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A Brief Introduction to Statistical Thermodynamics
A Web-Book by Rituraj Kalita, Dept. of Chemistry, Cotton College, Guwahati-781001 (Assam, India)
Preface   Ch. 1   Ch. 2   Ch. 3   Ch. 4   Ch. 5   Ch. 6   Ch. 7   Bibliog.
     Background topics/ vocabulary           General topics           Advanced (avoidable) topics
© 2006. Copyright reserved. The book or any portion of it can't be reproduced/ re-published/ circulated.

Ch. 1: Statistical Thermodynamics: A Brief Introduction and Tour
 

1.0 Some Background Talk:
Energy-States & Energy-Levels of a Thermodynamic System and of the Molecules

Quantum mechanics tells us that we may consider (exactly speaking, be bold enough to consider) the Schrödinger equation (HY = EY) of even a given thermodynamic (obviously non-microscopic) system, and get (i.e., wonder about getting) its allowed system-states (i.e., system-wavefunctions) Y and system-energies E. Such a system-state is called a microstate (complexion), and the total number of system-states associated with a definite value of system-energy is obviously the degeneracy G of the system energy-level.

Coming down to a more usual application of quantum mechanics, we easily recognize that for a system made of N independent (non-interacting) molecules (e.g., an ideal gas), the above Schrödinger equation may be separated into N one-molecule (i.e., one-particle) Schrödinger equations hf = ef (as H is a sum of one-molecule Hamiltonians h). Here f is a molecular state (also called a molecular energy-state), and e is the molecular energy. The innumerable molecular energy-states may be classified into a lesser number of molecular energy-levels l (l = 0, 1, 2, 3 etc.), where such an energy-level by definition consists of the molecular energy-states with a definite energy value e. The number of molecular-states within an energy level l (i.e., with a definite value of molecular energy e_l) is obviously the degeneracy (also called statistical weight) g_l of the molecular energy-level. We also recognize that for a system of identical molecules in identical environments (e.g., a pure gas system), the molecular Hamiltonian h is identical for each molecule, and so the set of molecular states and molecular levels are identical. Thus we have a common set of molecular levels with their characteristic degeneracies, the set being shared by all the molecules. (If there are two kinds of molecules in the system, as in the case of a mixture of nitrogen and argon gas, there'll be two common sets of such levels.)
Note: The term molecule here may be replaced by the general term particle. Unfortunately in English, there is no corresponding adjective particle-ar to talk of particle-ar energy levels, particle-ar states etc.
Some of you might be wondering: How is the time-independent Schrödinger equation applicable to a non-isolated system, or to the molecules within that (instead of the more general time-dependent one)? Well, those of you know quantum mechanics too much! There're, actually,  some simplifications involved here!

So the question now arises about how many molecules are occupying a common molecular energy level. The number of molecules occupying a molecular energy level l is called the occupation  number or population N_l of the energy level. Similarly we can also talk of the occupation  number or population N_s of the energy state s. It is obvious that sum of the molecular populations of all energy levels equal the total number of molecules N i.e., S_l N_l = N. Similarly, S_s N_s = N. Also, for a system of non-interacting particles, the system (internal) energy U equals the sum of energies e of the particles. As a molecule in a molecular energy level l has the energy e_l, the N_l molecules occupying the level have the energy N_l e_l , and so this means that S_l N_l e_l = U. Similarly, summing over the states, we get S_s N_s e_s = U.
Note: The thermodynamically observed system internal energy U is, exactly speaking, a time-average of the quantum-mechanical system-energies E. You may, as well, overlook this distinction for the time being!

 
1.1 What is Statistical Thermodynamics?

(Classical) Thermodynamics discusses thermodynamic behaviour of macroscopic systems involving empirically-observable thermodynamic properties (e.g., pressure, temperature, energy, enthalpy, entropy) & thermodynamic interactions (e.g., work done, heat absorbed). It never deals with the microscopic constitution of the macroscopic system (e.g., no idea of atoms or molecules is involved within classical thermodynamics).
Notes: (i) 'Macroscopic' has an opposite meaning of 'microscopic': a macroscopic system is a system that consists of a very large, say 10¹⁸ or more, number of microscopic molecules/ atoms/ ions etc., and is of a size visible to the naked eye (ii) 'Empirically' means experimentally: any thermodynamic property or interaction (say, the pressure of the  system, or the heat absorbed by the system) may be simply observed and measured by an (macroscopic-level) experiment.

