Debye heat capacity

The S(T) on the left-hand side of Eq. (39) could be an entropy 5vib,atp(7 ) determined from experiments or it could be the result of a detailed model calculation based on a certain E((o) in Eq. (27). There is a solution 0d to Eq. (39) for each temperature T. We shall call that 0d a Debye temperature. Because we represent the property of a system that has 3rL degrees of freedom with a single parameter 0d, it is obvious that we have to pay a price. In this case, the price is that 0d varies with T and also with the physical property that is modeled. (Here 0d refers to the entropy.) One may therefore introduce one Debye temperature 0 that describes the vibrational entropy, another Debye temperature 0c that describes the vibrational heat capacity, etc. The heat capacity Debye temperature 0c would be the solution to... 

Figure 4 The entropy Debye temperature Qs T) and the heat capacity Debye temperature Qc(T) for TiC, as obtained from recommended thermodynamic data (18), using Eqs. (39) and (40).

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... 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

Its value at 25°C is 0.71 J/(g-°C) (0.17 cal/(g-°C)) (95,147). Discontinuities in the temperature dependence of the heat capacity have been attributed to stmctural changes, eg, crystaUi2ation and annealing effects, in the glass. The heat capacity varies weakly with OH content. Increasing the OH level from 0.0003 to 0.12 wt % reduces the heat capacity by approximately 0.5% at 300 K and by 1.6% at 700 K (148). The low temperature (<10 K) heat capacities of vitreous siUca tend to be higher than the values predicted by the Debye model (149). 

Figure 3.6 The heat capacity of a solid as a function of the temperature divided by the Debye temperature...

These heat capacity approximations take no account of the quantal nature of atomic vibrations as discussed by Einstein and Debye. The Debye equation proposed a relationship for the heat capacity, the temperature dependence of which is related to a characteristic temperature, Oy, by a universal expression by making a simplified approximation to the vibrational spectimii of die... 

Equation (4.2) requires that the total area above 0 Kelvin be obtained, but heat capacity measurements cannot be made to the absolute zero of temperature. The lowest practical limit is usually in the range from 5 K to 10 K, and heat capacity below this temperature must be obtained by extrapolation. In the limit of low temperatures, Cp for most substances follows the Debye low-temperature heat capacity relationship11 given by equation (4.4)... 

For Cy/T to approach zero as T approaches zero, CV must go to zero at a rate at least proportional to T. Earlier, we summarized the temperature dependence of Cy on T for different substances and showed that this is true. For example, most solids follow the Debye low-temperature heat capacity equation of low T for which... 

Thus Cp m and CVm differ little from one another at low temperatures. The Debye low-temperature heat capacity equation (and other low-temperature relationships) we have summarized calculates Cp.m, as well as CV. m, without significant error. 

E4.1 Show that at very low temperatures where the Debye low temperature heat capacity equation applies that the entropy is one third of the heat capacity. 

Graph the above data in the form Cp,m/T against T2 to test the validity of the Debye low-temperature heat capacity relationship [equation (4.4)] and find a value for the constant in the equation, (b) The heat capacity study also revealed that quinoline undergoes equilibrium phase transitions, with enthalpies as follows ... 

Intermediate values for C m can be obtained from a numerical integration of equation (10.158). When all are put together the complete heat capacity curve with the correct limiting values is obtained. As an example, Figure 10.13 compares the experimental Cy, m for diamond with the Debye prediction. Also shown is the prediction from the Einstein equation (shown in Figure 10.12), demonstrating the improved fit of the Debye equation, especially at low temperatures. 

Figure 10.14 Graph showing the limiting behavior at low temperatures of the heat capacity of (a), krypton, a nonconductor, and (b). copper, a conductor. The straight line in (a) follows the prediction of the Debye low-temperature heat capacity equation. In (b), the heat capacity of the conduction electrons displaces the Debye straight line so that it does not go to zero at 0 K.

The Debye temperature, can be calculated from the elastic properties of the solid. Required are the molecular weight, molar volume, compressibility, and Poisson s ratio.11 More commonly, do is obtained from a fit of experimental heat capacity results to the Debye equation as shown above. Representative values for 9o are as follows ... 

Figure 10.15 Comparison of the fit of the Debye heat capacity equation for several elements. Reproduced from K. S. Pitzer. Thermodynamics. McGraw-Hill, Inc., New York, 1995, p. 78. Reproduced with permission of the McGraw-Hill Companies.

Dalton s law of partial pressure 264, 406 Davies, C. A. 449, 456, 507 Debye heat capacity equation for solids 572-80, 651-4... 

Debye heat capacity equation 572-80 Einstein heat capacity equation 569-72 heat capacity from low-lying electronic levels 580-5 Schottky effect 580-5 statistical weight factors in energy levels of ideal gas molecule 513 Stirling s approximation 514, 615-16 Streett, W. B. 412... 

The discovery of a transition which we identify with this has been reported by Simon, Mendelssohn, and Ruhemann,16 who measured the heat capacity of hydrogen with nA = 1/2 down to 3°K. They found that the heat capacity, after following the Debye curve down to about 11°K, rose at lower temperatures, having the value 0.4 cal/deg., 25 times that of the Debye function, at 3°K. The observed entropy of transition down to 3°K, at which the transition is not completed, was found to be about 0.5 E.U. That predicted by Eq. (15) for the transition is 2.47 E.U. 

Figure 11. Displayed are the TLS heat capacities as computed from Eq. (29) appropriate to the experiment time scales on the order of a few microseconds, seconds, and hours. A value of c = 0.1 was used here. If one makes an assumption on the specific value of Aq, it is possible to superimpose the Debye contribution on this graph, which would serve as the lowest bound on the total heat capacity. As checked for Aq = cod. the phonon contribution is negligible at these temperatures.

UT/6D . This limiting expression is known as Debye s third-power law for the heat capacity (problem 15). It is employed in thermodynamics to evaluate the low-temperature contribution to the absolute entropy. 

Fig. 3.1. Debye temperature versus temperature for some materials. The values of Debye temperatures are obtained by heat capacity measurements [10,11].

Since in our temperature range, the Debye temperature of Ge is 370K [47], the phonon contribution to the heat capacity can be neglected. Hence, the heat capacity of our samples is expected to follow the equation ... 

U. Piesbergen, Heat Capacity and Debye Temperatures G.Giesecke, Lattice Constants J.R. Drabble, Elastic Properties... 

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