Disorder in Atoms

Entropy is a quantitative measure of the disorder in a system. The more disordered a system is, the higher its entropy. To see what is meant by the disorder in a system, consider the following situation.

Two 500 mL flasks are connected by a short length of tubing. The tubing is fitted with a stopcock that can block or allow access through the tubing between the two flasks. The flasks and tubing are empty except for a single atom of a gas. The atom is free to move back and forth between the flasks through the tubing. The atom spends about half its time in the left flask and about half its time in the right flask. The probability of finding the atom in the left (or right) flask at any particular time is thus 1/2. Click on the radio buttons at right to see each of these possibilities. Click on the Moving radio button to resume the animation.

Now imagine that there are two atoms in the setup. Each one is free to move back and forth between the two flasks, spending half of its time (on average) in each one. There are four possible arrangements of the two atoms in the two flasks: both can be in the left flask, both can be in the right flask, the blue one can be in the left flask and the red can be in the right flask, or the blue one can be in the right flask and the red one can be in the left flask. The probability of finding both atoms in the left (or right) flask) is 1/2 x 1/2 or 1/4 and the probability of finding one atom in each flask is 1/2. Thus, the probability of finding an atom in each flask is greater than the probability of finding both atoms in one flask. Click on the radio buttons at right to see each of these possibilities. Click on the Moving radio button to resume the animation.

When there are three atoms in the setup there are eight possible arrangements of the atoms. The probability of finding all three atoms in the left (or right) flask is 1/2 x 1/2 x 1/2 or 1/8. The probability of finding at least one atom in each flask is 6/8 or 3/4. Click on the button at right to cycle through all eight possibilities. Click on the Moving radio button to resume the animation.

When the number of atoms becomes large, the probability of finding them all in one flask becomes vanishingly small. If there are 100 atoms, the probability is 1/2¹⁰⁰ or 7.89 x 10^-31. If there is a mole (6.022 x 10²³) of atoms present, the probability of finding them all in one flask is so close to zero that you will never find them all in one flask.

What this means is that if you have a sample of gas confined to one of the flasks with the stopcock closed, when it opens the gas will flow into the other flask until equal numbers of atoms are in both flasks. Small numbers of atoms may move from one flask to the other, but the laws of probability guarantee that all of the atoms will never move into one flask or the other of their own accord. Click on the image at right to see how this happens.

The arrangement with equal numbers of atoms in both flasks is more disordered than (has more entropy than) one in which all atoms are in one or the other flask.

Disorder in Atoms