Chemistry Reference
In-Depth Information
where we have introduced the T-dependent negative quantity
a
¼
m
A
m
B
R
3
kT
<
0
ð
11
:
72
Þ
We then obtain the familiar formula
11
R
3
Ð
W
W
F
ðWÞ
exp
½
aF
ðWÞ
Ð
W
d
h
Vexp
ð
V
=
kT
Þi ¼
m
A
m
B
¼
m
A
m
B
R
3
d
da
ln K
ð
a
Þ
d
W
exp
½
aF
ðWÞ
ð
11
:
73
Þ
where
ð
K
ð
a
Þ¼
d
W
exp
½
aF
ðWÞ
ð
11
:
74
Þ
W
is called the Keesom integral.
We evaluate (11.73) for a
small (high temperatures and large
distances between the dipoles) by expanding the exponential in (11.74):
ð
ð
a
2
2
F
ðWÞ
2
W
exp
½
aF
ðWÞ
W
1
þ
aF
ðWÞþ
þ
ð
11
:
75
Þ
d
d
W
W
where we have just seen that, in the expansion, the second integral
12
vanishes, so that only the quadratic term contributes to the Keesom
integral.
We have
ð
ð
p
A
ð
p
0
2
p
Ð
W
2
d
W
F
ðWÞ
¼
d
w
d
u
A
sin
u
d
u
B
sin
u
B
0
0
ð
sin
2
A
sin
2
B
cos
2
wþ
4cos
2
A
cos
2
u
u
u
u
Þ
B
11
Relation (11.73) is much the same as that observed in the Debye theory of the orientation of
electric dipoles in gases, and in the Langevin (classical) or Brillouin (quantal) equations for the
paramagnetic gas. In the last case, summations replace integrations over the parameter a.
12
And all odd powers of it.
