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Lecture 6 – Virial Expansions and Critical Behavior
September 11, 2008
Using the virial expansion in pressure:
(1) at p = 0, Z = 1, perfect gas
(2) at T
→∞
, Z = 1 for all gasses
(3)
T
0
p
p
Z
lim
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
→
()
...
p
RT
a
b
2
RT
a
RT
a
b
RT
1
p
Z
3
T
+
⎟
⎠
⎞
⎜
⎝
⎛
−
+
⎟
⎠
⎞
⎜
⎝
⎛
−
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
−
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
→
RT
a
b
RT
1
p
Z
lim
T
0
p

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of
8
If
RT
a
b
>
then
0
p
Z
lim
T
0
p
>
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
→
(Repulsive interactions dominate)
(note the initial
slope is positive)
RT
a
b
=
then
0
p
Z
lim
T
0
p
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
→
(Boyle's condition)
(note the initial
slope is zero)
RT
a
b
<
then
0
p
Z
lim
T
0
p
<
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
→
(Attractive interactions dominate)
(note the initial
slope is negative)
b = a/RT
can be rewritten as
T
B
= a/Rb
This is the Boyle
Temperature!!!
Perfect gas behavior over a

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