Heat Debye theory

Both the Einstein and Debye theories show a clear relationship between apparently unrelated properties heat capacity and elastic properties. The Einstein temperature for copper is 244 K and corresponds to a vibrational frequency of 32 THz. Assuming that the elastic properties are due to the sum of the forces acting between two atoms this frequency can be calculated from the Young s modulus of copper, E = 13 x 1010 N m-2. The force constant K is obtained by dividing E by the number of atoms in a plane per m2 and by the distance between two neighbouring planes of atoms. K thus obtained is 14.4 N m-1 and the Einstein frequency, obtained using the mass of a copper atom into account, 18 THz, is in reasonable agreement with that deduced from the calorimetric Einstein temperature. 

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. 

We must also consider the conditions that are implied in the extrapolation from the lowest experimental temperature to 0 K. The Debye theory of the heat capacity of solids is concerned only with the linear vibrations of molecules about the crystal lattice sites. The integration from the lowest experimental temperature to 0 K then determines the decrease in the value of the entropy function resulting from the decrease in the distribution of the molecules among the quantum states associated solely with these vibrations. Therefore, if all of the molecules are not in the same quantum state at the lowest experimental temperature, excluding the lattice vibrations, the state of the system, figuratively obtained on extrapolating to 0 K, will not be one for which the value of the entropy function is zero. 

The heat capacity of a nonmetal at 7 K is 1.5 J/Kmol. Assuming that the Debye theory holds up to this temperature, what is the molar entropy of the substance at 5 K ... 

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. 

One example of an experimental problem that can usefully be solved by adjusting theory to yield linear equations is the example of the determination of heat capacities, Cp at low temperatures, especially temperatures where experimental values are simply inaccessible. Because Cp cannot be measured experimentally down to absolute zero then an appropriate extrapolation needs to be made (see Frame 16). This latter possibility arises because the Debye theory of heat capacities at low temperatures predicts that as T —> 0 ... 

Figure 3.20 Heat capacity of copper (0 = 343 K [22]), magnesia (0 = 946 K [23]), and diamond (0 = 2230 K [23]) as a function of temperature, as predicted by Debye theory ...

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),... 

Debye Theory of the Heat Capacity of Solids. Debye assumed that a cubic crystal of side L and volume V = Lr can be taken as a vacuum (German Hohlraum) that supports a set of standing waves, each with form... 

The Debye theory of the heat capacities of solid elements W) yields an expression for their entropies. 

Between 0 and Ta one frequently resorts to the Debye theory for the heat capacity of a nonconducting solids, and extended to metals by Sommerfeld. As a first approximation one uses the relation... 

The important conclusions, therefore, to be drawn from the Debye theory are that at low temperatures the atomic heat capacity of an element should be proportional to T, and that it should become zero at the absolute zero of temperature. In order for equation (17.4) to hold, it is necessary that the temperature should be less than about 9/10 this means that for most... 

Since Cv is equal to 464.4 (T/oy at very low temperatures, one heat capacity value under these conditions can be used to derive the characteristic temperature. With this known, the variation of Cv with temperature at higher temperatures can be obtained from equation (17.2). Alternatively, if 6 is found, as described in 17c, from heat capacity measurements at moderate temperatures, the values at low temperatures, i.e., less than /lO, can be estimated from equation (17.4). However, where the characteristic temperature 6 has been determined by two methods, that is, from low temperature and high temperature measurements, the agreement is not exact, showing, as is to be expected, that the Debye theory is not perfect. 

Values of CP may be measured down to a few degrees Kelvin, but a means of extrapolation must be found for the last few degrees. The Debye theory of the heat capacity of solids gives, at low temperatures ... 

The Debye theory assumes that there is a continuous distribution of frequencies from V = 0 to a certain maximum value v = Vj). The final expression obtained for the heat capacity is complicated, but succeeds in interpreting the heat capacity of many solids over the entire temperature range rather more accurately than the Einstein expression. At low temperatures, the Debye theory yields the simple result... 

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. 

