Statistical Thermodynamics and Microscale Thermophysics

Statistical Thermodynamics and Microscale Thermophysics Many of the exciting new developments in microscale engineering are based on the application of traditional principles of statistical thermodynamics. This book offers a modern view of thermodynamics, interweaving classical and statistical thermodynamic principles and applying them to current engineering systems. It begins with coverage of microscale energy storage mechanisms from a quantum mechanics perspective and then develops the fundamental elements of classical and statistical thermodynamics. Next, applications of equilibrium statistical thermodynamics to solid, liquid, and gas phase systems are discussed. The remainder of the book is devoted to nonequilibrium thermodynamics of transport phenomena and an introduction to nonequilibrium effects and noncontinuum behavior at the microscale. Although the text emphasizes mathematical development, it includes many examples and exercises that illustrate how the theoretical concepts are applied to systems of scientific and engineering interest. It offers a fresh view of statistical thermodynamics for advanced undergraduate and graduate students, as well as practitioners, in mechanical, chemical, and materials engineering. Van P. Carey is a Professor in the Mechanical Engineering Department at the University of California, Berkeley. The main focus of his research is development of advanced computational models of microscale thermophysics and transport in multiphase systems.

Statistical Thermodynamics and Microscale Thermophysics

VAN P. CAREY University of California, Berkeley

CAMBRIDGE UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK http://www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA http://www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 Typset in Times Roman 10/12 pt. in L9TEX2 £ [TB] A catalog record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data Carey, V. P. (Van P.) Statistical thermodynamics and microscale thermophysics / Van P. Carey. p. cm. ISBN 0-521-65277-4 (hb). - ISBN 0-521-65420-3 (pb) 1. Statistical thermodynamics. 2. Thermodynamics. I. Title. QC311.5.C36 1999 621.402'l-dc21 98-45449 CIP ISBN 0 521 65277 4 hardback ISBN 0 521 65420 3 paperback

Transferred to digital printing 2004

To Lee F. Carey, Bart Conta, and Dennis G. Shepherd, three of the best applied thermodynamicists I ever met

Contents

Nomenclature Preface

x xv

1

Quantum Mechanics and Energy Storage in Particles 1.1 Microscale Energy Storage 1.2 A Review of Classical Mechanics 1.3 Quantum Analysis Using the Schrodinger Equation 1.4 Model Solutions of the Time-Independent Schrodinger Equation 1.5 A Quantum Mechanics Model of the Hydrogen Atom 1.6 The Uncertainty Principle 1.7 Quantum Energy Levels and Degeneracy 1.8 Other Important Results of Quantum Theory

1 1 5 8 11 17 20 23 25

2

Statistical Treatment of Multiparticle Systems 2.1 Microstates and Macrostates 2.2 The Microcanonical Ensemble and Boltzmann Statistics 2.3 Entropy and Temperature 2.4 The Role of Distinguishability 2.5 More on Entropy and Equilibrium 2.6 Maxwell Statistics and Thermodynamic Properties for a Monatomic Gas

29 29 30 37 43 50

3

A Macroscopic Framework 3.1 Necessary Conditions for Thermodynamic Equilibrium 3.2 The Fundamental Equation and Equations of State 3.3 The Euler Equation and the Gibbs-Duhem Equation 3.4 Legendre Transforms and Thermodynamic Functions 3.5 Quasistatic and Reversible Processes 3.6 Alternate Forms of the Extremum Principle 3.7 Maxwell Relations 3.8 Other Properties

71 71 75 80 83 90 93 95 99

4

Other Ensemble Formulations 4.1 Microstates and Energy Levels 4.2 The Canonical Ensemble 4.3 The Grand Canonical Ensemble 4.4 Fluctuations 4.5 Distinguishability and Evaluation of the Partition Function vn

61

107 107 108 117 127 137

viii

Contents

5

Ideal 5.1 5.2 5.3 5.4 5.5 5.6 5.7

6

Dense Gases, Liquids, and Quantum Fluids 6.1 6.2 6.3 6.4 6.5

Gases Energy Storage and the Molecular Partition Function Ideal Monatomic Gases Ideal Diatomic Gases Polyatomic Gases Equipartition of Energy Ideal Gas Mixtures Chemical Equilibrium in Gas Mixtures

145 145 145 149 160 164 167 170 179

Behavior of Gases in the Classical Limit Van der Waals Models of Dense Gases and Liquids Other Models of Dense Gases and Liquids Analysis of Fluids with Significant Quantum Effects Fermion and Boson Gases

179 182 196 205 212

Crystals Monatomic Crystals Einstein's Model Lattice Vibrations in Crystalline Solids The Debye Model Electron Gas Theory for Metals Entropy and the Third Law

225 225 228 229 233 238 241

7

Solid 7.1 7.2 7.3 7.4 7.5 7.6

8

Phase Transitions and Phase Equilibrium 8.1 Fluctuations and Phase Stability 8.2 Phase Transitions and Saturation Conditions 8.3 Phase Equilibria in Binary Mixtures 8.4 Thermodynamic Similitude and the Principle of Corresponding States

