Editing Phase rule (section)

==Consequences and examples==

===Pure substances (one component)===
For pure substances {{nowrap|1=''C'' = 1}} so that {{nowrap|1=''F'' = 3 − ''P''}}. In a single phase ({{nowrap|1=''P'' = 1}}) condition of a pure component system, two variables ({{nowrap|1=''F'' = 2}}), such as temperature and pressure, can be chosen independently to be any pair of values consistent with the phase.  However, if the temperature and pressure combination ranges to a point where the pure component undergoes a separation into two phases ({{nowrap|1=''P'' = 2}}), {{nowrap|1=''F''}} decreases from 2 to 1.<ref name="Sonntag">{{cite book | last1=Sonntag | first1=Richard E. | last2=Wylen | first2=Gordon J. Van | title=Introduction to Thermodynamics | publisher=Wiley | publication-place=New York [u.a] | date=1982-06-21 | isbn=0-471-03134-8 | pages=515–516}}</ref> When the system enters the two-phase region, it is no longer possible to independently control temperature and pressure.
[[File:Carbon dioxide pressure-temperature phase diagram.svg|right|thumb|280px|Carbon dioxide pressure-temperature phase diagram showing the [[triple point]] and [[Critical point (thermodynamics)|critical point]] of carbon dioxide]] In the [[phase diagram]] to the right, the boundary curve between the liquid and gas regions maps the constraint between temperature and pressure when the single-component system has separated into liquid and gas phases at equilibrium.  The only way to increase the pressure on the two phase line is by increasing the temperature. If the temperature is decreased by cooling, some of the gas condenses, decreasing the pressure.  Throughout both processes, the temperature and pressure stay in the relationship shown by this boundary curve unless one phase is entirely consumed by evaporation or condensation, or unless the [[critical point (thermodynamics)|critical point]] is reached.  As long as there are two phases, there is only one degree of freedom, which corresponds to the position along the phase boundary curve.

The critical point is the black dot at the end of the liquid–gas boundary. As this 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 boundary curve, {{nowrap|1=''F'' = 2}} and the temperature and pressure can be controlled independently. Hence there is only one phase, and it has the physical properties of a dense gas, but is also referred to as a [[supercritical fluid]].

Of the other two-boundary curves, one is the solid–liquid boundary or [[melting point]] curve which indicates the conditions for equilibrium between these two phases, and the other at lower temperature and pressure is the solid–gas boundary.

Even for a pure substance, it is possible that three phases, such as solid, liquid and vapour, can exist together in equilibrium ({{nowrap|1=''P'' = 3}}).  If there is only one component, there are no degrees of freedom ({{nowrap|1=''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 {{nowrap|1=''μ''<sub>sol</sub>(''T'', ''p'') = ''μ''<sub>liq</sub>(''T'', ''p'') = ''μ''<sub>vap</sub>(''T'', ''p'')}}, which are sufficient to determine the two variables T and p. In the diagram for CO<sub>2</sub> the triple point is the point at which the solid, liquid and gas phases come together, at 5.2&nbsp;bar and 217&nbsp;K. It is also possible for other sets of phases to form a triple point, for example in the water system there is a triple point where [[ice I]], [[ice III]] and liquid can coexist.

If four phases of a pure substance were in equilibrium ({{nowrap|1=''P'' = 4}}), the phase rule would give {{nowrap|1=''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 allowed by the phase rule normally means that the  phases are not all in true equilibrium.

===Two-component systems===
For binary mixtures of two chemically independent components,  {{nowrap|1=''C'' = 2}} so that {{nowrap|1=''F'' = 4 − ''P''}}. In addition to temperature and pressure, the other degree of freedom is the composition of each phase, often expressed as [[mole fraction]] or mass fraction of one component.<ref name="Sonntag"/>

[[Image:Binary Boiling Point Diagram new.svg|thumb|320px|right|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 [[Vapor–liquid equilibrium#Boiling-point diagrams|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''<sub>1L</sub>), and mole fraction of component 1 in the vapour phase (''x''<sub>1V</sub>). However, since two phases are present ({{nowrap|1=''P'' = 2}}) in equilibrium, only two of these variables can be independent ({{nowrap|1=''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 ([[Contour line#Temperature and related subjects|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<sub>1L</sub>) and pressure. In that case the phase rule implies that the equilibrium temperature ([[boiling point]]) and vapour-phase composition are determined.

Liquid–vapour [[phase diagram]]s for other systems may have [[azeotrope]]s (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.

=== Aqueous solution of 4 kinds of salts ===
Consider an aqueous solution containing sodium chloride (NaCl), potassium chloride (KCl), sodium bromide (NaBr), and potassium bromide (KBr), in equilibrium with their respective solid phases. Each salt, in solid form, is a different phase, because each possesses a distinct crystal structure and composition. The aqueous solution itself is another phase, because it forms a homogeneous liquid phase separate from the solid salts, with its own distinct composition and physical properties. Thus we have P = 5 phases. 

There are 6 elements present (H, O, Na, K, Cl, Br), but we have 2 constraints:

* The stoichiometry of water: n(H) = 2n(O).
* Charge balance in the solution: n(Na) + n(K) = n(Cl) + n(Br).

giving C = 6 - 2 = 4 components. The Gibbs phase rule states that F = 1. So, for example, if we plot the P-T phase diagram of the system, there is only one line at which all phases coexist. Any deviation from the line would either cause one of the salts to completely dissolve or one of the ions to completely precipitate from the solution.