Heat capacity Debye theory

However, the possibility that might not go to zero could not be excluded before the development of the quantum theory of the heat capacity of solids. When Debye (1912) showed that, at sufficiently low... 

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. 

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. 

Tn the critical region of mixtures of two or more components some physical properties such as light scattering, ultrasonic absorption, heat capacity, and viscosity show anomalous behavior. At the critical concentration of a binary system the sound absorption (13, 26), dissymmetry ratio of scattered light (2, 4-7, II, 12, 23), temperature coefficient of the viscosity (8,14,15,18), and the heat capacity (15) show a maximum at the critical temperature, whereas the diffusion coefficient (27, 28) tends to a minimum. Starting from the fluctuation theory and the basic considerations of Omstein and Zemike (25), Debye (3) made the assumption that near the critical point, the work which is necessary to establish a composition fluctuation depends not only on the average square of the amplitude but also on the average square of the local... 

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 ... 

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. 

The heat capacity of EuS was measured to test the predictions of spin-wave theory from 1° to 38°K. by McCollum and Callaway (137) and independently from 10° to 35°K. by Moruzzi and Teaney (145). A sharp Neel peak was found at 16.2 °K. Magnetic and lattice contributions to the heat capacities were resolved on the assumption of a dependence for the lattice and a T dependence for the magnetic contribution at temperatures above the Neel point. A plot of CT vs. yields a straight line between 21° and 31 °K. and a Debye temperature of 208 °K. 

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 main value of Debye s theory is that it provides a reasonably satisfactory treatment of the heat capacity of solids. 

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. 

Heat Capacities and the Debye-Hfickel Theory.—By combining the general equation (44.46) with the expression for L2 [equation (44.39)3 derived from the Debye-Huckel theory, it is found that for a solution containing a single strong electrolyte,... 

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... 

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