ln
Q
N
C
C(^) C
ln
Q
N
D
D(^) D
ln
Q
N
A
A(^) A
ln
Q
N
B
B(^) B
0
In the next step, we will do two things: bring the A- and B-containing terms
to the other side of the equation, and combine the two logarithms on each side
into one (using the rule that ln a ln bln ab):
ln
Q
N
C
C(^) C
Q
N
D
D(^) D
ln
Q
N
A
A(^) A
Q
N
B
B(^) B
If the logarithms of two products are the same (as the above equation indi-
cates), then the arguments of the two individual logarithms are the same.
Another way to put this is that we can take the inverse logarithm of both sides
of the above equation and still have an equality:
Q
N
C
C(^) C
Q
N
D
D(^) D
Q
N
A
A(^) A
Q
N
B
B(^) B
(18.60)
At this point, we will rearrange equation 18.60 to bring all of the partition
functions Qito one side and all of the amounts Nito the other. The exponents
(^) iwill appear on both sides (as a consequence of the algebra of exponents). By
convention, we will write the expressions with product quantities in numera-
tors and reactant quantities in denominators. We get
(
(
Q
Q
C
A)
)
C
A(
(
Q
Q
D
B))
BD
((
N
N
AC
))
AC
(
(
N
N
BD)
)
D
B (18.61)Consider equation 18.61. The partition functions Qiare constants that arecharacteristic of each chemical species, and the coefficients (^) iare characteris-
tic of the balanced chemical reaction. Therefore, the left side of equation 18.61
is some constant that is characteristic of the chemical reaction. Equation 18.61
shows that this characteristic constant is related to the amounts of each chem-
ical species when the reaction reaches chemical equilibrium,even though each
individual Qiitself is defined in terms of the molecule, not any extent of reac-
tion! Since the fraction in terms of the Qivalues has a characteristic value, then
the fraction in terms of the amounts Niat equilibrium must also have a char-
acteristic value. This value is called the equilibrium constantfor the reaction.
For an ideal gas, the partition function Qis a simple function of volume
(again, from qtrans) times a more complicated function of temperature (from
several other q’s):
Qf(T) V
It is convenient to divide each molecular Qby volume to get a volume-
independent partition function:
Q
V
f(T)By substituting this volume-independent partition function into the partition
function expression for the equilibrium constant, we can get an equilibrium
constant, labeled K(T), which is characteristic of the chemical species in the
reaction and dependent solely on T:K(T) (18.62)
Q
V
C
(^) C
Q
V
D
(^) D
Q
V
A
(^) A
Q
V
B
(^) B
642 CHAPTER 18 More Statistical Thermodynamics