Gibbs' phase rule

# Overview
Gibbs' phase rule, stated by Josiah Willard Gibbs in the 1870s, is the fundamental rule on which phase diagrams are based.
where  ( or in some references P ) is the number of phases present in equilibrium (Types of solid, liquid, gas phases etc), N=number of non computational variables.
F is the number of degrees of freedom or independent variables taken from temperature, pressure and composition of the phases present.
C is the number of chemical components required to describe the system

# Example
Consider water, the H2O molecule, C = 1.
When three phases are in equilibrium,  = 3, there can be no variation of the (intensive) variables i.e. 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 = 2  3 + 1 = 0
When two phases are in equilibrium,  = 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 = 2  2 + 1 = 1
To study the phase diagram of a two component system, it is necessary to determine the composition of a mixture at different temperatures. So, a thermal analysis technique is used for this purpose. In this method, solids of different compositions are separately heated above their melting points. The resultant liquids are cooled slowly and cooling curves are constructed by plotting temperature against time. From this, it is possible to determine the Eutectic point (point where solidification of second component starts).
Away from the boundaries of the phase diagram of water, only one phase exists (gas,liquid, or solid),  = 1. So there are two degrees of freedom.  
At these points, Gibbs rule states: F = 2  1 + 1 = 2
Note that if you are considering three (intensive) variables:  pressure, temperature, and volume of a gas (ie. one phase,  = 1) then only two of the variables can be independent. This fact is illustrated by the universal gas law:
Another example: a balloon filled with carbon dioxide has one component and one phase, and therefore has two degrees of freedom: temperature and pressure. If one has two phases in the balloon, some solid and some gas, then one loses 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 (374C or 705F) and 22.064 MPa .

# Condensed phase rule
In many solids with high melting temperature; the vapour pressure of the solids and even that of the liquid is negligible in comparison with atmospheric pressure.

# 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 (1807-1884), 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 phase rule into (and from) Euler's polyhedral formula is: 
F\Leftrightarrow V; \quad N \Leftrightarrow E;\quad \pi \Leftrightarrow F.
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