Partial molar heat capacity, constant pressure

U, H, and S as Functions of T and P or T and V At constant composition, molar thermodynamic properties can be considered functions of T and P (postulate 5). Alternatively, because V is related to T and P through an equation of state, V can serve rather than P as the second independent variable. The useful equations for the total differentials of U, H, and S that result are given in Table 4-1 by Eqs. (4-22) through (4-25). The obvious next step is substitution for the partial differential coefficients in favor of measurable quantities. This purpose is served by definition of two heat capacities, one at constant pressure and the other at constant volume ... 

When the heat of solution of an electrolyte in water to form a dilute solution is measured calorimetrically at several temperatures, the standard partial molar (constant pressure) heat capacity of the electrolyte, Cpe , is obtained from the temperature coefficient of these heats, extrapolated to infinite dilution. Alternatively, the difference between the specific heat of a dilute solution of the electrolyte and that of water is obtained by flow microcalorimetry to yield the same quantity. Accurate density data at the appropriate temperature are required for the use of this technique. A recent description of the methods available for the determination of the heat capacities of aqueous electrolytes is presented by Hakin and Bhuiyan (2010). Such determinations are accurate to 1 to 4 J mol (Hepler and Hovey 1996)... 

Recalling d(dH dIf)/dT - d(0HldT)/d%, we have from Eq. 2.15 the heat of reaction at constant pressure as a function of the heat capacities, Cp, of all the reaction species. The temperature dependence of the heat of reaction at constant pressure is thus determined by the partial molar heat capacities, cp of the reaction species as shown in Eq. 2.29 ... 

A single homogeneous phase such as an aqueous salt (say NaCl) solution has a large number of properties, such as temperature, density, NaCl molality, refractive index, heat capacity, absorption spectra, vapor pressure, conductivity, partial molar entropy of water, partial molar enthalpy of NaCl, ionization constant, osmotic coefficient, ionic strength, and so on. We know however that these properties are not all independent of one another. Most chemists know instinctively that a solution of NaCl in water will have all its properties fixed if temperature, pressure, and salt concentration are fixed. In other words, there are apparently three independent variables for this two-component system, or three variables which must be fixed before all variables are fixed. Furthermore, there seems to be no fundamental reason for singling out temperature, pressure, and salt concentration from the dozens of properties available, it s just more convenient any three would do. In saying this we have made the usual assumption that properties means intensive variables, or that the size of the system is irrelevant. If extensive variables are included, one extra variable is needed to fix all variables. This could be the system volume, or any other extensive parameter. 

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... 

The transitions between phases discussed in Section 10.1 are classed as first-order transitions. Ehrenfest [25] pointed out the possibility of higher-order transitions, so that second-order transitions would be those transitions for which both the Gibbs energy and its first partial derivatives would be continuous at a transition point, but the second partial derivatives would be discontinuous. Under such conditions the entropy and volume would be continuous. However, the heat capacity at constant pressure, the coefficient of expansion, and the coefficient of compressibility would be discontinuous. If we consider two systems, on either side of the transition point but infinitesimally close to it, then the molar entropies of the two systems must be equal. Also, the change of the molar entropies must be the same for a change of temperature or pressure. If we designate the two systems by a prime and a double prime, we have... 

Both these considerations would be taken into accoimt if the activation process were assumed to occur at a constant pressure, p, such that the partial molar volume of the solvent is independent of the temperature, though this possibility does not appear to have been considered. A full discussion is beyond the scope of this chapter, but the resulting heat capacities of activation are unlikely to differ greatly from those determined at a constant pressme of, say, 1 atm. (see p. 137). Unfortunately, this approach requires the definition of rather clumsy standard states for solutes, e.g., hypothetically ideal, 1 molal, under a pressure such that a given mass of the pure solvent occupies a particular volume. 

On a single graph plot the constant-pressure partial molar heat capacity for both benzene and carbon tetrachloride as a function of composition. 

The effect of gas impurities is to increase the volume of gas without changing the partial pressure of water. The latent heat component of the heat duty is therefore inversely proportional to the dry-basis mole fraction of chlorine. The same proportionality would hold for the sensible heat component if the impurities had the same average molar heat capacity as chlorine. While this is not precisely the case, the differences are small in comparison with the latent heat duty, and the assumption of constant Cp causes very little error. Figure 12.6 can, therefore, be used as a first approximation to the chiller load by dividing the indicated number by the mole fraction of chlorine. 

Figure 16.35a and b shows stability assessments of proteins. The stability is analyzed in dilute buffer solution by determining the changes in the partial molar heat capacity at constant pressure, ACp. 

Finally, we can obtain a formula for Cp,i, the partial molar heat capacity at constant pressure of species i, by writing the total differential of H in the form... 

Thus, in an ideal gas mixture the partial molar internal energy and the partial molar heat capacity at constant pressure, like the partial molar enthalpy, are functions only of T. 

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