4.8: Phase Equilibria

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Equilibrium conditions

If the state of an isolated system is an equilibrium state, this state does not change over time We expect an isolated system that is not in an equilibrium state to undergo a spontaneous, irreversible process and eventually to reach an equilibrium state. Just how rapidly this process occurs is a matter of kinetics, not thermodynamics. During this irreversible adiabatic process, the entropy increases until it reaches a maximum in the equilibrium state.

Simple derivation of equilibrium conditions

A system with two degrees of freedom is called bivariant, one with one degree of freedom is univariant, and one with no degrees of freedom is invariant. For a system of a pure substance, these three cases correspond to one, two, and three phases respectively. For instance, a system of liquid and gaseous H\(_2\)O (and no other substances) is univariant (\(F = 3 - P = 3 - 2 = 1\)); we are able to independently vary only one intensive property, such as \(T\), while the liquid and gas remain in equilibrium.

Here is a simpler, less formal derivation of the three equilibrium conditions in a multiphase system of a single substance.

It is intuitively obvious that, unless there are special constraints (such as internal partitions), an equilibrium state must have thermal and mechanical equilibrium. A temperature difference between two phases would cause a spontaneous transfer of heat from the warmer to the cooler phase; a pressure difference would cause spontaneous flow of matter.

When some of the substance is transferred from one phase to another under conditions of constant \(T\) and \(p\), the intensive properties of each phase remains the same including the chemical potential. The chemical potential of a pure phase is the Gibbs energy per amount of substance in the phase. We know that in a closed system of constant \(T\) and \(p\) with expansion work only, the total Gibbs energy decreases during a spontaneous process and is constant during a reversible process (Eq. 5.8.6). The Gibbs energy will decrease only if there is a transfer of substance from a phase of higher chemical potential to a phase of lower chemical potential, and this will be a spontaneous change. No spontaneous transfer is possible if both phases have the same chemical potential, so this is a condition for an equilibrium state.

The number of independent variables

From this point on, unless stated otherwise, the discussions of multiphase systems will implicitly assume the existence of thermal, mechanical, and transfer equilibrium. Equations will not explicitly show these equilibria as a condition of validity.

In the rest of this chapter, we shall assume the state of each phase can be described by the usual variables: temperature, pressure, and amount. That is, variables such as elevation in a gravitational field, interface surface area, and extent of stretching of a solid, are not relevant.

How many of the usual variables of an open multiphase one-substance equilibrium system are independent? To find out, we go through the following argument. In the absence of any kind of equilibrium, we could treat phase \(\pha\) as having the three independent variables \(T\aph\), \(p\aph\), and \(n\aph\), and likewise for every other phase. A system of \(P\) phases without thermal, mechanical, or transfer equilibrium would then have \(3P\) independent variables.

We must decide how to count the number of phases. It is usually of no thermodynamic significance whether a phase, with particular values of its intensive properties, is contiguous. For instance, splitting a crystal into several pieces is not usually considered to change the number of phases or the state of the system, provided the increased surface area makes no significant contribution to properties such as internal energy. Thus, the number of phases \(P\) refers to the number of different kinds of phases.

Each independent relation resulting from equilibrium imposes a restriction on the system and reduces the number of independent variables by one. A two-phase system with thermal equilibrium has the single relation \(T\bph = T\aph\). For a three-phase system, there are two such relations that are independent, for instance \(T\bph = T\aph\) and \(T\gph = T\aph\). (The additional relation \(T\gph = T\bph\) is not independent since we may deduce it from the other two.) In general, thermal equilibrium gives \(P-1\) independent relations among temperatures.

By the same reasoning, mechanical equilibrium involves \(P-1\) independent relations among pressures, and transfer equilibrium involves \(P-1\) independent relations among chemical potentials.

The total number of independent relations for equilibrium is \(3(P-1)\), which we subtract from \(3P\) (the number of independent variables in the absence of equilibrium) to obtain the number of independent variables in the equilibrium system: \(3P - 3(P-1) = 3\). Thus, an open single-substance system with any number of phases has at equilibrium three independent variables. For example, in equilibrium states of a two-phase system we may vary \(T\), \(n\aph\), and \(n\bph\) independently, in which case \(p\) is a dependent variable; for a given value of \(T\), the value of \(p\) is the one that allows both phases to have the same chemical potential.

The Gibbs phase rule for a pure substance

The complete description of the state of a system must include the value of an extensive variable of each phase (e.g., the volume, mass, or amount) in order to specify how much of the phase is present. For an equilibrium system of \(P\) phases with a total of \(3\) independent variables, we may choose the remaining \(3 - P\) variables to be intensive. The number of these intensive independent variables is called the number of degrees of freedom or variance, \(F\), of the system: \begin{gather} \s{ F = 3 - P } \tag{8.1.17} \cond{(pure substance)} \end{gather}

The application of the phase rule to multicomponent systems will be taken up in Sec. 13.1. Equation 8.1.17 is a special case, for \(C = 1\), of the more general Gibbs phase rule \(F = C - P + 2\).