Reino W. Hakala Michigan Technological University Houghton, Michigan 49931
Simple Justification of the Form of Boltzmann's Distribution Law
Suppose that the various kinds of energy which a system possesses are independent of one another, hence additive: e h l = el er . . .
+ +
(where the subscripts 1, 2 . . . refer to d i e r e n t quantum levels, not to different energy levels). Since the probabilities of independent events are multiplicative, it then fo1lon.s that the probability of the system possessing the total energy e, e, . . . is the product of the probabilities that i t possesses the independent component energies el, ez, . . . P(h + el . . . ) = P(er) X P(e4 X . . . The exponential function has the same behavior as the probability, P: exp I-b(e, el . . .)I = exp (-bel) X erp (-bel) X . . .
+ +
+
We can measure directly the probability that a molecule has a certain energy, e,, by the fraction of molecules, N , / N , which possess that energy, whence we can change the last equation to NI = a exp (-be;) The constant a is readily evaluated by summing both sides over a11 of the quantum levels, ~(N~IN = )a C e x p (-be() i
i
Since the sum on the left-hand side equals unity, we find that
+ +
where
-b is a constant. Thus we can write
P(ei) = n exp ( - b e < ) where a is a proportionality constant. We have used a negative exponent because it is presumably more difEcult, hence less probable, for a system to reach a higher energy level than a lower one.
which is one form of Boltzmann's distribution law. It remains only t o evaluate b, which, by various alternative ways,' can be shown t o be equal to l / k T . 'See, for example, HAKALA, R. W . , J. CHEM.EDUC., 39, 525 (1962); ibid., 44,436 (1967).
Volume 44, Number 1 1 , November 1967
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