Biomedical Engineering Reference
In-Depth Information
This can be identified with the thermodynamic internal energy
U
th
N
n
i
l
i
h
2
8
mL
2
U
th
=
+
k
i
+
(12.3)
i
=
1
We do not yet know how the kinetic energy will be divided out amongst the particles, all we
know is that the summation will be a complicated function of the quantum numbers. For
the sake of argument, call the summation
A
. Since the box is a cube of side
L
, the volume
V
of the box is
L
3
and so we can write
h
2
8
m
AV
−
2/3
U
th
=
(12.4)
We know from elementary chemical thermodynamics that for a thermally isolated system
∂
U
th
∂
V
p
=−
N
and so the particle in a box model gives the following relationship between pressure and
volume:
2
3
h
2
8
m
AV
−
5/3
p
=
(12.5)
which can be rearranged together with the expression for
U
th
to give
2
3
U
th
pV
=
(12.6)
That is to say, the pressure times the volume is a constant at a given temperature. Not only
that, but if the internal energy of the sample (of amount
n
) is given by the equipartition of
energy expression
2
nRT
, we recover the ideal gas law.
12.1 Sharing Out the Energy
Statistical thermodynamics teaches that the properties of a system in equilibrium can be
determined in principle by counting the number of states accessible to the system under
different conditions of temperature and pressure. There is as ever some interesting small
print to this procedure, which I will illustrate by considering three collections of different
particles in the same cubic infinite potential well. First of all, I consider distinguishable
particles like the ones discussed above that have mass and quantized energy. I raised the
question in Chapter 11 as to whether or not we could truly distinguish one particle from
another, and there is a further problem in that some particles have an intrinsic property
called
spin
just as they have a mass and a charge.
We will then extend the ideas to photons (which have momentum but no mass), and
finally consider the case where the particles are electrons in a metal.
We noted in Chapter 11 two differences between a one-dimensional and a three-
dimensional infinite well: firstly, the three-dimensional well has degeneracies both natural
and accidental; secondly, the three-dimensional quantum states tend to crowd together as
the quantum numbers increase. If we consider a box of 1 dm
3
containing (for example)
20
Ne
