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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. 2: Distributions, Boltzmann Distribution Law, Molecular Partition Function and Expressions for the Thermodynamic Functions
 

2.1 Distributions, Most Probable & Average Distributions, and Distribution Laws:

The ideal (or practically ideal) pure gas system we are mostly interested in (in this book) consists of a set of indistinguishable molecules, and so must be either Bosons obeying B-E statistics or Fermions obeying F-D statistics. However ideal or nearly-ideal gases always obey the restriction g_l >> N_l, for which (as under that condition g_l + N_l ≈ g_l – N_l ≈ g_l)
either [(g_l + N_l – 1)!/{( g_l – 1)!.N_l!}]  = [{(g_l + N_l – 1).(g_l + N_l – 2)......(g_l + 1).g_l}/N_l!]
or [g_l ! / {( g_l – N_l )! . N_l!}] = [{g_{l .}(g_l – 1).(g_l – 2)........(g_l – N_l +1)}/N_l!]
simply equals / N_l!
So, either P_l [(g_l+N_l–1)!/{(g_l–1)!.N_l!}] or P_l [g_l!/{(g_l–N_l)!.N_l!}] equals P_l /N_l! giving W_d = P_l /N_l! which is the characteristic relation corrected-Boltzmann statistics. This means that ideal or nearly-ideal pure gas systems always obey this statistics.

For any macroscopic system in a given macrostate, there are many possible distributions consistent with the constancy requirements characteristic of a given macrostate, e.g., the constancy requirement S_l N_l = N for the total number N and the requirement S_l N_l e_l = U for the total molecular energy U (equal to the system internal energy). However these possible distributions have differing probabilities of being observed (depending, obviously, on their p_d = W_d/W values). The m. p. distribution has the highest individual probability of being observed. Next to this, (remember we're talking about macroscopic systems) it is only the distributions very similar to the m. p. distribution that have appreciable probabilities of being observed; the distributions significantly differing from the m. p. distribution have so miniscule probabilities of being observed that they are never observed in any observation. [people state this as: the W_d and the p_d functions are sharply peaked around the m.p. distribution] So, for macroscopic systems, we say that it is practically the m. p. distribution that is always observed! Averaging over all possible distributions (obviously weighing as per their respective probabilities) and noting that the odd type of distributions with negligible probabilities hardly contribute to this average, we immediately recognize that the average distribution is also practically the m. p. distribution!

A law that gives (for a macroscopic system) the m. p. distribution (and so also gives both the usually observed distribution-set and the average distribution, these three distributions being practically the same – as discussed above), is called a distribution law. As a distribution is characterized simply by the populations of all the molecular levels, so a distribution law is naturally an expression for the population N_l of the (arbitrary) molecular level l. An example is the well known Boltzmann distribution law, which will be discussed in detail in the next section.
 

2.2 The Boltzmann Distribution Law and One of Its Derivations:

The Boltzmann distribution law is the most popular distribution law, valid for both Maxwell-Boltzmann and corrected-Boltzmann systems, dictating populations in the m. p. (or in the average) distributions. According to this law, the population N_l of the l-th level (while in the m. p. distribution, which is denoted by *) is given as: N_l* = (N/q) g_l exp{(-e_l / (kT)} [where q is the sum S_l g_l exp {-e_l /(kT)}, referred to as the molecular partition function]. As for a macroscopic system, it is practically the m. p. distribution which is always observed,  the * sign may as well be withdrawn (it is OK if N_l* >> 1) giving N_l = (N/q) g_l exp{(-e_l / (kT)}

An obvious corollary of this law is about the ratio of populations in two levels l΄ & l as:
N_l΄ / N_l  =  (g_l΄ / g_l) exp{-(e_l΄ - e_l)/(kT )}     [N & q being same for any of the levels, cancels off].
Though derived (in this chapter) for complete molecular levels, both the law and its corollary can be successfully applied for specific-mode energies (e.g., rotational, vibrational etc.) of the molecules, to give populations or population-ratios in specific-mode energy levels (irrespective of other specific-mode energy levels occupied) [see section 3.6].

