 Images: Dust jacket Inside flaps Excerpts from Book (PDF files): Title Page Contents Preface Excerpts Appendix F For thermal diffusion and thermophoresis, see Appendix F. Buy it now at limited-time, introductory price of \$20 + S&H (Secure online order)  Retailers - Please contact publisher to request discount information Contact Links: Publisher Author Appendix F Define Universe and Give Two Examples 567 567 From Dahneke, Barton E., Define Universe and Give Two Examples, BDS Publications, Palmyra, New York, 2006, 567-587. (http://defineuniverse.com) Appendix F Thermodynamics, Statistical Mechanics, and Kinetic Theory# 1. Thermodynamic Laws, Systems, and Entropy In Chapter 10 we described the first law of classical or macroscopic thermodynamics as the conservation of energy in a system in which heat Q is recognized as a form of energy. In its most general form the first law of thermodynamics is written Q = ΔE + min× Ein – mout× Eout – W, where Q is heat addition to the system, ΔE is energy increase of the system, min and mout are masses added to and extracted from the system, Ein and Eout are the energy contents per unit mass of the added and extracted masses, and W is work energy extracted from the thermodynamic system. This form of the law applies for an open system in which mass, heat, energy, and work may all be exchanged. For a closed system, no mass exchange occurs and the law applies but with min = mout = 0. For an isolated system, no mass or heat or energy or work exchange occurs. However, energy may be internally converted into heat for which process ΔE = ΔEi + Qi = 0. We also mentioned in Chapter 10 that Rudolf Clausius et al discovered an equilibrium-state property of materials that he eventually named entropy. Change in entropy of a system due to heat Q transferred to the system is ΔS = Q/T + I, where T is the absolute temperature of the heat (source) and I is the irreversibility of the process due to, say, friction or electrical resistance or …. In ideal processes I = 0 and ΔS = Q/T, the only classical-thermodynamics type of process in which entropy change is exactly defined. For a real process wherein I ≥ 0 we write ΔS ≥ Q/T. Clausius stated the second law of thermodynamics as follows: “No spontaneous process in an isolated system causes system entropy to decrease.” This law has been stated in other forms and we soon consider a famous one due to L. Boltzmann. The universe exemplifies an isolated system, i.e., no transfer of energy, work, heat, or matter to or from the system. In such systems, evolution toward maximum entropy proceeds by internal processes by which, in the vicissitudes of spontaneous energy fluctuations even in the vacuum, some energy is inexorably converted to heat, a form of energy from which there is never a full or reversible return to any other. This universal, at-least-partially mono-directional process is relentless in its effect: isolated systems ever evolve toward their only possible stationary or equilibrium state, referred to as a “heat-death,” in which system entropy is maximized and all system energy is unusable heat.1 Clausius’ (and Boltzmann’s) statement conveys the relentless nature of entropy production in spontaneous generation and exchange of heat. Entropy extended the scope of thermodynamics, especially beginning in the 1870s when Austrian physicist Ludwig Boltzmann (1844-1906) began developing powerful statistical mechanics thereby extending James Clerk Maxwell’s introduction of statistical methods to physics in his 1859 and 1864 kinetic theories of gases. 568 Appendix F Traditional classical thermodynamics considers macroscopic characterization of systems by their bulk properties, such as system volume V, pressure P, and energy E. But in statistical mechanics we seek deeper understanding at a more fundamental, microscopic level in characterization of systems by atomic (or even subatomic) properties, such as atom velocities. Moreover, we seek a methodology in statistical mechanics useful for small numbers of atoms and for nonequilibrium systems, possibilities not contemplated in traditional, macroscopic thermodynamics. 2. Entropy in Statistical Mechanics Throughout Boltzmann’s life most scientists held a preconception against the atomic theory of matter (illustrated by J. J. Waterston’s experience described in Appendix A). Continuum theories of matter and energy were favored as superior to atomic theories. Moreover, since Newton’s mechanics are fully reversible in time, i.e., any solution for forward flowing (positive) time applies equally for backward flowing (negative) time, physicists objected to the concept of a thermodynamic property, entropy, that only increases in time until it reaches a stationary, maximum value. How could such a process be consistent with the fully reversible mechanics of the atoms involved? To justify the atomic-molecular theory of matter and the existence of an entropy-like property in time-reversible mechanics, Boltzmann sought and discovered a property of a gas he called H that continually decreases in time until it reaches a stationary, minimum value. Boltzmann’s H-theorem states that the time-rate-of-change of H is less than or equal to zero, i.e., dH/dt ≤ 0, with H defined by2 H = Σj pj loge(pj). In this sum, index j indicates a system state. The sum includes all states accessible to the system, corresponding to many microscopic configurations or microscopic states or microstates consistent with its bulk-properties state (i.e., everything we know about the system). Each such microstate is assumed to be equally probable. The probability pj that the system is in its jth discrete microstate is regarded as equal to the fraction of an ensemble – a huge number of (imaginary) identical replications of a prototype system – in microstate j or the fraction of time the prototype system is in microstate j. In quantum statistical mechanics all possible exchanges of identical atoms are counted as a single configuration (page 573), such exchanges being conceptually beyond any capability to detect. In addition, when system energy does not vary with position of an atom in system volume V, many geometric configurations form different but bulk-property-equivalent states. The number of possible microstates of a system is usually much larger than the number of atoms in the system.3 By its above definition and because probability pj must satisfy 0 ≤ pj ≤ 1 and pj loge(pj) = 0 when pj is zero or one and is otherwise negative, H is always negative. Boltzmann’s H and entropy S for a dilute (or perfect) gas are related by S = – k H = – k Σj pj loge(pj) ≥ 0, with k the Boltzmann’s constant. Boltzmann’s version of the second law, derived from his reversible-mechanics analysis of the evolving state of perfect-gas atoms, is that isolated-system entropy only increases until it reaches a maximum value, i.e., Define Universe and Give Two Examples 569 dS/dt ≥ 0. In 1902, American engineer-physicist-chemist Josiah Willard Gibbs (1839- 1903) entered the statistical-mechanics story.4 He introduced the powerful concept of the ensemble as a superior foundation of statistical mechanics (see endnote 11 of Chapter 2) and utilized it to correct and extend Boltzmann’s results. While Boltzmann ignored interactions, Gibbs included them. Thus, Boltzmann’s results apply only for a perfect gas while Gibbs’ sometimes identical expressions, such as the one for entropy of a system, are derived from the superior conceptual basis that allows their application to real gases, liquids, and solids in which strong molecular interactions occur.5 To illustrate the nature of entropy and the value of Gibbs’ approach, consider a closed thermodynamic system containing a solution in which a crystal (with strong atomic bonding) is forming. In a closed system, heat and energy but not matter may transfer into or out of the system. A crystal represents a highly ordered state with the crystal atoms purified and fixed in a regular structure. Before