Vapor-liquid equilibrium extensive variables

When we consider a one-component, two-phase system, of constant mass, we find similar relations. Such two-phase systems are those in which a solid-solid, solid-liquid, solid-vapor, or liquid-vapor equilibrium exists. These systems are all univariant. Thus, the temperature is a function of the pressure, or the pressure is a function of the temperature. As a specific example, consider a vapor-liquid equilibrium at some fixed temperature and in a state in which most of the material is in the liquid state and only an insignificant amount in the vapor state. The pressure is fixed, and thus the volume is fixed from a knowledge of an equation of state. If we now add heat to the system under the condition that the temperature (and hence the pressure) is kept constant, the liquid will evaporate but the volume must increase as the number of moles in the vapor phase increases. Similarly, if the volume is increased, heat must be added to the system in order to keep the temperature constant. The change of state that takes place is simply a transfer of matter from one phase to another under conditions of constant temperature and pressure. We also see that only one extensive variable—the entropy, the energy, or the volume—is necessary to define completely the state of the system. 

In the case of binary vapor-liquid equilibrium for example, Eq. 12.7.1 indicates that knowledge of two intensive variables, say T andxi, suffices to determine P and yj, but not the extensive properties, such as the amounts of liquid and vapor. The condition for their determination is provided by Duhem s Theorem. 

As we have already indicated, to completely fix the equilibrium thermodynamic state of a one-component, single-phase system, we must specify the values of two state variables. For example, to fix the thermodynamic state in either the vapor, liquid, or solid region of Fig. 7.3-6, both the temperature and pressure are needed. Thus, we say that a one-component, single-phase system has two degrees of freedom. In addition, to fix the total size or extent of the system we must also specify its mass or one of its extensive properties, such as total volume or total energy, from which the mass can be calculated. 

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