Fundamentals of equations of state

Shalom Eliezer9810248334, 9789810248338, 9789812778130

The equation of state was originally developed for ideal gases, and proved central to the development of early molecular and atomic physics. Increasingly sophisticated equations of state have been developed to take into account molecular interactions, quantization, relativistic effects and so on. Extreme conditions of matter are encountered both in nature and in the laboratory, for example in the centres of stars, in relativistic collisions of heavy nuclei, in inertial confinement fusion (where a temperature of 109 K and a pressure exceeding a billion atmospheres can be achieved). A sound knowledge of the equation of state is a prerequisite for understanding processes at very high temperatures and pressures, as noted in some recent developments. This book presents a detailed pedagogical account of the equation of state and its applications in several important and fast-growing topics in theoretical physics, chemistry and engineering. It should be of interest to upper-level undergraduates, graduate students, researchers and academics in theoretical physics, theoretical and nuclear chemistry, materials and mechanical engineering.

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Table of contents :
Front cover……Page 1
About the authors……Page 3
Title page……Page 5
Date-line……Page 6
CONTENTS……Page 7
Dedication……Page 11
Foreword……Page 13
Preface……Page 15
Preface to the Original Edition……Page 19
1.1 General remarks……Page 21
1.2 Phenomena at various densities and temperatures……Page 22
1.3 Quantum pressure and compressibility……Page 26
1.4 Pressure-temperature diagram……Page 28
1.5 Radiation effects……Page 31
2.1 Phenomenology……Page 35
2.2 Statistical picture……Page 41
2.3 Maxwell-Boltzmann distribution……Page 43
3.1 The partition function……Page 48
3.2 Thermodynamic functions……Page 50
3.3 The Gibbs’paradox……Page 51
4.1 Classical considerations……Page 54
4.2 The partition function……Page 60
4.3 The vibrational partition function……Page 62
4.4 The rotational partition function……Page 64
4.6 Summary……Page 67
5.1 Introduction……Page 69
5.2 Classical statistics……Page 70
5.3 Bose-Einstein statistics without restriction on the total number of particles: photons……Page 71
5.4 Bose-Einstein statistics for a constant number of particles……Page 75
5.4.1 Bose-Einstein condensation……Page 82
6.2 The grand partition function and other thermodynamic functions……Page 86
6.2.1 The Fermi-Dirac distribution function……Page 89
6.3 Relativistic considerations……Page 96
6.4.1 Non-relativistic case……Page 103
6.4.2 Extreme relativistic case……Page 104
7.2 The thermodynamic formulation……Page 106
7.3 The Saha ionisation formula……Page 109
8.1 Introduction……Page 116
8.2 Charged particle description……Page 117
8.3 Electrostatic energy……Page 119
8.4 Total free energy and equation of state……Page 121
9.1 Overview……Page 124
9.2 The Thomas-Fermi model at $T=0$……Page 125
9.2.1 Consideration of a gas of atoms……Page 128
9.2.2 Solution of the Thomas-Fermi equation……Page 129
9.2.3 Derivation of the Thomas-Fermi equation using variational principle……Page 140
9.2.4 The kinetic and potential energies of an atom……Page 141
9.2.5 Calculation of pressure……Page 148
9.3 Inclusion of exchange interaction: the Thomas-Fermi-Dirac equation……Page 152
9.3.1 Calculation of pressure……Page 156
9.4 Derivation of equation (9.103) using the virial theorem……Page 159
9.5 The Thomas-Fermi model at finite temperatures……Page 161
9.5.1 Calculation of thermodynamic functions……Page 164
9.6 Exchange and quantum corrections to the Thomas-Fermi model……Page 169
10.1 Introduction……Page 173
10.2 The Einstein model of solids……Page 175
10.3 The Debye model of solids……Page 177
10.4 The Griineisen relation……Page 180
10.5 Slater-Landau calculation of $gamma$……Page 181
10.6 Results and discussion……Page 184
11.1.1 Mass conservation equation……Page 185
11.1.2 Momentum conservation equation……Page 186
11.1.3 Energy conservation equation……Page 187
11.2 Sound waves and Rieman invariants……Page 189
11.3 Rarefaction waves……Page 193
11.4 Shock waves and the Hugoniot relation……Page 196
12.1 Foundations of hydromechanics……Page 204
12.2 Distribution functions and the Boltzmann equation……Page 205
12.3 Loss of information……Page 209
12.4 Derivation of macroscopic equations……Page 210
12.4.2 The equation of motion (momentum conservation)……Page 211
13.1 Introduction……Page 217
13.2 The Grueneisen coefficient $gamma(V)$ and an equation for the cold pressure $P_c$……Page 220
13.3 The specific volume $V_{0c}$ of the ‘zero point’ and the initial conditions for the $P_c$ equation……Page 224
13.4 Isentropic processes near the Hugoniot curve and the free surface velocity……Page 228
13.5 Equations of state for aluminum, copper and lead……Page 230
13.6 Semi-empirical interpolation equation of state……Page 237
14.1 Pellet fusion……Page 241
14.2 The limiting case of isentropic (shock-free) volume ignition (self- similarity model)……Page 243
14.3 Central core ignition with minimized entropy production……Page 252
14.4.1 The nonlinear-force pushing……Page 262
14.4.2 The cannon ball scheme……Page 264
14.5 The two-temperature equation of state……Page 266
14.5.1 Electronic contributions to the EOS……Page 267
14.5.2 The ion contributions to the EOS……Page 268
14.5.3 Results and discussion……Page 275
15.1 Overview……Page 277
15.1.1 The equation of state for an ideal gas……Page 279
15.1.2 The equation of state for a degenerate electron gas……Page 282
15.2 The equation of hydrostatic equilibrium……Page 287
15.3 Expressions for pressure and temperature inside a star……Page 289
15.4 Numerical estimates of $P_c$, $bar{P}$ and $bar{T}$ by assuming uniform density inside the star……Page 292
15.5 Some useful theorems……Page 293
15.6.1 The gravitational potential energy……Page 294
15.6.2 The virial theorem……Page 295
15.7 Qualitative understanding of the evolution of a star……Page 299
15.8 The contribution due to radiation pressure……Page 303
15.9 The polytropic model……Page 306
15.10 The standard model……Page 314
15.11 The white dwarf stars……Page 319
15.11.1 Solution of the equation of hydrostatic equilibrium for a completely degenerate gas in the extreme relativistic limit……Page 320
15.11.2 The general solution corresponding to a completely degenerate gas……Page 321
16.1 Overview……Page 325
16.2.1 Introduction……Page 326
16.2.2 The partition function……Page 328
16.2.3 The bootstrap condition……Page 330
16.2.4 The thermodynamic functions: pressure and energy……Page 335
16.2.5 Transverse momentum distribution……Page 337
1 A free particle inside a box and the density of states……Page 341
2 The Stirling formula……Page 345
3 Table of Fermi-Dirac functions……Page 346
4 Derivation of the virial theorem result……Page 353
5 Tables of Thomas-Fermi corrected equation of state……Page 357
6 Some mathematical relations for Chapter 13……Page 371
7 A note on the Lawson criterion……Page 373
8 Derivation of the equation describing hydrostatic equilibrium for a completely degenerate gas……Page 374
References……Page 375
Index……Page 382
Back cover……Page 387