Heat capacity empirical estimates

To use equation 2.10 correctly, we need to know how the heat capacities vary in the experimental temperature range. However, these data are not always available. A perusal of the chemical literature (see appendix B) will show that information on the temperature dependence of heat capacities is much more abundant for gases than for liquids and solids and can be easily obtained from statistical mechanics calculations or from empirical methods [11]. For substances in condensed states, the lack of experimental values, even at a single temperature, is common. In such cases, either laboratory measurements, using techniques such as differential scanning calorimetry (chapter 12) or empirical estimates may be required. 

Estimation Procedures. There are basically two ways which have been developed to deal with the fact that heat capacity terms are large in reactions involving ions. One is based on empirical relationships (the entropy correspondence principle) between ionic entropies at different temperatures which Criss and Cobble (62) developed and checked to 200 C. Lewis (63) has checked a number of its predictions against available experimental evidence and has found the method reasonably satisfactory for several... 

The lack of auxiliary data such as crystal field energies for the tribromides and triiodides further limits the value of the semi-empirical approach used here to estimate high temperature heat capacities. 

Kopp s rule is a simple empirical method for estimating the heat capacity of a solid or liquid at or near 20°C. According to this rule, Cp for a molecular compound is the sum of contributions (given in Table B.IO) for each element in the compound. For example, the heat capacity of solid calcium hydroxide. Ca(OH)2, would be estimated from Kopp s rule as... 

Use the empirical rule of Dulong and Petit stated in Problem 7 to estimate the specific heat capacities of vanadium, gallium, and silver. 

Fit empirical heat capacity data with a suitable function of temperature by estimating the values of the coefficients in the function. 

Table VII. Comparision of experimental heat capacity with values estimated from an empirical additivity approach.

This is in fair agreement with -log K2 = 10.18 at 50°C computed from the empirical temperature function in Table A1.1. The disagreement reflects the fact that the heat capacity of the reaction is not zero, as assumed in the calculation. If we refine our estimate by including the heat capacity correction term of the expanded integrated van t Hoff equation, the predicted value for K2 at 50°C is... 

As mentioned earlier, the constant-pressure heat capacity of solids is a function of temperature in fact, Cp goes to zero at the absolute zero of temperature and approaches a constant at high temperatures. An approximate estimate for Cp of solids for temperatures of interest to chemical engineers comes from the empirical law (or observation) of DuLong and Petit that... 

An experimental test of these equations requires additional parameters over those which have been used before. The correlation distance I will be required again and also the friction constant / , but an evaluation of H also requires a knowledge of yo and Cp. These can probably be estimated with fair reliability for the system n-hexane - - aniline for which the attenuation and dispersion were measured by Chynoweth and Schneider. Both components have heat capacities close to 50 cals/mole deg, and one can reasonably take this value for the mixture. The approximate empirical rule that Cp —Cy 5R allows an estimate of... 

Although surface areas and particle size distributions undoubtedly play a role in determining maximum loading levels, particle shape, density and heat capacity are also likely to play a role. The complex nature of the inter-relationships between these factors and the lack of suitable metrics has made it difficult to explore this phenomenon beyond a strictly empirical level, however. BET surface area estimates, oil absorption numbers and average particle... 

Specific heat capacities of solid substances near normal atmospheric temperature can be estimated with a reasonable degree of accuracy by combining two empirical rules. 

Another rough estimation method for the specific heat capacity of aqueous solutions is based on the empirical relationship... 

The strict additivity of the heat capacity contributions of the group vibrations, and the continuous change in 1 with chemical composition, led to the development of addition schemes for heat capacities. As long as the contribution which corresponds to the backbone grouping of a polymer is known empirically, it is possible to estimate heat capacities of such polymers and copolymers. The just completed discussion of the Tarasov analysis indicates that, indeed, the group vibrations are additive, but that for the low-temperature heat capacities deviations are to be expected as long as the intermolecular skeletal vibrations are not fuUy excited. In this case it is possible to estimate 3 from similar substances for an improved estimate. 

The first theoretical work providing information on the Debye temperature (Go) of intermetallic clathrates dates back to the year 1999 [33]. Molecular dynamics calculations for the carbon-framework of type-I and type-II clathrates used a Lennard-Jones potential (later on also for Si-based clathrates [34]). 0d for Ci36 [35] and for Siiae [34] were estimated from the calculated elastic constant Cn applying the empirical relation Qd = —11.3964 + 0.3475 x C — 1.6150 x 10 X Cj 1. Moriguchi et al. [36] used an empirical bond-order potential developed by Tersofif for the calculation of several thermodynamic properties, including the heat capacity, for the type-I and type-II Si networks. From the heat capacity data in the temperature range from 0 to 150 K 6d was extracted applying the Debye-model. The heat capacity, Cy, was calculated by the density functional theory (DFT),... 

Recently, semiempirical methods based on DFT calculations have been developed for catalyst screening. These methods include linear scaling relationships [41, 42] to transfer thermochemistry from one metal to another and Brpnsted-Evans-Polanyi (BEP) relationships [43 7]. Here, these methods and also methods for estimation of the surface entropy and heat capacity are briefly discussed. Because of their screening capabilities, semiempirical methods can be used to produce a first-pass microkinetic model. This first-pass model can then be refined using more detailed theory aided by analytical tools that identify key features of the model. The empirical bond-order conservation (BOC) method, which has shown good success in developing microkinetic models of small molecules, has recently been reviewed [11] and will not be covered here. 

To describe heat capacities of copolymers, it is naturally impossible to measure each and every composition. The heat capacities in the solid as well as the liquid states tend to be additive with respect to composition, which allows estimates to be made of the heat capacities of the copolymers (see also the ATHAS empirical addition schemes. Refs. 42 and 43). 

The corresponding values of AC are adduced in Table 6.3. The accuracy of calculations of ACp according to Equation 6.30 can be confirmed by using the estimation of the heat capacity jump AN value according to the Boyer empirical relationship [85] ... 

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