in use are the calorie in various forms, usually the thermochemical calorie of J, and the Angstrom, A, which is m. The word 'litre' is now regarded as a special name for the cubic decimetre, with the recommendation that neither the word litre nor its symbol, 1, should be used to express results of high precision. This is because the litre was previously defined as dm 3, this definition being rescinded in Physical constants Recommended values of the fundamental physical constants are published by the International Union of Pure and Applied Chemistry. Some of the values given in Pure and Applied Chemistry, 51, 1 (1979) are: Avogadro constant L Boltzmann constant k gas constant R Planck constant h Faraday constant F mass of proton charge on electron e mass of electron speed of light in vacuum permittivity of vacuum normal atmosphere zero of Celsius scale standard acceleration of free fall (31) X mol (44) x J K (26) J K- 1 mol (36) x J s (27) X 10 4 C mol (86) x kg (46) X C (47) X kg (1) X 108 m S-l (5) x C2 N- 1 m x 10 5 Pa exactly K exactly m s - 2 exactly The notation used is that the figure(s) in brackets is (are) the standard deviation of the last figure(s) quoted.

3 Sf units, physical constants, conversion factors 187 Conversion factors for units 1 amp = I A = I J s - I V - I 1 atm = kpa 1 cal = J 1 coulomb = 1 C = 1 A s = 1 J V-I 0.30 debye = C m 1 dyne = 10-5 N 1 electron volt = 1 ev = x J 1 erg = 10-7 J 1 farad = 1 F = 1 A S V-I = 1 m - 2 kg - I S4 A 2 1 gallon (UK) = litre = gallon (USA) 1 inch = cm 1 joule = I J = I N m 1m 3 = 10 6 cm 3 1 micron = 1 p. = I p.m = 10-6 m I newton = I N = 1 m kg s - 2 lpascai=1 Pa=INm- 2 =lm- I kgs- 2 1 pound (UK) = 1 Ib = kg I torr = 1 mmhg = kpa 1 volt = 1 V = 1 J A -I S - I = 1 m 2 kgs- 3 A-I I watt = 1 W = 1J s - I = I m 2 kg s - 3 The Greek alphabet A IX alpha B p beta r y gamma i\ (j delta E E epsilon Z, zeta H '1 eta e 0 theta I iota K K kappa i\ ). lambda M p. mu N v nu - ~ xi 0 0 omicron n 1t pi P p rho 1: (J sigma T T tau Y u upsilon <l> 4> phi X X chi 'I' psi n OJ omega '"

4 188 SI units. physical constants, conversion factors Summary of useful relations cos 2 A + sin 2 A = 1 cos 2 A - sin 2 A = cos 2A sin 2A = 2sinA cos A sin (A±B) = sin A cos B±cos A sin B cos (A ±B) = cos A cos B+ sin A sin B exey = ex + Y a-x = 1jaX if y = ex then x = In y In x = loge x = log x = og 10 X log1o lo-x = -x ax 2 +bx+c = 0; b 1 x = --±-.j(b 2-4ac) 2a 2a. x 3 X S SIn x = x-- 3 ' +, n(n - 1) (1 + xt = 1 + nx + 2! x n(n -1) (a+xt=an+nan-1x+ 2! an- 2 x x 2 f(x) = f(o) + xl' (0) + 2! f" (0) f(x) 1 I' (x) ff() () 0 1m -() = 1m ~ ( 1 ) a = 9 a = or CX) x ~ 9 a X x ~ 9 a X z = z(x, y); dz = (!:)Y dx+(;;)x dy dy dx dy du du dx

5 SI units, physical constants, conversion/actors 189 (;:)y = -(;;)x ( ; ~ ) % f udv = uv - f vdu e iy = cos y + i sin y e- iy = cos y -i siny