The thermodynamic probability W_d of the distribution d of molecules in an ideal gas is given  (as per the Corrected Boltzmann Statistics relation) as: W_d = P_l / N_l!
where N_l is the occupation number (population) in the l-th level, g_l is its degeneracy (statistical weight), and the product extends over all molecular energy-levels.

The Boltzmann distribution is the m. p. (most probable) distribution, and so for it the probability and the thermodynamic probability are the maximum. However, the gaseous isolated (or just in a definite macrostate) system considered must obey its two obvious restrictions: S_l N_l = N, a constant, and S_l e_l N_l = U, another constant. So, Lagrange method of undetermined multipliers (for maximization of a function under one or more constraints) must be used during maximization. Now, the two restrictions in the differential form are:
S_l dN_l = 0 and S_l e_l dN_l = 0.  For mathematical simplicity, lnW_d instead of W_d is maximized (when W_d is maximum, lnW_d is also maximum, so there is no problem about that).
So, for the m.p. distribution, Lagrange method¹ gives: 
d ln W* + a S_l dN_l* – b S_l e_l dN_l* = 0  for any set of values of dN_l*
(where a and –b are the Lagrange undetermined multipliers)
As ln W* = S_l (N_l* ln g_l - ln N_l!* ) = S_l (N_l* ln g_l - N_l* ln N_l* + N_l*) using the Stirling² approximation (according to which, for a large natural number y, ln y! = y ln y – y)
we get,  d ln W* = S_l (ln g_l dN_l* - N_l* (1/ N_l*) dN_l*  - ln N_l* dN_l* + dN_l*)
= S_l ln (g_l / N_l*) dN_l*
So Lagrange method gives:
S_l ln (g_l / N_l*) dN_l + a S_l dN_l* - b S_l e_l dN_l* = 0   (for any set of dN_l*)
or, S_l [ ln (g_l / N_l*) + a - b e_l ] dN_l* = 0    (for any set of dN_l*)

As this relation is true for any set of dN_l*, so every coefficient of dN_l* is zero³. This means,
ln (g_l / N_l*) + a - b e_l  = 0. This gives ln (g_l / N_l*) = -a + b e_l 
or_,  g_l / N_l* = exp (-a + b e_l)
or,  N_l* = g_l exp (a) exp (-b e_l)
Summing over both sides for all energy levels and noting that S_l N_l* = N, the total number of molecules, we get
exp (a) = N / S_l [ g_l exp (-b e_l)]. Noting that b could be found to equal 1/(kT), [here k is the Boltzmann constant, and T is the thermodynamic, i.e., absolute-scale temperature; see section 3.9 for a verification of this relation], the Boltzmann distribution law is thus⁴: 
        N_l* = (N/q) g_l exp {-e_l /(kT)}  
where the molecular partition function q is defined by q = S_l g_l exp {-e_l /(kT)}
(The summation here is over all the molecular levels l).

However, for greatly occupied levels in any macroscopic system, any observed occupation number N_l is very close to the occupation number N_l* in the m. p. distribution. So, the Boltzmann distribution law may be written in a simplified way as: N_l = (N/q) g_l exp (-e_l/ kT)

