Page 56 - 4498

microscopic calculation of the entropy of a system.
The microscopic quantity related to entropy is the relative probability of
different ways of sorting the molecules of the system. Let us first consider
some qualitative applications of this relationship:

1. Free expansion. In a free expansion the gas molecules confined to
one half of a box are permitted to fill the entire box. Let us consider the
entire box in its two circumstances: first, the partition is removed with the

molecules all occupying one half of the box, and second, with the
molecules filling the entire box. The first condition is a state of very low
probability; left on its own, it would be very unlikely for the system to sort
itself in this way. The second condition is a rather high probability. We

can regard the molecules in the free expansion as moving from a condition
of low probability to one of high probability. That is, given all the possible
ways of distributing or sorting the molecules randomly within the box, a

large number of those ways show a rather uniform distribution of
molecules, while a very small number show a nonuniform distribution. For
instance, let us consider a box with only 10 molecules and evaluate the

number of ways for a particular number n to be found in the left half of the
box at any instant. Since each molecule has two possible locations in the
box in this scheme (left half or right half), the total number of ways we can
10
distribute the molecules, with two .choices' for each, is 2 .= 1024. Only
one way is possible out of those 1024 ways, in which we will find all the
molecules in the left half (n = 10), while it can be shown that there are 252
ways of having a uniform distribution (n = 5). As the number of molecules

increases, the relative probability of a uniform distribution increases
dramatically. With 100 molecules, there is still only one way to sort them
29
all into the left half, but there are about 1Q ways to distribute them
equally between the two halves. The free expansion, in which there is an
increase in entropy, can, thus, be regarded as a transformation from a state
of a very low probability to a state of a very high probability.

2. Heat conduction. In this example two bodies of different temperatures
T and T come to a uniform intermediate temperature T when they are
1
2
c
placed in contact. This case is similar to the free expansion, except for the
case when we sort the molecules by their speed rather than by their
location. Again we consider the entire system in two circumstances: just

after contact, with the "hot" molecules (mostly faster moving ones) on one
side and the "cool" molecules (mostly slower moving) on the other, and at
a much later time, when the distribution of speeds between the two halves

56