An Introduction to Thermodynamics and Statistical Mechanics Second Edition pps

An Introduction to Thermodynamics and Statistical Mechanics Second Edition Keith Stowe California Polytechnic State University... 554 ideal gases cont.

An Introduction to

Thermodynamics and Statistical Mechanics

Second Edition

Keith Stowe

California Polytechnic State University

Contents

Preface

List of constants, conversions, and prefixes

Part I Setting the scene

1 Introduction

Part 11 Small systems

2 Statistics for small systems

3 Systems with many elements

Part III Energy and the first law

4 Internal energy

5 Interactions between systems

Part IV States and the second law

6 Internal energy and the number of accessible states

7 Entropy and the second law

8 Entropy and thermal interactions

Part V Constraints

9 Natural constraints

10 Models

11 Choice of variables

12 Special processes

13 Engines

14 Diffusive interactions

Part VI Classical statistics

15 Probabilities and microscopic behaviors

16 Kinetic theory and transport processes in gases

17 Magnetic properties of materials

18 The partition function

page vii xii

1

3

23

25

40

63

65

79

99

101

117

135

153-155

186

210

226

252

287

327

329

352

369

382

v

vi Contents

Index

absolute zero

approaching, 506-510

behaviors near, 175-176

acceptors, 483,488

activation energy, 297

adiabatic demagnetization, 508-509

adiabatic processes, 229-232

in ideal gas, 231-232

in photon gas, 449

temperature changes, 229

air, liquefaction of, 312-314

alloys, 314-315

angular momentum, 9,11-14,531

atomic magnets, 371-372

atomic vibrations, 457

average molecular speed, 355

band-edge equivalent states, 485-486

band structure and width, 477-479

bands

density of states, 486

in divalent metals, 480

overlapping, 479

unfilled, 479-481

valence and conduction, 192,477-479

Bardeen-Cooper-Schrieffer (BCS)

theory distribution, 518

bell curve, see Gaussian

beta (Ilk]), 332

bias, forward and reverse, 497

Big Bang, radiation from, 441,443

binary mixtures, 308-316

binomial expansion, 29,42

black holes, 520

blackbody emissivity, 442

blackbody radiation, 438-449

distribution, 439

energy density, 439-440

energy flux, 441-444

Bohrmagneton, 371

boiler, 273 Boltzmann statistics, 333 Boltzmann's constant, 72, 126, 138 Bose-Einstein condensation, 512-513 for fermion pairs, 517-518 Bose-Einstein statistics, 403, 423 bosons, 13

degenerate, 176,394,413,429,510, 512-519

occupation number, 403,423-425 relativistic and nonrelativistic, 403, 423-425

bound states, 15-17 Brownian motion, 342 canonical ensemble, 330 Carnot, Sadi, 265 Carnot

cycle, 265 efficiency, 266, 267 engine, 265-267 catalysts, 297 chain rule, 215 charge carriers, 483 mobility, 484 thermal excitation, 482 chemical equilibrium, 295-297 chemical potential, 81,83-84, 137, 288-291

and concentration, 288, 290 and heat released, 83-84 and nunlber of particles and osmosis, 293-294 and particle distributions, 84-86 and phase space, 84

and potential energy, 288, 290 and the second law, 288

at all temperatures, 431-433 calculation of, 391-392,411-414 classical limit, 431

dependence on T,p, 168,288 from partition function, 388,394, 406-407,413

in low- and high-density limits, 394, 412-414

of degenerate systems, 401,413-414, 429,430

bosons, 429, 513 fermions, 429, 430

of nearly degenerate fermions, 432-433

chemical reactions, heat transfer, 144 classical1irnit, 431

classical probabilities, 333-334,336-337 classical statistics, 329-342

examples, 336-337 limits of, 386,409 needed ingredients, 402,413 when to use, 333

Clausius-Clapeyron equation, 299-300 coefficient of performance, 258 coefficient of utility, 267 cold packs, 309 collapsed star, size of collision frequency, 357-359 compressibility, 196 compression ratio, 275 Compton scattering, 6 condenser, 273 conduction band, 192, 193,

