Biomedical Engineering Reference
In-Depth Information
Table 7.5
Enumeration for two dice A and B
Total score,
S
Possible individual
W
=
number
Probability
scores of A and B
of ways
2
1
+
1
1
1/36
3
1
+
2, 2
+
1
2
2/36
4
1
+
3, 3
+
1, 2
+
2
3
3/36
5
1
+
4, 4
+
1, 2
+
3,
4
4/36
3
+
2
6
1
+
5, 5
+
1, 2
+
4,
5
5/36
4
+
2, 3
+
3
7
1
+
6, 6
+
1, 2
+
5,
6
6/36
5
+
2, 3
+
4, 4
+
3
8
2
+
6, 6
+
2, 3
+
5,
5
5/36
5
+
3, 4
+
4
9
3
+
6, 6
+
3, 4
+
5,
4
4/36
5
+
4
10
4
+
6, 6
+
4, 5
+
5
3
3/36
11
5
+
6, 6
+
5
2
2/36
12
6
+
6
1
1/36
Total
36
1
7.3 Boltzmann Probability
I have mentioned the Boltzmann distribution several times already, so let us now consider
a very simple macroscopic system comprising a gas of
N
inert particles at low pressure in
a thermally insulated constant container of volume
V
.
I am going to look ahead and assume that you know about energy quantization and there-
fore assume that the particles have the very simple energy level diagramshown in Figure 7.4;
each energy level corresponds to a single quantum state, and the allowed energies are
ε
n
=
nD n
=
1, 2, 3 ...
(7.5)
I shouldmention that not all molecular systems showquantized energy; translational motion
is not quantized either in quantum mechanics or in classical mechanics. It just makes my
discussion simpler if we take the energy to be quantized.
Once again, I am assuming that you have come across such diagrams if you have studied
elementary atomic structure theory, and you might have been expecting me to add electrons
as up and down arrows (to denote the spin states). The meaning of these diagrams here is
a little different; the diagram shows the allowed quantum states for each particle and I add
particle labels to indicate their quantum states as in Figure 7.5. The left-hand arrangement
of the three particles A, B and C has particles A and B in the lowest quantum state, none in
the second and C in the third. I have chosen a different arrangement on the right-hand side
of the figure.
I labelled the particles A, B and C to distinguish one from another, but there is more to
this business of distinguishability than meets the eye. Keep this comment in mind as the
topic will reappear later in the text.