Notes: (1) The Lagrange method of undetermined multipliers, useful for maximization of a multi-argument function under inter-argument constraint(s), may be applied for maximization of the function f (y,z) = yz under the constraint y + z = 2 in the following way: Constraint is (dy + dz) = 0. Differential of function is df = y dz + z dy. So, df + l(dy+dz) = 0 => (y+l)dz + (z+l)dy = 0 (for any set of values of dy & dz). So y+l = z+l = 0 => y = z = –l => –2l = 2 => l = –1  and so y = z = 1, and maximum value of f = yz is 1.1 = 1. (Here l is a Lagrange multiplier) Cross-check it by maximizing f = y(2–y) in the straight way of making df/dy = 0. (2) Using a scientific calculator, check the validity of the Stirling approximation by finding, say, 63!, then finding ln63!, and then comparing this result with 63.ln63 – 63 (3) If ax+by+cz = 0 is true for any value of the variables x, y, z; then the coefficients a, b and c must be all zero. Isn't it so? Just think about it a little!
(4) This statement of the distribution law is in terms of the m. p. distribution. Another version exists that talks about the average occupation number of a level <N_l>, averaged over all possible distributions for the system (averaged, taking care of the differing probabilities p_d of the possible distributions), and finally states: <N_l> = (N/q) g_l exp (-e_l / kT).    (i.e., the average population <N_l> is just the same as the m. p. population N_l*). This version may be stated also in terms of average occupation number <N_s> of a molecular state s as: <N_s> = (N/q) exp(-e_s/kT). [The per-level form is obtainable just as a corollary of this per-state form (how?).]
(5) Calculated as (N/q) g_l exp (-e_l / kT), this N_l* is obviously, in general, a fraction! So, the m. p. distribution we're calculating is, exactly speaking, a hypothetical m. p. distribution (with fractional populations - an obvious absurdity) closest to the actual m. p. one. (When N_l* is a large number, this error is negligible; we'll always overlook this!)
 

2.3 Functional Form and Significance of the Molecular Partition Function:

The molecular partition function q expressed as a sum of terms over levels or over states (also denoted by the letter z, from the original German name Zustandsumme meaning 'sum over states') is a dimensionless quantity (pure number) of a central importance in corrected-Boltzmann and Maxwell-Boltzmann systems, as most of the thermodynamic functions (e.g., U, P, S) of such systems can be expressed in terms of q (and their value obtained from the knowledge of q). In addition to the sum over levels form
q = S_l g_l exp {-e_l /(kT)} (as previously mentioned), q may also be expressed
in a sum over states form q = S_s exp{-e_s /(kT)} (where the summation is over all the molecular states s, e_s being the energy of the molecular state s). The molecular p. f. is a function of only the system temperature T and the system volume V i.e., q = q(T,V) (it is not a function of the number N of constituent molecules). This function is of the simple form
q = V.f (T), where f (T) is a function of temperature T only.

The numerical value of q indicates the pattern of distribution of molecules among the different levels. Generally the energies of the levels are measured with the ground-level energy taken as zero. Under this popular convention, we find that at very low temperatures, q is small and the upper levels are hardly occupied. When the temperature is raised, q increases and the upper levels start getting occupied. When no level other than the ground one is occupied, q has the lowest possible value g₀, which is the degeneracy of the ground molecular level (this lowest value of q is exactly encountered at the absolute zero of temperature i.e., at T = 0, though this value may be practically encountered also at somewhat higher temperatures).
 

2.4 An ST-Form of 1st Law of Thermodynamics (for Independent-Particle Systems):

This 1st law tells us that dU = dq + dw, where U is the system (internal) energy, dq is the heat absorbed by the system and dw is the work done on the system. On the other hand, for a system of independent (non-interacting) particles, we have (from U = S_l N_le_l) that dU = S_l N_l de_l  + S_l e_l dN_l . Now, let us consider a situation in which no work is done (dw = 0) but some heat is absorbed (i.e., dU = dq + 0). As no work is done, the volume and/or surface area etc. haven't changed, and so the molecular energy levels remains constant (as known from quantum mechanical considerations). So the changes de_l in the energy levels e_l are zero, and so dU = S_l e_l dN_l + 0. This gives the value of dq as: dq = S_l e_l dN_l , and from which it is obvious that dw = dU – dq = S_l N_l de_l.