478 conduction electrons, 466-468 heat capacity, 468 thermal energy, 466-468 thermal properties, 457 conduction

electrical, 54-55 thermal, 79 conductors, 479-481 conserved quantities, 242, 362

551

I

I

I

552

constraints

first law, 167-168

natural, 155-178

second law, 158-165

second order, 160 -162, 164

third law, 166,175-177

types of, 157

zeroth law, 166

contents, v-vi

continuity equation, 242, 362

convection, 81

vertical, 229

cooling

by adiabatic demagnetization,

508-509

by throttling, 235-237

Index

by helium diffusion, 507

diffusive, 506-507,510,512-519

mechanical, 506-507

optical methods for, 510

through expansion, 506 507

cosmic background radiation, 441,443,

449

critical point, 298, 302

Curie law, 375

current density, 355

currents, drift and diffusion, 494-496

cycles

gas and vapor, 260

open and closed, 260

Debye

cutoff, 460-461

energy, 461

frequency, 461

function, 462

model, 459-465

temperature, 461

degenerate levels, 104

degenerate systems, 175-177, 504

bosons, 510,512-519

conditions for, 504

fermions, relativistic and

nonre1ativistic, 430

gases, 428

degrees of freedom, 71-72,74

and partition function, 389-390,

410-411

energy of, 342

density of states, 10 -11,105-110, 390-391

and internal energy, 105,390-391 and number of particles, 390-391 for gases, 402-405,422-432 Fermi gas, 486

phonon gas, 459 detailed balance of radiation, 440-441 diamagnetism, 369

diatomic gas molecules, 71 diesel engines, 271 differentials, exact and inexact, 88-91 diffusion, 83-87

across p-njunction, 494-496 diffusion equation, 362 diffusive cooling, 507-508,510, 512-519

diffusive equilibrium, 85-86 diffusive interactions, 83-87,287-316 and Gibbs free energy, 170-171 distinguishable subsystems, 386, 409 donors, 483,488

doped semiconductors, 482, 488-489 Doppler effect in cooling, 510 drift and diffusion currents, 494-496 Dulong-Petit law, 339

effective mass of charge carriers, 486 efficiency of engines, 254, 266, 267 and reservoir temperatures, 266 Einstein model

boson systems, 513-519 for lattice vibrations, 459 elastic constant, 16 electrical charge, 5-6 electrical circuits, thermal noise in, 450 electrical conductivity, 484, 485, 486-487

in semiconductors, 485, 486-487 temperature dependence of, 481 electrical current density, 483 electrical properties of materials, 497 electromagnetic waves, 6, 80 electronic devices, 494-497 electrons

and holes, 192-194 conduction, 192 valence, 192 emissivity, 442

for large systems, 120-124 energy

fluctuations, 123 harmonic oscillator, 16 internal, 74

potential, 65-69 radiation, 440-444 thermal, 72-73,74 transfer, 79 engines, 252-275 Carnot, 265-267 coefficient of utility, 267 constraints, 254, 262 cycles, 262 efficiency, 254, 266 gas piston, 255-256 gas turbine, 257-259 increasing efficiency of, 274 275 internal combustion, 272 model cycles, 254, 262 perfo~ance analysis of, 262-265

p-V diagrams, 254 reversible, 266, 267

T -S diagrams, 254 types of cycle, 260 ensemble average, 330 ensembles, 26, 329-330 enthalpy

calculation of, 264 definition, 168 dependence on (T, p), 264 inthrottling process, 236

in performance analysis, 262-265

of ideal gas, 264 properties of, 170 entropy, 126-130 and heat transfer, 136 and mixing, 308,309-315 and number of states, 126, 127 and reversibility, 233 and the second law, 128 and the third law, 142-145 and thermal interactions, 135-145

