It has been suggested that this article or section be merged with Gibbs phase rule. (Discuss)

It has been suggested that this article or section be merged with Phase equilibrium. (Discuss)

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. Wikipedia does not have an article with this exact name. ... It has been suggested that this article or section be merged with Gibbs phase rule. ... Wikipedia does not have an article with this exact name. ... A phase boundary describes the interface two substances that can remain in contact indefinitely (that is to say, at equilibrium) without mixing, as when oil meets water or air meets stone. ... Chemistry (in Greek: χημεία) is the science of matter that deals with the composition, structure, and properties of substances and with the transformations that they undergo. ... The phrase degrees of freedom is used in three different branches of science: in physics and physical chemistry, in mechanical and aerospace engineering, and in statistics. ... Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American physical chemist. ... Events and Trends Technology The invention of the telephone (1876) by Alexander Graham Bell. ...

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"). Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ... The degree Celsius (°C or ℃ (Unicode 0x2103)) is a unit of temperature named after the Swedish astronomer Anders Celsius (1701–1744), who first proposed a similar system in 1742. ... The pascal (symbol Pa) is the SI unit of pressure. ... In physics, the triple point of a substance is the temperature and pressure at which three phases (gas, liquid, and solid) of that substance may coexist in thermodynamic equilibrium. ... In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ...

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. Carbon dioxide is an atmospheric gas composed of one carbon and two oxygen atoms. ...

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. The ideal gas law, or universal gas equation, is an equation of state of an ideal gas. ...

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. A phase diagram or phase space is a useful construct used in mathematics and physics to demonstrate and visualise the changes in a given system. ... Leonhard Euler by Emanuel Handmann Leonhard Euler [oilər] (April 15, 1707–September 18, 1783) was a Swiss mathematician and physicist. ...

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. Leonhard Euler by Emanuel Handmann Leonhard Euler [oilÉ™r] (April 15, 1707–September 18, 1783) was a Swiss mathematician and physicist. ... In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ...

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