Relative Partial Molar Heat Capacities

In equation (7.70), we defined the relative partial molar enthalpy L, as 

The difference Cp. -C°pi is the relative partial molar heat capacity Jt. Thus 

The quantity J can be used to calculate the effect of temperature on Z, through equation (7.100). 

Solution First, we obtain an expression for L2 as a function of temperature 

Separating variables and doing an indefinite integration gives 

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Differentiation of equation (7.65) with respect to temperature gives an equation for Ji, the relative partial molar heat capacity, given by... 

This, of course, is the difference between the heat capacity of the solution and the sum of those of the unmixed liquid elements. Using Eq. (38) and defining relative partial molar heat capacities of the components as... 

Figure 17.5 Derived thermodynamic properties at T — 298.15 K and p = 0.1 MPa for (2Cic-CfiHi2 + X2n-CjHi4) (a) excess molar heat capacities obtained from the excess molar enthalpies (b) relative partial molar heat capacities obtained from the excess molar heat capacities (c) change of the excess molar volume with temperature obtained from the excess molar volumes and (d) change of the excess molar enthalpies with pressure obtained from the excess molar volumes.

Cp m of -1.4 J K-1 - mol-1 is again of moderate size. Figure 17.5b summarizes the relative partial molar heat capacity Jt = (CA m,- C t m,We note that the molar heat capacity of hexane in the infinitely dilute solution is 7.4 J-K 1 - mol-1 less than the molar heat capacity of pure hexane. 

The relative partial molar enthalpy and relative partial molar heat capacity are obtained from8... 

The relative partial molar heat contents (Li) in cal. mole and the relative, partial molar heat capacities ( pi — Cpi) in cal. deg. mole, of the water in hydrochloric acid solutions are as follows ... 

This standard textbook on chemical thermodynamics contains an Appendix (no. 4) of selected data for aqueous electrolyte solutions. Compiled are activity coefficients, Debye-Huckel parameters, relative partial molar enthalpies, and relative partial molar heat capacities for about 70 of the most common electrolytes in aqueous solution at 25 °C. More recent Debye-HUckel parameters are to be found 1n the pages of Pitaer, Pelper, and Busey and of Bradley and Pitaer (see item [121]) and the paper of Clarke and Glew, Hem [223. 

IR. Relative Partial Molar Heat Contents.—The partial molar thermal properties, namely, heat content and heat capacity, are of particular interest, as well as of practical importance, as will be seen from some of the examples to be given below. In accordance with the general definition ( 26a), the partial molar heat content of any constituent of a solution is represented by... 

The revised HKF model is constructed such that the non-solvation contribution to V° and C° dominates at low temperatures and becomes -oo at 228 K, and the solvation contribution dominates at high temperatures. The contributions of the solvation and non-solvation parts of the partial molar volume of NaCl are compared in Figure 17.11, and in Figure 17.12 the solvation and non-solvation contributions to the partial molar heat capacity of NaCl are shown as a function of temperature. This illustrates quite nicely how the two contributions combine to produce a maximiun, and it can easily be imagined how the shape of the combined curve is controlled by the fit parameters of the two contributions. Of course, the two contributions do not always cross in such a pedagogically convenient way. In Figure 17.13 we show the two contributions to the partial molar volume of HCl as a function of temperature the same features are present, but the relative contributions of the two parts of the model are quite different. 

The third unusual property of aqueous solutions of inert gas molecules is the relatively large partial molar heat capacity of the solute in water. This is equivalent to a large heat capacity of the solvation of the inert gases in water. The partial molar heat capacity of a solute s in mixtures of s and water is defined by... 

This handbook contains extensive tables of data for the more common Inorganic and organic aqueous electrolyte solutions. Properties covered include dielectric constants, activity coefficients, relative partial molar enthalpies, equilibrium constants, solubility products, conductivities, electrochemical potentials, Gibbs energies and enthalpies of formation, entropies, heat capacities, viscosities, and diffusion coefficients. Unfortunately, only a few of the tables contain references to the sources of the data. 

We now show that equations analogous to Eq. (34) follow for the enthalpy and entropy of mixing, AHM and ASM, but that, in contrast to the chemical potentials, the partial molar enthalpies and entropies for the components differ from those for the species. Finally we show that the equation for the constant pressure relative heat capacity is of a slightly more complicated form than Eq. (34). Equation (34) and its analogs for and ASM are necessary for comparison of model predicted quantities with experiment. From basic thermodynamic equations we have... 

Note that in contrast to the partial molar volume, this quantity is not a relative one. This follows from the fact that the absolute value of the partial molar enthalpy cannot be determined. In a thermodynamic system with constant T and P, the isobaric heat capacity can be regarded as the measure of the enthalpy fluctuations of the system ... 

This 268 page article is concerned with the prediction of the thermodynamic properties of aqueous electrolyte solutions at high temperatures and pressures. There is an extensive discussion of the fundamental thermodynamics of. solutions and a discussion of theoretical concepts and models which have been used to describe electrolyte solutions. There is a very extensive bibliography ( 600 citations) which contains valuable references to specific systems of interest. Some specific tables of interest to this bibliography contain Debye-Hiickel parameters at 25 C, standard state partial molar entropies and heat capacities at 25 °C, and parameters for calculating activity coefficients, osmotic coefficients, relative apparent and partial molar enthalpies, heat capacities, and volumes at 25 °C. 

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