Biomedical Engineering Reference
In-Depth Information
The value of β is not quite as straightforward. What we need to do is to use the expression
to calculate a quantity such as the heat capacity of a perfect monatomic gas which can be
compared with experiment. When this is done we find
N
exp
ε
i
k
B
T
−
N
i
=
i
=
1
exp
(7.16)
ε
i
k
B
T
p
−
The formula is correct even when the quantum states are degenerate, as it refers to the
individual quantum states. It also holds in an equivalent integral form when the energies
form a continuum.
7.4 Safety in Numbers
I should now make a few points about Boltzmann's law. First of all, the basic result
N
2
N
1
=
exp
(ε
2
−
ε
1
)
k
B
T
−
can only be an approximation, for if I take a system of 10 000 two-state particles in
equilibrium at a temperature
T
=
(ε
2
−
ε
1
)/
k
B
then we find
N
2
N
1
=
0.36788
which corresponds to
N
2
=
2689.4. These estimates cannot be true, since
the numbers must be integers. The particles inevitably collide with one another and interact
with the surroundings (technically referred to as the heat bath), and the picture that emerges
is one of ceaseless activity. The number of particles in each state fluctuates fromonemoment
to the next as energy flows into and out of the system via the heat bath. We therefore have
to understand that the numbers appearing are averages, which I ought to write as
7310.6 and
N
1
=
exp
N
2
(ε
2
−
ε
1
)
k
B
T
N
1
=
−
(7.17)
In order for Boltzmann's law to apply, the system must be in thermal equilibrium. The
particles must be distinct and independent, which means that it must be possible in principle
to distinguish one particle from another (by colouring, numbering or whatever means we
want to use). With all of that in mind, I will drop the '<...>' notation unless it matters.
The average number of particles with energy ε fluctuates at random; the average popu-
lation is given correctly by Boltzmann's law but the actual population fluctuates. Do these
fluctuations matter when we want to predict macroscopic properties? I want to show you
in this section that they are negligible.
First of all, let me establish that there is truly safety in numbers. Consider tossing a
number of coins a certain number of times. It is common in the UK to call one side of the
coin the 'head' side (H) and the other side the 'tail' side (T). In this experiment, I am going
to throw the coin(s) in the air and record whether the coin lands H or T uppermost. I then
repeat the experiment (say) 10 times. There is no particular reason for the coin to land either
