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# Derivation of Phase Rule

In thermodynamics, the phase rule (also called Gibbs’ phase rule) is a mathematical principle that decides the degree of freedom of a thermodynamic system. Thermodynamic systems are controlled by three major parameters that are essential for defining any such system. The three parameters are pressure, volume and temperature.

If F is the degree of freedom of a thermodynamic system, C is the number of components and P is the number of phases present in the system (in equilibrium), then the degree of freedom can be given by a simple mathematical equation given below: F = C - P + 2

This is called Gibbs’ phase rule. It was derived by Josiah Willard Gibbs in the latter part of the nineteenth century. Gibbs’ results were published in a paper called “On the Equilibrium of Heterogeneous Substances” between 1875 and 1878. Gibbs assumed that the different components of the system did not react with each other.

## Phase

Phase is an entity defined in the physical sciences. Phase is a region of space in which the matter has the same physical properties and other defined parameters that are used to describe it. Matter in the same phase has the same density, refractive index, magnetization and chemical composition. The phase should also be mechanically separable.

Let us consider the example of ice and water in a jar. The ice and water are two different phases of the same chemical material (H2O). The air above the jar is the third phase and the glass of which the jar is made is the fourth phase involved in this thermodynamic system.

Phase is sometimes used synonymously with state of matter. But both have some fundamental differences. Several different immiscible phases of the same state of matter can exist. As in the example above, ice and the glass jar have the same state of matter, but different phases.

## Component

Components are defined as the number of chemically independent constituents of a thermodynamic system. They are the minimum required entities that can tell us about the composition of all the phases involved in a system.

Components are calculated on the basis of the number of all chemical species present in a system minus the number of chemical reactions between them. These chemical reactions must occur in equilibrium with each other. Further, any number of constraints are also subtracted if they exist like charge neutrality and balance of molar quantities.

A simple example is the system of water, hydrogen and oxygen. Water is made up of hydrogen and oxygen, but the reaction between the given three chemical entities does not occur in normal circumstances. Since there is no reaction taking place, the number of independent components of the system is three.

In the derivation of Gibbs’ phase rule, however, the components are assumed to be not reacting to each other.

## Degree of Freedom

The degree of freedom is defined as the number of parameters that describe a thermodynamic system that can be varied without affecting the value of the other parameters. In other words, it is the number of independent parameters with which we can describe a system.

## Derivation

Any thermodynamic system in equilibrium can be described by a certain number of variables. The variables chosen should be enough to satisfactorily determine all the physical properties associated with that system. For such a system, which is in equilibrium with C number of components, the variables that describe it are:

• Pressure: Pressure is the same for all the components in the system and their various phases.
• Temperature: Temperature is also the same for all the components of the system.
• Concentration: The concentration of each component of the system can be determined separately.

The concentration variable of a single phase written in terms of the number of components C in the system is (C - 1). Therefore the concentration variable of P phases in the system is

P(C - 1)

Combining with temperature and pressure, we get the total number of variables as:

P(C - 1) + 2…………………{1}

Moreover, all the phases in the system can only be in equilibrium when the chemical potential of each component remains constant irrespective of the phase in which the component is. The number of equilibria of a single component written in terms of the number of phases P is (P - 1). It follows that the number of equilibria for C components in the system can be written as:

C(P - 1).................................{2}

Combining eq. {1} and {2} we get:

F = [ P(C - 1) + 2] - [ C(P - 1)]

F = [ CP - P + 2 - CP + C]

F = C - P + 2……………….{3}

Eq. {3} is called Gibbs’ phase rule.

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