crystallization, solute atoms are neither purified nor fixed (distinguishable) but are mixed and randomly drifting about in the solution. When crystal growth is slow and system temperature remains nearly fixed, the crystallization is essentially reversible. Nevertheless, system entropy decreases in such spontaneous crystallization, a claim justified in endnote 6. What happened? Isn’t entropy supposed to increase in spontaneous processes? Have we encountered an enigma? A reader might say “We were led to believe (by the reader’s induction) that a spontaneous process should always give a positive ΔS, either when heat is indirectly generated by inefficiency (irreversibility) in use of energy or generated directly from energy. But ΔS is negative in our crystallization-of-solute-atoms illustration! What kind of swindle is going on here?” No one is being swindled because the system is not isolated. For the closed system in our illustration no net internal-heat increase occurs. Heat slowly generated by crystallization in the system is slowly transferred out of the system so that system entropy decreases. But entropy of the universe inevitably increases by more than system entropy decreases because, for outward heat flow, the system-boundary temperature Tb is slightly smaller than internal system temperature T and environmental (universe) entropy increase ΔSe = Q/Tb > |Q/T| = |ΔS|. In applying thermodynamics, and especially the second law, it is essential to take account of system type as well as processes. Otherwise one quickly finds him- or her-self in deep tapioca (pudding). Our crystallization example illustrates a general principle: entropy change represents change in information required to fully specify a system state. Specifying a system of atoms fixed in a regular crystal structure requires less information than atoms randomly drifting in solution. In general, uncertainty in system state increases with heating (Q = T ΔS > 0) and vice versa with cooling. Heating extends the range of accessible “configurations” and requires more information to specify the system. Cooling reduces accessible configurations, ultimately to a single, ground state. But with increase of energy content of matter or space, energy content of other matter or space decreases, so a general implication of heating or cooling is not obvious except within an isolated system such as the universe. In communication and information theory an information entropy identical to Boltzmann’s entropy emerges and provides 570 Appendix F an identical function in specifying an information-system state.7 Entropy, then, is a measure of information required to specify a system state or information more generally. Using entropy, consequences of exchange or conversion of heat and other related processes in systems may be characterized in illuminating ways. Clausius’ and Boltzmann’s versions of the second law for isolated systems and their variations for other systems introduce subtle but powerful means for analyzing thermodynamic processes and predicting if and when they will occur naturally and spontaneously. The nature of thermodynamics and entropy and common pitfalls in their use are further indicated or implied in the microscopic-scale theories and mathematical tools provided by statistical-mechanics or -thermodynamics we next consider. 3. Characteristic Properties and Equilibrium Distributions We have already described three types of thermodynamic systems. (1) In an isolated system (no transfer of energy or matter), increase in system entropy dS occurs when incremental heat dQi = |dEi|  T dS is internally generated at absolute temperature T at cost of internal system energy dEi. (2) In a closed system (energy or work but not matter may enter or leave the system), heat may be generated in or transferred into or out of the system causing system entropy change ΔS  Q/T. (3) In an open system, entropy may additionally be changed by transfer of matter into or out of the system. The most common cause of confusion and error connected with entropy occurs in use of correct principles or expressions but for a wrong system type. Therefore, in our sketch of thermodynamics we utilize a principle valid for all systems and already suggested by the second law: the stationary, equilibrium, or most-probable state occurs at a maximum of the system’s information entropy subject to constraints characteristic of each type of system.8 Maximization of system entropy or “Maxent” subject to these characteristic constraints or fixed properties results in a characteristic thermodynamic property for each type of system which is minimum at equilibrium, providing a useful criterion for the equilibrium or most-probable state. Characteristic properties for various system types, determined by maximization of system entropy subject to the fixed-properties constraints, are listed in the following Table.8, 9 For isolated systems of volume V, containing N atoms and fixed system energy E, the characteristic property is – S so that equilibrium corresponds to maximum S. For closed systems at fixed volume V, containing N atoms at absolute System Type Traditional Name in Statistical Mechanics Fixed Properties Characteristic Function (Partition Function) Characteristic Property Isolated Microcanonical N, V, E Z =  (see text)  S =  k loge(Z) Closed Canonical N, V, T Z = j exp(Ej) F =  kT loge(Z) Open Grand Canonical , V, T Z = ij exp([nj  Ei])  PV =  kT loge(Z) Open Isothermal-Isobaric N, P, T Z = ij exp([PVj + Ei]) G =  kT loge(Z) Define Universe and Give Two Examples 571 system temperature T, the characteristic property is the Helmholtz free energy defined as F = E – TS so that equilibrium corresponds to a minimum F. For open systems at fixed V, T, and chemical potential μ (Greek “mu” = μ = G/N) the characteristic property is – PV, with P the pressure, so that equilibrium corresponds to a maximum PV. For open systems at fixed N, P, and T, the characteristic property is the Gibbs free energy defined by G = E + PV – TS so that equilibrium corresponds to a minimum G. Let distribution p = {p1, p2, p3, …, pj, …} be defined as the probability distribution of a system over its possible discrete quantum states denoted by quantum number j = 1, 2, 3, …, j, … for which each state has discrete system energy E = {E1, E2, E3, …, Ej, …}. Specification of vectors (i.e., quantities containing multiple values or elements) p and E statistically specifies the microscopic state of a system, i.e., a complete description of its statistical distribution over exact properties on a microscopic, and therefore also macroscopic, level of detail. In contrast, specifying only bulk properties such as N, V, and E or T specifies only the macroscopic state of the system, i.e., its state fully defined on only a bulk or macroscopic level of detail. To derive an equilibrium distribution vector p we employ the above-stated “Maxent” principle that at equilibrium a system’s Boltzmann or information entropy is a maximum subject to imposed constraints (or, equivalently, the characteristic system property is minimum). We derive distribution p for both isolated and closed systems as illustrations of statistical-mechanics methodology for all systems. In the following illustrations we utilize important contributions of Boltzmann, Gibbs, American physicist Edwin T. Jaynes (1922-1998), and many others.8 While Boltzmann and Gibbs lived in the age of classical physics, we use the more correct quantum physics in our illustrations but include classical-physics results when valid. In the interest of simplicity and brevity, we must ignore many interesting details. 