Statistical thermodynamics, on the other hand, attempts to explain and to predict the thermodynamic behaviour of the macroscopic system in terms of the behaviour of its very large number of microscopic constituents (i.e., molecules, atoms or ions etc.). As there are a very large number of such constituent particles involved, such interpretation must be made in a statistical way; so this branch is called by this name. As the properties of the microscopic constituents are governed by quantum mechanics, so statistical thermodynamics is said to be a bridge between quantum mechanics describing microscopic systems and thermodynamics describing macroscopic systems.

1.2 How Statistical Thermodynamics Works: A Brief History of Entropy

Entropy S, definable for a macroscopic system only, is a thermodynamic property that determines the spontaneous direction of a process (e.g., a reaction); and its infinitesimal change dS is defined as dS = dq_rev / T. Let us now see how it has been (and presently is) understood, calculated and predicted in statistical thermodynamics (ST).

It is basically an ST-concept that entropy is a measure of randomness or disorder within the system. In the early decades of ST, this randomness was understood as the number of ways in which the constituent molecules could occupy the spatial volume elements of molecular size; the entropy was considered to be proportional to the natural logarithm of this number W with the Boltzmann constant k (i.e., 1.381 x 10^–23 J K^–1) working as the proportionality constant. But in 20th century, with the advent of quantum mechanics, it became clear that it is rather the varying ways of occupation of the molecular energy states wherein the randomness of a gas-system actually lies (the spatial positions of the microscopic molecules are not exactly specifiable as per quantum theory). Within this new approach too, earlier it was thought that the relevant number W is the number of ways in which the molecules may occupy the molecular energy states while the macroscopic system exists with a definite system energy U, and so W is just the degeneracy G of the system energy level with system energy U, considering the Schrödinger equation for the macroscopic system itself. But now, we know that the system needn't exist with a very definite system energy, but may instead exist with a very slightly fluctuating narrow range of energies, while existing in just a definite thermodynamic state (giving a somewhat higher value of W). Thus, in the ST-endeavour of explaining and theoretically-calculating the value of entropy, the Boltzmann equation S = k.lnW inscribed in Boltzmann's tombstone remains the same, but its meaning has varied over the centuries and decades.
Are you thinking of proving or justifying that thermodynamically defined entropy S =  ∫(dq_rev / T) and this newly-encountered S = k.lnW are just identical? Well, we'll be doing in the next chapter!
Note: 'Spatial' means pertaining to space: it is an adjective form of space.

1.3 Some Basic Concepts of Statistical Thermodynamics

1.3.1 Assembly (of particles): In ST, the thermodynamic system is considered to be an assembly of the (very large number of) microscopic constituent particles (commonly, molecules). The particles may be physical or abstract (molecules in a gas are physical, normal modes of vibrations in a solid are abstract), identical or different (molecules of a pure substance are identical, of a mixture are different), interacting or non-interacting (ideal gas molecules are non-interacting, liquid molecules are interacting), and if identical may be indistinguishable (very common case, e.g., gas-molecules/ electrons) or distinguishable (e.g., normal modes of vibrations). In this course, we concentrate on the simplest cases of non-interacting identical-particle assemblies (e.g., ideal pure gas systems).

1.3.2 Microstate (Complexion): In general, a microstate or complexion (of the thermodynamic system) is a quantum mechanical state of the system, i.e., one particular solution (wavefunction) of the Schrödinger equation of the whole system. For assemblies of non-interacting molecules, a microstate (complexion) further means one particular way of arrangement of the constituent molecules among the allowed molecular energy states.

1.3.3 Macrostate (Thermodynamic State): A macrostate is a thermodynamic state (of the given macroscopic system), and is associated with definite values of enough number of thermodynamic properties (e.g., pressure, temperature, volume, energy, entropy, composition etc.) so as to completely characterize the macroscopic system in the classical thermodynamic sense. For example, when we keep the pressure, temperature and volume (here 3 properties) values fixed for a system consisting of only a pure non-reacting gaseous substance (say, oxygen), the system remains in a particular macrostate (and so, as per concepts of classical thermodynamics, the other parameters such as energy and entropy remains at fixed values). However, from a statistical-thermodynamic viewpoint, some thermodynamic properties (other than those kept fixed) actually fluctuate to a very slight degree (say, to an extent of around N^–0.5 times their mean values, where N is the number of constituent molecules). Thus, for a closed system of 1 mol of oxygen kept at constant pressure and temperature (i.e., N, P, T fixed), the internal energy U actually fluctuates slightly around some mean U₀ as: U = U₀ ± (1.289 x 10^–12).U₀ (i.e., to an extremely negligible extent).

