# Appendices: Mathematical Identities

## DERIVATIVES & GIBBS-DUHEM EQUATION

Internal energy, Helmholtz free energy, Gibbs free energy and enthalpy each have *natural* variables for which they obey useful partial derivative relations. Entropy does too, but gives the same information as the internal energy.

These four energy equations imply the following useful expressions:

If we confine our attention to a single phase, the four energy functions *E. S, A, G* and their derivatives will be continuous. Under these conditions, the mixed second derivatives must be equal. For example,^{1} *d(dE/dS)/DV = d(dE/dV)/dS* implies the *Maxwell equation,*

4’ve omitted the subscripts for constant *V, S* to reduce notational clutter.

Three other most common Maxwell relations are:

The equations above relate to changes in the *extensive* thermodynamic variables *E, A, G,* and *H* for a single phase. An important equation that relates changes in *intensive* variables is the Gibbs-Duhem equation,^{[1]}

This can be seen using the third lines in both Eqs. (11.1) and (11.2), where the Gibbs function *G(T. P. N*) is a function of the intensive variables temperature and pressure and the extensive number of particles *N.* Thus *G* is a function *f(T,P)* times *N,* but because of (11.2), *f(T,P) = fi(T, P).* Given *G = pN. dG* = *Ndfi + pdN.* Combining this with the third of Eqs. (11.1) results in the Gibbs-Duhem equation, 11.5.

# Subject Index

absolute zero (unattainability), 186

absorptivity, 166 adiabatic accessibility, 40, 296 adiabatic demagnetisation, 248 nuclear spin, 249 adiabatic process, 45 allotropes of carbon, 185 available energy, 83-85, 120, 230, 294

finite reservoirs, 87 Avogadro constant, 23

Baierlein theorem, 110 bank transaction analogy, 197 binding energy, 141 bit of information, 257 black hole, 91, 173

Hawking radiation, 174 heat engine, 175 missing information, 92 negative heat capacity, 93 Schwarzschild radius, 173 singularity, 174 blackbody radiation, 155 boiling point, 238-239 Boltzmann

constant, 23, 71 entropy, 69

number of accessible states *Cl,* 192 factor, 78 reservoir, 76 statistical mechanics, 79

Bose-Einstein statistics bosons, 98

ideal gas, 100, 117, 189 Brownian motion, 5-6 Brownian ratchets, 271-278 & second law of

thermodynamics, 277 flashing ratchet, 273 Btu, 22

caloric theory, 6-9 death, 8

calorie, 8, 13, 14, 22 Carnot cycle, 217

Novikov-Curzon-Ahlborn, 227 zero power, 45 Carnot theorem, 220 chemical potential, 102 Clausius-Clapeyron equation,

106, 246

CMBR radiation, 171 coefficient of performance cooling, 220 heating, 220

colour spectrum mnemonic ROY G BIV, 153 combined cycle efficiency, 232 composite system, 66, 68-77 concavity and entropy increase, 73 concavity of entropy, 71 conduction, 25 convection, 25 cooling, 237-250

adiabatic demagnetisation, 248

adiabatic expansion, 21, 162, 238

by pressure reduction, 238 Joule-Thomson process, 239 corresponding states (law of), 108 cryogenics

- 4He dilution refrigerator,
- 248

Pomeranchuk cooling, 247 Curie temperature, 134

Magneto-optical device, 138 cycles

combined cycle, 231 heat pump, 234 irreversible, 223, 224, 227 reversible, 216, 218, 221, 224 second law efficiency, 229 working substance, 214

dark energy, 90 Debye solid, 124-125

Debye temperature, 124 heat capacity, 125 degree of freedom, 20 deposition, 23 dewar bottle, 28 diffusion, Graham’s law, 118 disgregation, 58

& energy spreading, 210 disorder, 61, 207 Dulong-Petit law, 123, 125

efficiency

Carnot theorem, 220 second law, 223 thermal, 219 Einstein Nobel Prize, 6 Einstein solid, 120-123

Einstein temperature, 121 energy & entropy, 122, 123 heat capacity, 122 electricity generating plants, 228 electron volt (eV), 142 EM radiation spectrum, 13 emergent force (entropic), 89 energy

compared to momentum, 11 bond,15 dark energy, 91 electromagnetic, 1 free (available to do work, 80 gravitational, 1, 24 internal, 3 needs a partner, 40 nuclear, 1 stored, 1 units, 22 caloric, 8 compared, 22 vs. momentum, 11 world use, 22 energy dispersal, 295 energy equipartition, 123 energy flow direction, 205 energy spreading, xv, 18, 42, 59, 80, 91, 114, 209, 211,