atT=0,142

definition of, 126 dependence on (p, V), 219

dependence on (T,p), 219

in binary systems, 308-310, 315

of gases and solids, 129

of interaction, 308,309-310,315

of many systems, 128

of mixing, 308-315

of photon gas, 449

of solids, 187

equations of state, 187

for ideal gases and solids, 187-189

for liquids, 191

for real gases, 189-191

equilibrium, 102-103,158

diffusive, 85-86

the approach to, 158, 159-160

thermal, 13 7

equilibrium concentrations, 290-291

equilibrium constant, 296

equipartition, 71, 138-139,341-342

and van der Waals model, 218

heat capacities, 197-198

testing, 217-218

evaporation, 85

excitation temperature, 337

expansion

and internal energy, 82

free, 237-238

extrinsic variable, 136

factorials, 30-31

Fermi-Dirac statistics, 403, 423

Fermi energy, 177

degenerate fermions, 402, 413

Fermi gas model, 486-487

Fermi level, 176,458,489-490

in semiconductors, 485

temperature dependence of, 489-490

Fermi surface, 176,430

Fermi tail, 480-481

fermion gas, nearly degenerate, 467,

535-536

fermionpairs, 517-518

fermions, 13

degenerate, 176, 401-415

degenerate relativistic and

nonrelativistic, 430

nearly degenerate, 432-433

occupation number, 403, 423

relativistic and nonrelativistic, 403,

423-425

ferromagnetism, 370

first law, 88, 135, 137

constraints, 167-168

definition, 88 fluctuations, 40-44, 123

in occupation number, 404-405, 426-428

at equilibrium, 162-164

in ideal gases, 164 relative, 43, 53 forward bias, 497 Fourier analysis, 7 amplitudes, 7 free expansion, 237-238 freezing and boiling points, 293 friction, 234

fundamental postulate, 103-104 gas cycle, 260

gas laws, in quantum gases, 404-405, 426-428

gas piston engines, 255-256 gas turbine engines, 271-272 gases

degenerate, 428 density of states, 402-405,422-432 diatomic, 71

ideal, 187, 188 liquefaction, 312-314 molar heat capacities, 196 molecular velocities, 352-354 partition function, 389-393,409 phonon, 192

pressure in, 357 properties of, 198-200 quantum, 402,404-405,422, 426-428

real as opposed to ideal, 189-191 relativistic, 111

relativistic and nonrelativistic, 403,

423 root mean square speed, 342, 355 states for, 108, 11 0

gasoline engines, 269 gauges, 165 Gaussian distribution, 44-50 Gibbs free energy

and chemical equilibrium, 295 and phase transitions, 306 definition, 168

in binary systems, 309-315 properties of, 170-171 van der Waals, 303

grand canonical ensemble, 330 gravitational energy in collapsed star, 521 greenhouse effect, 446-448

gyromagnetic ratio, 14 harmonic oscillator, 15-17 heat, 79

heat capacity, 337-339

at low temperature, 144 changes with Vandp, 219-220

Debye model, 463 definition, 141 diatomic gases, 338

of photon gas, 449

of solids, 338-339,468-470 heat equation, 241-243,362 solution for, 243 heat flux, 239 heat function, 170 heat pumps, 259-260 heat shields, 445-449 heat transfer, 79-81 and accessible states, 140 and entropy, 136 and diffusive interactions, 83-84 direction of flow, 158, 162

in chemical reactions, 144 heat, inexact differential, 90 helium

expansive heating, 237 liquid, 506

phase diagrams, 515 superfiuid, 515-516 helium diffusion, 507-508 refrigeration by, 507 helium I and IT, 515,516 phase transition, 516 helium III, 506,507,517 Helmholtz free energy, 384-385 and partition function, 384-385 definition, 168

properties of, 169-170 van der Waals, 302-303 holes, 193,372,483 hydrogen, expansive heating, 237 hysteresis, 305

ice, melting point, 300 ideal gases

adiabatic processes, 231-232

'I

i

'I

I ,I ,I

I

554

ideal gases (cont.)

entropy, 187

equations of state, 188

identical particles, 105-107

and counting of states, 239

and entropy, 239

Index

and occupation number, 387-388,407

identical subsystems, 387-388,407

insulation by layered foils, 445-446

insulators, 482

integrals, standard, 534

interacting systems, 117-124

energy distribution, 534

interaction entropy, 308,309-310,315

interactions, 79-92

types of, 79

internal combustion engines, 272

internal energy, 74

and accessible states, 101-111

and chemical potential, 405-407,

428-433

dependence on (T,p), 218

fluctuations in, 384

from partition function, 394,412-414

in gases, 69, 73

in liquids, 67, 69

in solids, 68, 73

integrated, 167

mean value, 384

quantum effects, 69-71

three ways to change, 88

intrinsic materials, 482

intrinsic variables, 136

interdependence, 168

isobaric processes, 226-228

isothermal compressibility, 196, 197

isothermal processes, 228-229

jet engines, 271-272

J oule-Thompson process, 235-237

kinetic theory, 352-364

lapse rate, 229

large numbers, tools for, 121

latent heat, 299

lattice energy, 462

low-temperature fluctuations, 463

lattice vibrations, 459-464

Einstein model, 459 law of mass action chemical, 295-297

in semiconductors, 487 layered foils, 445-446 Lenz's law, 369 liquid helium, 506,515-516 phase diagrams, 515 production of, 506, 507 liquids, 191