1.3.4 Distribution (among energy levels): You know that in an ideal gas, the constituent gas molecules get themselves distributed among the various allowed (common) molecular energy levels: this gives us the idea of distribution. A particular way of arrangement of the molecules within the allowed molecular energy levels implies a particular distribution. Thus, for a system of identical and non-interacting molecules, a distribution means the specification of the occupation numbers (populations) of all the molecular energy levels. Among all possible distributions of a system (possible while in a given macrostate), the one with the highest probability of being observed is called the most probable (m. p.) distribution (this distribution is sometimes denoted by the superscript *).
For any macroscopic system, only distributions very similar to the m. p. distribution have significant probabilities of being observed, other ones have only very negligible probability (so are never observed).

1.3.5 Thermodynamic Probability: Thermodynamic probability W_d of a given distribution d within a macroscopic system is the number W_d of microstates (complexions) possible for the given distribution. Similarly, thermodynamic probability W of a macroscopic system existing in a given macrostate (thermodynamic state) is the number W of microstates (complexions) possible for the given macrostate. Obviously, summing over all the distributions possible for a macrostate, we get Σ_d W_d = W. (Thermodynamic probability is an extremely large natural number, whereas the actual probability p of any event is a number just within 0 and 1.)

1.3.6 Fundamental Postulate of ST (postulate of equal a priori probabilities): It states that all possible microstates of a system, corresponding to the same value of system energy, are equally probable. For practical purpose, it means that all possible microstates of a system existing in a given macrostate have practically the equal probability of being observed.

1.3.7 Some Corollaries of the Fundamental Postulate: (i) As there are W number of microstates of equal probability for a system in a given macrostate, so the probability of observing any particular microstate among them is p_mic = 1/W . (ii) As a distribution (within a given macrostate) is observed when any of its W_d number of constituent microstates are observed, so the probability of observing the distribution is p_d = W_d /W . Summing over all the possible distributions, the sum of p_d obviously equals unity: Σ_d p_d = 1. (iii) For the m. p. distribution, p_d is highest or maximum (compared to other distributions), so W_d  is a maximum (W being a constant for the given macrostate).

1.3.8 Boltzmann Equation (present version, of course): The entropy S of a macroscopic system at a particular macrostate (thermodynamic state) is given by the relation S = k ln W, where W is the thermodynamic probability of the system while existing in that macrostate.
[For macroscopic systems made of independent and identical particles, it has an additional, approximate form S = k ln W*, which of course gives practically the same value for S, even though W* < W (here W* means the thermodynamic probability W_d of the m. p. distribution).]
Note: W is a very large number. For a mole of He gas at 298 K & a low pressure, we have, S = 126.03 J K^–1.
So,   ln W = (126.03 J K^–1 / 1.381 x 10^–23 J K^–1) = 9.126 x 10²⁴.
This gives W = 3.266 x 10^{3963371441849076179144202}.    (What? After the 1, how many zeros are there!)

1.3.9 Ensemble and the Ergodic Hypothesis: An ensemble is a hypothetical collection of a very large number of replicas of the given macroscopic system in the given macrostate. (This is similar to a pure gas assembly i.e., system being a real collection of a very large number of identical constituent molecules - but this comparison should stop here!) The ergodic hypothesis states that for a macroscopic system, the time average (made over a sufficiently long time duration) of a thermodynamic property equals the ensemble average (made over the  very large number of hypothetical replicas of the system). This hypothesis helps, because any thermodynamic property is observed (by human science) for rather very long (!) durations (yes, thinking in molecular terms) of the order of, say, at least several hundred milliseconds; and so our observed thermodynamic property can be justifiably identified with the concept of time average. The ensemble average, then, is theoretically calculated via methods of ST.
 