217, 293, 295 energy transfer, 10 heat, 12

material transfer, 14 rate (power), 27 work, 10

energy-entropy duo

competition, 80, 81, 83, 89, 127, 129, 134, 135, 137 interdependence, xv ensemble, 275 enthalpy

change and bond strength, 31 change and heat of formation, 31

defined, 31 energy for solids, 182 free energy, 86 of formation, 31 entropic force, 89, 90, 93

accelerated expansion of universe, 94

dark energy alternative, 93 rubber band,89

entropy, 55

*vs.* energy graphs, 71 Baierlein’s theorem, 110 Boltzmann expression, 69 Clausius definition, 56, 73 concavity, 71 confusion, 61 dimensionless, 190, 191 dimensionless entropy, 191 disgregation, 210 non-concave examples, 71, 77, 93

numerical entropy, 179 principle of entropy increase, 216

principle of increase, 55 third law, 186 zero-point entropy, 98 entropy clock, 57 entropy metaphors disorder, 207 energy spreading, 209 freedom, 210

missing information, 59, 210 multiplicity, 210 optiony, 210 epicatalysis, 292 equal *a priori* probabilities, 70 equilibrium (see thermodynamic equilibrium), 3, 29 equipartition theorem, 123 equity, 41, 42, 75, 293 and thermodynamic equilibrium, 76 Clausius statement , 43 Kelvin-Planck statement, 43 exergy, 83-87, 230 expansion of the universe, 90 extensive function, 30

Fermi energy, 283 Fermi sea, 100 Fermi-Dirac statistics, 282 fermions, 98, 99, 245 ideal gas, 100, 103, 117, 189 ferromagnetism, 132-138 Curie temperature, 134 heat capacity, 136 spont aneous magnetisation, 137

Feynman ratchet & pawl, 269 first law of thermodynamics, 29, 30 fluctuations, 54, 64, 66, 76, 157, 171, 253, 265, 266, 272, 295

and Jarzynski equality, 147 nanoscale, 272 thermal, 146

Fokker-Planck equation, 276 friction, 18, 201 work by, 203 frictionless surfaces, 18 frozen-in state, 117, 187, 294 fuels require oxygen, 2, 16

gas

ideal (dilute), 96 nonideal, 104 van der Waals, 21, 107 vapour-liquid transition, 104 virial series, 111 gas in gravity, 286

“proof” of uniform T, 287-288 “proof” of variable *T,* 288 experimental results, 289 gas mixing

entropy of mixing, 113 Gibbs paradox, 117 mixing entropy function, 116 role of information, 118 Gibbs free energy, 79, 299 Gibbs paradox, 117, 261 Gibbs-Duhem equation, 299 gravitational potential energy, 24 gravity and non-extensivity, 30, 91

hard sphere mixture, 90 heat, 12

& work energy processes, 13 as a verb, 196 defined by work, 39

definition, 40, 198 description *vs.* definition, 198 language problem, 9 mechanical equivalent, 14 of combustion, 16 heat capacity

at low temperature, 187 misnomer, 197 thermal inertia, 187, 197 third law, 187

heat engines, combined cycle, 231

heat pump, 234

helium-3

Cooper pairs, 246 density maximum, 246 helium-4 (^{4}He), 99, 242 liquefaction, 242 superfluid, 243 helium-4 ^{4} He

superfluid entropy = 0, 244 Helmholtz free energy, 79-83, 299 Hubble constant, 94 Hubble parameter, 93

information

and available energy, 119 is physical, 257 measurement, memory, erasure, 263 information theory

bit (binary digit), 257 connections with physics, 257 infrared radiation discovery, 155 intensive function, 30, 107 internal energy, 3

duo with entropy, xv entropy-temperature linkage, 75

frame dependence of A*E,* 39 not in classical mechanics, xv irreversible heating, 45

Jarzynski equality, 143 examples, 149 experimental test, 151

Jensen’s inequality, 145 Joule [J] (energy unit), 22 Joule-Thomson process, 239

Kamerlingh Onnes ent halpy, 31

helium-4 liquefaction, 241 liquefaction of ^{4} He, 241, 242 superconductivity, 242 virial series, 50 Kelvin temperature [K], 51 kelvin unit as an energy, 191 kilowatt hour [kWh], 22 Kirchhoff’s law, 166

lambda temperature, 245 lambda transition, 243 Landauer principle, 260, 263 Langevin equation, 275 language issues entropy, 207 heat, 195-196 stored *vs.* transferred, 198 thermal energy, 197 laser light: work or heat?, 199 laws of thermodynamics compared, 188 interdependence, 206 liquefaction of gases, 239-242 luminous efficacy, 23

macroscopic system, 2 magnetic cooling, 249 magnetic moment, 126, 246 Markov process, 276 Maxwell’s demon

