The physical origins of entropy are explained. Configurational entropy in the point approximation was used previously, but Chapter 7 shows how configurational entropy can be calculated more accurately with cluster expansion methods, and the pair approximation is developed in some detail. Atom vibrations are usually the largest source of entropy in materials, and the origin of vibrational entropy is explained in Section 7.4. Vibrational entropy is used in new calculations of the critical temperatures of ordering and unmixing, which were done in Chapter 2 with configurational entropy alone. For metals there is a heat capacity and entropy from thermal excitations of electrons near the Fermi surface, and this increases with temperature. At high temperatures, electron excitations can alter the vibrational modes, and there is some discussion about how the different types of entropy interact.

Planck immediately set about attempting to understand the significance of his formula for black-body radiation. He began by using Boltzmann's procedure in statistical mechanics, an approach he had previously rejected, but then adopted empirically a definition of the entropy of the oscillators which introduced the concept of quantisation. HIs derivation was not understood by his contemporaries, including Einstein, because of the lack of a theoretical motivation for the definition of entropy. Despite a major effort to understand his formula, Planck found no classical solution to the meaning of h, Planck's constant.