Solid Heat Capacity

Example 15 Estimate Solid Heat Capacity of Dinenzothiophene.. . ... 

Solid Heat Capacity Solid heat edacity increases with increasing temperature, with steep rises near the triple point for many compounds. When experimental data are available, a simple polynomial equation in temperature is often used to correlate the data. It should be noted that step changes in heat capacity occur if the compound undergoes crystalline state changes at mfferent temperatures. 

There are no reliable prediction methods for solid heat capacity as a function of temperature. However, the atomic element contribution method of Hurst and Harrison,which is a modification of Kopp s Rule, provides estimations at 298.15 K and is easy to use ... 

Cps = solid heat capacity at 298.15 K, J/mol K n = number of different atomic elements in the compound N, = number of atomic elements i in the compound Agi = numeric value of the contribution of atomic element i found in Table 2-393... 

Example 15 Estimate solid heat capacity of dibenzothiophene, Ci2HsS. The required atomic element contributions from Table 2-393 are C = 10.89, H = 7.56, and S = 12.36. Substituting in Eq. (2-63) ... 

TABLE 2-393 Atomic Element Contributions to Estimate Solid Heat Capacity at 298.15 K... 

A relatively simple example of a group contribution technique is the method for estimating liquid and solid heat capacities (159). This method is a modification of Kopp s rule (160,161) which was originally proposed in 1864. Kopp s rule states that, at room temperature, the heat capacity of a solid compound is approximately equal to a stoichiometric summation of the heat capacities of its atoms (elements). The Hurst-Harrison modified equation is as follows ... 

The second class comprises conventional solids, defined by a chemical formula, but whose property requirements are very minimal. In this class we included lignin, cellulose, mannan, galactan, xylan, arabinan and the biomass. The properties specified in the database include molecular weight, heat of formation, solid molar volume, and solid heat capacity. 

When the solids heat capacity is higher (as is the case for most organic materials), the temperature reduction is inversely proportional to the heat capacity. 

The fluidized bed characteristics of high solids heat capacity, large interfacial heat transfer area, and good solids mixing allow the assumptions of thermal equilibrium between the solids and the gas, uniform bed temperature and negligible heat capacitance of the gas. An additional assumption required to use equation (9) is that the reactions do not change the gas volume. 

Low temperature heat capacity measurements by Anderson 10) y Bronson and MacHattie 42) y Keesom and van den Ende 176) y and Armstrong and Grayson-Smith 16) were used to calculate an entropy and enthalpy at 298 K. of 13.58 e. u. and 1536 cal./gram atom, respectively. From many sources, Kelley 186) derives an equation for the solid heat capacity above 298 K. Kubaschewski and coworkers 206) select 544.5 K. as the melting point and 2600 50 cal./gram atom for the heat of melting. Data on... 

Johnston, Hersh, and Kerr 168) have measured the heat capacity of the crystalline form from 13 to 305 K., and calculate the entropy at 298 K. to be 1.403 0.005 e. u. and the enthalpy at 298 K. to be 292 cal./gram atom. In the absence of definite information, we have estimated that the solid heat capacity will reach a value of 7.5 cal./degree/-gram atom at the melting point and have extrapolated the low temperature measurements in a reasonable manner to obtain this value. Cueilleron (77) has measured the melting point of the crystalline variety and reports a range of 2273 to 2348 K., which we have... 

Clusius and Schachinger (66) have measured the heat capacity from 12 to 273 K., and Clement and Quinnell (58) from 1.7 to 21.3 K., from which can be derived the entropy at 298 K. of 13.82 e. u. and an enthalpy at 298 K. of 1578 cal./gram atom. Roth, Meyer, and Zeumer (375) have reported data for the solid heat capacity, melting point, heat of melting, and liquid heat capacity. Oelsen (353) has measured the heat of melt-... 

The entropy at 298° K. has been estimated by Lewis and Gibson (313) to be 7.8 0.5 e. u. Kelley (185) has given an equation for the solid heat capacity from 298° to 1800° K. which we have extrapolated to the melting point. We have assumed that the heat capacity of the liquid is the same as that of the solid at the melting point. We have... 

Kelley 186) estimates the entropy at 298 K. as 12.5 =t 0.5 e. u. The solid heat capacity above room temperature was estimated by comparison with calcium. Eastman, Cubicciotti, and Thurmond 93) have reported a transition point at 862 K. and a melting point of 1043 K., in good agreement with the review of Kubaschewski, Brizgys, Huchler, Jauch, and Reinartz 306). Kubaschewski and coworkers 306) have estimated... 

Solids. Only very rough approximations of solid heat capacities can be made. Kopp s rule (1864) should only be used as a last resort when experimental data cannot be located or new experiments carried out. Kopp s rule states that at room temperature the sum of the heat capacities of the individual elements is approximately equal to the heat capacity of a solid compound. For elements below potassium, numbers have been assigned from experimental data for the heat capacity for each element as shown in Table 4.2. For liquids Kopp s rule can be applied with a modified series of values for the various elements, as shown also in Table 4.2. For example, the heat capacity at room temperature of Na2S04 lOHaO would be 2(6,2) + 1(5.4) + 14(4.0) + 20(2.3) = 119.8 cal/(g mol)( C). The heat capacity of coal can be estimated from equations in the Coal Conversion Systems Technical Data Book cited in the supplementary references. Consult Reid or Perry s Handbook fpr tables of heat capacity data for solids. 

Solids Solid heat capacity increases with increasing temperature and is proportional to V near absolute zero. The heat capacity at a solid-solid phase transition becomes large, and there can be a substantial dif-... 

For a quick estimation of sohd heat capacity specifically at 298.15 K, the very simple modification of Kopp s rule [Kopp, H., Ann. Chem. Pharm. Liebig, 126 (1863) 362] by Hurst and Harrison [Hurst, J. E., and B. K. Harrison, Chem. Eng. Comm., 112 (1992) 21] can be used. At other temperatures and to obtain the temperature dependence of the solid heat capacity, the method given below by Goodman et al. should be used. 

Example Estimate the solid heat capacity for p-cresol at 307.93 K. Structure ... 

TABLE 2-349 Group Values and Nonlinear Correction Terms for Estimation of Solid Heat Capacity with the Goodman et al. Method... 

TABLE 2-350 Element Contributions to Solid Heat Capacity for the Modified Kopp s Rule ... 

Solid density = 1150 kg/m3 Heat of vaporization = 2450 J/g Solid heat capacity 2.5 J/(g-K) Water heat capacity 4.184 J/(g-K)... 

The fact that a quantum oscillator of frequency u> does not interact effectively with a bath of temperature smaller than hu>/kg implies that if the low temperature behavior of the solid heat capacity is associated with vibrational motions, it must be related to the low frequency phonon modes. The Debye model combines this observation with two additional physical ideas One is the fact that the low frequency (long wavelength) limit of the dispersion relation must be... 

The Tg curves for the water-maltodextrin-sucrose system was plotted using the expanded Gordon-Taylor model (Equation 21.1) for ternary systems, considering the variation in heat capacity for water (ACpi) equal to 1.94 J/g °C (Kalichevsky and Blanshard, 1993) and for sucrose (ACp2) equals to 0.60 J/g °C (Roos, 1993). The value for ACps of 0.24 J/g °C used here for maltodextrin MOR-REX 1910 was estimated from considering the value for k (the ratio of changes in the water and solid heat capacities at Tg) to be equal to 8.055. Table 21.1 shows the parameters used to determine the glass-transition curves for the maltodextrins, with and without additives (sucrose). 

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