What do we mean when we say that entropy, S, can be calculated
from the expression S = k ln W, in which W is the number of equivalent
ways the molecules can be arranged to give the same observable result?
Why should a gas inevitably have a higher entropy than a crystal
of the same substance? It is hard to answer these questions in our
own universe without getting bogged down in mathematics. But it
is much easier in an imaginary universe with only four atoms in
it, and only nine places where the atoms can be.
Imagine that the nine places in our mini-universe are arranged
in the 3 x 3 grid shown at the upper left. All four atoms placed
in a closepacked square will constitute a "crystal" in our imaginary
space, and any other arrangement of the four atoms will be called
a "gas." Examples of crystals and gases are given in the margin.
If we examine every possible arrangement of four atoms in our nine-point
universe, how many of these arrangements will lead to crystals and
how many to gases?
First of all, how many total arrangements are there for both gases
and crystals? The first atom can go to any one of 9 places. The
second atom has 8 places left open, the third has 7 places, and
the last atom has only 6 unoccupied choices. The total number of
ways of placing four atoms on the nine locations is 9 * 8 * 7 *
6 = 3024 ways.