# The Phase Rule and Its Applications

__GIBBS PHASE RULE __

Phase rule may be defined as:

When a heterogeneous system in equilibrium at a definite temperature ans pressure, the number of degrees of freedom is equal to by 2 the difference in the number of components and the number of phases provided the equilibrium is not influenced by external factors such as gravity, electrical or magnetic forces, surface tension etc.

It is applicable for all the universally present heterogeneous systems.

Mathematically the rule is written as

F=C-P+2

Where

F = number of degree of freedom

C= Number of components

P = Number of phases of the system

**‘PHASE’**

A phase may be defined as : any homogeneous part of a system having all physical and chemical properties the same throughout. A system may consist of one phase or more than one phases.

(1) A system containing only liquid water is one-phase or 1-phase system (P = 1)

(2) A system containing liquid water and water vapour (a gas) is a two-phase or 2-phase system (P = 2).

(3) A system containing liquid water, water vapour and solid ice is a three-phase or 3-phase system. A system consisting of one phase only is called a homogeneous system. A system consisting of two or more phases is called a heterogeneous system

**‘COMPONENTS’**

A system ‘C’ in the Phase Rule equation stands for the number of components of a system in equilibrium. The term component may be defined as : the least number of independent chemical constituents in terms of which the composition of every phase can be expressed by means of a chemical equation.

For example-

- Water and sulphur systems are 1-component systems. Water system has three phases: ice, water, and water vapour. The composition of all the three phases is expressed in terms of one chemical individual H
_{2}O. Thus water system has one component only. Sulphur system has four phases: rhombic sulphur, monoclinic sulphur, liquid sulphur and sulphur vapour. The composition of all these phase s can be expressed by one chemical individual sulphur (S). Hence it is a 1-component system. As is clear from above, when all the phases of a system can be expressed in terms of one chemical individual, it is designated as a one-component or 1-component system - Mixture of gases. A mixture of gases, say O
_{2}and N_{2}, constitutes one phase only. Its composition can be expressed by two chemical substances O_{2}and N_{2}.

Phase Components

Gaseous Mixture = x O_{2} + y N_{2}

Hence a mixture of O_{2} and N_{2} has two components. In general the number of components of a gaseous mixture is given by the number of individual gases present.

**DEGREES OF FREEDOM**

The term Degree of Freedom represented by F in the phase Rule equation (F = C – P + 2) is defined as follows: the least number of variable factors (concentration, pressure and temperature) which must be specified so that the remaining variables are fixed automatically and the system is completely defined

A system with F = 0 is known as nonvariant or having no degree of freedom.

A system with F = 1 is known as univariant or having one degree of freedom.

A system with F = 2 is known as bivariant or having two degrees of freedom.

For example-

- For a pure gas, F = 2.

For a given sample of any pure gas PV = RT.

If the values of pressure (P) and temperature (T) be specified, volume (V) can have only one definite value, or that the volume (the third variable) is fixed automatically. Any other sample of the gas under the same pressure and temperature as specified above, will be identical with the first one. Hence a system containing a pure gas has two degrees of freedom (F = 2).

- For a mixture of gases, F = 3.

A system containing a mixture of two or more gases is completely defined when its composition, temperature and pressure are specified. If pressure and temperature only are specified, the third variable i.e., composition could be varied. Since it is necessary to specify three variables to define the system completely, a mixture of gases has three degrees of freedom (F = 3).

- For water ⇔water vapour, F = 1.

The system water in equilibrium with water vapour, has two variables temperature and pressure. At a definite temperature the vapour pressure of water can have only one fixed value. Thus if one variable (temperature or pressure) is specified, the other is fixed automatically. Hence the system water has one degree of freedom (F = 1).

**ONE–COMPONENT SYSTEM**

For a one-component system we can write the phase rule equation as :

F = C – P + 2 = 1 – P + 2 = 3 – P

Three cases may arise :

Case 1. When only one phase is present,

∴ F = 3 – 1 = 2

Thus the system is bivariant. It can be completely defined by specifying the two variables, temperature and pressure. Or that, both the temperature and pressure can be varied independently. Therefore a single phase is represented by an area on P, T-graph.

Case 2. When two phases are in equilibrium,

F = 3 – 2 = 1

The system then has one degree of freedom and is termed monovariant.

This means that the pressure cannot be changed independently if we change the temperature. The pressure is fixed automatically for a given temperature. A two-phase system is depicted by a line on a P, T-graph.

Case 3. When three phases are in equilibrium,

F = 3 – P = 3 – 3 = 0

∴ F = 0

The system has zero degree of freedom and is termed nonvariant or invariant. This special condition can be attained at a definite temperature and pressure. The system is, therefore, defined completely and no further statement of external conditions is necessary. A three-phase system is depicted by a point on the P, T-graph. At this point the three phases (solid, liquid, vapour) are in equilibrium and, therefore, it is referred to as the Triple point.