245 245 264 271 284

9

Nonequilibrium Thermodynamics 9.1 Properties in Nonequilibrium Systems 9.2 Entropy Production, Affinities, and Fluxes 9.3 Analysis of Linear Systems 9.4 Fluctuations and Correlation Moments 9.5 Onsager Reciprocity of Kinetic Coefficients 9.6 Thermoelectric Effects

297 297 298 301 304 309 311

Nonequilibrium and Noncontinuum Elements of Microscale Systems 10.1 Basic Kinetic Theory 10.2 The Boltzmann Transport Equation 10.3 Thermodynamics of Interfaces 10.4 Molecular Transport at Interfaces 10.5 Phase Equilibria in Microscale Multiphase Systems

325 325 337 346 351 357

10

Contents

ix

10.6 Microscale Aspects of Electron Transport in Conducting Solids 10.7 The Breakdown of Classical and Continuum Theories at Small Length and Time Scales Appendix I

Some Mathematical Fundamentals

375 384 391

Appendix II Physical Constants and Prefix Designations Appendix III Thermodynamics Properties of Selected Materials Appendix IV Typical Force Constants for the Lennard-Jones 6-12 Potential

397 399 409

Index

411

Nomenclature

normalized bulk velocity, = wo/{2NpJi^/M)1^2 Redlich-Kwong constant aR av van der Waals constant AE* interface area Redlich-Kwong constant ^R by van der Waals constant B second virial coefficient particle speed c molar specific heat at constant pressure cp molar specific heat at constant volume cv C third virial coefficient electron charge eQ E energy Ex electric field in the x direction fugacity f average occupancy of a microstate with energy e fie) /(W, 25, t) fractional particle velocity distribution distribution function for an ensemble of particles /eP ensemble number density /es particle velocity distribution /pv fraction of molecules with speeds greater than c f>c F Helmholtz free energy degeneracy 8 radial distribution function g(r) frequency distribution function g(v) distribution function for electron quantum states g*(e) gravitational acceleration 8e molar Gibbs function 8 degeneracy of energy level / gi G Gibbs function Planck's constant h h molar specific enthalpy molar specific enthalpy of saturated liquid h molar specific enthalpy of saturated vapor K = h/2ix n H enthalpy Hamiltonian H I moment of inertia number flux of molecules j mass flux of molecules jm a

Nomenclature Jax Js,x Ju,x Ja Js Ju &B kt kx K Kc Kp L Ltj L ms Lvr Lz m M M rij (ni) N Na NaJ A^A Na N'a p Pi px P p pj P Pa Pc PY Psat P q qa qe qi qmi #nuci qTOt

xi flux flux flux flux flux flux

of species a in the x direction of entropy in the x direction of energy in the x direction of species a of entropy of energy Boltzmann constant thermal conductivity force constant for linear restoring force wavenumber, = 2n/X equilibrium constant equilibrium constant characteristic system dimension kinetic coefficients mean spacing between particles temperature gradient length scale, = T/WT Lorentz number, = kt/a&T mass per particle mass per molecule for polyatomic molecules molecular mass number of ensemble members in energy level £j mean number of particles in microstate I' number of particles number of species a particles number of species a particles in energy level / Avogadro's number number of species a particles per unit volume instantaneous number of species a particles momentum generalized m o m e n t u m m o m e n t u m in the x direction =Pr~l fluctuation probability density function generalized momentum pressure partial pressure for species a critical pressure reduced pressure, = P/Pc equilibrium saturation pressure probability molecular partition function molecular partition function for particle species a partition function for electronic energy storage generalized coordinate partition function for internal energy storage partition function for nuclear energy storage partition function for rotational energy storage

xii

Nomenclature

*7rot, nucl D VE yc

=l/kBT coefficient of thermal expansion analysis parameter ratio of specific heats, = cp/cv activity coefficient for species a bulk motion correction factor for a > 0 bulk motion correction factor for a < 0 surface excess mass liquid film thickness differential heat interaction (positive into system) differential work interaction (positive out of system) molar latent heat of vaporization, = hy — h\ energy energy level i energy microstate /' Fermi energy characteristic energy in the Lennard-Jones 6-12 pair potential absolute Seebeck coefficient of material A thermoelectric power for a thermocouple of materials A and B angular coordinate in spherical polar coordinates Debye temperature, = hv^/k^ Einstein temperature, = Fermi temperature, = rotational temperature, = /z 2/(87T 2/& B) = 9TOt for linear molecules, = (#rot,A#rot,##rot,c)1/3 for nonlinear polyatomic molecules vibrational temperature, = hv/kB isothermal compressibility wavelength mean free path between collisions thermal de Broglie wavelength lattice wave amplitude chemical potential (per molecule) for species a Fermi energy molar chemical potential for species a, = /jLaNA absolute viscosity frequency Debye cutoff frequency Einstein characteristic frequency mean collision frequency

