A general field of physical chemistry dealing with the various situations in which two or more phases (or states of aggregation) can coexist in thermodynamic equilibrium with each other, with the nature of the transitions between phases, and with the effects of temperature and pressure upon these equilibria. Many superficial aspects of the subject are largely qualitative, for example, the empirical classification of types of phase diagrams; but the basic problems always are susceptible to quantitative thermodynamic treatment, and in many cases, statistical thermodynamic methods can be applied to simple molecular models.
Thermodynamics requires that when two phases, α and β, are free to exchange heat, mechanical work, and matter (chemical species), the temperature T, the pressure P, and the chemical potential (partial molar free energy) μi of each particular component i must be equal in both phases at equilibrium. Algebraically, equilibrium exists when Tα = Tβ, Pα = Pβ, μi,α = μi,β, and μj,α = μj,β.
These conditions of thermal, mechanical, and material equilibrium need not all be present if the equilibrium between phases is subject to inhibiting restrictions. Thus, for a solution of a nonvolatile solute in equilibrium with the solvent vapor, the condition of equality of solute chemical potentials μ2,α = μ2,β need not apply, since there can be no solute molecules in the vapor phase. Similarly, in osmotic equilibria, in which solvent molecules can pass through a semipermeable membrane, whereas solute molecules cannot, μ1,α = μ1,β and T1,α = T2,β, but the solute chemical potentials μ2 are unequal, as are the pressures on opposite sides of the membrane. See also Osmosis; Solution.
If a system consists of P phases and C distinguishable components, there are C + 2 thermodynamic variables (C chemical potentials μi, plus the temperature and pressure) which are interrelated by an equation for each phase. Since there are P independent equations relating the C + 2 variables, only F = C + 2 − P variables need be fixed to define completely the state of the system at equilibrium; the other variables are then beyond control. This relation for the number of degrees of freedom F, or variance, is called the phase rule. It has proved to be a powerful tool in interpreting and classifying types of phase equilibria.
When chemical changes may occur in the system, the number of components C is the number of independent components whose amounts can be varied by the experimenter; this is equal to the total number of chemical species present less the number of independent chemical equilibria between them.
An invariant system has no degrees of freedom (F = 0), for which the number of phases P = C + 2. For a one-component system, such an invariant point is a triple point at which three phases coexist at a single temperature and pressure only; for a two-component system, a quadruple point (four phases) would be invariant. See also Triple point.
