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On page 53, example 3.9, we consider why energy exchanges between two systems from the point of view of the 2nd law.

We consider two separate systems. Each has ten particles, and each particle has two possible energy states. System A has total energy [itex]U{a}[/itex] = 2, and [itex]U{b}[/itex] = 4. Thus binomial statistics predicts the multiplicities of these systems:

W([itex]U{a}[/itex]) = [itex]\frac{10!}{8!2!}[/itex] = 45

W([itex]U{b}[/itex]) = [itex]\frac{10!}{6!4!}[/itex] = 210

Now the confusing part, to me, is the math when these two systems come into thermal contact. Then the author asserts that the initial multiplicity is

W([itex]U{total}[/itex]) = [itex]\frac{10!}{8!2!}[/itex][itex]\frac{10!}{6!4!}[/itex]

And that maximum multiplicity is found at

W([itex]U{total}[/itex]) = [itex]\frac{10!}{7!3!}[/itex][itex]\frac{10!}{7!3!}[/itex] = 14,400

But why consider the systems in this way, as opposed to thinking of a new system, with 20 particles, having [itex]U{a+b}[/itex] = 6?

Then we get

W([itex]U{a+b}[/itex]) = [itex]\frac{20!}{14!6!}[/itex] = 38760 ≠ W([itex]U{total}[/itex])

I'm trying to develop a sense of the difference, I suppose, between two systems in thermal contact and one system. After they've equilibrated, how are they not treatable as one system? They clearly aren't, because if they were, then the total multiplicity of that one system must = the total multiplicity of the two systems A and B.