xiv

Nomenclature

pN ac ae 0 4>a , . . . , a system will macroscopically attain an equilibrium state that is unchanging with time. At the microscopic level, the system is constantly changing its configuration. The system thus is in a state of dynamic equilibrium. We will see in later sections of this text that the nature of this dynamic equilibrium strongly influences how the system reacts to changes in the constraints on the system. This issue is central to thermodynamic analysis of systems, since as engineers and scientists, we often want to predict how a system will change if a change occurs in one or more of the macroscopic constraints on the system (such as the specified values of U, V, or particle numbers). Example 1.1 The mean speed of a helium atom in helium gas at 27°C is about 1,020 m/s. Use this information to estimate the internal energy of the gas per kmol. Solution The mass per helium atom m is known to be 6.65 x 10~~27 kg. Assuming the internal energy is due only to kinetic energy of the molecules, we can estimate the mean kinetic energy per atom as ek = \mc2 = (±) 6.65 x 10- 27 (l,020) 2 = 3.46 x 10~21 J. Multiplying by Avogadro's number, we obtain the internal energy per kmol: u = (6.02 x 1026)3.46 x 10~21 = 2.082 x 106 J/kmol = 2,082 kJ/kmol

Pioneering efforts to construct statistical theories of the behavior of large collections of particles were made by Maxwell, Boltzmann, and Gibbs in the last half of the nineteenth century. Because these efforts preceded the development of quantum theory, simple classical mechanics treatments of molecule energy storage mechanisms were incorporated into these initial statistical mechanics models. In fact, the statistical mechanics models of Maxwell and Boltzmann were highly controversial when they were first proposed because the existence of atoms and molecules was not widely accepted by scientists at that time. The work of Planck, Einstein, and others in the early part of the twentieth century established the existence of atoms, molecules, and eventually the subatomic structure of matter. Quantum theory provided the means for predicting the manner in which energy is stored in atoms, molecules, and subatomic particles. This made it possible to extend the statistical mechanics concepts developed by Boltzmann and Gibbs to a wide variety of system types. In this text, the current knowledge of the atomic structure of matter is taken as the starting point for development of macroscopic thermodynamics. Macroscopic equilibrium thermodynamics is a consequence of the nature of microscale energy storage and the statistical behavior of very large numbers of particles. We will therefore first establish the basic elements of quantum theory necessary to determine the rules of energy storage and energy exchange in atoms, molecules, and subatomic particles. We will then construct a statistical analysis of a system composed of a large number of particles that obey such rules. Once these two components are blended together to form a statistical thermodynamic framework, we will explore the application of the framework to a variety of systems.

1.2 IA Review of Classical Mechanics

1.2

A Review of Classical Mechanics

Classical mechanics analysis of the motion and mechanical energy storage of a particle can be cast in at least three ways. The Newtonian formulation of classical mechanics can be stated as

where p is the particle momentum and F is the force exerted on the particle. The Lagrangian formulation of classical mechanics is stated in terms of the Lagrangian L defined as

L(x, y, z, u, v, w) = K(u, v, w) - U(x, y, z),

(1.2)

where U(x, y, z) is the potential energy of the particle and K is the kinetic energy of the particle given by K(u, v, w) = ™[u 2 + v2 + w2].

(1.3)

In the above relation, m is the mass of the particle. Noting, for the x direction, that dL dK — = — =mu = p x, au au

dL dU — = -— = F x, ox ox

(1.4)

Newton's equation can be written as

dt\duj

dx

If we generalize the notation as x=q\,

y = q2,

z = q$,

u = q\,

v = q 2,

w = q?>,

(1.6)

the equations of motion can then be written as dt\dqj

= ^ ,

7 = 1,2,3.

(1.7)

An extraordinary and very useful property of Lagrange's equations of motion is that they have the same form in any coordinate system (useful since it is sometimes easier mathematically to define U in a particular coordinate system). A set of equations like (1.7) can be written for each particle in a multiparticle system. A third formulation of classical mechanics is the Hamiltonian formulation. To construct the Hamiltonian formulation of classical mechanics, we begin by defining generalized momenta as dL

Pj^—r-, dqj

7 = 1,2, ...,37V

(for a system of TV particles).

(1.8)

Each generalized momentum pj is said to be conjugate to coordinate qj. Since each momentum is linearly proportional to the velocity qj, it is a simple task to replace any velocity terms in the formulation with the corresponding generalized momentum pj. We do so throughout

11 Quantum Mechanics and Energy Storage in Particles the definition of the Lagrangian and define the Hamiltonian H for a system of N particles to be 37V

i

L(

p2, •.., P3N,q\,42, .. (1.9)

In general, for systems of particles we expect that the total kinetic energy of the particles is given by 3/V

^(