4. Equilibrium or Most-probable Distribution for the Isolated System The isolated system has characteristic property – S so its most probable state occurs at maximum S (equivalent to minimum – S) subject to fixed N, V, and E. This agrees with the statements of Clausius and Boltzmann that maximum S corresponds to the stationary, equilibrium condition or dS/dt ≥ 0. Thus, we seek the distribution p that maximizes system entropy, subject to specified N, V, and E. For an isolated system with N, V, and E fixed, many (imagined) macroscopically- identical replications of the system (the Gibbs ensemble of the system) contain many different microscopic configurations or states, each having the same macroscopic state, i.e., identical macroscopic or bulk properties. Each microstate, being equally consistent with the known bulk properties, is regarded as equally probable, a fundamental assumption in statistical mechanics called the assumption of equal a priori probabilities. Let Ω (upper case Greek “omega”) be the number of microstates giving N, V, and E. Thus, Ω is a measure of the degree of ignorance of the system’s microstate. That is, the probability of observing any one microstate is p = {p1 = p2 = ... = pj = 1/Ω} and the information entropy of the ensemble is Ω Ω S = – k Σj=1 pj loge(pj) = – k Σj=1 (1/Ω) loge(1/Ω) = k loge(Ω). 572 Appendix F But an ensemble is, equivalently, a single system at many different times. Thus, an ensemble average of a system property is a time average of the property10 and S is the experimental system entropy. When Ω = 1, the system microstate is fully known and S = 0. When S is large, Ω is much larger, the system distributes over Ω microstates, and its microstate is poorly known. That is, entropy of an isolated system, having known bulk properties N, V, and E, can be regarded as a measure of uncertainty in its microstate. Equation S = k loge(Ω) is Boltzmann’s principle, so named by Albert Einstein even though it was first written by Max Planck in 1906. This equation was carved on Boltzmann’s headstone in the Central Cemetery in Vienna. 5. Equilibrium or Most-probable Distribution for the Closed System Consider now an ensemble of macroscopically identical closed systems at fixed N, V, and T. System-j energy Ej may vary between different accessible Ej values at different times, because energy fluctuations of mean variance σE 2 occur in a system. (We shall shortly write an expression for σE 2.) A description of system-quantum-state distribution p must include dependence on T and Ej, with E = = Σj pj Ej the average or “expectation value” of the prototype system energy. We maximize the information entropy of the ensemble subject to this energy constraint (E = Σj pj Ej) and “normalization” (Σj pj = 1) to find the most-probable distribution p for the ensemble of systems over their accessible states consistent with the known bulk properties of the prototype system (everything we actually know about the system). We find the Maxent condition using Lagrange’s method of undetermined multipliers.11 In Lagrange’s method we form a sum Λ (upper case Greek “lambda”) containing the quantity to be maximized (information entropy) and the constraints to be applied, each multiplied by a Lagrange multiplier λi (lower case Greek “lambda” sub i). Λ = – k Σj pj loge(pj) + λ1 {Σj pj – 1} + λ2 {Σj pj Ej – }. By Lagrange’s method, the Maxent or most-probable distribution p occurs when ∂Λ/∂pj = 0 for all j and ∂Λ/∂λi = 0 for i = 1 and 2. Distribution p must therefore satisfy – k {loge(pj) + 1} + λ1 + λ2 Ej = 0 or pj = exp(– α – β Ej), with new constants alpha = α = 1 – λ1 / k and beta = β = – λ2 / k. The first constraint requires that exp(– α) Σj exp(– βEj) = 1 from which exp(α) = Z = Σj exp(– β Ej) and the maximally-probable (equilibrium) distribution p is, for every state j, [F1] pj = exp(– β Ej) / Z. The value of β is determined using the energy constraint Σj pj Ej = together with entropy S =  k Σj pj loge(pj) and Helmholtz free energy F = E  TS. We write F = E  TS = Σj Ej exp( β Ej)/Z + kT/Z Σj exp( β Ej) {loge[exp( β Ej)]  loge(Z)} =  βkT  kT loge(Z). Define Universe and Give Two Examples 573 But F and E are independent thermodynamic properties; the result holds if and only if β = 1/kT and F = – kT loge(Z). The most-probable or equilibrium distribution p for a closed system is, for every j, pj = exp(– Ej / kT) / Z, with Z = Σj exp(– Ej / kT). Z is called the partition function. It describes the partitioning of ensemble systems over their accessible energy states. When the system is one molecule, Z describes the partitioning of ensemble molecules over their accessible energy levels. Z may be evaluated by replacing summation with integration using the equality (page 202) dx dpx dy dpy dz dpz = h3 to obtain the μ-space (“mu”-space is singlemolecule space) element h3 = dx dy dz dpxdpydpz with state-j energy Ej = Ej translation + Ej internal + 1/N Σk ≥ j Σm ≥ n Σn ϕm n j k (rm n), with ϕn n j j (rn n) = 0. Ej translation = (px 2 + py 2 + pz 2)/2m, Ej internal is energy contained in rotation, vibration, and electronic excitation of a molecule, and ϕm n j k(rm n) is the atomic interaction potential energy for atom pair j-k, in which atom-n in molecule j is separated from atom-m in molecule k by rm n, with m and n = 1,2,3,4,... For dilute, noble-gas atoms, Ej internal = 0 for absolute temperature T < 10,000 K and ϕm n j k is negligible. For a single such atom    Z1 = h3 ∫dx ∫dy ∫dz ∫dpx ∫dpy ∫dpz exp{β(px 2 + py 2 + pz 2)/2m} = V (2πm kT / h2)3/2. over volume V    When ϕm n j k is negligible, the N gas atoms of the system behave independently (except during brief collisions) in system volume V so that the partition function is Z = Z1 N or Z = Σj exp(– Ej / kT) = {(2πmkT / h2)3/2 V}N / N!. N! (N factorial) is introduced to obtain the correct quantum statistics in the classical limit, i.e., to correct for N! exchanges of N identical atoms giving the same quantum microstate. For interacting molecules (non-negligible ϕm n j k), Z is more complicated. Partition function Z fully characterizes the equilibrium-system state, i.e., P = 1/β ∂loge(Ζ) / ∂V, E =  ∂loge(Z) / ∂β, F =  kT loge(Z), S =  β2 ∂[k/β loge(Z)] / ∂β, and Cv = kβ2 ∂2[loge(Z)] / ∂β2 = kβ2 σE 2, with P the system pressure, Cv its specific heat at constant system volume V, and σE 2 = <(E  )2> =  2 the mean variance in fluctuations of system energy E. Mixing of discrete and continuous variables in the preceding derivation and a following one is not consequential for the usual case when N is large. The above distribution p is justified only for the equilibrium state because we have invoked a thermodynamic relation to evaluate β. While thermodynamics strictly applies only to stationary, equilibrium systems, we suppose that the Maxent or maximum-entropy-at-most-probable-distribution principle may be used to determine most-likely, stationary distributions over accessible states for stationary, nonequilibrium systems as well. Adopting this principle to define a most-likely distribution in nonequilibrium systems, wherein deviations or fluctuations 574 Appendix F from equilibrium may be neither rare nor relatively small, provides probability distributions over even rarely-populated microscopic states.12 Characterization of nonequilibrium systems allows characterization of nonequilibrium processes and provides most-probable transition pathways over an energy or other barrier inhibiting formation of a new equilibrium state as it becomes more stable than a previously more-stable one. Common transport processes and most-probable process pathways and even rate constants for transition processes have been and can be determined from such distributions, capabilities beyond the scope of equilibrium thermodynamics. Example processes so characterized by statistical mechanics include transport of heat, momentum, and mass, phase-change nucleation kinetics, such as formation of droplets or crystals in vapors, liquids, or solids, and chemical reactions.12 6. Velocity Distribution of Gas Particles in a Temperature Gradient We illustrate the utility of statistical mechanics by deducing a microscopic, statistical, nonequilibrium velocity distribution of gas particles, that is, of atoms, molecules, and particles suspended in a gas. Nonequilibrium systems lie beyond the scope of thermodynamics. In