heat capacities, 197-198 potential energy in, 67 states for, 110 low temperatures, 505-519 attaining, 506-510 measuring, 510-511 magnetic interaction energy, 15 magnetic moment, mean, 373 magnetic moments, 14-15,371-372,531 magnetic properties of materials, 369 magnetism, 369

low-temperature limit, 375 magneton, Bohr and nuclear, 371 mass, effective, 486

mass action, law of, 295-297,487 materials

electricaJ properties of, 497 magnetic properties of, 369 Maxwell, James Clerk, 6 Maxwell's relations, 172-175 Maxwell velocity distribution, 352-354 derivation, 172

meaning, 173-174 mean field models, 301-303 mean free path, 358 mean values, 26-27,31,43,354-355, 383-385,403,423

mechanical cooling, 506-507 mechanical interaction, 81-82 metals

divalent bands, 480 s- and d-bands, 481 microcanonical ensemble, 330 minerals and alloys, 314-315 miscible fluids, phase transitions, 312-314

mixing, 238-239 mixing entropy, 308-315

mobility, 484 models, 200 molar heat capacity, 141, 196 molecular diffusion, 359,361 momentum and wavelength, 6 motion, Brownian, 342 mUltiple occupancy, 388, 406-407 natural constraints, causes of, 158 nearly degenerate fermions, 432-433 neutron stars, 520

Niagara Falls, 289 nonequilibrium processes, 234-244 nuclear magneton, 371

Nyquist theorem, 450 occupation number, 402-405, 422-432 and fluctuations, 404-405,426-428 for bosons, 403, 423-425 for fermions, 403, 423 order parameter, 305 osmosis, 293-294 osmotic pressure, 294 Otto cycle, 269 paramagnetism, 369,373-375 parameters and constraints, 157 partial derivatives, 211-212,213-216 ratios, 216

particle distributions, 84-86,390-391 and second law, 288

particle flux, 355-357 particle transfer, 83-87 and changes in temperature, 86 particle waves, 6

particles direction of flow, 158, 162 distinguishable or identical, 105-107 partition function, 382-394

definition, 383 distinguisqable subsystems, 386,409 examples of, 394,412-414 for a gas, 389-393,409 for many subsystems, 386-388 for rotation, 390-391 for vibrations, 391-392,411-414 identical subsystems, 387-388,407 translational motion, 390, 411 phase diagrams, 298-299

phase equilibrium, 298-307

phase space, 9-10

and chemical potential, 84

and particle densities, 389-393,409

phase transitions, 73

first order, 304, 306

higher order, 304, 306

in minerals and alloys, 314-315

in miscible fluids, 312-314

Landau theory, 306

phonon gas, 459-465

density of states, 459

phonons, 191-192,459-465,518

distribution, 460

maximum energy, 460-461

photoelectric effect, 6

photon gas

adiabatic processes, 449

energy distribution, 439

entropy of, 449

heat capacity, 449

inside Sun, 443

photons, 438-440

chemical potential, 439

in oven, 438-440

occupation number, 439

Planck's constant, 6

p-njunctions, 494-497

diffusion across, 494

potential energy shift, 494

postulate, fundamental, 103-104

potential energy

and forces, 65-69

and phase transitions, 67,68

potential energy reference level, 67, 72

potential wells, 67, 68

in liquids, 67

pressure, 137

principle of detailed balance of radiation,

440-441

probabilities

air molecules in a room, 27, 31, 33,

36,44,49-50

and accessible states, 103

and configurations, 27

and entropy, 401,413-414

classical statistics, 333-334

closely spaced states, 340

coins, 27,29,33

criteria, 27,31

dice, 27,28

in small systems, 27-32

in statistics, review, 401,413-414 quantum statistics, 334

probability distributions, gases, 352-354 probability of being in a state, 330-332 processes, quasistatic, 103

p-Vand T-S diagrams, 254 p-Vdiagram

for engines, 254 van der Waals model, 301 quantum confinement, 458 quantum effects, 5-17, 69-71 quantum gases, 402,422 chemical potential, 405-407,428-433 energy and particle distribution, 403, 423-425

gas laws, 404-405,426-428 internal energy, 404-405,426-428 mean values, 403,423

quantum probabilities, 334, 335 quantum states, 9-10 quantum statistics, 401-415 examples, 335

needed ingredients, 402, 422 when to use, 333

quarks, charge of, 6 quasistatic processes, 103 R-value, 240

radiation, 80 emission, absorption, reflection, 445 random walk, 50-55

Rankine cycle, 273-274 reaction rate, 297 reactions, chemical, 290 real gases, 189-191 refrigerators, 258 coefficient of performance, 258 reheat cycle, 275

relativistic gases, states for, 111 relaxation time, 102, 235 reservoir, 331

hot and cold, 253 reverse bias, 497 reversibility, 232-234 and heat transfer, 233 root mean square, 44 rotation