1.4 A Simple-Minded Derivation of the Boltzmann Equation S = k.lnW:

Let us now present a simple derivation of the Boltzmann equation, based on the assumption that entropy S of any macroscopic system, being a universal measure of randomness of the system, should be a function of only the thermodynamic probability W of the system. [As W is the number of microscopic ways (i.e., microstates) in which the system may exist, W is clearly another universal measure of randomness]. The derivation also utilizes the fact of thermodynamics that entropy is an extensive thermodynamic property.

Let us consider two isolated systems A & B, placed side by side so that we can visualize a combined system AB, but existing yet without any interactions between these systems (A & B). Entropy S being an extensive thermodynamic property, entropy of the combined system is a sum of that of the systems A & B, i.e., S_AB = S_A + S_B. As we assume S to be a function of only W, we may write S = f (W). This means S_A = f (W_A), S_B = f (W_B) and
S_AB = f (W_AB), and so we have: f (W_AB) = f (W_A) + f (W_B). As the systems A & B do not interact, so any of the W_A microstates in system A may coexist with any of the W_B microstates in system B. So the number of microstates W_AB available for the combined system AB is W_AW_B, and so we have:  f (W_AW_B) = f (W_A) + f (W_B).
Differentiating this w.r.t. W_A keeping W_B constant, we get,
f ΄(W_AW_B).W_B = f ΄(W_A) + 0  [where f ΄(W) = (df /dW)].
Similarly, differentiating w.r.t. W_B keeping W_A constant, we get,
f ΄(W_AW_B).W_A = 0 + f ΄(W_B). So we have,
f ΄(W_AW_B) = f ΄(W_A) / W_B = f ΄(W_B) / W_A. This gives W_A f ΄(W_A) = W_B f ΄(W_B).
As A & B are arbitrarily considered systems, this clearly W f ΄(W) is a universal constant for any system, i.e.,
W f ΄(W) = k΄   [where k΄ is that universal constant]
Or,     W (df /dW) = k΄   or,    W (dS /dW) = k΄     [as f (W) = S] .
Or,  dS = k΄dW /W  or,  dS = k΄d lnW
Integrating we get,  S = k΄ lnW + C

It can be easily shown that this constant of integration C is zero. To do so, we note that the function S = f (W) (found now to be k΄ lnW + C) was considered to be a function of only W. This means that the constants k΄ & C in that function are universal constants for any systems. So for the two systems A & B and for their combination-system AB we get 
S_A = k΄ lnW_A + C, S_B = k΄ lnW_B + C  and  S_AB = k΄ lnW_AB + C.
As W_AB = W_A W_B, we have:
S_AB = k΄ lnW_AB + C = k΄ lnW_A + k΄ lnW_B + C.
However,  S_AB = S_A + S_B = (k΄ lnW_A + C) + (k΄ lnW_B + C) = k΄ lnW_A + k΄ lnW_B + 2C
Comparing these two values of S_AB , we immediately get C = 2C or, C = 0.
This means S = k΄ lnW.

It can be shown (Section 3.9) that the above-mentioned universal constant k΄ is nothing other than the Boltzmann constant k, and so we have: S = k lnW. This is the Boltzmann equation.
 

1.5 The Three Types of Non-Interacting Identical-Particle Assemblies:

There are only three basic possible types of assemblies of identical particles. If the particles are distinguishable (remember that two identical gas molecules are indistinguishable, just as two electrons in an atom are also so), they are called Boltzons or Maxwell-Boltzmann particles; the assembly of non-interacting identical Boltzons (e.g., the normal modes of vibration in an elemental solid as per Einstein theory of solids) is governed by the Maxwell-Boltzmann (MB) Statistics characterized by the W_d–relation (two such relations are displayed in picture format below) W_d = N! P_l / N_l!. On the other hand, if the particles are indistinguishable (e.g., the identical molecules within a ideal pure gas), they may be either Bosons (Bose-Einstein particles) with system-wavefunction symmetric w.r.t. particle exchange, or Fermions (Fermi-Dirac particles) with system-wavefunction anti-symmetric w.r.t. particle exchange. An assembly of non-interacting identical Bosons obey the Bose-Einstein (BE) Statistics characterized by the relation W_d = P_l [(g_l + N_l – 1)! / {( g_l – 1)! . N_l!}]. Similarly, an assembly of non-interacting identical Fermions obey the Fermi-Dirac (FD) Statistics characterized by the relation W_d = P_l [g_l ! / {( g_l – N_l )! . N_l!}]. However, under the simplifying condition g_l >> N_l (this condition is, for example, valid for any nearly-ideal pure gases) both the above-mentioned BE and FD assemblies approximately obey (how? see next chapter) rather a Corrected Boltzmann Statistics (with their particles called corrected Boltzons) characterized by the common relation W_d = P_l / N_l!. Should we call it a 4th type?

