Gibbs' phase rule,^{[1] [2]} was proposed by Josiah Willard Gibbs in the 1870s as the equality

F = C − P + 2 ,

where P ( alternatively π or Φ) is the number of phases in thermodynamic equilibrium with each other and C is the number of components. Typical phases are solids, liquids and gases. A system involving one pure chemical is an example of a one-component system. Two-component systems, such as mixtures of water and ethanol, have two chemically independent components. F is the number of degrees of freedom, which means the number of intensive properties such as temperature or pressure, which are independent of other intensive variables.

[edit] Foundations

• A phase is a form of matter that is homogeneous in chemical composition and physical state. Typical phases are solid, liquid and gas. Two immiscible liquids (or liquid mixtures with different compositions) separated by a distinct boundary are counted as two different phases, as are two immiscible solids.
• The number of components (C) is the number of chemically independent constituents of the system.
• The number of degrees of freedom (F) in this context is the number of intensive variables which are independent of each other.

The basis for the rule (Atkins and de Paula,^[2] justification 6.1) is that equilibrium between phases places a constraint on the intensive variables. More rigorously, since the phases are in thermodynamic equilibrium with each other, the chemical potentials of the phases must be equal. The number of equality relationships determines the number of degrees of freedom. For example, if the chemical potentials of a liquid and of its vapour depend on temperature (T) and pressure (p), the equality of chemical potentials will mean that each of those variables will be dependent on the other. Mathematically, the equation μ_liq(T, p) = μ_vap(T, p), where μ = chemical potential, defines temperature as a function of pressure or vice versa. (Caution: do not confuse p = pressure with P = number of phases.)

[edit] Consequences

[edit] Pure substances

For pure substances C = 1 so that F = 3 – P. In a single phase (P = 1) condition of a pure component system, two variables (F = 2), such as temperature and pressure, can be controlled to any selected pair of values. However, if the temperature and pressure combination ranges to a point where the pure component undergoes a separation into two phases (P = 2), F decreases from 2 to 1. When the system enters the two phase region, it becomes no longer possible to independently control temperature and pressure.

A typical phase diagram for a pure chemical compound.

The phase diagram to the right shows a phase line (blue) that maps the constraint between temperature and pressure when the single component system has separated into liquid and gas phases. If the pressure is increased by compression, some of the liquid condenses and the temperature goes up. If the temperature is decreased by cooling, some of the liquid condenses, decreasing the pressure. Throughout both processes, the temperature and pressure stay in the relationship shown by the blue phase line unless one phase is entirely consumed by evaporation or condensation, or the critical point is reached. As long as there are two phases, there is only one degree of freedom, which corresponds to position along the phase line.

As the critical point is approached, the liquid and gas phases become progressively more similar until, at the critical point, there is no longer a separation into two phases. Above the critical point and away from the phase line, F = 2 and the temperature and pressure can be controlled independently.

The red curve is the phase line for equilibrium between the solid and gas phases, and the two green curves represent two possible phase lines for solid-liquid equilibrium. The solid curve with positive slope represents a typical substance whose melting point increases with pressure, and the dotted curve with negative slope represents water whose melting point decreases with pressure.

Even for a pure substance, it is possible that three phases, such as solid, liquid and vapour, can exist together in equilibrium (P = 3). If there is only one component, there are no degrees of freedom (F = 0) when there are three phases. Therefore, in a single component system, this three phase mixture can only exist at a single temperature and pressure, which is known as a triple point. Here there are two equations μ_sol(T, p) = μ_liq(T, p) = μ_vap(T, p), which are sufficient to determine the two variables T and p.

If four phases of a pure substance were in equilibrium (P = 4), the phase rule would give F = -1 which is meaningless, since there cannot be -1 independent variables. This explains the fact that four phases of a pure substance (such as ice I, ice III, liquid water and water vapour) are not found in equilibrium at any temperature and pressure. In terms of chemical potentials there are now three equations, which cannot in general be satisfied by any values of the two variables T and p, although in principle they might be solved in a special case where one equation is mathematically dependent on the other two. In practice, however, the coexistence of more phases than the phase rule allows normally means that the phases are not all in equilibrium, i.e. that one or more is metastable.

[edit] Two-component systems

For binary mixtures of two chemically independent components, C = 2 so that F = 4 – P. In addition to temperature and pressure, other variables are the composition of each phase, often expressed as mole fraction or mass fraction of one component.

Boiling Point Diagram

As an example, consider the system of two completely miscible liquids such as toluene and benzene, in equilibrium with their vapours. This system may be described by a boiling-point diagram which shows the composition (mole fraction) of the two phases in equilibrium as functions of temperature (at a fixed pressure).

Four thermodynamic variables which may describe the system include temperature (T), pressure (p), mole fraction of component 1 (toluene) in the liquid phase (x_1L), and mole fraction of component 1 in the vapour phase (x_1V). However since two phases are in equilibrium, only two of these variables can be independent (F = 2). This is because the four variables are constrained by two relations: the equality of the chemical potentials of liquid toluene and toluene vapour, and the corresponding equality for benzene.

For given T and p, there will be two phases at equilibrium when the overall composition of the system (system point) lies in between the two curves. A horizontal line (isotherm or tie line) can be drawn through any such system point, and intersects the curve for each phase at its equilibrium composition. The quantity of each phase is given by the lever rule (expressed in the variable corresponding to the x-axis, here mole fraction).

For the analysis of fractional distillation, the two independent variables are instead considered to be liquid-phase composition (x_1L) and pressure. In that case the phase rule implies that the equilibrium temperature (boiling point) and vapour-phase composition are determined.

Liquid-vapour phase diagrams for other systems may have azeotropes (maxima or minima) in the composition curves, but the application of the phase rule is unchanged. The only difference is that the compositions of the two phases are equal exactly at the azeotropic composition.

[edit] Phase rule at constant pressure

Condensed systems have no gas phase. When their properties are insensitive to the (small) changes in pressure which occur, one fewer variable needs to be specified, which results in the phase rule at constant pressure

F = C − P + 1 ,

This is sometimes misleadingly called the "condensed phase rule", but it should be noted that it is not applicable to condensed systems which are subject to high pressures (for example in geology), since the effects of these pressures can be important. The rule is useful for some applications in materials science.

[edit] See also

Phase diagram

[edit] References

[edit] Further reading

Mogk, David: Teaching Phase Equilibria. Gibbs' Phase Rule: Where it all Begins (The phase rule in geology)

Predel, Bruno; Hoch, Michael J. R; Pool, Monte. Phase Diagrams and Heterogeneous Equilibria : A Practical Introduction. Springer. ISBN 3540140115. 

White, Mary Anne. Properties of Materials. Oxford University Press (1999). ISBN 0195113314.  Chapter 9. Thermodynamics Aspects of Stability