Thus we get two generalized relations dq = S_l e_ldN_l and dw = S_l N_lde_l with dU = dq+dw. These relations give the molecular interpretations of heat and work, according to which (for independent-molecule systems): Heat absorbed by the system gets expressed through changes only in the populations of the molecular energy levels; work done on the system gets expressed in changes only in the energies of the molecular energy levels.
 

2.5 Expressions for the Thermodynamic Functions in Terms of q (for Ideal Gases):

2.5.1 Expression for system's internal energy, U:
The defining relation of q is: q = S_l g_l exp {-e_l /(kT)}. Differentiating q w.r.t. T at constant volume V, and noting (from quantum-mechanics considerations) that at constant volume the degeneracies g_l and energies e_l are constants, we get
(∂q/∂T)_V = S_l g_l exp {-e_l /(kT)}.(-e_l /k).(-1/T² )
or, S_l g_l e_l exp {-e_l /(kT)} = k T² (∂q/∂T)_V
Now, as U =  S_l N_l e_l
Applying the Boltzmann distribution law N_l = (N/q) g_l exp {-e_l /(kT)}, we get
U =  S_l N_l e_l  = (N/q) S_l g_l e_l exp {-e_l / (kT)} = (N/q) k T² (∂q/∂T)_V = NkT² (∂ ln q/∂T)_V
So we have this expression for the internal energy U, U = NkT² (∂ ln q/∂T)_V

2.5.2 Expression for pressure, P:
Using the expression dw = S_l N_l de_l  for work done on the system, and noting that for a simple system with only P-V work allowed dw equals (–P dV), we get  –PdV = S_l N_l de_l
Applying the Boltzmann distribution law N_l = (N/q) g_l exp (-e_l / kT) for N_l, we get:
 –PdV = (N/q) S_l g_l exp (-e_l / kT) de_l
or, –P = (N/q) S_l g_l exp (-e_l / kT) (de_l/dV)       or,  S_l g_l exp (-e_l / kT) (de_l/dV) = –qP/N
Differentiating the expression for q w.r.t. V at constant T, and noting that (∂e_l/∂V)_T is same as (de_l/dV), as the molecular level energy-value e_l (it is dictated by quantum mechanics alone) is independent of temperature T, we get (∂q/∂V)_T = S_l g_l exp (-e_l / kT) (∂e_l/∂V)_T.{-1/( kT)}
or, S_l g_l exp (-e_l / kT) (de_l/dV) = -kT.(∂q/∂V)_T
This gives, –qP/N = -kT.(∂q/∂V)_T , or, P =  (NkT/q).(∂q/∂V)_T = NkT (∂ ln q/∂V)_T
So we have this expression for pressure, P = NkT (∂ ln q/∂V)_T 

2.5.3 Expression for entropy, S:
Using relations dS = dq_rev/T and dq = S_l e_l dN_l* _{(as the system practically exists in the m.p. distribution) ,} and noting that for a system of non-reacting pure substance, dq = dq_rev , we have:
dS = (1/T).S_l e_l dN_l* = k S_l (be_l).dN_l*              [as 1/T = kb]
For the corrected-Boltzmann system of indistinguishable particles, ln (g_l / N_l*) = -a + b e_l 
or, be_l = ln (g_l / N_l*) + a 
So, dS = k S_l {ln (g_l /N_l*) + a}.dN_l*
 = k S_l ln (g_l /N_l).dN_l* + k S_l a dN_l*
 = k S_l ln (g_l /N_l).dN_l* + ka S_l dN_l* = k S_l ln (g_l /N_l*).dN_l* + ka dN       [as S_l N_l* = N]
But S_l ln (g_l /N_l*) was nothing other than dW*, while for the system in a given macrostate (or for an isolated system) N is constant i.e., dN is zero. This means that dS = k (d lnW*)
giving S = k lnW*.
Note: Mathematics-experts among you may surely utter: where's the constant of integration C [while integrating from dS = k (d lnW*)]? Well, 'it can be shown' that here C has to be zero. [I like to talk about such situations as: whatever can't be shown (by the author, or right now) is stated as 'it can be shown'!]

