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 constituents (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 relative mole fractions X of the components in each phase ie 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 constituents, and P is the number of phases that cannot be shared.
Examples
The phases of matter are solid, liquid, gas. At a temperature of 0.01 degree Celsius and pressure of 611.73 pascals, water exists in all three phases at the same time, and that temperature-pressure pair is called the triple point for that reason. Gibbs' rule related the number of phases to the number of degrees of freedom in the thermodynamic system, modeling it on the Euler characteristic (pronounced "oiler").
For instance, a balloon filled with carbon dioxide has one component and one phase, and therefore has only 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.
For the trivial chemical case of a box of gas (single-phase, single component) Gibbs' formula is in some ways a restatement of the universal gas law that had first been developed in the 1830s, which relates pressure, volume, temperature and the number of particles involved. Gibbs' version simplifies the law for quickly understanding specific cases.
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 understand 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:
- <math>F\Leftrightarrow V;\quad C \Leftrightarrow E;\quad P \Leftrightarrow F.<math>