Heat capacity calculation methods

Example 4.1 illustrates this kind of calculation and compares the result with that obtained by taking the steam to behave as an ideal gas. For nonideal gases with known PVT equations of state and low pressure heat capacities, the method of calculation is the same as for compressors which is described in that section of the book. 

Table 1.2. Values of entropy and heat capacity calculated by the MINDO/3 [16] method and obtained experimentally at 298 K...

In the case of lanthanide trihalides, we endorse the computational method of heat capacity calculations in a wide interval of temperatures (from about T = 0 to T ), based on the analysis of the experimental values of the low-temperature heat capacities, Cp,exp(71- High-temperature enthalpy increments are used for correcting Cp,cai(T) values in the range of temperature T > 0.57. Such an approach removes the restriction of quasi-harmonic approximation of a heat capacity in this temperature range. 

To reliably select separate thermodynamic parameters for the compoimds under consideration, we must improve the suitable calculation procedures and/or criteria that allow the validity of the available experimental data to be estimated. To this end, we have selected lanthanide trifluorides as example. They are much less sensitive to hydrolysis although they have noticeable corroding action on container materials at high temperatures. In view of this, we have chosen the method of heat capacity calculations over a wide temperature range, up to the melting point, based on an analysis of the experimental low-temperature heat capacities and high-temperature enthalpy increments. This method is described below. Its application to lanthanide di- and trichlorides will be presented later on. 

This chapter presents quantitative methods for calculation of enthalpies of vapor-phase and liquid-phase mixtures. These methods rely primarily on pure-component data, in particular ideal-vapor heat capacities and vapor-pressure data, both as functions of temperature. Vapor-phase corrections for nonideality are usually relatively small. Liquid-phase excess enthalpies are also usually not important. As indicated in Chapter 4, for mixtures containing noncondensable components, we restrict attention to liquid solutions which are dilute with respect to all noncondensable components. 

Note This method of temperature regulation does not give all properties of the canonical ensemble. In particular, you cannot calculate Cy, heat capacity at constant volume. 

The implicit Crank-Nicholson integration method was used to solve the equation. Radial temperature and concentrations were calculated using the Thomas algorithm (Lapidus 1962, Carnahan et al,1969). This program allowed the use of either ideal or non-ideal gas laws. For cases using real gas assumptions, heat capacity and heat of reactions were made temperature dependent. 

Although these potential barriers are only of the order of a few thousand calories in most circumstances, there are a number of properties which are markedly influenced by them. Thus the heat capacity, entropy, and equilibrium constants contain an appreciable contribution from the hindered rotation. Since statistical mechanics combined with molecular structural data has provided such a highly successful method of calculating heat capacities and entropies for simpler molecules, it is natural to try to extend the method to molecules containing the possibility of hindered rotation. Much effort has been expended in this direction, with the result that a wide class of molecules can be dealt with, provided that the height of the potential barrier is known from empirical sources. A great many molecules of considerable industrial importance are included in this category, notably the simpler hydrocarbons. 

A considerable variety of experimental methods has been applied to the problem of determining numerical values for barriers hindering internal rotation. One of the oldest and most successful has been the comparison of calculated and observed thermodynamic quantities such as heat capacity and entropy.27 Statistical mechanics provides the theoretical framework for the calculation of thermodynamic quantities of gaseous molecules when the mass, principal moments of inertia, and vibration frequencies are known, at least for molecules showing no internal rotation. The theory has been extended to many cases in which hindered internal rotation is... 

A number of other thermodynamic properties of adamantane and diamantane in different phases are reported by Kabo et al. [5]. They include (1) standard molar thermodynamic functions for adamantane in the ideal gas state as calculated by statistical thermodynamics methods and (2) temperature dependence of the heat capacities of adamantane in the condensed state between 340 and 600 K as measured by a scanning calorimeter and reported here in Fig. 8. According to this figure, liquid adamantane converts to a solid plastic with simple cubic crystal structure upon freezing. After further cooling it moves into another solid state, an fee crystalline phase. 

Approximate values can be calculated for solids, and liquids, by using a modified form of Kopp s law, which is given by Werner (1941). The heat capacity of a compound is taken as the sum of the heat capacities of the individual elements of which it is composed. The values attributed to each element, for liquids and solids, at room temperature, are given in Table 8.2 the method illustrated in Example 8.6. 

The test is primarily a screening tool relative to reactivity of substances and reaction mixtures and is highly useful for that purpose. The determined initiation temperature is approximate. The energy calculations based on temperature increase and heat capacities are semi-quantitative because of the quasi-adiabatic mode of the system operation. The method of insulating the test cell results in moderate reproducibility of temperature rise and related pressure rise. Another disadvantage is the relatively small sample quantity with respect to full scale quantities thus, there could be a problem in that the sample may not be truly representative. 

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. 

Figure 4.2. Variation of heat capacity with temperature as calculated from the equations of Frenkel et al. [4]. The differences observed between isotopic species and the way heat capacity depends on molecular size and structure can be described thermodynamically, but they must be explained by the methods of quantum-statistical thermodynamics. The right-hand scale is for H2 and D2 the left-hand scale is for the other compounds.

We can calculate AH from thermal data alone, that is, from calorimetric measurements of enthalpies of reaction and heat capacities. It would be advantageous if we could also compute AS from thermal data alone, for then we could calculate AG or Ay without using equilibrium data. The requirement of measurements for an equilibrium state or the need for a reversible reaction thus could be avoided. The thermal-data method would be of particular advantage for reactions for which AG or AT is very large (either positive or negative) because equilibrium measurements are most difficult in these cases. 

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. 

Heat contents can be measured accurately by a number of techniques based on a drop method. This involves heating the sample to a high temperature and dropping it directly into a calorimeter held at a lower temperature. The calorimeter then measures the heat evolved while the sample cools to the temperature of the calorimeter. The temperature at which the sample is initially heated is varied and a plot of Ht — 298.15 vs temperature is drawn (Fig. 4.1). Heat capacities can then be calculated using Eq. (3.9). A popular calorimeter for this is the diphenyl ether calorimeter (Hultgren et al. 1958, Davies and Pritchard 1972) but its temperature range is limited below about 1050 K. 

Flame temperatures can be measured directly, using special high-temperature optical methods. They can also be calculated (estimated) using heat of reaction data and thermochemical values for heat of fusion and vaporization, heat capacity, and transition temperatures. Calculated values tend to be higher than the actual experimental results, due to heat loss to the surroundings as well as the endothermic decomposition of some of the reaction products. Details regarding these calculations, with several examples, have been published [5]. 

Exploiting the principles of statistical mechanics, atomistic simulations allow for the calculation of macroscopically measurable properties from microscopic interactions. Structural quantities (such as intra- and intermolecular distances) as well as thermodynamic quantities (such as heat capacities) can be obtained. If the statistical sampling is carried out using the technique of molecular dynamics, then dynamic quantities (such as transport coefficients) can be calculated. Since electronic properties are beyond the scope of the method, the atomistic simulation approach is primarily applicable to the thermodynamics half of the standard physical chemistry curriculum. 

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