Biomedical Engineering Reference
In-Depth Information
The canonical partition function for an ideal gas is therefore
2π
mk
B
T
h
2
3
N
/2
V
N
N
Q
=
(8.19)
!
The partition function for a real system is often written as the product of an ideal part and
an excess part due to nonideal behaviour:
Q
ideal
Q
excess
Q
=
where
exp
d
q
1
V
N
Φ
k
B
T
Q
excess
=
−
(8.20)
The point of doing this is that thermodynamic properties such as
A
are often measured
experimentally as an ideal and an excess part:
A
ideal
A
excess
A
=
+
The ideal part can be related to
Q
ideal
and the excess part to
Q
excess
.
8.8 Virial of Clausius
Let me focus attention on one particular particle
i
moving in the box shown in Figure 8.1.
As this particle moves it will be subject to some varying force
F
i
and
m
d
v
i
d
t
F
i
=
(8.21)
Taking the scalar product of both side of this equation with
r
i
we get
m
r
i
.
d
v
i
d
t
r
i
.
F
i
=
(8.22)
Consider now the vector identity
d
d
t
(
r
i
.
v
i
)
r
i
.
d
v
i
d
r
i
d
t
=
d
t
+
.
v
i
(8.23)
which can also be written
d
d
t
(
r
i
.
v
i
)
r
i
.
d
v
i
=
d
t
+
v
i
(8.24)
On comparison of Equations (8.22) and (8.24), we have
m
d
v
i
r
i
.
F
i
=
−
d
t
(
r
i
.
v
i
)
(8.25)
or
1
2
r
i
.
F
i
=−
1
2
m
d
1
2
mv
i
−
d
t
r
i
.
v
i
+
(8.26)
The next step is to sum corresponding terms on both sides of the equation for each particle
in the box. For
N
particles each of mass
m
, this gives
N
N
N
1
2
1
2
m
d
1
2
m
−
r
i
.
F
i
=−
r
i
.
v
i
+
v
i
(8.27)
d
t
i
=
1
i
=
1
i
=
1
