Debye theory of specific heats

DEBYE THEORY OF SPECIFIC HEAT. The specific heal of solids is attributed to the excitation of thermal vibrations of the lattice, whose spectrum is taken to be similar to that of an elastic continuum, except that it is cut off at a maximum frequency in such a way that the total number of vibrational modes is equal to the total number of degrees of freedom of the lattice. 

The Einstein equation was the first approximation to a quantum theoretical explanation of the variation of specilic heat with temperature. It was later replaced by the Debye theory of specific heat and its modifications. 

In this chapter we shall discuss those properties of ice crystals which derive essentially from the thermal motions of water molecules within the crystal structure. In broad outline the theory describing these phenomena is simple and well known and leads to simple generalizations like the Debye theory of specific heats. However, because of the structure of the water molecule and, deriving from it, the structure of the ice crystal, such theories in their simple form represent only a first approximation to the observed behaviour. The coefficient of thermal expansion, for example, is negative at low temperatures and the specific heat is only poorly described by a Debye curve. It will be in tracing the reasons for some of these deviations from simple behaviour that most of our interest will lie. 

The SI unit for heat capacity is J-K k Molar heat capacities (Cm) are expressed as the ratio of heat supplied per unit amount of substance resulting in a change in temperature and have SI units of J-K -moC (at either constant volume or pressure). Specific heat capacities (Cy or Cp) are expressed as the ratio of heat supplied per unit mass resulting in a change in temperature (at constant volume or pressure, respectively) and have SI units of J-K -kg . Debye s theory of specific heat capacities applies quantum theory in the evaluation of certain heat capacities. 

The actual dependence of pATsp on the temperature is rather complicated because of the dependence of the specific heat Cp on T, which is given by Debye s theory of specific heat for the reacting oxides and corresponding lattice dynamical model for crystalline solids. Simple assumptions regarding the net change in specific heats of the components involved in the dissolution reactions, however, allow one to avoid these complications [3]. 

Debye, P. Zur Theorie der spezifischen Warmen. [On the theory of specific heats.] Annalen der Physik 39, 789-839 (1912). English translation in Collected Papers of Peter J. W. Debye, pp. 650-696. Interscience New York (1954). 

Theories of this kind have been developed by Tetrode,f Sommerfeld, and Keesom, all on the assumption that a gas behaves at low temperatures like a solid and that therefore Debye s well-known theory of specific heats becomes applicable in its essentials. ... 

Fig. 3.2 According to the Debye theory the specific heat of many solids is, to a good approximation, a universal function of the reduced temperature T/ d [82]. FVomRef. [48].

Fig. 4.—Graph of specific heats at low temperatures according to Debye the small circles show observed points, the continuous curves correspond to Debye s theory. (-) is a temperature characteristic of the substance, such that C(= Cv) is a function of...

In fact, new points of view and new methods had to be discovered for the examination of chemical equilibria. Of late years experimental knowledge has been extended, especially in the field of specific heats, both of solid and gaseous bodies, and as a result of this work theory has made great advances, expressed particularly in Debye s TMaw, and in the complete elucidation of the time-honoured law of Dulong and Petit. 

Tarassov (1955) and also Desorbo (1953) have considered these ideas in relation to a onedimensional crystal in which case the one-dimensional frequency distribution function predicts a T dependence of the specific heat at low temperatures. In the case of crystalline selenium, however, it has been found necessary to combine the one-dimensional theory with the three-dimensional Debye continuum model in order to obtain quantitative agreement with the data below about 40° K. Tem-perley (1956) has also concluded that the one-dimensional specific heat theory for high polymers would have to be combined with a three-dimensional Debye spectrum proportional to T3 at low temperatures. For a further discussion of one-dimensional models see Sochava and TRAPEZNrKOVA (1957). 

The specific heat of Si3N4 ceramics is in the temperature range 293 up to 1200 K [Cp (293 K) = 0.67 KJ (K kg)-1] nearly independent of the composition of the additives. The isobaric specific heat values agree well with the isochoric specific heat calculated by Debye s theory. Also the Dulong Petit s rule can applied as an approximation of the Cv values [25 J(K mol)-1] at temperatures >1100 K [371]. From the Cp values at around 100 K the amount of the amorphous grain boundary phase can be calculated [371]. 

Specific heat can be predicted fairly accurately by mathematical models through statistical mechanics and quantum theory. For solids, the Debye model gives a satisfactory representation of the specific heat with temperature. Difficulties, however, are encountered when the Debye theory is applied to alloys and compounds. Plastics and glasses are other classes of solids that fail to follow this theory. In such cases, only experimental test data will provide sufficiently reliable specific heat values. 

The relations between experimental quantities mostly concern the temperature effects. First, let us consider the specific heat. In Chap. XV, Sec. 3, we have seen that it should be fairly accurate to use a Debye curve for the specific heat of an alkali halide, using the total number of ions in determining the number of characteristic frequencies in that theory. It is, in fact, found that the experimental values fit Debye curves accurately enough so that we shall not reproduce them. We can then determine the Debye temperatures from experiment, and in Table XXIII-5 we give these values for NaCl and KC1, the two alkali halides... 

This is a heat capacity proportional to the temperature, and in Sec. 5, Chap. V, we computed it for a particular case, showing that it amounted to only about 1 per cent of the corresponding specific heat of free electrons on the Boltzmann statistics, at room temperature. In Table XXIX-2 we show the value of the electronic specific heat at 300° abs., computed from the values of Wi which we have already found, in calories per mole. We verify the fact that this specific heat is small, and for ordinary purposes it can be neglected, so that the specific heat of a metal can be found from the Debye theory, considering only the atomic vibrations. At low temperatures, however, Eq. (2.4) gives a specific heat varying as the first power of the temperature, while Debye s theory, as given in Eq. (3.8),... 

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