Heat capacity equations for

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... 

The recommended heat capacity equations for the lanthanide trichlorides are listed in table 9. 

Since C = 0 in each heat-capacity equation for the product gases (Table 4.1), Eq. (4.7) yields... 

Two gram moles of carbon dioxide are heated from 400°C to 1100°C. Calculate AH by integrating the heat capacity equation for carbon dioxide. Compare your result with the value calculated from the enthalpy tables for the combustion gases. 

L E.l Heat Capacity Equations for Organic and Inorganic Compounds (at Low Pressures) ... 

Noting that the last term in the heat capacity equation for SO2 is in the equation analogous to (4 26) is... 

In the following we shall give enthalpy and heat capacity equations for some typical folding models. For each model the relative partition function is given as well as the enthalpy and the heat capacity functions resulting from the temperature derivatives according to equations 35 and 36. In this... 

Diakonov, I.I., Tagirov, B.R., and Ragnarsdottir, K.V. (1998) Standard thermodynamic properties and heat capacity equations for rare earth element hydroxides. I. La(OH)j(s) and Nd(OH)3(s). Comparison of thermochemical and solubility data. Radiochim. Acta, 81, 107-116. 

Appendix C-3 gives constants for the ideal-gas, heat-capacity equation... 

Hea.t Ca.pa.cities. The heat capacities of real gases are functions of temperature and pressure, and this functionaHty must be known to calculate other thermodynamic properties such as internal energy and enthalpy. The heat capacity in the ideal-gas state is different for each gas. Constant pressure heat capacities, (U, for the ideal-gas state are independent of pressure and depend only on temperature. An accurate temperature correlation is often an empirical equation of the form ... 

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... 

Figure 10.12 Comparison for diamond of the experimental Cr.m (circles) and the prediction of the Einstein heat capacity equation with = 1400 K (solid line). The experimental results below T = 300 K are closely spaced in temperature, and not all are shown in the figure.

PRINT "INPUT HEAT CAPACITY DATA FOR EQUATION A+BT+CT 2+DT 3"... 

For the compounds listed below, estimate the constants in the equation for ideal gas heat capacity, equation 3.19, using the method given in Section 8.9.2. 

Repeat the calculation in Exercise 8 for equilibrium conversion and equilibrium concentration, but taking into account variation of AH° with temperature. Again assume ideal gas behavior. Heat capacity coefficients for Equation 6.42 are given in Table 6.196. 

To calculate for this transition, it is necessary to have heat capacity data for both glassy and crystaUine glycerol from near 0 K to the melting point and the heat of fusion of both glass and crystal. Such data [7] lead to a ASm for Equation (11.7) of 19.2 J K mol. Thus, glassy glycerol cannot be assigned zero entropy at OK rather, it possesses a residual entropy of 19.2 J moP. ... 

As we mentioned, it is necessary to have information about the standard enthalpy change for a reaction as well as the standard entropies of the reactants and products to calculate the change in Gibbs function. At some temperature T, A// j can be obtained from Af/Z of each of the substances involved in the transformation. Data on the standard enthalpies of formation are tabulated in either of two ways. One method is to list Af/Z at some convenient temperature, such as 25°C, or at a series of temperatures. Tables 4.2 through 4.5 contain values of AfZ/ at 298.15 K. Values at temperatures not listed are calculated with the aid of heat capacity equations, whose coefficients are given in Table 4.8. 

The procedure followed in the use of the tables of Andersen et al. [1], and Yoneda [4] is illustrated below for the estimation of standard entropies. These tables also include columns of base structure and group contributions for estimating fHm,298.i5K> thc Standard enthalpy of formation of a compound, as well as columns for a, b, and c, the constants in the heat capacity equations that are quadratic in the temperature. Thus it is possible to estimate AfGm gg.isK by appropriate summations of group contributions to Af7/ 298.i5K and to 5m,298.i5K- Then, if information is required at some other temperature, the constants of the heat capacity equations can be inserted into the appropriate equations for AG, as a function of temperature and AGm can be evaluated at any desired temperature (see Equation 7.68 and the relation between AG and In K). 

The procedure for numerical integration (7) is analogous to that for differentiation. Again we will cite an example of its use in thermodynamic problems, the integration of heat capacity data. Let us consider the heat capacity data for solid n-heptane listed in Table A.4. A graph of these data (Fig. A.3) shows a curve for which it may not be convenient to use an analytical equation. Nevertheless, in connection with determinations of certain thermodynamic functions, it may be desirable to evaluate the integral... 

Besides equilibriumconstants, additional thermodynamic data were included, if available, although little emphasis was put on their completeness. The data for primary master species comprise the standard molar thermodynamic properties of formation from the elements (AfG standard molar Gibbs energy of formation AfH°m standard molar enthalpy of formation ApSm- standard molar entropy of formation), the standard molar entropy (5m), the standard molar isobaric heat capacity (Cp.m), the coefficients Afa, Afb, and Afc for the temperature-dependent molar isobaric heat capacity equation... 

Equation (4.36) provides a simple method for estimating an important heat transfer dimensionless group called the Prandtl number. Recall from general chemistry and thermodynamics that there are two types of molar heat capacities, C , and the constant pressure heat capacity, Cp. For an ideal gas, C = 3Cpl5. The Prandtl number is... 

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