Notes: (i) P means a product just as S means a sum; so here P_l means a product over the levels l (ii) w.r.t. is an abbreviation for with respect to. (iii) If the particles are indistinguishable, it means that the system wavefunction either remains same (Bosons, e.g., photons) or just gets its sign changed (Fermions, e.g., electrons) for interchange of a (any arbitrary) pair of particles, as you have probably learnt during discussion of spin-included electronic wavefunction of He-atom. (v) Derivations of the three fundamental W_d-relations are found in most textbooks dealing with ST. Let's derive only the FD-statistics relation here: For FD systems, a molecular state could be occupied by at most one particle (in the two other system-types, there exists no such restrictions), so, in the l-th molecular level, there are N_l occupied states and (g_l – N_l) unoccupied states among the g_l states (obviously N_l ≤ g_l). As the states are distinguishable (the molecules, however, aren't), the number of ways in which the l-th level may be occupied is same as the number of ways in which N_l chairs may be selected out of g_l chairs, which is [g_l ! / {( g_l – N_l )! . N_l!}]. So, all the levels could be occupied by the molecules in P_l [g_l ! / {( g_l – N_l )! . N_l!}] ways.
(iv) For a distribution in a Maxwell-Boltzmann (non-Macroscopic, distinguishable-particles) system with N = 10, let g₀ = 6, g₁ = 12, g₂ = 40 and N₀ = 1, N₁ = 3, N₂ = 6, N₃ = N₄ = N₅ = ....... = 0. For this distribution, W_d = 10! [(6¹/1!).(12³/3!).(40⁶/6!).(g₃⁰/0!).......] ≈ (10  x 9 x 8 x 7 x 6!) x 6 x 288 x (4096000000/6!) x 1 x 1......
(= 10 x 9 x 8 x 7 x 6 x 288 x 4096000000 = 35672555520000000).
 

1.6 Towards an ST-Understanding of the Mystifying 2nd Law of Thermodynamics:

This mystifying law that mysteriously states 'a process spontaneously occurs in the direction in which the total entropy of system+surroundings increase' can be immediately understood even within the above limited knowledge of ST! When a system transforms from a state of lesser total-entropy to another state of more total-entropy, we can, in general, clearly visualize a distinctly great increase in the number (W) of possible microstates available. For example, when two gases such as argon and nitrogen are placed in two containers separated by a wall, and then a closed-hole is opened between the walls, the number of available microstates obviously greatly increase after the hole is opened. (With increase of lnW, S has increased as per S = k.lnW.) Out of these new total number of microstates, only a very miniscule fraction (say, 10^–80 or 10^–90 fraction) corresponds to the old, pure-compartments type microstates. (Had the gas molecule numbers been 4 or 5 instead of 10²⁰ or 10²¹, these pure-gases type microstates wouldn't have been such an absurd minority. Thus, 2nd law is a statistical magic played by the very large numbers of molecules!) As all the microstates available for the transformed system are equally probable, so, after the mixing occurred, only the vast majority of properly mixed-compartments type microstates will be observed by any observer, while the miniscule minority of pure-gases type microstates will not be at all observed, just because they now have very very small probability of being observed. So the mixing process does not get spontaneously reversed now (i.e., preferred direction is not violated)! Thus, ST converts the mystifying 2nd-law issue of preferred direction of thermodynamic processes to a simple probabilistic question, similar to a question of 'why don't I see all the dice with just a particular face up, when I throw ten thousand dice simultaneously on the ground?' Similarly, the upper part of your cup of tea doesn't spontaneously become hot at the expense of the lower part spontaneously getting cold (obeying 1st law of thermodynamics, nevertheless) simply because such an occurrence would have a very very low probability of being observed!