Biomedical Engineering Reference
In-Depth Information
in vacuo
.
B
is called the rotation constant. Each energy level is (2
J
+
1)-fold degenerate,
and so all 2
J
+
1 individual quantum states must be counted in the Boltzmann formula:
1) exp
∞
Bhc
0
J
(
J
+
1)
q
rot
=
(2
J
+
−
k
B
T
J
=
0
The sum cannot be found in a closed form, but we note that for molecules with small
rotational constants, the rotational energy levels crowd together and the summation can be
replaced by an integral, treating
J
as a continuous variable:
1) exp
d
J
∞
Bhc
0
J
(
J
+
1)
q
rot
=
(2
J
+
−
k
B
T
(17.19)
0
k
B
T
hc
0
B
=
Equation (17.19) works with complete reliability for all heteronuclear diatomic
molecules, subject to the accuracy of the rotational energy level formula and the applicab-
ility of the continuum approximation. For homonuclear diatomics we have to think more
carefully about indistinguishability of the two nuclei which results in occupancy of either
the odd
J
or the even
J
levels; this depends on what the nuclei are, and need not concern us
in detail here. It is dealt with by introducing a
symmetry factor
σ that is 1 for a heteronuclear
diatomic and 2 for a homonuclear diatomic. Parameter
q
rot
is written
1
σ
k
B
T
hc
0
B
q
rot
=
(17.20)
The so-called
rotational temperature
θ
rot
is often used in discussions of statistical
mechanics; we rewrite Equation (17.20) as
1
σ
T
θ
rot
q
rot
=
(17.21)
17.6.4
q
vib
Vibrational energy levels have separations that are at least an order of magnitude greater
than the rotational modes, which are in turn some 20 orders of magnitude greater than
the translational modes. As a consequence, the spacing is comparable to
k
B
T
for everyday
molecules and temperatures. If we consider a single harmonic vibrational mode for which
hc
0
ω
e
v
2
ε
v
=
+
1
k
s
μ
1
2π
c
0
ω
e
=
and if we take the energy zero as that of the energy level
v
=
0 we have
exp
∞
hc
0
v
ω
e
k
B
T
q
vib
=
−
v
=
0