his 1859 and 1864 kinetic theories of gases, James Clerk Maxwell deduced the equilibrium velocity distribution of gas atoms and molecules. While his methods are simple and elegant, they only apply for gases in equilibrium. But nonequilibrium gases can be in a stationary state characterized by a stationary distribution. Using Maxent with quantum and classical statistical mechanics we deduce velocity distributions for one-component gas particles and for all species in a multicomponent mixture in a stationary temperature gradient. Consider a closed system containing N particles of dilute or perfect gas in volume V = A×L, with A the uniform, x-y-plane cross-sectional area of the system of z-direction length L lying between z = z0 = 0 and z = z1 = L. We impose a z-direction temperature gradient so that gas temperature near plane z = z, i.e., for small [z  z], is T(z ) = Tz (1 + [z  z] γT / Tz), with γT (= Greek “gamma” sub T) = dT/dz the imposed temperature gradient at z. We assume gas pressure to be P = n(z) k T(z) with n(z) the local number density of all gas particles at altitude z, T(z) the temperature at the same plane, and k the Boltzmann’s constant. For mechanical stability we require uniform pressure P (an isobaric system). We illustrate notation in the relations nz Tz = n(z) T(z) = nz Tz and, with z near z, nz = n(z ) = nz / (1 + [z  z] γT / Tz). The probability any selected gas particle is between planes z and z+dz is p(z) dz = nz A dz / N = PV / NkTz dz / z1 = / Tz dz / z1. The probability density p(z) is probability per unit altitude z (or, in other cases, per unit change in another property, e.g., a molecular velocity component). Let < εϕTz > (Greek “epsilon” sub “phi” Tz) be total kinetic energy of species- ϕ gas particles at z due to random, thermal motions u,v,w, and systematic motion Vϕz <εϕTz> = <εϕz >Tz / Tz + mϕVϕz 2 / 2 = 3/2 kTz Tz / Tz + mϕVϕz 2 / 2 = 3/2 kTz + mϕVϕz 2 / 2 Define Universe and Give Two Examples 575 where index ϕ = 1,2,3,... indicates the gas-particle species in a mixture. Then gas kinetic energy due to random thermal motions <εϕz> is proportional to local gas temperature. This assumption defines local, nonequilibrium-gas temperature.13 We impose three constraints at every altitude z on each species of gas particles. (i) The sum of probabilities of system state j over all possible j-states equals one, [F2a] Σj pϕj = 1. (ii) Total kinetic energy of a particle due to random, thermal motions u,v,w and to systematic motion Vϕz averaged over all possible j-states is [F2b] Σj pϕj εϕj = < εϕTz > = 3/2 kTz + mϕVϕz 2 / 2. (iii) Number flux of all gas particles in the γT direction (the z-direction) is constant [F2c] jz net = Σϕ jϕz net = nz Σϕ ζϕz = nz Σϕ ζϕz Σj pϕj wϕj = constant with local number density of all gas particles nz and local number fraction of species- ϕ particles ζϕz. [F2c] prevents local accumulation or depletion of gas particles, required in an isobaric, stationary-temperature-field gas. In closed, stagnant-gas systems, jz net = 0. In writing [F2b] and [F2c] we include systematic velocity Vϕz in εϕz and in pϕ. In [F2a-c], summation over a single quantum number, subscript j, represents sums over all quantum numbers or integration over three-dimensional velocity space.14 In its classical form our result shall explicitly contain the three velocity components. The following derivation is, for clarity, described for one and two species but it applies to mixtures of any number of species. We seek the most probable quantumstate distributions pϕ and pη (η = Greek “eta”), where pϕ = {pϕ1, pϕ2, pϕ3,…} with pϕ1 the probability of ϕ-gas-particle energy state εϕ1, pϕ2 the probability of ϕ-gas-particle energy state εϕ2, etc. Ensemble systems are ϕ and η gas particles (in μ-space) of energies εϕj and εηi. The strategy is to maximize information entropy subject to constraints [F2].15 Mixed-species information entropy14 in Lagrange’s method gives Λ = – k Σi Σj (pηipϕj) loge(pηipϕj) + {λη1(Σi pηi – 1) + λϕ1(Σj pϕj – 1) + λη2(Σi pηi εηi – <εηz>) + λϕ2 (Σj pϕj εϕj – <εϕz>) + λη3 (nz Σi pηi wηi  jηz net) + λϕ3 (nz Σj pϕj wϕj  jϕz net)} =  k Σi pηi loge(pηi)  k Σj pϕj loge(pϕj) + {λη1(Σi pηi – 1) + λϕ1(Σj pϕj – 1) + ...} which is maximized to maximize information entropy by selecting values of elements of vector pϕ and of the three Lagrangian multipliers λϕ1, λϕ2, and λϕ3 so that ∂Λ/∂pϕj = 0 for j = 1,2,3,4,5,6,... and ∂Λ/∂λϕj = 0 for j = 1,2,3, with like expressions for pηi, to obtain (for both pϕj and pηi but showing only pϕj results) [F3] loge(pϕj) = – α – βzεϕj  γϕ wϕj or pϕj = exp(– α – βzεϕj  γϕ wϕj), [F3a] Σj pϕj = exp(– α) Σj exp(– βz εϕj  γϕ wϕj) = 1, [F3b] Σj pϕj εϕj = exp(– α) Σj εϕj exp(– βz εϕj  γϕ wϕj) = 3/2 kTz + mϕVϕz 2 / 2, [F3c] Σj pϕj wϕj = = exp(– α) Σj wϕj exp(– βz εϕj  γ ϕwϕj) = jϕz net / nϕz. 576 Appendix F To evaluate αz, βz, γϕ, and Vϕz we transform from quantized energy εϕ j to continuous energy εϕz = mϕ(u2 + v2 + w2 + Vϕz 2)/2. Sums in [F3] become integrals of the form e αz ∫ du∫ dv∫ dw g e[– βz mϕ(u2 + v2 + w2 + Vϕz 2)/2  γϕ w] = e αz ∫ du∫ dv∫ dw g e[– βz mϕ/2 (u2 + v2 + (w + Vϕz)2) ] with g = 1, εϕz, or w and γϕ = γϕz = βz mϕ Vϕz. To solve these integrals we use a truncated Taylor’s series: p(u,v,w;V) = p(u,v,w;0) + V (∂p/∂V) V = 0 = {1  βz mϕV w} p(u,v,w;0). This approximation has been demonstrated16 both theoretically and experimentally to be excellent if V  / 3, which V generally is. Via [F3a] (i.e., g = 1), integration over u,v,w from   to +  gives Zz = exp(αz) = ∫ du∫ dv∫ dw {1  βz mϕ V ϕzw} e[– βz mϕ/2 (u2 + v2 + w2) ] = (2π /βz mϕ)3/2. We determine βz using [F3b] (i.e., g = ε). Let εϕz = mϕ(u2 + v2 + w2 + Vϕz 2) / 2, where Vϕz is a possible systematic, z-direction velocity of species-ϕ particles. Then,    <εϕz> = mϕ / 2Zz ∫ du∫ dv∫ dw {u2 + v2 + w2 + Vϕz 2}{1  βz mϕVϕz w}e[– βz mϕ(u2 + v2 + w2 )/2]    = 3 / 2βz + mϕ Vϕz 2 / 2 = <εϕTz> = 3kTz / 2 + mϕ Vϕz 2 / 2 so that βz = 1 / kTz and Vϕz, if it is not zero, is a systematic velocity associated with γT. A one-way flux of g is the quantity of g passing unit area of an imaginary control surface (in plane z) per unit time, where g is particle number, mass, momentum, energy, or other quantity. Net flux is the difference between two opposing fluxes. Z-direction flux components and net flux of g due to species-ϕ particles are    (0) jϕgz = nϕz λ / Zz λ ∫ du∫ dv∫ dw w g {1  βz mϕ Vϕz w} e[– βz λ mϕ(u2 + v2 + w2) / 2 ] [F4]   0 () jϕgz net = nϕ zλϕ zλ / 2  nϕ z λz λ / 2  nϕz βz mϕ Vϕzz. We now determine pϕj for a one-component and a two-component gas. Case 1: the One-component Gas. In a pure, one-component gas, average net velocity (i.e., g = 1) is [F4a-1] jz net / nz = = λz γT / 4Tz  Vz. In a closed system = 0 and distribution p1 = p in a temperature gradient implies two offsetting velocities in : an implicit thermal-diffusion velocity given by the first term on the right-hand side of [F4a-1] and a balancing counterflow velocity wz cntr =  Vz. Both derive from γT and, since they are opposed and balanced, neither is readily detectable in a single-species gas. When jz net = 0, [F4b-1] wz cntr =  Vz =  λz γT / 4Tz. For a closed, stagnant, pure-gas, isobaric or stationary system (jz net = 0) with γT ≠ 0 to be possible, local counterflow velocity wz cntr must balance thermal diffusion. The probability pj of state j of a gas particle at altitude z is therefore Define Universe and Give Two Examples 577 [F5-1a] pj(εj;z,γT) = exp{– βzεT j} / Zz. In its classical-physics form, the microscopic, probability density p is [F5-1b] p(u,v,w;z,γT) = (βzm / 2π)3/2 {1 βz m wz cntr w} exp{– βzm(u2 + v2 + w2) / 2}. Even though [F5] is stationary, particles are not equilibrated at any altitude  thus the thermal diffusion and balancing counterflow of [F4a-1]. But stationary probability density p(u,v,w;z,γT) gives the most-probable distribution for nonequilibrium just as Maxwell’s result gives the most-probable distribution for equilibrium. And in both cases p(u,v,w;z,γT) du dv dw is the