This approximate form of Boltzmann equation S = k.lnW*, in contrast to the exact Boltzmann equation S = k ln W,  gives an mathematically easy way to calculate the entropy S as:
S = k lnW* = k S_l (N_l* ln g_l - ln N_l* ! )
= k S_l (N_l* ln g_l - N_l* ln N_l* + N_l*)  = k S_l N_l* ln (g_l /N_l*) + k S_j N_l* 
= k S_l N_l* ln (g_l /N_l*) + k N
As    N_l* = (N/q) g_l exp (-e_l / kT)      So, g_l /N_l* = (q/N).exp(e_l / kT)
Or,   ln (g_l /N_l*) = ln (q/N).+ e_l / kT
So,   S = k S_l N_l* {ln (q/N).+ e_l / kT} + k N
= k ln (q/N) S_l N_l* + S_l N_l*e_l / T + k N = Nk ln(q/N) + U/ T + Nk
So the expression for entropy (for corrected Boltzmann system of indistinguishable particles, e.g., gases) in terms of q, N, T and V is [using the above expression for U]:
S = Nk ln(q/N) + U/T + Nk  = Nk ln(q/N) + NkT (d lnq/dT)_V + Nk 
Note: The two expressions k lnW* and k lnW for S give practically the same value for S simply because both W* and W are so extremely large numbers that the difference between lnW* and lnW is negligible in comparison with either lnW* or lnW, even though W may be several times larger than W* [to get an idea of the situation, try comparing values of, say, ln 10²³ and ln (5x10²³). W  is unthinkably larger than 10²³.] This means we've verified the Bolzmann equation, starting from the thermodynamic definition of entropy!

For the proper M-B system of identical distinguishable particles (e.g., normal modes of vibration in solid as per Einstein theory), the relation S = k lnW* is similarly obtained. However, because of an additional N! factor in the expression of W*, the additional term (k.lnN! = Nk.lnN – Nk) appears in S, giving S = (Nk.lnN – Nk) + Nk ln(q/N) + U/ T + kN
So the expression for entropy for Maxwell-Boltzmann system of distinguishable particles is
S = Nk ln q + U/T = Nk ln q + NkT (d lnq/dT)_V
Note: Don't forget to use only the earlier expression (the blue one) for entropy of gases (not this one!).

2.5.4 Expression for the Helmholtz free energy, A:
Using above relations for U & S, an expression for the Helmholtz Free Energy A (where A = U – TS) for gas-systems may be immediately obtained as: A = U – TS =
U – T{Nk ln(q/N) + U/T + Nk}  = –NkT ln(q/N) – NkT   i.e., A =  –NkT ln(q/N) – NkT
Note: This expression for A is, obviously, not valid for a system of distinguishable particles.

2.5.5 Expression for the chemical potential per molecule, u:
In statistical thermodynamics we come across frequent references to the chemical potential per molecule in contrast to the chemical potential per mole (m) popular in classical thermodynamics. To distinguish this quantity from m, this author prefers using a different symbol u to indicate this molecular chemical potential. Just as m is defined mathematically by m = (∂A/∂n)_V,T = (∂G/∂n)_P,T , u is also defined as u = (∂A/∂N)_V,T = (∂G/∂N)_P,T [obviously, noting that N = n N_A, this means that m = u N_A (N_A is the Avogadro number)].

From the above expression for A we get:
       A = –NkT ln q + NkT ln N – NkT
So,  u =  (dA/dN)_V,T = –kT ln q + kT ln N + NkT (1/N) – kT
= –kT ln q + kT ln N + kT – kT = –kT ln (q/N). So, m = uN_A =  –RT ln(q/N)  [as kN_A = R]
So we have,  u = –kT ln (q/N) and m =  –RT ln(q/N)
Note: These expressions for u and m are, obviously, not valid for a system of distinguishable particles.