Biomedical Engineering Reference
In-Depth Information
since molecules have rotational and vibrational energies in addition to translational kinetic
energy.
Apart from the density of states mentioned above,
D
(ε)
=
B
ε
1/2
, we have to consider
the average occupancy of each quantum state. This is given by the Boltzmann factor
AN
exp
ε
k
B
T
−
where
A
is a constant and
N
the number of atoms. Combining the two expressions we get
1
N
A
exp
B
ε
1/2
d
N
dε
=
ε
k
B
T
−
We can eliminate the constants
A
and
B
in terms of
N
by use of the equations above and
we find after rearrangement
1
N
√
π (
k
B
T
)
3/2
exp
ε
1/2
d
N
dε
=
2
ε
k
B
T
−
(12.11)
which gives exact agreement with the experimental curves. It can be established by dif-
ferentiation of Equation (12.11) that the peak occurs at ε
1
2
k
B
T
, and it can also be
established by integration that the average energy per particle <ε>is
=
3
2
k
B
T
, in accord
with the equipartition of energy principle.
12.4 Black Body Radiation
Astudy of black body radiation was a milestone in the path to our modern theory of quantum
mechanics. The term black body radiation refers to the electromagnetic radiation emitted
by a 'perfect emitter' (usually a heated cavity).
The curves in Figure 12.5 relate to the energy
U
emitted by a black body of volume
V
per wavelength λ at temperatures of 1000 K (bottom curve), 1500 K (middle curve) and
2000 K (top curve). The quantity plotted on the
y
axis is
1
V
d
U
dλ
and it is observed that:
•
each of the curves has a peak at a certain maximum wavelength λ
max
;
•
λ
max
moves to shorter wavelength as the temperature increases.
These curves have a quite different functional form to the atomic kinetic energy curves
shown in Figure 12.4. Max Planck studied the problem and was able to deduce the following
expression that gives a very close fit to the experimental data:
1
V
8π
hc
0
λ
5
d
U
dλ
=
1
exp
hc
0
λ
k
B
T
(12.12)
−
1
For present purposes, it proves profitable to think of the problem in the following way.
We know that the heated cavity contains photons, and that the energy of a photon of