fraction of all particles at z having velocity components between u and u+du, between v and v+dv, and between w and w+dw. When γT = 0, wz cntr = 0, T is uniform, particles are uniformly equilibrated, and Maxwell’s distribution  written directly from [F1]  is recovered. Case 2: the Two-component or Binary Gas Mixture. In a closed, binary-gas system, average-net-mixture velocity (i.e., g = 1) is [F4a-2] jz net / nz =  [λ1z / 2  λ2z / 2] γζ1 + [ζ1z λ1z / 2 + ζ2z λ2z / 2] γT / 2Tz  ζ1z V1z  ζ2z V2z = 0 with ζϕz the number fraction of species-ϕ particles at z, their mean velocity, and λϕz their mean-free-path length. In binary mixtures, ζ1z + ζ2z = 1 and dζ1z/dz + dζ2z/dz = γζ1 + γζ2 = 0. From [F4a-2], with Dϕz = λϕz / 2 and εϕ = mϕ (uϕ 2 + vϕ 2 + wϕ 2) / 2, [F4b-2] ζ1z w1z cntr + ζ2z w2z cntr = [D1z  D2z] γζ1  [ζ1z D1z + ζ2z D2z] γT / 2Tz, and [F5-2a] p1(u1,v1,w1;z,γT) = (βzm1 / 2π)3/2 {1 + βz m1 w1z cntr w1} exp{– βzε1}, [F5-2b] p2(u2,v2,w2;z,γT) = (βzm2 / 2π)3/2 {1 + βz m2 w2z cntr w2} exp{– βzε2}. In cases 1 and 2 the velocity distribution for each gas-particle species is [F5]. This result is correct for all species in any mixture. Constraint [F2c] gives binarysystem condition [F4b-2]. That condition, ζ1z + ζ2z = 1, and the stationary conditions j1z net = j2z net = 0 require ζ1z w1z cntr = [D1z  D2z] γζ1  [ζ1z D1z  ζ2z D2z] γT / 2Tz and w2z cntr =  D2z γT / Tz in stagnant, binary mixtures. These binary-system equations describe thermal diffusion (the Soret effect) and diffusion thermo (the Dufour effect), the latter being the inverse of thermal diffusion, i.e., a γT (heat flow) due to a γζ (diffusion). Distribution [F5] provides the flux of any quantity g. For purely translational energy of gas particles, g = <εϕz>. Molecules also carry internal energy of vibration and rotation and sufficiently hot atoms and molecules carry electronic excitation energy. (To make [F5] a general distribution over internal-energy states as well, internal energy is added to translational energy, as on page 573.) But internal energy of, e.g., noble-gas atoms below 10,000 K, predominantly in their ground electronic state, is fixed as atoms carry neither vibrational nor rotational energy. Total molecular energy is calculated by one of three equivalent assumptions: (1) a quasi-equilibrium process, (2) equipartition of energy, or (3) the correction factor due to A. Eucken.17 Eucken wrote for thermal conductivity of gas ϕ, ϕz = ( 9 cpϕ / cvϕ  5 ) ηϕz cvϕ / 4.17 578 Appendix F After Maxwell we use the mean-free-path length λϕz (Greek “lambda” sub ϕz [Greek “phi” z]) as the average distance-from-last-“re-equilibration.” Dynamic viscosity ηϕ (Greek “eta”) of species-ϕ gas gives the frequently-used λϕz = ηϕ / (0.499 ρϕz ) with gas mass density ρϕz (= Greek “rho” sub ϕz) = nϕz mϕ and mean speed of mass-mϕ atom, molecule, or particle = (8kTz / πmϕ) in the gas at temperature Tz. Maxwell’s distribution provides useful comparisons. At γT = 0 number flux is    (0) [F6] jz = nz  du  dv  dw pMax(u,v,w) w = () nz /4.   0 () A crucial property of equilibrium expression [F6] is that jz net = jz+ + jz = 0. Using Maxwell’s distribution with γT ≠ 0 and the nz λ and described below on this page gives upward or downward flux at z (upward flux being positive) [F7a] jz = nz λ / 4 = nz / 4 {1 λzγT / 2Tz}. [F6] and the first term of [F7a] are molecular effusion, the origin of molecular diffusion. The second term of [F7a] is thermal effusion which is additive and is itself thermal diffusion, a net flux driven by γT. [F7a] predicts nonzero flux jz net in a closed, stationary system; jz net properly vanishes when we use [F5] in place of the Maxwell distribution, i.e., when we add nz wz cntr,  wz cntr being the thermal-diffusion velocity. Thermal-diffusion flux depends on control-surface orientation relative to γT since p(u,v,w;z,γT) is defined for γT parallel to the z axis. If we specify a control surface inclined at angle θ (Greek “theta”) from γT (because control-surface “direction” is specified by the perpendicular to it) the net [F7a]-type number flux is [F7b] jnet(z,θ) = jzθ+ + jzθ = nz cos θ λzγT / 4Tz. We now illustrate kinetic theory with a few applications of distribution [F5]. 7. Simple Kinetic Theory of Gas Transport Processes We determine the transport coefficients in gases, namely, (a) thermal conductivity (“kappa”) that characterizes heat or energy transfer rate, (b) viscosity η (“eta”) that characterizes momentum transfer rate, and (c) diffusivity D (script D) and thermal-diffusion diffusivity DT (D superscript T) that characterize particle number (or mass) transfer rate. Since thermal diffusion was confirmed in 1916, no mechanism or theory, simple or elaborate, has successfully characterized it.18 We shall provide both. We use temperature notation T(z ) = Tz  = Tz(1 γT / Tz). In calculating number flux at z we use number density and average speed of gas particles at their last “collision” before crossing plane z, i.e., at z λz. Expressions nϕz λ = nϕz (1 λϕzγT / Tz) and = (1 λϕzγT / 2Tz) shall often be used, where γT = dT/dz at z. Number flux components at z are denoted, e.g., jϕ (z) = jϕz . (a) Energy transfer or heat flux qz due to translational energy of atoms and molecules in a pure, stationary gas is given by the sum of the opposing components    qz+ = nz –  ∫du ∫dv ∫dw p(u,v,w;z,γT) w m (u2 + v2 + w2) / 2   0 Define Universe and Give Two Examples 579 = nz kTz / 2 {1  λz γT / 2Tz + 5 wcntr / 2}   0 qz– = nz +  ∫du ∫dv ∫dw p(u,v,w;z,γT) w m (u2 + v2 + w2) / 2    = – nz kTz / 2 {1 λz γT / 2Tz  5 wcntr / 2}. With wcntr =  λz γT / 4Tz and z (Greek “kappa” sub z) the thermal-conductivity or heat-transfer coefficient, net heat transfer qz net = qz+ + qz is [F8] qz net = – z γT, z = 9/8 nz λz k, and λκz = 8 z / (9 nz k). For atomic gases (cp / cv = 5/3) Eucken’s formula17 correctly gives dimensionless Prandtl number = η cp / = 0.67. [F8] and ληz = ηz/(0.499ρz) (page 578) give ληz / λκz = 3/5. (b) X-direction shear stress (shear force per unit area) occurs in a fluid undergoing x-direction shear. X-direction shear stress at a plane normal to the z axis, denoted τxz (Greek “tau” sub xz), is a force per unit area due to net x-directionmomentum flux carried by z-direction gas-particle motions across the x-y plane at z. We consider laminar flow, i.e., local, x-direction velocity is Uz  = Uz{1 γU / Uz} in which the shear rate is γU = dU/dz at z. We retain a temperature gradient to determine its effect, if any. Components of total z-direction transfer rate of x-direction momentum are, with m nz = ρz (Greek “rho” sub z) the mass density of the gas at z,    τxz+ = nz –  ∫du ∫dv ∫dw p(u,v,w;z,γT) w mUz(1  λγU/Uz)   0 = 1/4 ρzUz (1  λzγU / Uz + λzγT / 2Tz + 2 wcntr(1 + δ) / )   0 τxz– = nz +  ∫du ∫dv ∫dw p(u,v,w;z,γT) w mUz(1 + λγU/Uz)    =  1/4 ρzUz (1 + λzγU / Uz  λzγT / 2Tz  2 wcntr(1  δ) / ), δ is terms that cancel and τxz net = τxz+ + τxz gives gas viscosity ηz (Greek “eta” sub z) [F9] τxz net =  ηz γU, ηz = ρzλzz/ 2 {1  Uz γT / 2Tz γU  2Uz wcntr /λzγU} and ληz = 2ηz / (ρz). In a pure gas our curly-bracketed factor in ηz is unity. The result [F9] agrees with measured data for laminar flow in Newtonian fluids, defined as those fluids consistent with [F9] which all but a few liquids are, and this ληz result compares well with ληz calculated for rigid spheres given above. Spontaneous laminar-to-turbulent flow transitions were reported in 1883 by Osborne Reynolds19 who discovered a critical Reynolds number Rec = ρd /η = 2,300 in circular ducts exists above which flows eventually became turbulent (with d the circular duct diameter and the volumetric flow divided by πd2/4). Smoothing 580 Appendix F of duct-inlet flow raises Rec to at least 40,000. Turbulence first appears after an “entrance length” as large as order 1000 d. No present theory predicts these phenomena. Our analysis suggests influence of a temperature gradient is negligible, but the assumptions (no net flux in a stationary temperature field) do not