The Phase Rule

copyright by Philip A. Candela, 2000.
Liquid water is stable over a range of temperatures and pressures - that is, within certain well-defined limits, we can arbitrarily choose a pressure and temperature, and the system will still be in the liquid water field on the phase diagram for water. On the other hand, the equilibrium (i.e., mechanically, thermally and chemically balanced) coexistence of pure liquid water AND pure ice is rather restricted: at an arbitrarily chosen pressure, there is only one temperature at which liquid and solid H2O can coexist in equilibrium. When only one phase is stable, there is a greater degree of freedom in choosing P and T; when two phases are stable, the system has *less freedom*. We say that the number of degrees of freedom has been reduced by one when there is an additional phase in an assemblage. Examining this relationship further, we find that ice/water coexistence at any given pressure is expanded from a unique single temperature at the given pressure to a RANGE of temperatures by adding a second component (such as NaCl) to the system. That is, the degrees of freedom can be increased by increasing the number of components.

In general, the number of degrees of freedom in a system of phases is equal to the number of system components + 2 minus the number of phases:

F = C + 2 - p

the greater the number of components C, the greater the number of degrees of freedom; the greater the number of coexisting phases, p, the fewer the number of degrees of freedom.

Let's explore this further. The aim of chemical thermodynamics is to define, fully and quantitatively, the potential reactivity of materials. The fundamental quantity that expresses reactivity is the chemical potential. We define chemical potentials (reactivities) for components of phases. For every independently variable constituent of the phases in a system (the phase components) there is a corresponding chemical potential. The stoichiometric relationships (the so-called " balanced chemical reactions") among the phase components are R in number, where R is given by the difference between the total number of phase components (the sum of the components of each phase, summed over all the phases) minus the number of system components.

For example, in the simple case of a two phase binary CO2 - H2O system (e.g., bottled sparkling water, or a naturally carbonated spring near e.g., Saratoga NY, USA) there are four phase components (CO2 and water in the gas, and CO2 and water in the liquid. We can write two conditions of chemical equilibrium for this system:

u{CO2(g)} = u{CO2(liq)}


u{H2O(g)} = u{H2O(liq)}

where, e.g., u{CO2(g)} is the chemical potential of CO2 in the gas phase. The *components of the system* are CO2 and H2O, and are two in number. If we subtract the number of system components (2) from the total number of phase components(TPC) (4), we get the number of "reactions":

R = 4 - 2 = 2.

We can then re-write the phase rule as:

P+F = TPC +2 - R

When the number of degrees of freedom is ZERO, we have NO freedom to choose the T and P; (for example, when gas, ice and liquid coexist in the pure H2O system) nature chooses a unique point in T,P space. We refer to such a (F=0) assemblage as Invariant.

In an invariant phase assemblage, the number of degrees of freedom is zero. This is really a linear algebraic realtionship among a set of variables [the sum of the chemical potentials of the mineral and fluid-phase *end-members*, (the phase components) and the temperature and pressure of each phase] and a set of constraining equations [comprising the thermal and mechanical conditions of equilibrium (i.e, T(liquid) = T(gas), P(liquid) = P(gas)), the Gibbs-Duhem equations (which are "P" in number since there is one for each phase), and the number of linearly independent *"reactions"* (actually, the number of independent conditions of chemical equilibrium, expressed in terms of the chemical potentials of phase components).

Note that the "2" in the phase rule represents the net difference between {the number of thermal + mechanical variables}, and {the number of independent thermal and mechanical conditions of equilibrium}); i.e., it represents the fact that the number of conditons of thermal and mechanical equilibrium is one less than the number of phases!

Author: P.A. Candela