Page 102 - Modern physical chemistry

92 Entropy and the Second Law

Since the mixture is ideal, gas A behaves as if gas B is not present, and vice versa. The
entropy change for gas A in expanding from pressure P to pressure P A is

P
llSA = nXARln- = -nXARlnXA, [5.35]
P A
according to equation (5.26). Similarly, the entropy change for gas B in expanding from
pressure P to pressure P B is
P
llSB =nXBRln-=-nXBRlnXB· [5.36]
P B
For the total entropy change on mixing, we obtain

llS=-nR(XAlnXA +XBlnXB)=- nR(lnXfXA +lnX:X B )=kln 1 . [5.37]
N XfXAX:X B
Here k is Boltzmann's constant, XA the mole fraction of A, XB the mole fraction of B, NXA
the molecules of A, and NXB the molecules of B in the mixture.
5.7 Probability Change on Mixing

Let us now go over the mixing process of figure 5.4, comparing a statistical weight
for state 1 with that for state 2. The result will be used to relate the entropy to an appro-
priate statistical weight for a system.
In the final state, each molecule is somewhere in volume V. So its contribution to the
statistical weight for the state can be taken as 1. But since statistical weights multiply
when combined, the net weight of state 2 on the chosen scale is
[5.38]

In the state immediately after the shutter is opened, molecules of A occupy only
volume VA of V and molecules of B occupy only volume VB of V. If each element of V is
exactly like any other element of the same size, the probability of finding a given mole-
cule in one is proportional to its volume. So the statistical weight for finding a molecule
of A in VA' rather than in V, is V JV. Since statistical weights multiply, the weight for NXA
molecules of A being in VA is
[5.39]

The second equality arises because in the ideal-gas solution

~= ~ ~ ~~
VA +VB V

Similarly, the statistical weight for finding NXB molecules of B in volume VB, rather
than in the total volume V, is
NXB
W - ( VB ) _XNXB . [5.41 ]
B
-
B -
V
Consequently, the combined weight for state 1 is
[5.42]