Heat capacity Debye equation

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 ... 

The heat capacity and entropy of TiBr Ccr) have been measured over the temperature range 51 to 800 K by King et al. (2). Heat capacities above 800 K are estimated from graphical extrapolation. The value of S"(298.15 K) is derived from these data, based on S (51 K) - 8.60 cal K mol. The value of S (51 K) is estimated from a Debye-Einsteln extrapolation of the measured heat capacities, the equation being C - D(70.0/T) + E(120/T) + 2E(306/T). It is assumed that all electronic entropy is... 

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. 

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... 

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 ... 

Although the Debye model reproduces the essential features of the low- and high-temperature behaviour of crystals, the model has its limitations. A temperature-dependent Debye temperature, d(F), can be calculated by reproducing the heat capacity at each single temperature using the equation... 

This is, of course, the reciprocal of the time taken for an atom to move a lattice site distance into a vacancy. um can be estimated from the heat capacity of the crystalline material using the Einstein or Debye models6 of atoms as harmonic oscillators in a lattice. Combining Equations (2.30) and (2.31) gives the number of atoms moving per second as... 

Data for a large number of organic compounds can be found in E. S. Domalski, W. H. Evans, and E. D. Hearing, Heat capacities and entropies in the condensed phase, J. Phys. Chem. Ref. Data, Supplement No. 1, 13 (1984). It is impossible to predict values of heat capacities for solids by purely thermodynamic reasoning. However, the problem of the solid state has received much consideration in statistical thermodynamics, and several important expressions for the heat capacity have been derived. For our purposes, it will be sufficient to consider only the Debye equation and, in particular, its limiting form at very low temperamres ... 

The symbol 9 is called the characteristic temperamre and can be calculated from an experimental determination of the heat capacity at a low temperature. This equation has been very useful in the extrapolation of measured heat capacities [16] down to OK, particularly in connection with calculations of entropies from the third law of thermodynamics (see Chapter 11). Strictly speaking, the Debye equation was derived only for an isotropic elementary substance nevertheless, it is applicable to most compounds, particularly in the region close to absolute zero [17]. 

Use of Debye Equation at Very Low Temperatures. Generally, it is assumed that the Debye equation expresses the behavior of the heat capacity adequately below about 20 K [9]. This relationship [Equation (4.68)],... 

Putnam and Boerio-Gates [19] have measured the heat capacity of pure, crystalline sucrose from 4.99 K to above 298.15 K. Their smoothed results up to 298.15 K are shown in Table 11.7. Use the Debye equation and numerical integration of the experimental data to calculate at 298.15 K. 

In deriving the Debye heat capacity equation, one assumes that the atoms in an atomic solid are vibrating with a range or frequencies v varying from u = 0 to a maximum u — vm. The resulting equation for calculating Cv, m is... 

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 ... 

Our discussion of the specific heat capacity of polymers on the preceding pages has been quite empirical. There are, in fact, few fundamental rules that can be used for the prediction of specific heat capacity. At very low temperatures, the equations of Debye and Einstein may be used. 

The melting point of n-hexanol is —47.2°C (225.8 K), and its enthalpy of fusion is 3676 cal/ (g-mol). The heat capacity of crystalline n-hexanol (Cp)erysiai at temperatures below 18.3 K may be estimated using the Debye-Einstein equation ... 

There is some literature data on heat capacities and transformed heat capacities, but not enough to justify including them in the general treatments here, but they are of interest and may not be negligible. The adjustment of the heat capacity of a species for the ionic strength depends on both the first and second derivatives of the coefficient alpha in the Debye-Huckel equation. 

Molar heat capacities Cp "(crystalline reactant) can be determined down to about 10 K, and the Debye equation that applies at very low temperatures can be used to estimate heat capacities below 10 K. The Debye equation is Cpm ° = kT Heat of combustion measurements can be used to obtain AfH ° (298.15 K) of the crystalline substance at 298.15 K, and the heat of solution makes it possible to calculate AfH ° (aq soln,298.15 K). When third law measurements have been made, the standard Gibbs energy of formation of the substance in dilute aqueous solution can be calculated using... 

It will be seen from the Debye equation (17.2) that Cr is a function of 9/T only, and hence the plot of Cy against T/0 (or log T/9) should yield a curve that is the same for all solid elements.f The nature of the curve is shown in Fig. 9, and it is an experimental fact that the heat capacities of many elements, and even of a few ample compounds, e.g., ionic cr stals such... 

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