**PHASE DIAGRAMS **

A phase diagram is a plot showing the conditions of pressure and temperature under which two or more physical states can exist together in a state of dynamic equilibrium. The diagram consists of : (a) the Regions or Areas; (b) the Lines or Curves; and (c) the Triple point.

**(1) Regions or Areas **

The diagram is divided into three regions or areas which are labelled as ‘solid’, ‘liquid’, and ‘vapour’. These areas in Fig. are COB, COA and AOB. Each of the three areas shows the conditions of temperature and pressure under which the respective phase can exist. Applying the phase rule to the system when only one phase is present, we have F = 1 – 1 + 2 = 2 i.e., each single phase has two degrees of freedom.

**(2) Lines or Curves **

There are three lines or curves separating the regions or areas. These curves show the conditions of equilibrium between any two of the three phases i.e., solid/liquid, liquid/vapour, solid/vapour.

(a) Solid/liquid line (OC) which represents the equilibrium Solid ⇔ Liquid, is referred to as the Melting curve or Fusion curve.

(b) Liquid/vapour line (OA) which represents the equilibrium Liquid ⇔ Vapour, is referred to as the Vapour Pressure curve or Vaporisation curve for the liquid.

(c) Solid/vapour line (OB), which represents the equilibrium Solid ⇔ Vapour, is referred to as the Sublimation curve.

Applying phase rule to a one-component two-phase system.

F = C – P + 2 = 1 – 2 + 2 = 1

Thus phase rule predicts that the two phase equilibria stated above will have one degree of freedom. Along any of three lines on the phase diagram when one variable (pressure or temperature) is specified, the other is fixed automatically

**(3) Triple Point **

The three boundary lines enclosing the three areas on the phase diagram intersect at a common point called the Triple point. A triple point shows the conditions under which all the three phases (solid, liquid, vapour) can coexist in equilibrium. Thus the system at the triple point may be represented as :

Applying the phase rule equation, we have

F = C – P + 2 = 1 – 3 + 2 = 0

which predicts that the system has no degree of freedom.

**Metastable Equilibrium**

The vapour pressure curve AO of the liquid phase terminates at O, when the liquid freezes (or solidifies). However by careful cooling of the liquid under conditions that crystals do not form, the curve AO can be extended to A'. This means that the liquid can be cooled far below the freezing point or ‘supercooled’ without separation of the crystals. The supercooled liquid is in an unstable condition. On the slightest disturbance as introduction of a seed crystal, the entire liquid solidifies rapidly. Thus the dashed curve OA' represents a metastable equilibrium,

This system at once reverts to the true stable system

under suitable conditions. It is noteworthy that the dashed curve of the metastable liquid lies above the normal sublimation curve (BO). This implies that the vapour pressure of the metastable phase is always higher than that of the stable phase at the same temperature.

**PHASE DIAGRAMS FOR ONE COMPONENT SYSTEMS**

**Water system **

Water is a one component system which is chemically a single compound involved in the system.

The three possible phases in this system are: ice(solid), water (liquid phase) and vapour (gaseous phase).

Hence water constitutes a three phase, one component system.

Since water is three phase system it can have following equilibria,

The existence of these equilibria at a particular stage depends upon the condition of temperature and pressure, which are the variables of the system. If the values of vapour pressure at different temperatures are plotted against the corresponding temperature the phase diagram of the system is obtained.

(a) **Curves**

The phase diagram of the water system consists of three stable curves and one metastable curve which are explained as follows:

(i) **Curve OB:**

The curve OB is known as vapour pressure of water and tells about the vapour pressure of water at different temperatures. Along this curve the two phases water and vapour access together in equilibrium.

At point D the vapour pressure of water becomes equal to that much very pressure which represents the boiling point of water. The curve finishes at point B where the liquid water and the vapour are in distinguishable and the system has only one phase. This point is also called the critical point.

Applying the phase rule on this curve,

Hence, The curve represents a Univariant system first of these explains that only one factor the temperature of pressure is sufficient to be fixed in order to define the system

(ii) **Curve OA**

It is known as the sublimation curve of ice and gives the vapour pressure of solid ice at different temperatures. Along the sublimation curve the two phases ice and vapour exist together in equilibrium. The lower end of the curve OA extends to absolute zero where no vapour exists.

Thus , for every area contains

C=1 and P=1

Therefore, applying phase rule on area

F= C-P+2

=1-1+2 = 2

Hence, each area is a bivariant system. So. It becomes necessary to specify bith the temperature and pressure to define a one-phase system.