The Laws for Open Nonreacting Systems

The form (7.1.48) is also obtained when the fundamental equation (3.2.25) for dU is substituted into the constraint (7.1.41) for fixed N, V, S systems and into the constraint (7.1.18) for fixed N, V, U systems. It is also obtained when the fundamental equation (3.2.26) for dH is substituted into the constraint (7.1.42) for fixed N, P, S systems. As with other constraints derived in 7.1.4-7.1.5, the equality in (7.1.48) applies to reversible changes, while the inequality applies to irreversible processes. 

Although the one form (7.1.48) applies to several kinds of processes, the quantity on the Ihs is identified with the Gibbs energy only when T and P are the quantities held fixed. When other quantities are fixed, the Ihs takes other names, and for this reason Prigogine and Defay identify the Ihs of (7.1.48) as proportional to the ajfinity [1]. However, we reserve this name for the analogous quantity that arises in chemical reaction equilibria ( 7.4.4). 

In addition to its generality, the form (7.1.48) is important because it leads to a computational strategy for analyzing phase-equilibrium situations. In that strategy, a phase-equilibrium problem is treated as a multivariable optimization in which the Ihs of (7.1.48) is the quantity to be minimized. An alternative strategy, in which the computational problem is to solve a set of coupled nonlinear algebraic equations, arises from the constraints on open-system processes developed in 7.2. 

The interface itself has negligible mass compared to the masses of the phases, and during processes, states of the interface may be undefined or undefinable. We will treat the interface as an open system and interpret each phase as a port for the other phase that is, the open-system energy and entropy balances from 2.4 will apply. In what follows, we first derive the combined first and second laws ( 7.2.1). Then we find limits on the directions ( 7.2.2) and magnitudes ( 7.2.3) of mass and energy transfers between phases a and p. 

The energy balance (7.2.2) represents the open-system form of the first law (2.4.15), which can be written here as 

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In this section we develop the combined laws for nonreacting systems that are open to mass transfer. Consider a heterogeneous system composed of three parts bulk phases a and P plus an interface I between them, as in Figure 7.4. Each part contains C components, and the state of each is identified by a temperature, a pressure, and a set of mole numbers. Specifically, phase a has T , P , and total moles N phase P has TP, PP, and total moles N the interface has T, P, and total moles N. The component chemi-... 

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