Heat capacities of multiphase, closed systems

In this section we consider the heat capacity of a complete system rather than that of a single phase. Equation (9.2) continues to be the basic equation with the condition that d W = 0. The development of the appropriate equations requires expressions for the differential of the enthalpy with a sufficient number of conditions that the heat capacity is defined completely. There are two general cases univariant systems and multivariant systems. 

The state of a closed, univariant system is defined by assigning values to the temperature, the volume, and the mole numbers of the components. For a closed system the mole numbers are constant. Then, to define the heat 

The simplest univariant system is a one-component, two-phase system. The development of the expression for (8H/dT)V n for such a system is discussed in Section 8.3. The next-simplest system is a two-component, three-phase system. The appropriate equations for this system are 

The quantities n and n can be expressed in terms of n, n°, n2, and the indicated mole fractions with the use of Equations (9.22) and (9.23). Equation (9.21) can then be used to evaluate n in terms of the volume of the system, the molar volumes of the phases, and the mole fractions. The derivative (dn /dT)v can also be evaluated from the resultant equations. The derivative (dH/dT)v is obtained by appropriate differentiation of Equation (9.19) or (9.20), and can be evaluated with the use of expressions obtained by the methods discussed below. 

The state of a multivariant system is defined by assigning values to either the temperature, volume, and mole numbers of the components or the temperature, pressure, and mole numbers. Thus, we define heat capacities at constant volume or heat capacities at constant pressure for such closed systems. The equations and method of calculation are exactly the same as those outlined for univariant systems when the heat capacity at constant volume is desired. For the heat capacity at constant pressure, Equation (9.14) or (9.15) and the set of equations, one for each component, illustrated by Equation (9.18) are still applicable. The method of calculation is the same, with the exception that the volume of the system is a dependent variable 

---