A Statistical Mechanical Derivation of the van der Waals Equation of ...

Q
vdw
=
1 ⎛V
⎜
N!
⎝
N
− Nb ⎞ ⎛ NE ⎞ ⎛ − ⎞ ⎛
⎜ −
vdw 1 V Nb N
⎟ exp
⎟ = ⎜ ⎟ exp
⎜
3 3
Λ ⎠ ⎝ k
BT
⎠ N!
⎝ Λ ⎠ ⎝Vk
N
2
B
a ⎞
⎟
T ⎠
(5)
As always, we require the natural log of the partition function
2
⎛V
− Nb ⎞ N a
ln ( Qvdw
) = −ln( N!
) + N ln⎜
+
3
⎟
(6)
⎝ Λ ⎠ Vk T
We use Stirling’s approximation for the natural log of the factorial of a large number
B
2
⎛V
− Nb ⎞ N a
ln ( Qvdw
) = −N
ln( N ) + N + N ln⎜
+
3
⎟
(7)
⎝ Λ ⎠ Vk T
The first thermodynamic property that we will calculate is the pressure because that is what
everyone associates with the van der Waals equation of state.
( Q)
⎛ ∂ ln ⎞ k
BT
a k
BT
a
p = k
BT⎜
⎟ = − = −
(8)
2
2
⎝ ∂V
⎠ V
N , T
V Vm
− b V
− b ⎛ ⎞
m
N
⎜ ⎟
⎝ N ⎠
V
where we have defined a molecular volume, V m
≡ . This is the van der Waals equation of
N
state.
Now let’s derive other thermodynamic functions. We will start with the molecular
Helmholtz free energy,
A
=
⎡
⎛V
− Nb ⎞ Na ⎤
( Q) = −k
T − ln( N ) + 1+
+ ⎥ ⎦
A k
BT
= − ln
N , T B ⎢
ln⎜
⎟
(9)
N N
⎣
⎝ Λ ⎠ Vk
BT
m 3
The molecular internal energy is
B
E
m
=
E
N
=
k
BT
N
2
( Q)
⎛ ∂ ln
⎜
⎝ ∂T
⎞
⎟
⎠
N , V
=
3
2
k T −
B
Na
V
=
3
2
k T −
B
a
V
m
(10)
Unlike the ideal gas, the internal energy of a vdW fluid is not only a function of temperature but
also a function of molar volume. The chemical potential is
µ = −k
⎛ ∂ ln
T⎜
⎝ ∂N
( Q)
The molecular entropy is
B
⎞
⎟
⎠
T , V
= −k
B
⎡ ⎛Vm
− b ⎞
T ⎢ln⎜
⎟ −
3
⎣ ⎝ Λ ⎠ V
m
b 2a
+
− b V k T
m
B
⎤
⎥
⎦
(11)
2