Page 103 - Modern physical chemistry

S.8 Effectively Contributing Microstates 93

Substituting (5.38) and (5.42) for the numerator and the denominator in the operand
in formula (5.37) leads to

[5.43]

For this equation to hold, we must have
S = kin W + const' [5.44]
where S is the entropy of the given state, W the statistical weight for the state, and k
Boltzmann's constant.
An isolated system generally moves from a state of lower to a state of higher proba-
bility. In such a change, weight w increases and entropy S increases. Thus, the second
law of thermodynamics is vindicated.

Example 5.4
In an isolated system, fluctuations from the equilibrium state occur. What entropy change
is associated with going to a state whose statistical weight is 0.001 times that of the equi-
librium state? It is given that
W2 =0.001.
WI
So formula (5.43) yields

23
23
1
M' = kin :: = (1.3807 x 10- J K-1)In 0.001 = -9.5 x 10- J K- .
This change is not significant except in a very small system (one of colloidal size).

5.8 Effectively Contributing Microstates

Fixing the thermodynamic variables of a given system does not determine what indi-
vidual molecules do. The number of variables is too small by many orders of magnitude.
As a consequence, numerous disjoint (completely distinct) molecular states contribute
to a given thermodynamic state. And the amount that a molecular state contributes may
vary between 0 and 1 over time.
In the absence of evidence to the contrary, we presume that the average contribution
of each of the disjoint molecular states is the same. In effect, each is equally probable.
If we let Wbe the effective number of such microstates that contribute to the macrostate,
the statistical weight w for the macrostate is proportional:

w=aW. [5.45]
This relationship transforms formula (5.44) to
S = klnaW +const'= kInW +const. [5.46]
Equation (5.46) has been established for ideal gases. Does it hold in general?
Consider a system C possessing entropy Se and state number We in contact with an
ideal-gas system D possessing entropy SD and state number W D . The entropy S of the
combined system equals the sum of the entropies of the parts:

[5.47]