Biomedical Engineering Reference
In-Depth Information
refers only to the energy states of a representative particle, and it is a quantity of great
interest in our theories. It is called the
molecular partition function
and given the symbol
q
(sometimes
z
).
17.6.1
Ideal Monatomic Gas
Consider now an even more restricted case, a monatomic ideal gas. Each atom has mass
m
and the gas is constrained to a cubical container whose sides are
a
,
b
and
c
(so that the
volume
V
is
abc
).We will ignore the fact that each atom could well have different electronic
energy, and concentrate on the translational energy. Elementary quantum mechanics texts
(and Chapter 11) show that the available (translational) energy levels are characterized by
three quantum numbers
n
x
,
n
y
and
n
z
which can each take integral values ranging from 1
to infinity. The translational energy is given by
n
x
a
2
+
n
y
b
2
+
h
2
8
m
n
z
c
2
ε
n
x
,
n
y
,
n
z
=
and so each particle will make a contribution
exp
exp
n
x
a
2
+
n
x
n
y
n
z
ε
n
x
,
n
y
,
n
z
k
B
T
=
n
x
n
y
n
z
n
y
b
2
+
1
k
B
T
h
2
8
m
n
z
c
2
−
−
(17.13)
exp
n
y
exp
n
z
exp
n
y
b
2
=
n
x
1
k
B
T
h
2
8
m
n
x
a
2
1
k
B
T
h
2
8
m
1
k
B
T
h
2
8
m
n
z
c
2
−
−
−
to the translational molecular partition function. Each of the three summations can be treated
as follows. A simple calculation shows that, for an ordinary atom such as argon at room
temperature constrained to such a macroscopic container, typical quantum numbers are of
the order of 10
9
by which time the allowed energy states essentially form a continuum.
Under these circumstances, we replace the summation by an integral and treat
n
x
as a
continuous variable. So for example
exp
exp
d
n
x
∞
1
k
B
T
h
2
8
m
n
x
a
2
1
k
B
T
h
2
8
m
n
x
a
2
−
≈
−
n
x
0
2π
mk
B
T
a
h
=
The translational partition functions are then given by
2π
mk
B
T
h
2
3/2
q
trans
=
V
2π
mk
B
T
h
2
V
N
(17.14)
3/2
1
N
Q
trans
=
!
