Advanced Topics in Equilibrium Statistical
Mechanics
Glenn Fredrickson
3. Theory of Gases
A. The Ideal Gas
As we shall see, gases are much easier to deal with than liquids, because the low
density of gases makes collisions and pair interactions of molecules infrequent.
The strategy will be to derive an expansion in powers of particle density known
as the virial expansion
. The leading term, accurate at inFnite dilution, is the
ideal gas
and follows by setting
U
(
r
N
)=0.
We shall begin by pursuing the thermodynamic properties of a monatomic
ideal gas in the
canonical ensemble
.
Q
c
(
N,V,T
)=
Z
d
r
N
e
−
βU
=
V
N
Q
=
V
N
N
!
λ
3
N
T
βA
=
−
ln
Q
=ln
N
!+
N
ln(
λ
3
T
/V
)
We are interested in the ”thermodynamic limit”
of this expression, which corre
sponds to considering a system of macroscopic extent, i.e.
with Avogadro’s
number of atoms.
We take this limit by simultaneously taking
N
→∞
,
V
, while holding the
average particle number density
ρ
=
N
V
constant.
Stirling’s asymptotic approximation for the logarithm of large factorials, ln
N
!
∼
N
ln
N
−
N
+
...
, gives
∼
N
ln
N
−
N
ln
V
+
N
(ln
λ
3
T
−
1)
∼
N
ln(
N/V
)+
N
(ln
λ
3
T
−
1)
The ideal gas
equation of state
follows from:
p
=
−
∂A
∂V
)
N,T
βp
=
N
·
1
V
≡
ρ
1
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View Full DocumentThus, the ideal gas equation of state can be written in any of the alternate
forms:
βp
=
ρ, pV
=
Nk
B
T, pV
=
nRT
where
n
is the number of moles of
atoms.
Next, we repeat this calculation in the
grand canonical ensemble
:
Q
G
(
µ, V, T
)=
∞
X
N
=0
z
N
N
!
Q
c
(
N,V,T
)=
∞
X
N
=0
(
zV
)
N
N
!
=
e
The thermodynamic connection is
βpV
=ln
Q
G
=
or
=
z
=
e
βµ
λ
−
3
T
.
We thus have the pressure expressed as
p
(
z, T
), but a conventional equation of
state is of the form
p
(
ρ, T
). Thus, we need an expression relating the activity
z
to the average density
ρ
, i.e. we need the function
z
(
ρ
). Recall that
h
N
i
=
∂
ln
Q
G
∂
ln
z
)
β,V
=
V
∂z
∂
ln
z
)
=
Vz
So
z
=
h
N
i
V
≡
ρ
is the functional relationship between
z
and
ρ
in the ideal gas
limit. It follows immediately that the equation of state for an ideal gas in the
grand canonical ensemble is given as before by
=
ρ
.
Notice that we needed Stirling’s approximation in the canonical ensemble,
but not in the grand canonical ensemble. Thus, results from the two agree only
in the thermodynamic limit! This, however, is as expected, since ﬂuctuations in
the two ensembles are diFerent for systems of ±nite size and these will contribute
small corrections to thermodynamic quantities that go to zero in the limit of an
in±nite system.
Lecture 4
Reading: McQuarrie — Chp. 12, Andersen handout
B. Nonideal gases — Cluster & Virial Expansions
Now we are ready to talk about the problem of a real ﬂuid with intermolecular interactions
.
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 Spring '09
 Statistical Mechanics, Qg, Virial Expansions

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