Application the specific heat of crystals

We can use the concept of phonons to determine the thermal properties of crystals, and in particular their specific heat. This is especially interesting at low temperatures, where the quantum nature of excitations becomes important, and gives behavior drastically different from the classical result. We discuss this topic next, beginning with a brief review of the result of the classical theory. 

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Blackman, M. Contributions to the theory of the specific heat of crystals II. On the vibrational spectrum of cubic lattices and its application to the specific heat of crystals. Proc. Roy. Soc. London 148 A, 384 (1935). 

Chapter IX, page 99.—An interesting case of a condensed system, namely, the affinity of the transformation of amorphous silica (quartz glass) into crystallized quartz, has been worked out by R. Wietzel (" Zeitsch. anorg. Chem., 116, p. 71, 1921) in my old laboratory. Since, however, the specific heat of quartz glass falls off so slowly that it was not possible to get anywhere near the region of the T3-Law, the experimental investigation of this case is not yet complete. This is one of the cases where it is very desirable that accurate measurement of specific heat should be continued down to helium temperatures. This is the more important in that no direct experimental test of the application of the Heat Theorem to amorphous substances has yet been made with satisfactory reliability. The transformation from quartz into cristobalite could, however, be followed with adequate accuracy from the standpoint of the Heat Theorem. 

The heat of crystallization is the heat that has to be supplied or removed during crystallization at constant temperature. It is equal to the negative value of the heat of solution during the dissolution of crystals in an almost saturated solution. The heat of crystallization is accounted for in the enthalpy values. Processes in crystallizers can easily be tracked, if an enthalpy concentration diagram is available for the respective system. The pure component s enthalpy is zero at reference temperature, not the enthalpy of real mixtures however. In such diagrams, the lever rule is applicable. This is shown for the system calcium chloride/water in Fig. 8.3-5, where the specific enthalpy is plotted vs. the mass fractions. 

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