Molar heat capacities of saturated phases

The heat capacities that have been discussed previously refer to closed, single-phase systems. In such cases the variables that define the state of the system are either the temperature and pressure or the temperature and volume, and we are concerned with the heat capacities at constant pressure or constant volume. In this section and Section 9.3 we are concerned with a more general concept of heat capacity, particularly the molar heat capacity of a phase that is in equilibrium with other phases and the heat capacity of a thermodynamic system as a whole. Equation (2.5), C = dQ/dT, is the basic equation for the definition of the heat capacity which, when combined with Equation (9.1) or (9.2), gives the relations by which the more general heat capacities can be calculated. Actually dQ/dT is a ratio of differentials and has no value until a path is defined. The general problem becomes the determination of the variables to be used in each case and of the restrictions that must be placed on these variables so that only the temperature is independent. 

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 molar heat capacity of a gas is of the order of 20-32 J K-1 for many gases and AH/T is approximately 84 J K-1 for many normal liquids. Thus, Csat is approximately —50 to — 63 J K 1 that is, approximately 50-63 J of heat must be removed from 1 mole of gas, which is saturated with respect to a liquid or solid, in order to increase its temperature by 1 K. The negative values arise because the molar volume of the saturated gas decreases with increasing temperature. 

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, 

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