Notes on Phase Rule

Introduction: Phase rule is an important generalisation or tool which helps us to predict the condition that must be specified for a system to exhibit equilibrium. 

            Most importantly by the application of this rule it is possible to predict quantitatively the effect of changing pressure, temperature or concentration on a heterogeneous system at equilibrium by means of a diagram called the phase diagram.

             By about 1874, G.W Gibbs formulated this rule which is F = C-P +2 or F+P = C+2, where F = number of degrees of freedom, C= Number of components and P = number of phases. Thus phase rule states that the sum total of the number of degrees of freedom and the number of phases exceeds the number of componets by two. 

1.Phase: A phase is defined as any homogeneous and physically distinct part of the system separated by definite boundary surface and mechanically separable from other  part of the system. 

 

Examples illustrating phase are;

System	Phases
Ice and Water	2 phases: Ice (Solid) + Water (Liquid)
Water and Water vapour	2 phases: Water (L) and Water vapour (G)
Ice , Water and Water vapour	3 phases: Ice (Solid) + Water (Liquid) + Water vapour (G)
2 immiscible liquids CS2/H2O Or CCl4/H2O	2 phases: Both liquids
2 miscible liquids Alcohol + Water	1  phase
CacO3 , CaO , CO2	3 phases: CaCO3 (S), CaO(S) , CO2(G)
Monoclinic and Rhombic Sulphur	2 Phases (Both Solids)

2.Component: The number of components of a system at equilibrium is defined as the smallest number of independently variable constituents (molecular species) by means of which the composition of each phase can be expressed either directly or in terms of chemical equations using +Ve, -Ve or zero sign.

Example i: Water exists as 3 phases, Ice, Liquid and Vapour. The composition of each phase can be expressed in terms of H₂O. Hence it’s a one component system.

ii. Sulphur exists in 4 phases: 2 Solid phases (Rhombic and Monoclinic), a liquid phase and a gaseous phase. All these phases can be expressed in terms of single chemical species, sulphur (S). Hence it’s also a one components system.

iii. CO₂ is the chemical formula using which all three phases of CO₂ (S+ L+ G) can be expressed. Thus it’s also a one component system

iv. Consider the decomposition of CaCO₃ into CaO and CO₂. CaCO₃ ⇌ CaO + CO₂. Though there three different constituents , composition of each phase can be expressed in terms of any two constituents as follows:

Phase	CaO and CO2 as Components
CaCO3	Cao + CO2
CaO	Cao + 0CO2
CO2	0Cao + CO2

 

Phase	CaCO3 and CO2 as Components
CaCO3	CaCO3 + 0CO2
CaO	CaCO3 - CO2
CO2	0CaCO3 + CO2

 

Phase	CaCO3 and CaO as Components
CaCO3	CaCO3 + 0CaO
CaO	0CaCO3 + CaO
CO2	CaCO3 - CaO

v. In general the number of components of a gaseous mixture is given by the number of individual gases present.

vi. NaCl solution (Unsaturated): It’s a one phase system. Its composition can be expressed in terms of two chemical individuals, NaCl and H₂O.

Phase	Components
Aqueous Solution of NaCl	X NaCl + y H2O

 vii. NaCl solution (Saturated) in contact with excess solid sodium chloride: It’s a 2 phase system (Namely aq. NaCl solution + Solid NaCl). The composition of both phases can be expressed in terms of 2 chemical individuals, NaCl and H₂O.

Phase	Component
Aq. Nacl Solution	X NaCl + y H2O
Solid NaCl	NaCl + 0 H2O

 viii.Dissociation of NH₄ Cl .

 NH₄Cl(S) ⇌NH₃+HCl(g)

 Two phase state But the constituent of mixture are in the same proportion in solid and gas phase. Thus it’s a one component system.

Phase	Component
Solid	NH4Cl
Gas	X NH3+XHCl or X NH4Cl

 

3.Degrees of freedom or variance: It is defined as the least number of variable factors such as temperature, pressure and concentration which should be arbitrarily fixed in order to define the state of the system completely.

