Phase Equilibria and Phase Rule (Part -1) - Thermodynamics, Physical Chemistry, CSIR-NET - Government Jobs PDF Download

Phase Rule

The ‘phase rule’ generalization was given by J.W. Gibbs in 1874 and further studied by H.W.B. Roozeboom 1884. The phase rule is able to predict the conditions necessary to be specified for a heterogeneous system to exhibit equilibrium. During the study of chemical systems, we usually deal with the systems containing two or more phases in equilibrium, which are called heterogeneous or polyphase systems. Phase rule was based on the basis of the principles of thermodynamics. The phase rule is able to predict qualitatively, by means of diagram, the effect of changing temperature, pressure, or concentration on a heterogeneous system in equilibrium. In this chapter we will study the phase rule and its various applications in the daily life.

1 Gibb’s Phase Rule

Phase rule may be defined as:

When a heterogeneous system in equilibrium at a definite temperature and pressure, the number of degrees of freedom is equal to by 2 the difference in the number of components and the number ofphases 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

Where    

F = Number of degrees of freedom,
C = Number of components
P = Number of phases of the system

For understanding the various applications of phase rule a clear understanding of the various terms, phases (P), components (C) and degrees of freedom (F) present in the phase rule, is essential which have their specific meanings.

Components (C)

The minimum number of independently variable constituents in terms of which the composition of each phase of a heterogeneous system can be expressed directly or in the form of a chemical equation are called the components of system (C).

For example, a system consisting of a solution of sugar in water (P = 1 i.e. solution phase) is a two-component system because the solution phase present in the system consists of two constituents—water and sugar.

Some common examples related to the components are as follows:

(a) Consider the following system consisting of ice, water and vapour in equilibrium.

The system consists of three phases ice, water and vapour phase. The chemical substance present in each phase is HO- Therefore, the composition of each phase is expressed in terms of HO- Hence, it is called one-component system.

(b) The saturated solution of sodium chloride consists of three phases—solid sodium chloride, salt solution and water vapour in equilibrium.

The chemical composition of each phase of the system can be expressed if we consider two chemical constituents NaCI and water as shown below.

Phase Components
(i)    NaCI (s) = NaCI + 0H₂O
(ii)    NaCI(aq) = yNaCI + xH₂O
(iii)    H₂O(g)    = ONaCI + H₂O

(c) The system, CuSO₄.5H₂O (s)  CuSO₄.3H₂O (s) + 2H₂O (g) is a three-phase and two component system. It requires two constituents CuSO₄ and H₂O to express the composition of each phase of the system.

Degree of Freedom (F)

The smallest number of independently variable factors such as temperature, pressure and concentration which must be required in order to define the system completely are called the degree of freedom. Degree of freedom of a system is also known as variance.

When a system having no degree of freedom
F = 0 it is called non-variant system or invariant system.

When a system having only one degree of freedom
F = 1 it is a univariant or a monovariant system.

Similarly, a system having two degrees of freedom
F = 2 is a bivariant system and so on.

The term degree of freedom can be understood with the help of following examples:

1. The system ice water vapour has no degree of freedom (i.e., F = 0). The three phases of water i.e. ice, liquid water and vapour can exist together in equilibrium only at a particular-temperature and pressure (corresponding to the freezing point) and no factor is necessary to be specified to define the system. Hence, a system consisting of ice, water and vapour in equilibrium has no degree of freedom i.e. it is a nonvariant system.
2. For a mixture of gases, the number of degrees of freedom is three (i.e., F = 3). Such a system can be completely defined when the temperature, pressure and composition are fixed. In this case, the remaining factor i.e., volume gets automatically fixed. For example, a gaseous mixture consisting of 70% N₂ and 30% O₂ at 22°C and 1 atm pressure is completely defined and does not require any other information for its description. Hence, a system consisting of a mixture of gases has three degrees of freedom i.e. it is a trivariant system.
3. For a saturated LiCl solution, the number of degrees of freedom is one (i.e., F = 1). This is because the system can be completely defined by specifying the temperature only. The other two factors i.e. composition and vapour pressure get automatically fixed when the temperature is fixed. Hence, a system consisting of a saturated LiCl solution is a univariant system.

2. Derivation of Phase Rule Equation 

The Gibb’s phase rule can be derived on the basis of thermodynamic principle as follows.

Let us consider a heterogeneous system consisting of P(P₁, P₂, P₃ ...P) number of phases and C(C₁, C₂, C₃ ... C) number of components in equilibrium. Let us assume that the system is non-reacting i.e. the passage of a componentfrom one phase to another does not involve any chemical reaction. When the system is in equilibrium state it can be explained completely by specifying the following variables:

1. Pressure    
2. Temperature
3. Composition of each phase.

(a) Total number of variables required specifying the state of system:

1. Temperature: same for all phases
2. Pressure: same for all phases
3. Concentration 

Independent concentration variables for one phase with respect to the C components = C - 1    [ ∵ Conc. of last component is independent]

∴ Independent concentration variables for P phases with respect to the C components = P(C - 1) 

Total number of variables = P(C - 1) + 2    ...(1)

(b) The total number of equilibria:

The various phases present in the system can remain in equilibrium only when the chemical potential (m) of each component is the same in each phases, i.e.

(a) For each component the no of equilibria for P phases = (P - 1)

(b) For C component the no of equilibria for P phases = C(P - 1)

Total no. of equilibria involved (E) = C(P - 1)    ...(2)

From eq. 1 & 2 we get

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

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

This above equation is Gibb’s phase rule equation.

Some conclusions from the phase rule equation:

(a) For a system having a specified number of components, the greater the number of phases, the lesser is the number of degrees offreedom. For example,

(i) When the system consists of only one phase, we have

C = 1 and P = 1

So, according to the phase rule,

F = C - P + 2 = 1 - 1 + 2 = 2. The system has two degrees of freedom.

