4.1 Canonical Ensemble: Closed Systems
183
From our statistical mechanical relation,
k B d ln = dS =
1
T
dU +
P
T
dV ,
(4.1.39)
we see that
1
k B T
=
∂ ln
∂U
V
,
(4.1.40a)
P = k B T
∂ ln
∂V
U
,
(4.1.40b)
as the relationships between the microcanonical partition function N, V ) and
β and the pressure P .
Example 4.3 Entropy of Mixing.
In Example 2.6, Chap. 2, we examined the entropy difference, ((S) i→f , between
an initial thermodynamic state of an isolated system in which two ideal gases A and
B are separated by an impermeable membrane into compartments of volumes V 1
(with N A particles A) and V 2 (with N B particles B) and a final thermodynamic state
in which the membrane has been removed, with each gas accessing the total system
volume V 1 + V 2 . From Eq. (4.1.38), we have S = k B ln , so that the entropy for
the initial thermodynamic state may be expressed as
S i ≡ S A + S B = k B ln V 1 , N A ) + k B (U, V 2 , N B ) ,
while the entropy for the final thermodynamic state may be written as
S f = S A
B = k B ln (U, V 1 + V 2 , N A + N B ) ,
and hence ((S) i→f becomes
((S) i→f = k B [ln (U, V 1 + V 2 , N A + N B ) − ln (U, V 1 , N A )
− ln (U, V 2 , N B )]
= k B ln
(U, V 1 + V 2 , N A + N B )
(U, V 1 , N A ))(U, V 2 , N B )
.
The number of microstates V 1 + V 2 , N A , N B ) available to the N A + N B
ideal gas particles in the final volume V 1 + V 2 will be the product of the numbers
of microstates available to the N A ideal gas particles in volume V 1 and N B ideal gas
particles in volume V 2 times the number of additional microstates associated with
the distinguishability of the A and B particles. We may therefore write