Phase rule

Phase rule is an important tool for the quantitative treatment of heterogeneous systems in equilibrium. It deals with the study of equilibrium or with the study of equilibrium conditions in heterogeneous systems substances are present in different phases. Such a system can be studied with the help of phase rule. It enables us to predict the conditions that must be specified for a system to exist in equilibrium. It was introduced by W.J Gibbs in 1876, and it is based on thermodynamic principles.

DEFINITION OF TERMS
1) System It is defined as the part of the universe which is under investigation. It may be simple or complex and may be made up of subsystems.

2) SurroundingsAll the matter which are capable of interacting with the system are referred to as surrounding.

3) Phase (P) It is the homogeneous and physically distinct part of a system which is bounded by a surface and is mechanically separable from other parts of the system. Some examples are:

(i)A gas mixture constitutes a single phase, since gases are completely miscible.
(ii)Two miscible liquids (water and ethyl alcohol) constitute only one phase because it is homogeneous.
(iii)Two immiscible liquids (e.g. benzene and water) constitute two phases.
(iv) At freezing point, water consists of 3 phases Wl, Ws and Wv
(v)A saturated solution of NaCl in water consists of three phases. NaCl (s), NaCl solution and vapors.
(vi) A Heterogeneous mixture like CaCo3CaO+CO2 consists of three phases (two solids and one gas)

COMPONENTS(C)
The number of components of a system at equilibrium is defined as the smallest number of independently variable constituents, by means of which the composition of each phase can be expressed either directly or in terms of an algebraic equation. Some examples are:

(i)Water exists in three phases Wl, Ws and Wv, but the composition of each phase can be expressed in terms of H2O. So it is a one component system.

(ii)Sulphur exist in four phases SR, SM, SL and SV, but the composition of each phase can be expressed in terms of sulphur only, so it is one component system.

(iii)If we mix sodium sulphate and water, there will be a different phases
Na2SO4.H2O(s)
Na2SO4+7H2O(s)
Na2SO4.10H2O(s)
Na2SO4 solution (l)
Vapors (g)

The composition of different phases can be expressed by taking two constituents namely Na2SO4(i.e. the minimum number)
Na2SO4+H2O Na2SO4.H2O(s)
Na2SO4+7H2Oreversible reaction gives Na2SO4.7H2O(s)
Na2SO4+10H2O reversible reaction givesNa2SO4.10H2O(s)
Na2SO4+xH2O reversible reaction gives Na2SO4.xH2O(l)
ONa2SO4+xH2O—xH2O (g)
Hence sodium sulphate –water system is a two component system
(iv) Consider the equilibrium in thermal decomposition of calcium carbonate
CaCO3(s) CaO(s) +CO2 (g)
The composition of each phase can be expressed in terms of minimum two of the three independently variable constituents CaCO3, CaO and CO2 as follows:
Phases CaCO3 and CaO as Component CaCO3 and CO2 as component CaO and CO2 as Component
CaCO3(s) CaCO3+OCaO CaCO3+OCO2 CaO+CO2
CaO(s) OCaCO3+CaO CaCO3-CO2 CaO+OCO2
CO2(g) CaCO3-CaO OCaCO3+CO2 OCaO+CO2


Even though it appears like a three components system, according to the definition, we have to take minimum number of components to represent the composition of each phase in the system and hence it is a two component system

DEGREE OF FREEDOM (F):
Degree of freedom of a system at equilibrium is defined as the minimum number of independently variable factors such as temperature, pressure and composition which must be specified in order to define the condition of the system completely.

Consider a system having a single phase (e.g. O2 gas alone). Both temperature and pressure should be mentioned to state oxygen system. Being a one component system, a composition does not vary. Therefore the degree of freedom taken is taken as two (i.e. , temperature and pressure). The system is called bivariant system. Consider a one component system having two phases (e.g. water is equilibrium with water vapors. Being a one components system, composition does not vary at all. Out of the other two variable i.e., temperature and pressure, if we specify temperature of the water, the vapor pressure of the water gets fixed (vapors pressure have definite value at definite temperature). Therefore if we specify vapor pressure of water, the temperature of the water gets fixed. Hence by fixing either temperature or pressure the conditions of the system will be defined. Hence the degree of freedom for the system is one. The system is called univariant and monovariant.

Consider a one component system having three phases ( eg. Ice water == water vapor). All the three phases of water will exist only at particular temperature and particular vapor pressure and being a one component system, the composition does not vary. When all the three phases are in equilibrium, both temperature and pressure get fixed. Therefore the degree of freedom is zero. The system is called in-variant and non-variant.
For a two component system having one phase, e.g. an unsaturated solution of NaCl, the degree of freedom is three. For a two component system the having two phases in equilibrium, e.g. an unsaturated sodium chloride solution in equilibrium with water vapor, the degree of freedom is two. A two component system having three phases in equilibrium with each other, e.g. a saturated solution of sodium chloride in equilibrium as water vapor, the degree of freedom is one. A two component system having four phases, in equilibrium with one another. e.g Na2SO4.H2O, Na2SO4.7H2O, Na2SO4 solution and water vapor, the degree of freedom is zero.

PHASE RULE
Provided a heterogeneous system at equilibrium is not influenced by electric, magnetic or gravity forces, the number of degrees of freedom (F) of the system at equilibrium is related to the number of components (C) and the number of phases (P) of the system by the relationship F = C – P + 2 which is known as phase rule equation. The terms P, C and F are of special significance and hence require explanation.

PHASE DIAGRAM
Phase diagram is a diagram depicting the condition for equilibrium difference between different phases. The study of phase diagram is important for a proper understanding of the relative stability and equilibrium between the phases.

DERIVATION OF PHASE RULE

Let us consider a heterogeneous system in equilibrium having P and C components. The degree of freedom states that it is the number of independent variables fixed arbitrarily to define the system completely. The number of such variables is equal to the total number of variable minus the number of relation between them at equilibrium.
The number of variables:

(i)Pressure: There is one pressure variable for the whole system since each phase has the same pressure at equilibrium.

(ii)Temperature: There is one temperature variable for the whole system since each phase has the temperature at equilibrium. In other words, when a system is in equilibrium there can be only P and T.

(iii)Composition: When there is two components A and B in a phase and the molar concentration of A is known ( Consider 0.3 ) that of B can be calculated (1-0.3=0.7) Hence for a system with two components if the composition of one is known, the system can be defined. Thus, for a system of three components, if the molar concentration of two components is known the system can be defined. In short for C components the composition or concentration of (C-1) composition variables, therefore P phases have P(C-1) composition variables.
Total number of variables
=P(C-1)+1+1
For composition Temperature pressure
=P(C-1)+2