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 NV
V
∂
⎛⎞
=−
=
⎜⎟
∂
⎝⎠
y our definition of ensemble averaging:
By our definition of ensemble averaging:
,
j
E NV
j
j
E
e
V
NV
β
−
∂
−
∂
∑
( ) ( )
,,
,
N
jj
j
PP
N
V
p NV
QNV
==
∑
Math stuff:
j
j
ENV
j
j
E
Q
recall Q
e
so
e
VV
ββ
−−
∂
∂
−
∂∂
⎠
∑∑
,
,
N
N
,
j
j
E
e
V
−
∂
−
∂
⎠
⎞
∑
Lecture 7
1
,
,
1
j
N
N
Q
P
QV
Q
∂
∂
1l
n
l
n
So, we have shown
P
B
N
T
QQ
kT
V
∂
⎠
NN
And we equate the ensemble average <P> to the experimentally
observed pressure P.
p
( )
3/2
2
,
2
For our ideal monatomic gas:
Q N,V,
;
,
!
N
qV
m
V
Nh
π
⎡⎤
⎣⎦
ln
1
ln
:
B
NT
N
so
P
k T
11
ln
ln
!
ln
terms with no V
Nq N
=
+
ideal gas eqn.
B
Nk T
nRT
N
V
=
Note that for any ideal gas (monatomic, diatomic, and polyatomic),
(N,V) is proportional to V, so the ideal gas equation applies to any
Lecture 7
2
q( , )
p p
,
g
q
pp
y
ideal gas.
Let’s look at things just a bit more deeply.
In statistical mechanics the
energies, E (N,V), are eigenvalues of the N-particle Hamiltonian.
g,
j
(,)
,
g
p
Remember, we couldn’t find exact
eigenvalues even for a 3-body system
(except H
2
+
in the B.O. approximation) so what can we do for N-bodies?
1
st
we can assume that the N-particles are independent.
( ) ( )
Then
,
N
i
V
ε
=
1
ji
i
=
∑
If the particles are distinguishable, we know which is which
xamples would be our perfect crystal of N atoms at specific
2nd:
(examples would be our perfect crystal of N atoms at specific
coordinates).