“demon” is inappropriate, 254 2nd law is statistical, 252-254 birth, 252

capsule summary, 264 measurement, 257 measurement and memory, 257

memory and erasure, 257 named by Thomson, 253

Maxwell-Boltzmann speed distribution, 68 membrane, semipermeable, 33, 114, 118, 120, 261 metastable state, 187 missing information, 59, 210 momentum

vs. energy, 11

negative temperature, 129 Newton’s second law, 11, 274 Newton’s third law, 38 nonequilibrium systems, 57 Novikov-Curzon-Ahlborn cycle, 227

nuclear

binding energy, 141 fusion & fission, 143 nuclear force, 1, 142 nuclear magnetic resonance, 248 nuclear spin, 98, 261

ortho and para hydrogen cycle measurement, 262 memory erasure, 262 Otto cycles, 221-224

paramagnetism, 126, 248 energy & entropy, 128 partition function, 78 Pauli exclusion principle, 99 phase diagram, 105, 133, 239, 244 phase transition

para- to ferromagnetic, 133 vapour-liquid, 104 photon, 6, 154 colour, 154 photon gas, 156

fluctuations, 157 number density, 157 quantum mechanical,

relativistic, thermal, 156 thermodynamics functions, 159

physics of computation, 259 Planck constant, 23, 71 point particles, 17 power, 26 pressure units, 21 probabilities, 67-71 purist *vs.* utilitarian views, 200, 202

PVT plot, 104

radiation, 25

blackbody, 165 CMBR, 171 Hawking radiation, 172 incandescent lamp, 167 infrared, 3, 155 photon gas, 156 Planck radiation law, 167 radiative cooling rates, 176 radiative heating & cooling, 177

spectral absorptivity, emissivity, 165 spectral radiance, 165 recurrence paradox (Zermelo), 287 regenerator, 225 relaxation time

spin-lattice, 132 reversibility paradox

- (LoschLoschmidtidt),
- 287

reversible isothermal volume change, 200 reversible process, 46

isothermal volume change,

201

Richardson equation, 282 rigid bodies, 17 rubber band, 138 experiment, 138 model & graphs, 140

Sackur-Tetrode equation, 96 second law challenges

thermally emitted electrons in B-field, 281

second law of thermodynamics

Camtheodory. 295 Clausius statement , 42 Kelvin-Planck statement, 43 principle of entropy increase, 56, 216

statistical nature, 287 violation (Fu & Fu), 281-284 violation counterargument, 166

violation predicted (D’Abramo), 286 violation proposed (Sheehan, *et al.),* 292

Shannon’s missing information, 59 SI System of International Units, 23

Smoluchowski trapdoor, 267 computer models, 268 statistical mechanics, 78, 79 Stefan-Boltzmann constant, 167 Stirling approximation, 130 Stirling cycles, 224-227 regenerator, 225 sublimation, 23, 105 Szilard engine, 254

target system, 69 temperature

ideal gas definition, 20 international scale, 49, 51 Kelvin scale, 51 thermodynamic definition, 73 What is it?, 52

temperature-entropy diagram, 216-218, 220, 224, 235 tempergy, 71, 77, 144, 191, 294 temporal spreading, 192 thermionic emission, 282 thermodynamic equilibrium, 3, 5, 7, 37, 39, 42, 55, 76, 293

near-equilibrium, 200 thermodynamics

& computation, 265 language issues, 195 postulation-based, 295 third law of thermodynamics, 185, 186, 189

ground state degeneracy, 189 heat capacity approaches zero, 187

Nernst theorem, 184, 185 Planck statement , 186 residual entropy, 187 unattainability of absolute zero, 186

triple point, 51, 105, 190, 242

universal gravitational constant, 92, 173

utilitarian view, 200

van der Waals gas, 21 violation, 281 virial series, 50

walls (system boundaries) adiabatic, 33

diathermal (diathermic), 33 permeable, 33 work, 10, 11

defining heat, 39 work function, 282 work-energy theorem

(inadequacy), 35

zero-point energy, xv, 31, 74, 121, 122

zero-point entropy, 98, 185 zeroth law of thermodynamics, 49, 204

- [1] Pierre Duhem was a thermodynamicist, hydrodynamicist, and historian of science.