Reino W. Hakala
Howard University Washington 1, D. C.
I
A New Derivation of the Boltzman Distribution Law
T h e derivation to be presented is identical in most respects with the usual one. It differs from the latter mainly in that the Stirling approximation is not employed; an approximation is necessary, but the particular approximation which is used here has the merit that it can be introduced a t the very end of the derivation. The discussion of "thermodynamic" probahility is also somewhat different; it is based on "ordinary" concepts of probability, which are a t least somewhat familiar to students; and further clarity is achieved, it is felt, by bringing in Heisenberg's uncertainty principle, by which is made apparent the need for the concept of "phase space," in terms of which the thermodynamic probability can be rigorously defined. Let us consider a macrosystem which consists of numerous particles such as molecules. These particles need not be independent of one another; that is, they may have mutual potential energy. Since the system is made up of a large number of parts, there are a great many different ways of arranging its parts to give the same thermodynamic state, although a t any one instant the system has just one of these arrangements. The probability that the system exists in a given thermodynamic state is then equal to the probability that it exists in any one of the many possible arrangements that correspond to this particular thermodynamic state. Because only one arrangement can occur a t a time, the probability that the system exists in any of the arrangements equals the sum of the probabilities of each separate arrangement. By considerations of symmetry, each separate arrangement is equally probable. Thus the sum of the probabilities of each sepa. rate arrangement is equal to the total number of separate arrangements, which is called the "thermodynamic probability" of the system, times the probahility of each separate arrangement. That is,
t,ional coordinates corresponding t o the components of momentum, (2) divide this space into "cells" of equal volume, and (3) note which cell a given particle is in. This "observation" specifies the position and the momentum of the particle within the limits of the location and dimensions of its cell in phase space, in complet,e harmony with Heisenberg's uncertainty principle. On this basis, we can define the total possible number of separate ways, W N ,in which the system of N particles can be arranged, to give the same thermodynamic state, as the total number of distinguishable permutar t,ions of the N particles among the cells of phase space. (This is a reasonable definition because a permutation must be distinguishable to be observable.) (Alternatively, we may consider the phase space of the macrosystem. This is a 6N-dimensional abstract space, the coordinates of which are the position and momentum coordinates of each of the N particles of the macrosystem. Then W , is equal t o the number of cells occupied in the phase space of the macrosystem. In this 6N-dimensional phase space, each point is called a "representative point" because it represents the entire macrosystem.) If each particle of the macrosystem is different, as by possessing a different amount of energy, then each permutation of the particles is distinguishable; for a system of N particles, the total number of distinguishable permutations would then be N! If the particles differ only in having different amounts of energy, but N, particles have energy el, Nz particles have energy en, etc., then the total number of distinguishable permutatlionswill be less than N! because permutations of particles having identical energy are not distinguishable; in this case, the total number of distinguishable permutations is given by N!/Nl! X N2! X . . . . Therefore the pr~bahilit~y of the system is
P N = WNP!
where pN = the probahility of the system, WN = the total number of different possible arrangements of the N particles which comprise the system, and p, = the probability of each separate arrangement. We are now faced with the question as to what is actually meant by the "separate arrangements" of the system. To avoid any ambiguity, we should rely on definitions which are related t o what we are able, a t least in principle, to observe. Heisenberg showed that it is not possible to observe simultaneously both the position and the momentum of an object with complete precision. Accordingly, it is customary to (1) consider an abstract six:dimensional space, called the "phase space" of the particles, of which the coordinates are the three ordinary position coordinates and three addi-
It is necessary to distinguish between the thermodynamic state and the "statistical state" of a system. In classical thermodynamics, the state of a system is defined only a t equilibrium. If the system is not a t equilibrium, its state can be defined statistically by enumerating thc particles having various energies. (The number of particles N r having the same energy el is called the "population," "occupation number," or "occupancy" of energy level er.) According to the second law of thermodynamics, a system which is not at equilibrium will tend toward its equilibrium state, that is, its thermodynamic state. Thus the thermodynamic state of a system is the same as its most probable statistical state. Volume 38, Number 1, January 1961
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Another way of phrasing this conclusiou is that the probability of the thermodynamic state is equal to the maximum probability of the statistical state, meaning p , is a t a maximum for the thermodynamic state. Equivalently, since p, is a constant, the number of distinguishable permutations, W,, is at a maximum. The condition for the thermodynamic s t a t e t h a t is, for equilibrium--is then given by either
whence
Hence
for equilibrium. For mathematical convenience, we shall use the logarithm inst,ead. We then have the equilibrium condition Since N (the total number of particles in the system) is a constant, 6 In N ! = 0, whence the condition for equilibrium becomes simply The fact that N further condition
=
