Biomedical Engineering Reference
In-Depth Information
It is easy to demonstrate that the potential energy is
1
2
k
s
(
x
2
−
R
e
)
2
U
=
x
1
−
and the total energy ε
vib
of the harmonically vibrating diatomic is therefore
2
m
1
d
x
1
2
2
m
2
d
x
2
2
1
1
1
2
k
s
(
x
2
−
R
e
)
2
ε
vib
=
+
+
x
1
−
(3.15)
d
t
d
t
3.4 Three Problems
This simple treatment suggests three problems. First, how do we determine the spring
constant for a simple molecule such as
1
H
35
Cl or
12
C
16
O? Second, how good is the harmonic
approximation? Third, have we missed anything by trying to treat a molecular species as
if it obeyed the laws of classical mechanics rather than quantum mechanics?
The three questions are interlinked, but let me start with the third one. The experimental
evidence suggests that we have made a serious error in neglecting the quantum mechanical
details. If we irradiate a gaseous sample of
1
H
35
Cl with infrared radiation, it is observed
that the molecules strongly absorb radiation of wavenumber 2886 cm
−
1
. With hindsight we
would of course explain the observation by saying that the molecular vibrational energies
are
quantized
. Amajor flaw of the classical treatment is that the total vibrational energy is
completely unrestricted and quantization does not arise.
The quantum mechanical treatment of a harmonically vibrating diatomic molecule
is given in all the elementary chemistry texts. The results are quite different from the
classical ones.
1. The vibrational energy cannot take arbitrary values, it is
quantized
.
2. There is a single quantum number
v
which takes values 0, 1, 2, ... This is called the
vibrational quantum number
.
3. Vibrational energies ε
vib
are given by
k
s
μ
v
h
2π
1
2
ε
vib
=
+
where
h
is Planck's constant.
The results are usually summarized on an energy level diagram, such as that shown
in Figure 3.5. I have just drawn the first four vibrational energy levels, but there are an
infinite number of them. According to the harmonic model, the spacing between the levels
is constant.
A careful investigation into the mechanism by which electromagnetic radiation interacts
with matter suggests that transitions between these vibrational energy levels are allowed,
provided the vibrational quantum number changes by just 1 unit. So molecules with
v
=
0
can absorb radiation of exactly the right energy for promotion to
v
=
1. Molecules with
v
=
1 can either absorb radiation with exactly the right energy for promotion to
v
=
2or
they can emit radiation and fall to
v
=
0, and so on.