6 Index Abscissa, 12 Action, 133 Angular momentum, 71, 91, 135 Angular velocity, 67, 91 Anharmonicity, 21 Approximation, numerical, 53, 55, 57 Arc length, 1, 67, 118 Area by integration, 99, 115, 119 differential element of, 117 Argand diagram, 25 Argument of complex number, 26 Asymptotes, 18, 62 Axes principal, 70 rectangular cartesian, 21 Binomial series, 56 Boltzmann distribution, 48 Calculus of variations, 46, 129 Central limit theorem, 176 Centrifugal force, 67, 135 Chain rule partial derivatives, 75, 84 total derivatives, 39 Circle, 17, 116 Complex conjugate, 25 Complex numbers, 10, 24, 149, 154 exponential form, 27 Concave upwards, 34 Conjugate, complex, 25, 95 Conservative field, 96, 136 Continuous function, 31, 32, 33 Convergence of series, Coordinates cartesian, 21 cylindrical polar, 121 plane polar, 21, 118, 150 rectangular, 21 spherical polar, 121 Correlation analysis, 169 Correlation coefficient, 180, 182, 183 Couple, 70 Covariance, 171, 179, 183 Critical damping, 154 Critical point, 43 Cross differentiation, 82, 97, 122, 144, 145 Curl of vector field, 97 Curvature, 33 Curve fitting, 60, 173, 184 Cylindrical polar coordinates, 121 Damped harmonic motion, 153 Damping, critical, 154 Dependent and independent variables, 10, 12, 48, 80, 171 Derivative mixed second, 75 partial, 73 reciprocal of, 40, 74 second, 33 total, 30, 31, 74

7 Index 191 Differentiable function, 31 Differential exact, 82, 144 total, 36, 78 Differential element of area, 117 of volume, 120 Differential equations, 138 exact, 140, 144 first order homogeneous, 140, 142 linear, 140, 145, 147 linear, homogeneous, 148 order and degree of, 140 separable variables, 140 Differentiation by substitution, 39 cross, 82, 97, 122, 144, 145 implicit, 36, 75 of integrals, 108 rules for, 34 Dimensional analysis, 5 Dipole moment, 71, 94, 152 Direction cosines, 16, 91 Discrete variables, 168 Distribution function, pair, 127 Divergence of vector field, 97 Dynamics generalised, 132 Newtonian, 63 Ellipse, 17 End point maximum, 42 Energy heat, 66, 154 kinetic and potential, 65, 132, 152 Enthalpy, 8, 12, 100 Equations cubic, 9 quadratic, 8 second degree, 17 Equilibrium constant, 10, 24, 57 Error in arithmetic mean, 172 normal distribution of, 166, 174 random sign, 166, 176 root-mean-square, 166, 168, 169 small-sample, 171 systematic, 165 Euler-Maclaurin theorem, 107, 109 Euler's theorem, 86 Euler's equation, 132 Exact differential, 82, 144 Exponential functions, 1,27, 38 Extensive and intensive variables, 86 Factorial, 22, 161 Field electric, 65, 94, 152 gravitational, 65 scalar and vector, 95 First order reaction, 13 Fluids, theory of, 127 Force, 63, 64, 66, 70, 89, 152 generalised, 137 Fourier transform, 163 Functional notation, 8 Functions, even and odd, 33 Gamma function, 22, 161 Gas real, 59, 127 perfect, 6, 11, 18, 80, 86, 124 Gas constant, 6, 186 Gaussian distribution of error, 166 Generalised dynamics, 132 Generalised momentum, 136 Geometric series, 51 Gibbs equations, 83, 86, 88 Gibbs function, 12, 83, 88 Gradient of scalar field, 95 Graphical methods, 12 Gyroscopic motion, 91 Heat, 66, 154 capacity, 63, 85, 100 of vaporization, 39 Hamiltonian mechanics, 132 Harmonic motion damped,153 forced, 158 simple, 18, 150 Helmholtz free energy, 12 Hyperbola, 17,62 Hyperbolic functions, 55 Imaginary number, 25 root, 10, 24, 149, 154 Implicit differentiation, 36, 75 higher order, 37 Impulse, 64 Independent variable, 10, 12, 48, 80, 171 Indices laws of, 3 Miller, 17 Inertia moment of, 7, 68, 134 principal moments of, 70 product of, 69