ZNi is a constant int,roduces the
Since the statistical state of a system is defined in terms of the number of particles N , having a given energy e f , enumerated over the various values of the energy possessed by the particles of the system, the energy is also a variable that we should investigate. The energy of the system, being the sum of the energies of the particles which comprise the system, depends on the occupation numbers N iof the energy levels e , as follows:
The thermodynamic properties of a system, being statistical averages, are unaffected by the occurrence of any permutations of the particles of the system, as long as all of these permutations correspond to the same thermodynamic state. Hence the energy of the system, which is a thermodynamic property of the system, is a constant mheu the system is a t equilibrium. This int,roduces a third condition for equilibrium:
(In words, the energy of a system a t equilibrium is not affected by fluctuations in the occupation numbers of the energy levels of the system. This is merely another way of saying that the permutations of the particles do not affectthe energy of the system, which is a statistical average.) We have found three conditions for equilibrium. What single condition for equilibrium do they represent? To find this single condition, it is clearly necessary to combine the three separate conditions. Two of them involve the fluctuation 6N, while the remaining one involves 6 In N,! instead, whence the three conditions do not all have a common basis. To combine all three, it is necessary t o express 2 6 in N,! in terms of 6Ni. 34
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By calculus we have'
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Journal o f Chemical Education
We can combine the three conditions I: 6N, = 0 , I: e, 6N, = 0,and I : @ l / N , ) 6Ni = 0 by Lagrange's method of undetermined multipliers. I n this method, all but one of the conditions are multiplied by unknown parameters, a, b, . . . , and t,hen the various conditions are added t,ogether. Thus we have
This is the single condition for equilibrium. Since the fluctuations in t,he various occupancies, 6 N f , i = 0, 1, 2, . . . , are presumably independent, it follows that (a
+ be* + Z l/Ni) 6Ni = 0
for each i, that is, for each separate energy level. Granting that fluctuations do occur, 6 N i f. 0, we then lind that
This i s an exact result, provided that the assumption of the independence of the fluctuations is correct: the form of this result also depends upon our definition of the thermodynamic probability, W N . We shall now use this result t o find N i explicit,ly in terms of ei: Unfortunately, as there is no known simple expression for the sum of a harmonic series, it. is not possible to solve this equation explicitly for N , . It is therefore necessary to find an approximate sum for 2 1 / N , in terms of N , which can be solved explicitly for N*. The terms in the infinite seriesz
+ + +
+
I/& . . . l/z) dz is not The formula d in z! = (1 listed in anv hooks that the author has seen even though - it is a very simple and aesthetically appealing result. This ~erienexpansion of In N, can he derived in s, simple manner. By ordinary long division of ( 1 2) into 1, we find that
-
1/(1-2)=1+z+z2+za+
...
where the series is infinite. The same result is obtained by Newton's binomial expansion of ( 1 - z)F1. Then
Integration of the lefthand side, term by term, gives the familiar expansion -In ( I - v) =
16
+ ut/2 +
+ ...
1 ~ ~ / 3
Letting u = (Nr - l)/Nr in this expansion then gives the de sired result.
approach the corresponding terms in the harmonic series
as (N, - 1)/N, approaches unit,y, that is, as N, approaches infinity. The number of terms in the harmonic series approaches infinity as Ni approaches infinity. Thus the two series approach one another as N i approaches infinity. For very large values of N,, therefore, we have the convenient approximation, which does not appear t,o have been given previously, 1
+ '/, + '/,+ .. . + 1 / N i = In Ni
where Q is called the "partition function" of the part,icles. (We see that the absolute activity and the partition function are related hy A = N/Q.) If the same value of the energy, e,, can be achieved in g, different ways, then t,he energy level ei is called "degenerate." (A more civilized and more descriptive name would he "multiple.") And g, is called the '(degeneracy," "multiplicity," or "statistical weight?' of energy level ei. When this is the case, the Boltzmann distribution law must be modified. The occupancy of energy level et corresponding to each one of the various ways of obtaining the same value of the energy e, is then Nilg,
A exp ( -be;)
; .
hence the two forms of the Boltzmann distribution law are
from which a
+ bei + In N i
=0
and N d N = gi exp ( - b e ; ) / &
This approximate result is one form of the Boltzmann distribution law. It gives the number of particles, Ni,having the energy e,. The coefficient A = exp (-a) is called the "ahsolute activity" of the macrosystem. The other form of t,he Boltzmann distribution law gives the fraction of particles, NI/N, having the same energy e,: N , / N = A exp ( - b e i ) / N = A exp ( - b e i ) / Z N i i3 Q e x p ( - b e i ) / Z A exp ( - h e ; ) e e e p ( -bei)/Z exp ( - bed
or
N i / N = esp ( -be.)/Q
where Q = 2 g, exp (-be,). These are the most general ways of writing the Bolt,zmanndistribution law. (We still have A = N/Q.) I t remains only to show that b = l/kT, where k is Boltzmann's constant (the ideal gas constant R divided by Avogadro's number) and T is the absolute temperature. It seems that the only way to do this a t the present time is by the usual method: derive equations with the aid of the distribution law and compare the results with those of classical thermodynamics. Some authors prefer to define the absolute temperature by T = llkb. The present author would rather rely on classical thermodynamics hecause the Boltzmann distribution law is only an approximation, albeit a mathemat.ically convenient one. (In particular, it holds poorly a t low temperatures.)
Volume 38, Number
I , January 1961
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