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.