It should be noted that, for all the resins considered, the specific heat does not depend on cross-linking or the chemical structure below 100 K. This can be explained by the Debye theory, which states that the specific heat is a function only of the oscillator density, N, and 0/T. 6 is the Debye temperature, which can be determined by elastic parameters, such as Young s modulus, E, N is approximately equal for all resins, since they have nearly equal densities. At low temperatures, roughly the same value of E is asymptotically reached by the epoxy resins. 

Lattice specific heat, according to the Debye theory, is given by Ciatuce.r = 3R DF(xt) J/mole-deg, where R = 8.314 J/mole-deg, and the Debye function at temperature T, given by... 

If a solid were classical, the heat capacity would be 3 Alfc. This is indeed the case at high temperatures and is called the law of Dulong and Petit. However, the experimental heat capacity goes to zero at low temperatures. This can be explained by regarding the solid as a collection of quantized oscillators. The only difficulty is to determine the spectrum of frequencies of the oscillators. For many purposes, the solid can be regarded as an elastic continuum. The result is the Debye theory. If something more sophisticated is needed one must solve for the normal modes of the crystal, i.e., the method of lattice dynamics. 

As mentioned in 13 6, the transition metals have heat capacities which rise considerably above 3R per mole, and the same applies to their ionic salts. Germanium and hafnium also have heat capacities which are not in agreement with the Debye theory. Both metals have peaks in their curves at about 70 K, and in the case of hafiiium the heat capacity at the peak is as much as 46 J mol . At higher temperatures the heat capacity settles down to the Dulong and Petit value. 

Although the Debye theory predicts a heat-capacity curve which is often in good agreement with experiment, the assumptions in the theory with regard to the frequency spectrum are not necessarily correct. For certain simple types of lattice, Blackman (1937) was able to make a detailed calculation of the frequencies of the normal modes, allowing for the atomic structure of the system. It was found, as is assumed in the Debye theory, that there is an upper limit, to the possible frequencies, and it was also confirmed that the law should hold at very low temperatures. On the other hand, Blackman found that the frequency distribution does not have such a simple form (equation (13 54)) as was assumed by Debye and, in fact, it may have two or more peaks. In view of these results it seems that the agreement of the Debye theory with experiment is better than might reasonably have been expected. 

The heat capacity and transition enthalpy data required to evaluate Sm T ) using Eq. 6.2.2 come from calorimetry. The calorimeter can be cooled to about 10 K with liquid hydrogen, but it is difficult to make measurements below this temperature. Statistical mechanical theory may be used to approximate the part of the integral in Eq. 6.2.2 between zero kelvins and the lowest temperature at which a value of Cp,m can be measured. The appropriate formula for nonmagnetic nonmetals comes from the Debye theory for the lattice vibration of a monatomic crystal. This theory predicts that at low temperatures (from 0 K to about 30 K), the molar heat capacity at constant volume is proportional to Cv,m = aT, ... 

According to the Debye theory, the heat capacity Cy of a pure solid has the form... 

FIGURE 18.6 The Debye theory of heat capacity of crystals agrees better with experimental values of heat capacity at low temperatures. 

In electrical conductors, there is also a contribution to the heat capacity from the electronic motion (see Section 28.3). Heat capacities are hard to measure at low temperatures, and data for temperatures below 15 K are hard to find. Equation (28.2-29) is commonly used as a substitute for experimental data between 0 K and 15 K. Modifications to the Debye theory have been devised that use a temperature-dependent Debye temperature and give improved agreement with experiment. ... 

In this chapter, we have discussed the structure of solids and liquids. Many solids are crystalline, with molecular units arranged in a regular three-dimensional lattice. There are two principal theories for the vibrations of lattices of atoms, the Einstein and the Debye theories. In the Einstein theory the normal vibrational modes of the lattice are assumed to vibrate with the same frequency. In the Debye model, the normal modes of the lattice are assumed to vibrate with the same distribution of frequencies as would a structureless solid. In each theory, the formula for the heat capacity of the solid lattice conforms to the law of Dulong and Petit at high enough temperature. 

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