exactly match the process. After a sufficient entrance length, viscous heating might cause quantity Uz γT / 2Tz γU to reach a value at which ηz → 0. Vanishing of ηz is certainly a possible mechanism for onset of turbulence in duct flows and in clear-air-turbulence cells. “Smoothness” of flow or scale of instabilities also seems to be involved. The problem needs to be addressed in more detail, taking more space than is available here. (c) Diffusion and thermal diffusion in a binary mixture are the final transport topics we address. While [F5] correctly characterizes both thermal conductivity and viscosity, its greater transport-theory value is describing thermal diffusion and diffusion thermo in gases and thermophoresis and diffusiophoresis of particles in gases. In a binary-gas or a particle-in-gas mixture (i.e., ϕ = 1 or 2), both concentration and temperature gradients may exist, i.e., we may consider either or both of molecular diffusion and thermal diffusion. We consider the general case of both. In a consequential correction of the traditional method, we isolate variation in concentration from variation in temperature by using for local species-ϕ number density nϕz the product nz ζϕz with nz the total number density nz = n1z + n2z containing the complete temperature dependence and number fraction (Greek “zeta” sub ϕz) ζϕz = nϕz / nz containing the complete concentration dependence for each species. By [F5-2] the net-number-flux components are, with ϕ = 1 or 2,      0 jϕz net = nϕz–λ ∫du ∫dv ∫dw pϕz–λ w + nϕz+λ ∫du ∫dv ∫dw pϕz+λ w,   0    [F10a] j1z net = – nz D1z γζ1 + nz ζ1z D1z T γT / Tz = – nz D2z γζ1 + nz ζ2z D2z γT / 2Tz, [F10b] j2z net = – nz D2z γζ2 + nz ζ2z D2z T γT / Tz = – nz D2z γζ2  nz ζ2z D2z γT / 2Tz. Because γζ1 (=  γζ2) and γT are independent, the two molecular-diffusion terms must cancel one another as must the two thermal-diffusion terms to give jz net = 0. The in effect values that result are D1z = D2z and ζ1z D1z T =  ζ2z D2z T, with D2z T =  D2z / 2. While jz net = 0 and D1z = D2z were recognized in earlier analyses, without the abovementioned correction these analyses were hobbled by incorrect equations.20 Counterflows determined on page 577 are incorporated in [F10a-b] as the basis of the in effect values. Microscopic-level, nonequilibrium statistical mechanics thus provides correct governing equations [F10a-b], the transport coefficients Dϕz and Dϕz T, and deeper understanding of diffusion, thermal diffusion, and diffusion thermo. For instance, coupling of γζϕ and γT is most evident in the stationary condition: [F10a-b] with j1z net = j2z net = 0. In this condition, change Δ(γζ1 / ζ2z) =  Δ(γζ2 / ζ2z) is accompanied by change Δ(γT / 2Tz), which is the diffusion-thermo effect. For molecules or particles having friction coefficient fz (page 160) in a gas, [F11] Dϕζz = [λϕζz / 2] = kTz / fϕζz and λϕζz = 2 Dϕζz / , with subscript ϕζz denoting species ϕ (= 1 or 2) in medium ζ (see [F12]) at location z. Condensation-/deposition-rate use of λϕζz is described by Dahneke.21 In a binary mix, Define Universe and Give Two Examples 581 [F12] D1ζz = D2ζz = [D21zζ1z+ D22zζ2z], ζ1z D1ζz T =  ζ2z D2ζz T = ζ2z [D21zζ1z + D22zζ2z ] / 2. Mathematical results [F10] are interpreted in terms of the process physics. In closed, isobaric, binary, stagnant-gas mixtures, fully free transport is precluded by jz net = 0; free diffusion occurs for only one species, the one having limiting diffusivity D2z < D1z, i.e., D2z alone appears in [F10a-b]. Stronger thermal diffusion of species 1 preempts that of species 2 but motion of 1 is controlled by motion of 2 and by w1z cntr. The thermal-diffusion-molecule or thermophoretic-particle velocity of the species-2 particles is written directly from the pertinent term of [F10b] [F13] w2z T = D2z T γT / Tz =  D2z γT / 2Tz =  k γT / 2f2z, with k the Boltzmann’s constant. Note that as ζ2z → 0, w1z T =  ζ2z w2z T / ζ1z → 0. [F13] agrees within a few percent of the few measured data22 to which it has been compared, but broader testing is needed. In the stationary state, a closed, binary system satisfies j1z net = j2z net = 0 as well as j1z net + j2z net = 0, nz D2z divides out of [F10a-b], which then have solutions [F14] ζ1z = 1  (1  ζ10) / (Tz / T0) and ζ2z = ζ20 / (Tz / T0). [F14] are universal stationary solutions valid for any different-mobility-species pair. We obtain Tz / T0 for real gases using the stationary-state requirement q =  γT = constant or γT = 0 γT0 / . We take = (T) = 0 + γ ΔTz with ΔTz = Tz  T0, γ = d /dT and κ0 fitted constants that provide excellent approximations for and γT over a substantial range of T, illustrated in Figure F1. Solving γT = constant gives [F15] Tz / T0 = ((1 + B z / z1) + C) / (1 + C). Fitted values of 0 and γκ, giving the lines in Figure F1, are listed for three gases in the above table (for T0 = 273.2 K and ΔT1 = 300 K) with values of A, B, and C. Figure F1. Thermal conductivity (watts/m/K) versus temperature. Temperature, K Helium: Nitrogen: gas 0  Nitrogen Helium 2.3998 14.012 .0064025 .031881 A = 0/ 1 1.2494 1.4650 B = (2A + 1)/A2 2.2414 1.8311 C = T0/AT1 – 1 – 0.2711 – 0.3784 Argon 1.6402 .0043825 1.2475 2.2458 – 0.2700 582 Appendix F Figure F2 shows binary-mixture, stationary solutions ζ1z and ζ2z versus z/z1 obtained via [F14] and [F15] for several ζ10 and corresponding ζ20. Nitrogen is the species-1 gas (we assume z = 1z), T0 = 273.2 K, and ΔT1 = 300 K. Although not shown in Figure F2, species-2 particles can be purged from a hot gas region or flow. For example, consider a flow of gas or vapor containing low-level concentrations of impurities in a duct and through a heated, porous-plate “filter” in the duct with selectable velocity wface at the filter face, where γT / Tz = [γT / Tz]face. Species-2 (impurity) gas particles obtain net velocity w1z + w2z T at z and all species of particles with diffusivity D2 < D1 obtain velocity w1z  D2zγT / 2Tz and can be concentrated and held in the flow upstream of the filter face as long as wface < D2z [γT / 2Tz]face. The thermal diffusion ratio kT,18, 20, 23(a) long a popular “measure of thermal diffusion” based on stationary-state condition γζ2 = kT2 γT / Tz and kT2 =  D2z T / D2z, does not provide the information previously thought.18, 20, 23(a) These equations are supposed to characterize the two-bulb experiment23 wherein one of two connected bulbs is held at temperature Tz and the other at T0. Measured stationary-state bulbnumber- fraction differences Δζ2 = ζ2z  ζ20 give kT2 = Δζ2 / loge(Tz / T0) values. At some intermediate temperature it has been supposed D2z T =  kT2 D2z. However, by the present theory the stationary equation on which the analysis should be based is [F10b] at the stationary condition: γζ2 = kT2 ζ2z γT / Tz and kT2 =  D2z T / D2z = 1 / 2. Error and confusion in thermal diffusion date to and have propagated from its early investigations in “exact” or “rigorous” kinetic theory. Hence, thermal diffusion is the worst-predicted process of that theory. We have corrected the error in past analyses18, 20, 23(a) by using quantities nz and ζϕz in our equations. We conclude our analysis of thermal diffusion by describing its mechanism. The natural direction of thermal diffusion is that of γT (see [F4a-1]), the thermaldiffusion flux being balanced by a counterflow in a stagnant, one-component gas. In binary mixtures (D1 > D2), thermal diffusion of less-mobile species-2 particles is preempted by that of species-1 particles having stronger thermal diffusion. That is, while species-1 particles thermally diffuse in the direction of γT, species-2 particles are simply counterflowed in the opposite direction, as [F10] indicates. All transport is restricted by the limiting D2z. Species 1 thus concentrates in the hot region and 2 in the cold. In mixtures of more species, each species in turn tends to concentrate in its own region, each being in turn the most-mobile of the remaining species. 