1
Going Beyond U and C
V
(17.6, 7, 8)
So far we have considered the energy and heat capacities of two systems:
perfect gases
(monatomic & diatomic) and perfect atomic crystal.
Again, looking ahead a couple of weeks, we note
(
)
th
,
Pressure in the j
energy state
j
j
N
E
P
N V
V
∂
⎛
⎞
= −
=
⎜
⎟
∂
⎝
⎠
By our definition of ensemble averaging:
(
)
(
)
(
)
(
)
,
,
,
,
,
,
j
E
N V
j
j
N
j
j
j
E
e
V
P
P
N V
p
N V
Q N V
β
β
β
−
∂
⎛
⎞
−
⎜
⎟
∂
⎝
⎠
=
=
∑
∑
(
)
(
)
,
,
,
,
Math stuff:
j
j
E
N V
E
N V
j
j
j
N
N
E
Q
recall Q
e
so
e
V
V
β
β
β
β
β
−
−
∂
⎛
⎞
∂
⎛
⎞
=
=
−
⎜
⎟
⎜
⎟
∂
∂
⎝
⎠
⎝
⎠
∑
∑
(
)
,
,
,
1
j
E
N V
j
j
N
N
E
e
V
Q
P
Q
V
Q
β
β
β
β
−
∂
⎛
⎞
−
⎜
⎟
∂
∂
⎝
⎠
⎛
⎞
=
=
⎜
⎟
∂
⎝
⎠
∑

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Lecture 7
2
,
,
1
ln
ln
So, we have shown
P
B
N
N T
Q
Q
k T
V
V
β
β
∂
∂
⎛
⎞
⎛
⎞
=
=
⎜
⎟
⎜
⎟
∂
∂
⎝
⎠
⎝
⎠
And we equate the ensemble average <P> to the experimentally
observed pressure P.
(
)
(
)
(
)
3/2
2
,
2
For our ideal monatomic gas:
Q N,V,
;
,
!
N
q V
m
q V
V
N
h
β
π
β
β
β
⎡
⎤
⎛
⎞
⎣
⎦
=
=
⎜
⎟
⎝
⎠
,
,
ln
1
ln
:
B
N T
N
Q
Q
so
P
k T
V
V
β
β
∂
∂
⎛
⎞
⎛
⎞
=
=
⎜
⎟
⎜
⎟
∂
∂
⎝
⎠
⎝
⎠
(
)
(
)
,
,
1
1
ln
ln
!
ln
terms with no V
N
N
N
q
N
N
V
V
V
β
β
β
β
∂
∂
=
−
=
+
∂
∂
(
)
1
1
ideal gas eqn.
B
Nk T
nRT
N
V
V
V
β
=
=
=
Note that for any ideal gas (monatomic, diatomic, and polyatomic),
q(N,V) is proportional to V, so the ideal gas equation applies to any
ideal gas.