8 192 Index Inflection, point of, 41 Integration, 98 area by, 99, 115, 119 by partial fractions, 105 by parts, 102, 120 by substitution, 104, 120 by using polar coordinates, 119 curve length by, 123 multiple, 107, 117, 124 replacing summation by, 113 volume by, 100, 120, 122 Intercept, 13, 14 Inversion of a series, 56, 58 Joule-Thomson coefficient, 85 Kinetic energy angular, 68, 72 linear, 65, 132, 152 Kinetic theory of gases, 124 Kinetics, reaction, 13, 44, 139, 146 Lagrange's method of undetermined multipliers, 45, Lagrangian mechanics, 132 Laplace transform, 160 Least squares, method of, 13, 176, 184 error in both variables, 178, 183 straight line, equations for, 179 weighted, 183 Leibnitz's Theorem, 108 Length of arc, 1, 67, 118 of curve, 123 L'Hopital's rule, 61, 120 Limit calculus, 30 expansion in series, 63 L'Hopital's rule for, 61, 120 of sum or product, 35, 40 Line integral, 122 Logarithm, 38 Logarithmic series, 56 Maclaurin's theorem, 54 Maxima and minima, 41 calculus of variations, 129 subject to constraint, 45 Maxwell's relations, 83, 86 Mean arithmetic, 166 error in, 172 Miller indices, 17 Modulus of complex number, 26 Molecular properties, 48, 66, 68, 70, 72, 94,134,153 Moment of a force, 66, 70 Moment of inertia, 7, 68, 134 principal, 70 Momentum angular, 71, 91, 135 linear, 64, 71, 125 Multiple integration, 107, 117, 124 Nabla operator, 95 Napierian logarithm, 38 Newtonian mechanics, 63 Newton's method, 10, 53 NMR,92,164 Normal distribution of error, 166, 174 Numerical quadrature, 112 Operator differential, 34 nabla, 95 vector, 95 Ordinate, 12 Orthogonal, 16,92,95 Parabola, 17,21 Partial derivatives, 73 chain rule for, 75, 84 reciprocal of, 74 relations between, 84 Taylor's theorem in, 88, 170 Partial fractions, 105, 162 Partition function rotational, 113 translational, 112 Parts, integration by, 102, 120 Path, dependence on, 76, 78, 83, 122 Perfect gas, 6, 11, 18, 80, 86, 124 ph, 42 Phase rule, 11 Planck's constant, 6, 186 Polar coordinates cylindrical, 121 plane, 21, 118, 119, 150 spherical, 121 Potential energy, 65, 96, 133, 152 Power series, 51,60 Precession of gyroscope, 91 Principal axes of molecules, 70 Probability, 23, 25, 49, 168, 174 Projectile, motion of, 21 Propagation of error, 170, 181

9 Index 193 Quantity calculus, 5 Quantum theory, 72, 95, 136, 162 Radian, I, 67 Random sign error, 166, 176 Reaction kinetics, 13, 44, 139, 146 Real and imaginary parts, 25, 149, 155 Rectangular coordinates, 21, 121 Reduced mass, 68, 134 Relaxation time, 139 Residual, 178, 184 Resonance, 159 Resultant, 89 Root-mean-square error, 166, 168, 169, 170, 181 Root of an equation, 8 imaginary, 10,24, 149, 154 Rotation dynamics, 72 sign of, 21 Scalar, 89 product, 92 Series, binomial, 56 convergence of, cosine, 3, 55 exponential, 3, 38, 55 geometric, 51 inversion of, 56, 58 logarithmic, 56 power, 51, 60 sine, 2, 55 sinh and cosh, 55 SI units, 4, 185 Similar triangles, 14 Simple harmonic motion, 18, 150 damped, 153 forced, 158 Simultaneous equations, 11, 45 Single-valued functions, 32 Solid angle, 2, 125, 128 Spectroscopy, 7, 152, 164 Spherical polar coordinates, 121 Standard deviation, 168, 172, 173 Standard forms of integral, 106, 119 Stationary values, 41, 45, 129 Stirling's theorem, 23, 49 Stokes's law, 4 Straight line equation of, 13 least-squares, 176, 179 Substitution differentiation by, 39 integration by, 104, 120 Successive approximation Newton's method, 53 substitution in series, 59 Symmetry, circular, 21 Systematic error, 165, 166 Taylor's theorem, 51,61, 109, 148 in partial derivatives, 88, 170 Torque, 70, 94 Thermodynamic function of state, 83, 122 Thermodynamic properties, 12, 66, 83 Thermodynamic relations, 83, 85, 87 Time constant, 139 Total derivative, 30, 31, 74 chain rule, 39 Total differential, 36, 78 Transformation, integral, 160 Triangular graph, 15 Trigonometric identities, 27, 188 Turning point, 41 Units, SI, 4, 185 Variables dependent and independent, 10, 12,48, 80, 171 extensive and intensive, 86 Variance, 171 Variations, calculus of, 46, 129 Vector, 89 components of, 90 differentiation of, 90 localized, 91 operator, 95 product, 92 Volume by integration, 100, 120, 122 Weighting in least-squares analysis, 183 Work, 65, 83, 96