8. Thermophoresis and Diffusiophoresis Finally, we consider phoretic velocities of particles suspended in gas (i.e., an aerosol). Thermophoretic velocity due to γT is defined by [F10] and [F13]. Diffusiophoretic velocity due to γζ is also defined by [F10], which couples the two velocities. For small suspended particles, Knudsen number = Kn = λ/a >> 1, with a the spherical-particle radius or characteristic half-length for another shape and λ the meanfree- path length of the suspending gas molecules. Flow about such particles is called free-molecule flow. L. Waldmann24 predicted phoretic motions of spheres in such flow. Temperature, velocity, and concentration jumps21 occur at particle surfaces at lower Kn. Brock and Jacobsen,25 among others,24 considered w2 T for 0  Kn  1. Define Universe and Give Two Examples 583 Thermophoretic velocity of species-2 particles at any Kn (if f2z dependence on Kn is included) is given by [F13]. As shown by the minus sign in [F13], this velocity is negative or opposite the direction of γT (the positive z-axis direction). Until now we have ignored influence of a particle on local temperature gradient, but the data collected by Waldmann and Schmitt24 suggest that w2 cntr and w2z T depend on actual, local γT. Carslaw and Jaeger26 determined the temperature gradient γTz across a sphere at z having Kn = 0 and thermal conductivity p in a gas of thermal conductivity g and temperature gradient γT at z but far from the particle, [F16] γTz / γT = 1 + C (Kn ) (1  p / g) / (2 + p / g) where C (Kn ) (added here to the Carslaw-Jaeger result) is 1 at Kn = 0. Dahneke21 wrote correction factor C (Kn ) by analogy. For a spherical particle of radius a, [F17] C (Kn ) = (Kn + 1) / [2 Kn (Kn + 1) / α + 1] with Kn = λ / a, λ is given in equations [F8], and α  1 is the thermal accommodation coefficient or average fraction of thermal-energy difference transferred per molecular collision. At large Kn , C (Kn ) = α / 2Kn → 0 and thermophoresis is independent of p/ g. Solving [F4b-2] using γT and γTz for species 1 and 2 gives, in place of [F13], [F18] w2z T =  k γTz / 2f2z =  k {1 + C (Kn ) (1  p / g) / (2 + p / g)} γT / 2f2z. Notes and References for Appendix F. 1 How the second-law-predicted heat death of the universe will eventually occur has been described by Fred Adams and Greg Laughlin in their popular-level book The Five Ages of the Universe (The Free Press, New York, 1999). However, one should not lose sleep over the eventual demise of the universe. Our sun’s brightness will not begin to diminish perceptively for another six billion years. Events described in Book III will precede and supercede that event, are more urgent, and imminent. ζϕ z z / z1 ζ20 = 0.5 0.4 0.3 0.2 0.1 ζ10 = 0.5 0.6 0.7 0.8 0.9 Figure F2. Number fraction versus altitude z / z1. Legend solid dots, ϕ = 1 open diamonds, ϕ = 2 584 Appendix F 2 Boltzmann’s original H-theorem expression is modernized here using quantum theory concepts. For a description of the logarithm function, see endnote 27 of Chapter 8. 3 The number of distinct quantum states of a system containing N identical atoms includes N! / Πj nj! indistinguishable exchanges of the N= Σj nj atoms divided into distinguishable groups of nj atoms at energy levels Ej, with j = 1,2,3,…, where symbols Πj and Σj (upper case Greek “pi” sub j and “sigma” sub j) indicate, respectively, the product and sum over all j values and “n factorial” = n! = n (n-1) (n-2) … 2 1 with 0! = 1. This number of possible states of a system is usually much larger than its number of atoms; only when all or most atoms have one energy Ej are the two numbers comparable. But even when all or most atoms have identical energy Ej, differences in atom locations give a similarly high number of different, distinguishable states all having essentially identical Ej. (For further details see Tolman, Richard C., The Principles of Statistical Mechanics, Oxford University Press, Oxford, England, 1938, Chapter 13 and Hirschfelder, J. O., C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, John Wiley, New York, 1954, Chapter 2.) 4 Gibbs, J. Willard, Elementary Principles in Statistical Mechanics, Yale University Press, New Haven, CT, 1902; reprinted by Dover Publications, Inc., New York, 1960. 5 See Jaynes, E. T., “Gibbs’ vs Boltzmann’s Entropies,” American Journal of Physics 33, 1965, 391-398. This same author is quoted pertinently in endnote 11 of Chapter 2. The former paper is found with other salient articles in E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics, R. D. Rosenkrantz (editor), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1983. All Jaynes’ papers are available at http://bayes@wustl.edu/etj/node1.html. 6 We demonstrate entropy decrease in slow crystallization in a closed system by use of the characteristic property for such systems (Helmholtz free energy F = E  TS, Section 3), which is minimum at equilibrium. Then for slow, quasi-equilibrium crystallization, ΔF = {E2  TS2}  {E1  TS1} = ΔE  TΔS = 0. It follows that for slow, quasi-equilibrium crystallization, ΔE = TΔS. In crystal formation from solute atoms, each molecule added to the crystal gives up a latent heat of crystallization q. Without work extraction from a closed system the first law of thermodynamics is Q = ΔE, where ΔE is increase in system energy and heat Q is heat transferred to the system. (Positive or negative ΔE and Q represent energy and heat received by or released from the system.) Thus, crystallization of N molecules gives ΔE = Q =  Nq so that ΔS = Q/T =  Nq/T < 0. That is, as N molecules slowly crystallize they slowly release into the system latent energy of crystallization Nq which is subsequently released to the environment. The net effect is system energy decrease in the form of heat released from the system. Since system-boundary temperature Tb is slightly lower than internal system temperature T, increase in entropy of the environment (universe) Se = Q/Tb slightly exceeds system entropy decrease S = Q/T. 7 See, for example, Shannon, Claude E., A Mathematical Theory of Communication, University of Illinois Press, Urbana, IL, 1949. 8 System type and its traditional designation in statistical mechanics are given in the first two columns of the Table on page 570. The third column indicates properties of the system regarded as fixed, known, or required for specification of system state, with N, V, E, T, μ, and P being, respectively, number of system molecules, system volume, system energy, absolute temperature, chemical potential, and pressure. The partition function Z is shown in column 4 for each system type and column 5 shows the characteristic property that characterizes system state, i.e., the property minimized at equilibrium. Isolated systems evolve spontaneously toward minimum negative entropy or maximum S. Closed systems evolve spontaneously toward minimum Helmholtz free energy F = E – TS. Open systems evolve toward either minimum – PV or minimum Gibbs free energy G = E + PV – TS, depending on the type of open system. Like entropy for the isolated system, each of these properties characterizes system state for the indicated system type. The chemical potential is defined as μ = G / N so that specifying either one of μ or G also specifies the other if N is known. Spontaneous evolution of system types are characterized by d(–S)/dt ≤ 0, dF/dt ≤ 0, d(– PV)/dt ≤ 0, and dG/dt ≤ 0, with the equilibrium state defined by the minimum Define Universe and Give Two Examples 585 condition at which the characteristic property is constant. A useful concept for all system types is that minimum characteristic property – S, F, – PV, or G defines maximum system entropy or level of ignorance in specification of the system’s microscopic state given its known macroscopic properties, i.e., everything actually known about the system. The Maxent principle derives from the work of Boltzmann, Gibbs, R. T. Cox, Claude Shannon, and, especially, E. T. Jaynes (loc. cit.). 