(ii) When the system consists of two phases in equilibrium, we have

C = 1 and P = 2

F = C - P + 2 = 1 - 2 + 2 = 1. The system is monovariant.

(b) A system having a given number of components and the maximum possible number of phases in equilibrium is non-variant. For a one component system, the maximum possible number of phases is three. When a one-component system has three phases in equilibrium, it has no degree of freedom or non-variant system.

(c) For a system having a given number of phases, the larger the number of components, the greater will be the number of the degrees of freedom of the system.

For example,
For one-component system: C = 1, P = 2

∴ F = C - P + 2 = 1 - 2 + 2 = 1

For two-component system: C = 2, P = 2

∴  F = C - P + 2 = 2 - 2 + 2 = 2

The two-component system has a higher number of degrees of freedom.

3 Phase Diagrams

The graphical presentation giving the conditions of pressure and temperature under which the various phases are existing and transform from one phase to another is known as the phase diagram of the system. A phase diagram consists of areas, curves or lines and points.

4 Phase Rule for One-Component Systems

The least number of phases possible in any system is one. So, according to the phase rule equation, a one-component system should have a maximum of two degrees of freedom.

When    C = 1, P = 1

So,   F = C - P + 2 = 1 - 1 + 2 = 2

Hence, a one-component system requires a maximum of two variables to be fixed in order to define the system completely. The two variables are temperature and pressure. So, phase diagrams for one component system can be obtained by plotting P vs T.

In case of a one-component system, phase diagram consists of areas, curves or lines and points which provide the following informations regarding the system:

Point on a phase diagram represents a non-variant system.

Area represents a bivariant system

Curve or a line represents a univariant system.

Water system and the sulphur system are the example of one component systems.

[1] 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 phase), water (liquid phase) and vapour (gaseous phase).

Hence, water constitutes a three-phase, one-component system. Since water is a three-phase system, it can have the following equilibria:

ice vapour,

ice water;

water vapour

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

The phase diagram of the water system is shown in Fig. 2.1. The explanation of the phase diagram of water system is as follows:

Fig. 2.1 Phase diagram of water system

(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 curve of water and tells about the vapour pressure of water at different temperatures. Along this curve, the two phases—water and vapour exist together in equilibrium.

At point D, the vapour pressure of water become equal to the atmospheric pressure (100°C), which represents the boiling point of water. The curve OB finishes at point B (temp. 374°C and pressure 218 atm) where the liquid water and vapour are indistinguishable and the system has only one phase. This point is called the critical point.

Applying the phase rule on this curve,

C = 1 and P = 2

F = C - P + 2 = 1 - 2 + 2 = 1

Hence, the curve represents a univariant system. This explains that only one factor (either temperature or pressure) is sufficient to be fixed in order to define the system.

(ii) Curve OA: It is known as sublimation curve of ice and gives the vapour pressure of solid ice at different temperatures. Along sublimation curve, the two phases ice and vapour exist together in equilibrium. The lower end of the curve OA extends to absolute zero (-273°C) where no vapour exists.

Thus, for every area contains

C = 1 and P = 1

Therefore, applying phase rule on areas

F = C - P + 2 = 1 - 1 + 2 = 2

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

Table 2.1: Some salient features of the water system

5 Two-Component Systems

When the two independent components are present in a heterogeneous system, the system is referred to as a two-component system. Hence, according to the phase rule, for a two-component system having one phase,

F = C - P + 2 = 2 - 1 + 2 = 3

Therefore, the two component system having one phase will have three degrees of freedom or three variables would be required to define the system. The three variables are pressure (P), temperature (T) and concentration (C). This will require a three-dimensional phase diagram for the study of a two-component system. However, in order to simplify the study, a two-component system is usually studied in the form of a condensed system. A condensed system can be studied by reducing a comparatively less important variable. This reduces the degree of freedom of the system by 1 and the system can easily be studied with the help of a two-dimensional phase diagram.

It can have a maximum of following four phases:
Solid lead, Solid silver, Solution of molten silver & lead and Vapours The boiling points of silver and lead are considerably high and the vapour pressure of the system is very low. So, the vapour phase can be ignored and the system can be studied as a condensed system. This system thus can be easily studied with the help of a two dimensional T - C diagram and the reduced phase rule equation, F' = C - P + 1, can be used. This system is generally studied at constant pressure (atmospheric). The phase diagram of Lead-Silver system is shown in Fig. 2.2.

Fig. 2.2 Phase diagram of Pb-Ag system

(a) Curves

The phase diagram of the lead-silver system consists of following curves, which are explained as follows:

(i) Curve AC (Freezing point curve of lead): The AC curve shows the variation of the melting point of lead on addition of silver. The pure lead melts at 327°C (point A). Addition of silver lowers its melting point along curve AC. The added silver dissolves in molten lead to form Ag-Pb solution with the separation of some part of solid lead. Therefore, the two phases, solid lead and Ag-Pb solution remain together in equilibrium along the curve AC.

Hence,    

P = 2, (solid Pb and melt of Ag-Pb)

C = 2(Pb and Ag)

So,    C = 2 and P = 2,

On applying the reduced phase rule

F' = C - P + 1 = 2 - 2 + 1 = 1 The system is univariant.

(iii) Area BCF: The area consists of two phases—solid Ag and a solution of Pb and Ag. Hence it is also univariant.

(iv) Area DCFH: This area also has the two phases which are solid Ag crystals and solid eutectic crystals. Hence C = 2 and P = 2, the system is univariant.

(iv) Area CEGD: The area also has the solid Pb crystals and solid eutectic crystals phases. The system is univariant.

Table 2.2: Some salient features of the Pb-Ag system.