Question

We have claimed that for two subsystems A and B, that the density of states of the combined system AB is equal to the product of the individual densities of states: ρ_AB = ρ_Aρ_B in order to make it the case that S_AB =S_A+S_B.Let’s test this in the following model. We have two boxes A and B of equal size. We will use our primitive “left-right” model for counting microstates. Let there be n particles distributed in box A and m particles distributed in box B.How many microstates W_A for system A with n particles? How many microstatesW_B for system B with m particles? How many microstatesW_AB for system AB with n+m particles?

Answer 1

For two subsystems A and B, the density of states of the combined system AB is equal to the product of an individual densities of states which are given as :

_AB = _A . _B

In order to make it the case that,   S_AB =S_A + S_B

In the following model, we have two boxes A and B of equal size.

Let there be n particles distributed in box A and m particles distributed in box B.

(a) How many microstates W_A for system A with n particles?

W_A = 2ⁿ

(b) How many microstates W_B for system B with m particles?

W_B = 2^m

(c) How many microstates W_AB for system AB with n+m particles?

W_AB = W_A . W_B

W_AB = (2ⁿ) . (2^m)

W_AB = 2^n+m