Suppose that a system of N atoms of type A is placed in diffusive contact with a system of N atoms of type B at the same temperature and volume. Show that after diffusive equilibrium is reached the total entropy is increased by 2N log 2. The entropy increase 2N log 2 is known as the entropy of mixing. If the atoms are identical (A ≡ B), show that there is no increase in entropy when diffusive contact is established. The difference in the results has been called the Gibbs paradox.