Index

molecular, 69-70 partition function for, 390-391 second law, 85,125-126,288 statements of, 125, 128 semiconductor devices, 494-497 semiconductors, 193,482-494 doped, 482,488-489 Fermi level in, 489-490 intrinsic, 482, 483-487 law of mass action, 487 n- and p-type, 488 transition to intrinsic, 491-492 small numbers, tools for, 121 small systems, 25-35,40-55 solar energy flux, 443 solar spectrum, 441,443 solids

atomic vibrations in, 191-192 equations of state for, 189

555

heat capacities of, 196,457,468-470 modeling of, 191-194

states for, 108, 109, 110 thermal properties of, 457-470 solubility gap, 312

solutions and chemical potential, 291-292 and vapor pressure, 292-293 colligative properties, 291-294 freezing and boiling points, 293 special processes, 226-244 specific heat, 141 spectra of accessible states, 389-390, 4·10-411

spin, 12,13 down or up, 13 spin entropy, 508 spin quantum number, 13 spin-orbit coupling, 375 staged compressors and turbines, 275 standard deviation, 42-44,46,47,48,49 standard integrals, 534

standing waves, 15 stars, death of, 519 states

accessible, 532-533 and energy distribution, 117-124 and heat transfer, 140

closely spaced, 340 correction for identical particles, 106

556

states (cont.)

description of, 16, 17

for a system, 101, 105-1lO

Index

for interacting systems, 117-124

for liquids, 11 0

for macroscopic systems, 120-124

for monatomic gas, 109

for polyatomic gas, 109

for solids, 108, 109, 11 0

per particle, 107-lO9

probability of being 41, 330-332

spacing of, 104-105

spectrum of, 402, 422

statistical independence, 32-34

statistical mechanics, 4, 5

statistics, quantum and classical,

405-407,428-433

Stefan-Boltzmann constant, 442

stellar collapse, 519

steps, random walk, 50-55

Stirling's formula, 30,32,46,47, 107

stoichiometric coefficients, 295-296

storms, 230

stress, 361

sum over states, 390,411

Sun, 520

superconductivity, 517

supercooling, 303

superfluid, 516

superheating, 303

system, states for, 532-533

Taylor series, 45, 65, 531-532

temperature, 72,73, 135-139

and mixing, 309

and occupation number, 403, 423

and particle transfer, 86

definition, 135

scales, 166 transition to intrinsic, 491 thermal conduction, 239-241 thermal conductivity in gases, 359,361,

430 thermal energy, 72-73,74 thermal equilibrium, 13 7 thermal interaction, 79-81 thermal inversion, 229-232 thermal motion and diffusion, 85,86 thermal noise, 450

thermal properties of solids, 457-470 thermal resistance, 240

series and parallel, 240 thermodynamical potentials, 168 thermodynamics, 4, 5

thermometers and gauges, 165 third law, 142

and Helmholtz free energy, 385 constraints due to, 166, 175-177 statement of, 142

throttling process, 235-237,

506 thunderheads, 230 translational motion, partition function, 390,411

transport processes, 359-362 triple point, 165

turbines, 257-259,273

uncertainty principle, 7-9,16,17

vacancies, 483 valence band, 192, 193, 478 van der Waals' model, 190, 198-200, 301-303

and equipartition, 218 vapor cycle, 260,272-274

vapor pressure, 292-293 variables

changing, 211-217 choice of, 210-221 dependent and independent, 91, 156, 186,210

intrinsic and extrinsic, 136 velocity distributions in gases, 352-354

vibration molecular, 69-70 partition function for, 391-392, 411-414

viscosity, 359,361 volume expansion, 196,197

water freezing, 198 special properties of, 298 triple point of, 165 wave functions, 6

wave nature of particles, 6

wave number, 7 wavelength and momentum, 6 waves

particle, 6 superposition, 7 white dwarfs, 177,520 work, 81-82 and internal energy, 81, 82 direction of, 158,162 inexact differential, 90 types of, 81, 83-84 work function, 170

Z, partition function, 383 zero-point energy, 16 zeroth law constraints, 166