Maier-Kelley equation

This is the Maier-Kelley equation for heat capacities (Maier and Kelley, 1932). Several other equations are also commonly used to fit the heat capacity (see for example Tables 7.1, 7.2 and 7.3). Rather than try to present the rest of the equations in this chapter for all current versions of this equation, we continue to use the Maier-Kelley equation as an example. Commonly used equations derived using other heat capacity expressions are presented in Appendix B. 

In machine computations, enthalpies are most conveniently calculated by storing values of o, b, and c for various substances and using the integrated form of the Maier-Kelley equation. 

Combining this with the Maier-Kelley equation... 

This equation is equivalent to equation (1) in Ulbrich and Merino (1974). The heat capacity integrals are evaluated from the integrated form of the Maier-Kelley equation ... 

Because equation 3.76 is valid at the various T conditions, by application of the additivity of polynomials, it follows that the coefficients of Maier-Kelley-type functions for mineral i can also be derived from the corresponding coefficients of the constituent oxides—i.e.,... 

We remind the reader at this point that all the expressions involving a, b, and c in this chapter are derived using the Maier-Kelley expression for the variation of Cp as a function of T. Several other equations are in common use, which will of course change the derived equations. Obviously we cannot derive all the equations for each case the important thing is to see the dependence of the various functions discussed here on the heat capacity. In Appendix B we present the essential equations for several versions of the heat capacity expression. 

The equations derived thus far have been based on an explicit formulation of the variation of heat capacity with temperature (that is, the Maier-Kelley formula). It has led to some rather lengthy equations. 

Equation (8.9) would be the procedure used with the HDNB data set, or any other set that gives standard thermodynamic parameters at 298 K, 1 bar, plus Maier-Kelley heat capacity coefficients. The RHF tables hst AjG° at various temperatures above 298 K at one bar, so that these can be used directly to give the first two terms on the right-hand side of equation (8.6). In other words... 

This, combined with the more complex integration of dG over a temperature interval at one bar pressure, allowed us to calculate the position of phase boundaries at high pressures and temperatures. The next question is how to evaluate the pressure integral (11.1) when a fluid such as H2O or CO2 is involved, either in the pure form, mixed with other fluid components, or reacting with solid phases Obviously, assuming that the molar volume of a fluid is a constant is not even approximately true, and is unacceptable. A possible way to proceed would be to express U as a function of P in some sort of power series, just as we did for Cp as a function of T (equation 7.12). V dP could then be integrated, and we could determine the values of the power series coefficients for each gas or fluid and tabulate them as we do for the Maier-Kelley coefficients. 

Another widely used heat capacity equation was suggested by Berman and Brown (1985). They claim it reproduces calorimetric data better than does Maier-Kelley, and also ensures that Cp approaches the high temperature limit predicted by lattice vibration theory. However, the main reason for knowing... 

The Maier-Kelley and the Berman-Brown equations are intended for temperatures above 298.15K, while the Shomate equation is valid down to OK. The upper temperature limit for aU three equations varies depending on the experimental data available. 

These equations are a bit awkward if you are using a calculator, but are simple to program, and values for many reactions can be obtained directly from the software, e.g., supcrt92 in the case of the Maier-Kelley coefficients. As an example, consider the reaction... 

But now let s discuss something that is confusing. The numbers we calculated in Table 3.6 for the wollastonite reaction are exactly the same numbers you get by mnning supcrt92 for this reaction. Now let s try the same comparison for reaction (3.19) that is, we want to know the standard heat of formation of anhydrite from the elements at temperatures above 25 °C, both from Equation (3.32) and from supcrt92. The Maier-Kelley coefficients are shown in Table 3.7. 

In 3.5.3 we introduced both the Maier-Kelley and the Berman-Brown equations to describe the variation of heat capacity with temperature, and in Equation (5.30) showed an equation for the variation of A G° with T, using the Maier-Kelley formulation. This equation is... 

This rather long equation is simple to use if the Maier-Kelley a, b, and c values are available for all products and reactants, and gives accurate results. 

For reactions in which the aqueous species are isocoulombic but minerals or other phases are also present, one simply uses the Maier-Kelley expression for log A, Equation (9.25), whether or not the pure phases in the reaction are compositionally balanced. For the density model, two expressions for the variation of log AT are combined, giving... 

Maier C. G. and Kelley K. K. (1932). An equation for the representation of high temperature heat content data. J. Amer. Chem. Soa, 54 3243-3246. 

This equation was suggested by Maier and Kelley (1932), and is used in the program supcrt92 (Johnson et al., 1992) (except for a different sign for the c term) for minerals and gases, to be described later. It is... 

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