Saturated phases, molar heat capacities

As a first example for saturated phases, we consider one phase of a two-phase, single-component system that is closed. The molar enthalpy, and hence the molar heat capacity, of a phase is a function of the temperature and pressure. However, the pressure of the saturated phase is a function of the temperature because, in the two-phase system, there is only one degree of freedom. The differential of the molar enthalpy is given by... 

The molar heat capacity of a saturated phase is thus determined to be... 

The determination of the molar heat capacity of a phase saturated with respect to other phases in a multicomponent system requires the application of sufficient conditions to define the heat capacity. Although expressions are developed here for the molar heat capacity of a saturated phase in general, the expressions can be evaluated only if the phase is pure. The molar enthalpy of a phase is a function of the temperature, pressure, and (C — 1) mole fractions, where C represents the number of components. Thus,... 

Equations (9.9), (9.10), (9.11), or (9.12) in conjunction with Equation (9.2) give expressions for the molar heat capacity of a saturated phase. However, each equation contains the quantity (dS/dXi)TtPtX, which in turn contains terms such as (H — Hk) when xk is taken to be the dependent mole fraction. Evaluation of such quantities requires the knowledge of the absolute values of the enthalpies. Therefore, such terms cannot be evaluated, and the values of the molar heat capacities cannot be calculated. The necessity of knowing the absolute values of the enthalpies arises from the fact that a number of moles of some components must be added to, and the same number of moles of other components must be removed from the 1 mole of saturated phase in order to change the mole fractions of the phase. However, if the saturated phase is pure, even though it is in equilibrium with other phases that are solutions, the molar enthalpy of the phase is not a function of the mole fractions and Equations (9.9)—(9.12) reduce to Equation (9.3). 

However, from the extensive property of heat capacity, we can recognize that the molar two-phase heat capacity Cy must just be the mole fraction-weighted sum of the saturation heat capacities of the individual phases ... 

For miscible blend phases, these parameters need to be described as a function of the blend composition. In a first approach to describe the behavior of the present PPE/PS and SAN/PMMA phases, these phases will be regarded as ideal, homogeneously mixed blends. It appears reasonable to assume that the heat capacity, the molar mass of the repeat unit, as well as the weight content of carbon dioxide scale linearly with the weight content of the respective blend phase. Moreover, a constant value of the lattice coordination number for PPE/PS and for SAN/PMMA can be anticipated. Thus, the glass transition temperature of the gas-saturated PPE/SAN/SBM blend can be predicted as a function of the blend composition (Fig. 17). Obviously, both the compatibilization by SBM triblock terpolymers and the plasticizing effect of the absorbed carbon dioxide help to reduce the difference in glass transition temperature between PPE and SAN. 

Numerical calculations of phase equilibria require thermodynamic data or correlations of data. For pure components, the requisite data may include saturation pressures (or temperatures), heat capacities, latent heate, and volumetric properties. For mixtures, one requires a PVTx equation of state (for determination of d/), and/or an expression for the molar excess Gibbs energy (fw determination of yt). We have discussed in Sections 1.3 and 1.4 the correlating capabilities of selected equations of state and expressions for g, and the behavior of the fugacity coefficients and activity coefficients derived ftom them. 

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