Page 192 - Chemical equilibria Volume 4

168 Chemical Equilibria
ε ,
electronic
e
r
v
interactional ε , so: vibrational ε , rotational ε , translational ε t and
I
ε = ε + ε + ε + ε + ε + ε I [A2.18]
v
e
n
t
r
The partition function of the molecule becomes:
⎛ ε n i ⎞ ⎛ ε e i ⎞ ⎛ ε v i ⎞
⎜∑
⎜∑
⎜∑
z = exp − ⎟ exp − ⎟ exp − ⎟
B ⎠
B ⎠
B ⎠
n i ⎝ k T e i ⎝ k T v i ⎝ k T [A2.19]
⎛ ε r i ⎞ ⎛ ε t i ⎞ ⎛ ε I i ⎞
⎜∑
⎜∑
exp − ∑ exp − ⎟ exp − ⎟
⎜ ⎟
B ⎠
B ⎠
B ⎠
r i ⎝ k T t i ⎝ k T I i ⎝ k T
Thus, we see the emergence of partial partition functions, which pertain
to the different forms of energy:
⎛ ε n i ⎞ ⎛ ε e i ⎞ ⎛ ε v i ⎞
⎜∑
⎜∑
⎜∑
z = exp − ⎟ , z = exp − ⎟ , z = exp − ⎟
v
e
n
B ⎠
B ⎠
B ⎠
n i ⎝ k T e i ⎝ k T v i ⎝ k T
⎛ ε r i ⎞ ⎛ ε t i ⎞ ⎛ ε I i ⎞
⎜∑
⎜∑
⎜∑
z = exp − ⎟ , z = exp − ⎟ , z = exp − ⎟ [A2.20]
t
r
I
B ⎠
B ⎠
B ⎠
r i ⎝ k T t i ⎝ k T I i ⎝ k T
The global partition function then takes the form of a product of the
partial partition functions:
z = z z z z z z I [A2.21]
ne v r t
Sometimes, we use the term internal contribution to speak of the product:
z int = z z z z [A2.22]
ne v r
This is the product of all the contributions other than those of translation
and interaction.
The global molecular partition function then becomes:
z = z z z I [A2.23]
int t