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# Gibbs' phase rule

Gibbs' phase rule, stated by Josiah Willard Gibbs in the 1870s, is the fundamental rule on which phase diagrams are based. It provides the number of degrees of freedom for a given thermodynamic condition, that is: how many control variables (pressure, temperature, composition, etc ) can be altered while maintaining this condition. Mathematically, this number is

F = C + 2 − π ,

where π (or in some references P ) is the number of phases in equilibrium coexistence (solid, liquid, gas phases etc) and C is the number of chemical components in the system. The number 2 arises from the two main thermodynamic control parameters, usually temperature and pressure, so that C+2 may be considered as the total number of control parameters necessary to specify a thermodynamic state for a mixture with C components.

Any other thermodynamic variable can be obtained from these two by the use of the equation of state :$\left\{ f\left(p,V,T\right) = 0\right\}$. For example, gases obey approximately the universal ideal gas law:

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

## Application

For example, a pot of water under normal atmospheric pressure can be heated and cooled to any desired temperature from 0 to 100 Celsius without restriction, and likewise a block of ice might exist at any temperature below 0 Celsius. The ability to change either the atmospheric pressure or the temperature separately gives the system two degrees of freedom. But a block of ice in equilibrium with water can exist only at the melting point. In order to change the temperature without losing one component of the mixture, one must simultaneously change the pressure - only one degree of freedom exists rather than two.

In terms of the formula, consider water, the H2O molecule. Clearly, as for all one-component systems, C = 1.

When three phases are in equilibrium, π = 3, and Gibbs rule states: F = 2 − 3 + 1 = 0. That is, there can be no variation of the (intensive) variables: temperature and pressure must be at exactly one point, the triple point (experimentally, at a temperature of 0.01 degree Celsius and a pressure of 611.73 pascals). Only at the triple point can three phases of water coexist.

When two phases are in equilibrium, such as along the melting or boiling boundaries, π = 2, and Gibbs rule states F = 2 − 2 + 1 = 1: one variable can be varied independently, e.g., the temperature, with the other ones varying accordingly.

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

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).

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 (374°C or 705°F) and 22.064 MPa. Since criticality occurs at the termination of coexistence, the relevant dimension for critical objects is always one less than the corresponding coexistence. E.g., we saw F=1 for water, a one-component system. Criticality occurs when this line terminates, therefore F=0, and we can only have critical points for these systems.

## Use In Phase Diagrams

The common characteristics a phase diagram are classified geometrically by the phase rule.

F system geometry
< 0 overdetermined --
0 invariant point
1 univariant line
2 divariant area

## 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. If we do not consider changes in pressure as a relevant degree of freedom, the count is reduced by one and

F = N + 1 − π

## 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 (1707-1783), 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 + 2 − F. For the familiar cubic polyhedron: V = 8, E = 12, F = 6, so that 8 = 12 + 2 − 6, which checks.

The phase rule into (and from) Euler's polyhedral formula is: $FLeftrightarrow V; quad N Leftrightarrow E;quad pi Leftrightarrow F.$

## References

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