For example, if we consider a gaseous mixture of N₂ and O₂, we need to specify the temperature, pressure and the concentration in order to define the state of the mixture completely. Thus the degree of freedom of the system is 3.

If we will consider only one phase of the water, say liquid, then it may exist at two different pressures at the same temperature. Thus using only temperature, the state of the water can’t brghe predicted. Hence we need to specify two variables (T and P) in order to define the state of the water completely, and water is said to have 2 degrees freedom. 

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Advantages of Phase Rule:

1. It gives simple methods of classifying equilibrium states of system.

2. It confirms that, different systems having the same degrees of freedom behave in a similar way.

3. It let us predict the behaviour of a system when subjected to various changes like temperature, pressure and concentration.

4. There is no need of considering the molecular structure as phase rule only considers macroscopic systems.

5. It can be applied both to physical and chemical phase reactions.

Limitations of Phase Rule:

1. As the pass rule is applicable to heterogeneous system in equilibrium, there is less hope from phase rule for systems which are slow in reaching the equilibrium state.

2. The various variables involved in phase rule are temperature, pressure and concentration. Other factors such as Magnetic and electric influences have not been considered.

3. All phases must be present under similar temperature and pressure.

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One component systems:

A system in which the composition of any of the phases can be expressed by a minimum of one chemical species, is called an one component system.

Maximum number of phases in one component system: When F is minimum, P becomes maximum. The minimum value of F is 0. Since F = C - P + 2, P(max) = C - F + 2 = 1 - 0 + 2 = 3. This the maximum number of phases in one component system is 3.

Maximum number of degrees of freedom: When P is minimum F becomes maximum. Minimum number phase in a system is 1. Thus F(max) = C - P + 2 = 1 - 1 + 2 = 2.

The water system:

Since only water is the chemical species here, it is an one component system. 

Phases: Solid, Liquid and vapour. The phase diagram of water is given below:

The diagram consists of:

Areas: 

Only liquid phase present in the Area AOC.

Only vapour phase present in the area AOB

Only solid phase is present in the area BOC.

Curves: 

OA: Vapour pressure curve,

OB: Sublimation curve, 

OC: Fusion curve 

OA' : Metastable curve

Points:

Triple point O (4.58mm and 0.0098 degree centigrade) : When all the above curves meet.

Point L' (1 atm and 0 degree centigrade): Melting point

Point F (1 atm and 100 degree centigrade) : Boiling point

Point A (218 atm and 374 degree centigrade) : Critical Pressure & temperature

Curve OA: (Known as Vapour pressure curve): Along this curve liquid water and water vapour (2 phases) remain in equilibrium.

Liquid water ⇌ Water Vapour 

 It gives the vapour pressure of water at different temperatures.

We come across a point F on curve OA, which represents the bong point of water. Boiling point is defined as the temperature tl at which the vapour pressure of water becomes equal to the atmospheric pressure.

The curve ends at point A, which represents the critical temperature and pressure of water. At this point both liquid water and vapour merge into each other and water is said to remain in a critical state.

Along this curve the degrees of freedom F = C - P + 2 = 1 - 2 +2 = 1. That means for any given temperature, there exist a fixed pressure and need not to be mentioned.

The vapour pressure increases as the temperature increases which is indicated by the curve OA and by the formula below:

Now the curves OB and OC can similarly be explained.

Triple Point O : The three curves OA, OB, and OC meet at the point O. All the three phases co-exist at this point. 

Thus the degrees of freedom,

F = C - P + 2 = 1 - 3 + 2 = 0. Thus point O is invariant. It means that three phases can co-exist in equilibrium only at a definite temperature and pressure which correspond to the point O.

If temperature and pressure are varied from the value of this point, different castes may arise:

a. If pressure is increased keeping the temperature constant only liquid phase will be there. Solid and vapour phases will be converted into the liquid.

b. If pressure is lowered keeping the temperature constant, liquid and solid phase will be converted into value phase.