9 This table is adapted from Lloyd L. Lee (Molecular Thermodynamics of Nonideal Fluids, Butterworths, Boston, 1988, 34) who provides several examples of the Maxent method. 10 Why an ensemble average? Two equivalent ensembles may be used. The first is many replications of a prototype system. The second is a single prototype system observed at many times. The latter corresponds to an actual system evolving in time. The most-probable state in the ensemble of macroscopically identical systems is the most probable state of the single system evolving in time. When probability of a particular or set of microstates dominates, it dominates both averages. 11 See, e.g., Taylor, Angus E., Advanced Calculus, Ginn and Company, Boston, 1955, 198-201. 12 Consider a system containing a molecular species that may cluster, such as H2O (water) or Ar (Argon). Let distribution vector p = {p1, p2. ..., pj, ...} define the distribution in number of clusters per unit volume containing j molecules in the cluster, with j = 1, 2, 3, …, j, … The questions we pose are (1) does clustering occur? and, if so, (2) what is the distribution p of cluster sizes? and (3) what is the rate of formation of clusters that survive and grow? Associated with each cluster size is a molecular Helmholtz free energy (endnote 6) per molecule fj, with subscript j because cluster free energy per molecule is cluster-size dependent. Hence, f1 is the free energy of a free, isolated molecule, f2 is the free energy per molecule for a clustered pair, etc. The free energy difference that occurs when j molecules cluster together is therefore ΔFj = j (fj – f1). Cluster-size dependence of ΔFj is adequately approximated by use of surface free energy or surface tension γ (Greek “gamma”). Spherical clusters of radius r(j) contain j = 4/3 π r3/v molecules of bulk molecular volume v and contribute surface free energy ΔFS = 4π r2 γ per cluster. Excess free energy associated with a cluster of size r over that of a cluster-free system is ΔF(r) = – 4/3 π r3/v δF + 4π r2 γ, where δF (“delta” F) is increase in system free energy caused by removal of a single molecule from the center of a large cluster (bulk state) to the isolated (vapor) state. The mathematical form of this expression requires a maximum in ΔF or a surface-free-energy barrier at radius rmax = 2vγ/δF of height ΔFmax = 16π v2γ3/(3δF2). Beyond this maximum a cluster is stable and grows spontaneously and rapidly. Similar expressions apply for other cluster shapes (crystals). Since F is the characteristic function for a closed system, we write pj(r) / p1 = exp(– βΔF(r)) and pcrit = p1 exp(– βΔFmax) with β = 1/kT, where pcrit is the number concentration of critical-size clusters, i.e., clusters at size corresponding to the maximum of the free-energy barrier. To determine number concentration distribution p explicitly, the constraint N = Σj pj is used with N as the upper j-limit of the sum, N being the total number of clustering-species molecules per unit volume. But, in practice, a lower limit may be used since relatively few large clusters occur before nucleation of the new phase. These expressions answer the first two questions posed above and we now turn to the third: what rate of nucleation of stable nuclei of a new phase will occur? We make the reasonable assumption that growth occurs predominantly in one-molecule increments because p1 is generally larger than all other cluster populations and single molecules transport through the medium at higher speeds. (This assumption may be checked and corrected, if necessary, by including growth due to larger cluster assimilations by critical-size clusters.) The net growth rate of stable clusters is Icrit = (dp/dt)crit = [p1 η R1crit – Rcrit-] p1 exp(– βΔFmax) where p1 is the concentration of single-molecules, p1R1critpcrit is the rate of collisions per unit volume of single- and critical-size clusters, η (Greek “eta”) is a sticking efficiency or fraction of colliding molecules that stick (η appears to be nearly unity), and Rcrit-pcrit is the rate per unit volume at which critical-size clusters spontaneously shrink by loss of one or more molecules. We have not specified the medium to be solid, liquid, or gas, although transport kinetics that control R1crit as well as Rcrit-, γ, and η strongly depend on both the clustering and the medium materials. 586 Appendix F In production of small droplets from a vapor of the clustering species we estimate the rate of formation of stable, critical-size clusters by estimating R1crit, Rcrit- and η. A suitable estimate for η is unity; we take Rcrit- ≅ p1 η R1crit /2 as an approximate rate constant, i.e., about half the droplets at the maximum of a smooth free-energy barrier will pass over the barrier and half will not. Effusional flux of vapor molecules at number concentration p1 in a gas (Equation [F6]) is p1/4, where = √(8kT/πm1) with m1 the molecular mass of the clustering species and T the absolute temperature. This flux multiplied by the effective cluster surface area gives the critical cluster growth rate p1 R1crit which, multiplied by the concentration of critical size clusters p1 exp(– βΔFmax), gives the stable-droplet formation rate per unit volume Icrit+, Icrit+ ≅ 2π v2γ2/δF2 √(8kT/πm) p1 2 exp(-16βπ/3 v2γ3/δF2). Dahneke (in Theory of Dispersed Multiphase Flow, Richard E. Meyer (editor), Academic Press, New York, 1983, 97-133) provides expressions for droplet growth rates in gases and vapors. Chemical kinetics illustrations are given by Eyring, H., and E. M. Eyring, Modern Chemical Kinetics, Reinhold, New York, 1963. 13 (a) Hirschfelder, J. O., et al, loc. cit., 455. (b) Chapman, Sydney, and T. G. Cowling, The Mathematical Theory of Non-uniform Gases, Cambridge University Press, 1960, 37. (c) Reed, Thomas M., and Keith E. Gubbins, Applied Statistical Mechanics, Butterworth-Heinemann Reprint, Boston, 1973, 354. (d) Jeans, Sir James, An Introduction to the Kinetic Theory of Gases, Cambridge University Press, Cambridge, 1962, 28. 14 Hill, Terrell L., Introduction to Statistical Thermodynamics, Addison-Wesley, Reading, MA, 1960. 15 Maximizing information entropy (Maxent principle) subject to macroscopic constraints imposes on a system only what is actually known about the system. It leads directly to characteristic properties for equilibrium systems of various sorts. Macroscopic constraints and Maxent together with known equilibrium-thermodynamic relations define a characteristic thermodynamic property for each type of system. For nonequilibrium systems, thermodynamics does not apply so that (1) the Lagrange multipliers cannot be determined by use of thermodynamics and (2) no characteristic thermodynamic property of a nonequilibrium system emerges from the analysis. Otherwise Maxent treatments of equilibrium and steady-state, nonequilibrium systems are identical. 16 Dahneke, B., Aerosol Science 4, 1973, 147-161. 17 Eucken, A., Physik. Zeitschrift 14, 1913, 324; reference 19, 237ff. 18 Chapman, Sydney, and T. G. Cowling, loc.cit., 142-144, 252ff, 399-404. 19 Reynolds, Osborne, Transactions of the Royal Society (London) 174, 1883, 935-982. 20 D1z = D2z and D1z T =  D2z T are standard results, the latter wrong (see [F12]). Hirschfelder, et al, loc. cit., 518ff, Bird, R. B., W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, Wiley, New York, 1960, 502, 568. Error can occur in use of γni since γni = ∂ ni / ∂ z ≠ dni / dz = γni + (∂ ni / ∂ T) γT. 21 Dahneke, 1983, loc. cit. 22 Talbot, L., Rarefied Gas Dynamics, Sam S. Fisher (editor), AIAA, New York, 1980, 467-488. 23 (a) Grew, K. E., and T. L. Ibbs, Thermal Diffusion in Gases, Cambridge University Press, 1952. (b) Bird, R. B., et al, loc. cit., 568, 574-575. 24 Waldmann, L., Zeitschrift für Naturforschung 14a, 1959, 376. See also Waldmann, L., and K. H. Schmitt, in Aerosol Science, C. N. Davies (editor), Academic Press, New York, 1966, 137-162. 25 Jacobsen, S., and J. Brock, Journal of Colloid and Interface Science 20, 1965, 544. 26 Carslaw, H. S., and J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, 1959, 426.