Gibbs' phase rule

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In chemistry, Gibbs' phase rule describes the possible number of degrees of freedom (F) in a closed system at equilibrium, in terms of the number of separate phases (P) and the number of chemical components (C) in the system. It was deduced from thermodynamic principles by Josiah Willard Gibbs in the 1870s.

The (intensive) variables needed to describe the system are Pressure, Temperature and the Chemical Potential (as may be related to the relative mole fractions X ) of the components in each phase, i.e. PC+2-P in total.

The key thermodynamics result is that at equilibrium the Gibbs free energy change for small transfers of mass between phases is zero. This requires the chemical potentials for a component to be the same in every phase. There are thus C(P-1) such thermodynamic equations of constraint on the system.

Gibbs' rule then follows, as:

F = C − P + 2.

Where F is the number of degrees of freedom, C the number of chemical components, and P is the number of phases that cannot be shared.

[edit] Examples

Consider water, the H₂O molecule, C = 1.

. when three phases are in equilibrium, P = 3, there can be no variation of the (intensive) variables ie. F = 0. Temperature and pressure must be at exactly one point, the 'triple point' (temperature of 0.01 degree Celsius and pressure of 611.73 pascals). Only at the triple point can three phases of water exist at the same time. At this one point, Gibbs rule states: F = 1 - 3 + 2 = 0

. when two phases are in equilibrium, P = 2, such as along the melting or boiling boundaries, the (intensive) variable pressure is a determined function of (intensive) variable temperature, ie. one degree of freedom. Along these boundaries, Gibbs rule states: F = 1 - 2 + 2 = 1

. Away from the boundaries of the phase diagram of water, only one phase exists (gas,liquid, or solid), P = 1. So there are two degrees of freedom. At these points, Gibbs rule states: F = 1 - 1 + 2 = 2

Note that if you are considering three (intensive) variables: pressure, temperature, and volume of a gas (ie. one phase, P = 1) then only two of the variables can be independent. This fact is illustrated by the universal gas law:

pressure * volume  = nR * temperature      (where nR is a constant)

Another Example - For instance, a balloon filled with carbon dioxide has one component and one phase, and therefore has two degrees of freedom - in this case temperature and pressure. If you have two phases in the balloon, some solid and some gas, then you lose a degree of freedom - and indeed this is the case, in order to keep this state there is only one possible pressure for any given temperature.

It is important to note that the situation gets more complicated when the (intensive) variables go above critical lines or point in the phase diagram. At temperatures and pressure above the critical point, the physical property differences that differentiate the liquid phase from the gas phase become less defined. This reflects the fact that, at extremely high temperatures and pressures, the liquid and gaseous phases become indistinguishable. In water, the critical point (thermodynamics) occurs at around 647K (374°C or 705°F) and 22.064 MPa .

[edit] Relation to Euler's formula

Once the form of the phase diagram is known from thermodynamics principles, Gibbs' phase rule can be syntactically transformed into the polyhedral formula of Leonhard Euler (1707-1784), so that chemical students knowledgeable in Gibbs' phase rule can learn to memorize Euler's polyhedral formula, and vice versa.

Euler's polyhedral formula states a relation between the number of a polydedron's vertices, V, with the number of the polyhedron's faces, F, and the number of the polyhedron's edges, E. In the ordering of Gibb's rule, Euler's formula can be written: V = E − F + 2. For the familiar cubic polyhedron: V = 8, E = 12, F = 6, so that 8 = 12 − 6 + 2, which checks.

The syntactic transformation of Gibbs' phase rule into (and from) Euler's polyhedral formula is: