Partition Functions and Ideal Gases
(Chapter 18.1, 2)
s we discussed last time, for gases where the number of available
As we discussed last time, for gases where the number of available
eigenstates is greater than N we can write
,
N
q V
β
⎡⎤
⎦
( )
( )
, ,
indistinguishable!
!
q
QNV
N
⎣⎦
=
e will explicitly determine the molecular partition function for an
We will explicitly determine the molecular partition function for an
ideal gas, using our available knowledge from Q.M.
For an ideal monatomic gas, there can only be two
contributions to the energy:
.
trans
elec
ε
εε
=+
These energies are independent, so:
ans
elec
qq q
=⋅
.
trans
()
First,
,
i
trans
i
qV
e
βε
−
=
∑
or an atom in a cube of side
2
222
h
nn
=
+
+
For an atom in a cube of side
a
:
( )
,,
2
8
,
,
1,2,3.
..
xyz
in
n
n
x
y
z
nnn
ma
==
=
Lecture 8
1
2
2
exp
trans
x
y
z
h
so q
n
n
n
a
∞
⎛⎞
=−
+
+
⎜⎟
⎠
∑
,, 1
22
8
p
exp
exp
y
x
z
ma
hn
=
∞∞∞
⎝⎠
=−⋅−
⋅−
⎟
⎟
⎟
∑∑
111
exp
888
ma
ma
ma
===
∑∑∑
ince the box is a cube each sum is identical!
Since the box is a cube, each sum is identical!
( )
3
,e
x
p
ans
q
V
∞
⎥
⎟
2
1
8
trans
n
ma
=
⎢⎥
∑
As we have seen in Chem 442, the energy level spacings are
very small for macroscopic values of
a
, the box length.
Change
from discrete n to continuous n:
3
2
0
x
p