Alternative Derivations of the
Reino W. Hakala Howard University
Washington, D. C.
Boltzmann Distribution Law
In a previous article,' a derivation was given of the Boltzmann distribution law which had the main virtue that the approximation used (in place of the usual Stirling approximation) was introduced a t the very end of the derivation instead of a t the beginning as is customary. This derivation involved the proof of the relationship
made where 6 In N 1 !first makes its appearance, an approximation is thereby introduced a t an earlier stage of the derivation-before the use of Lagrange's method of undetermined multipliers instead of several steps after. Nevertheless, Wyatt's modification, involving an elegant proof of the above approximation, is quite beautiful. It seems worth while in this connection to show that Stirling's approximation can also be used at this particular stage of the derivation. Following the earlier derivation, it is first proved that
which holds for very large values of Nr. Another way to find this relationship is as follows: This relationship was not explicitly stated in the previous article although use was directly made of it. Then Stirling's approximation is employed to derive the above approximation for 6 in Nt!:
Since
8
and as
in Ni!
8(Ni In N; - Ni)
; .
&?Ii In Ni
; .
- 6Ni
+
= Ni6 In Ni In Ni8Ni - 8Ni = bN, In Ni8Ni - 6Ni = In Ni6Ni
+
it follows that
Substitution of this result into the previous equation then gives the approximate relationship x = 1
Keeping Ax
=
A
(L) Ni
Hence N
Ni
lim
C =
which, assuming independent, non-zero variations of the occupancies 6N, (i = 0, 1,2, 3, . . .) leads to
1,
9 -oasNi-t Ni
=
-
In Ni
m
A ( r / N i l lim0
=
JIN'(%)d($)
=
In Nt ( =
m
c (+) (k) A
r = 1
= JlnNi
d In
(e)
at the limit)
1:or very large values of N i , therefore,
which is one form of the Boltzmann distribution lam. If this approach utilizing Stirling's approximation is preferred, use can be made of Wall's simple geometric proofa of Stirling's approximation which, in algebraic form, is essentially as follows: InN! = l n ( 1 . 2 . 3 . . . N ) =
The earlier approach offers some practice with series; the present approach illustrates a proper technique for the approximation of series by integrals. Wyattz has pointed out that the essence of the author's previous article was the demonstration that 8 In Ni I
In NibNi
/
Inl+InZ+
. . . +I n N
Insdz=NlnN-N+1 =NInN-N;
N>>1
To prove that
; .
when Nt is large. If, however, this substitution is ' HAKALA, R. W., J. CHEM.EDUC., 38.33 (1961). 39, 27 (1962). 'WYATT, P. A. H., J. CHEM.EDUC., 526
0
; .
Ni = exp ( - a - bet)
Ni
z =
+ a + bei
Journal o f Chemical Educofion
-
TALL, F. T.,"Chemical Thermodynamics," W. H. Freeman and Co., San Francisco, 1958, p. 233.
which is involved here, it is only necessary to show that
distribution law, it may be derived as follows. From
which is quite straightforward. If, instead, one of the author's other two approaches, involving the proof of
we obtain
1
+
'/2+1/5+
... + 1/Ni
=lnNi
is followed, then, when Stirling's approximation is needed later on in various applications of the Boltzmann
S d l n N ! =.flnNdN
whence
InN!=NInN-N+c = A' In N
- A':
N
>> e
Volume 39, Number 10, October 1962
/
527