Areas:

The areas bounded by curves OA, OB, OC are AOB, AOC and BOC. Since these areas contain only a single phase, the degrees of freedom, F = C - P + 2 = 2. These areas are bivariant. That means any single phase can exist at two different pressures for the same temperature and vice versa.

Metastable equilibrium:  It is possible to cool water below its freezing point without the Separation of solid ice. Thus we extend the curve OA to OA'. along OA' liquid phase remain in Metastable equilibrium with water vapour. If we put a small piece of ice into the super cooled liquid, it at once changes into solid and the curve merges into OB.

Effect of changing temperature and pressure along curve OA.

Consider the point ‘L’ (bivariant at 1 atm and at certain temperature) in the region BOC.  Now if we increase the temperature slowly from point ‘L’ keeping the pressure constant, the system will shift along the line LL’. At L’ (univariant) fusion of ice takes place. The system will have two phases but one degree of freedom. At this 1 atm pressure the whole of the solid continues to melt into liquid water on further heating. Temperature remains constant until only one phase (liquid water) remains. Now on further heating the system shifts along the line L’ F and at F vaporization begins. At this point F, again the temperature remains constant until whole of liquid water vaporize.

Similarly the change along any line can be explained in the phase diagram.

Sulphur System:

Sulphur system is a 4 phase one component system. It has two crystalline forms, Rhombic and Monoclinic. That means it can exist in two solid forms with 95.6⁰c as the transition temperature at 1 atm. At this temperature and pressure the two forms can be interconverted into each other. Rhombic form is stable below 95.6⁰c and above it the monoclinic form is stable.

            All the four phases cannot coexist at the same time as in one component system the maximum number of phases coexisting at the same time is three. The phase diagram is given below:

The diagram consists of:

Areas: 

Only rhombic sulphur (S_R) present in the Area left to AOCD

Only Monoclinic sulphur (S_M) present in the Area OCB.

Only liquid phase (SL) present in the area CBE

Only vapour phase (S_V) is present in the area right to AOBE.

Curves: 

OA: Sublimation curve of Rhombic sulphur,

OB: Sublimation curve of Monoclinic sulphur, 

OC: Transition curve of rhombic and monoclinic sulphur

BC: Fusion curve of monoclinic sulphur

CD: Fusion curve of rhombic sulphur

BE: Vapour pressure curve of liquid sulphur

OM ' : Metastable curve for rhombic sulphur and sulphur vapour

EM '  : Metastable curve for liquid sulphur and sulphur vapour

M 'C : Fusion curve of metasable rhombic sulphur and liquid sulphur

Points:

Triple point O (95.8 degree centigrade): Where three curves meet to have equilibrium between S_R, S_M, and S_V. Also this point O is the transition temperature of rhombic sulphur at which it changes into monoclinic sulphur.

Triple point B (120 degree centigrade): Where three curves meet to have equilibrium between S_M, S_L and S_V. Also this point B is the melting point of monoclinic sulphur.

Triple point C (150 degree centigrade): Where three curves meet to have equilibrium between S_R, S_M, and S_L. Also at this point rhombic sulphur changes into liquid without changing into monoclinic.

Point M' (114 degree centigrade): Where three metastable curves meet to have equilibrium between S_R, S_L, and S_V. Point M‘ represents the melting point of metastable rhombic sulphur.

Now you can explain the sulphur system as we explained water system.

Two component system:

If we apply phase rule to a 2 component system, then we get, 

F = C – P+ 2 

= 2 - P + 2 

= 4 – P. 

Now since minimum number of phase, P = 1, the maximum degrees of freedom, F is 3. This indicates that a minimum of three variables would be necessary to describe a system which is difficult to graph. Therefore we keep one of the variables, the pressure, constant and choose the rest two, the temperature and concentration. Thus the degree of freedom of the system is reduced by 1. In such case we apply the modified or reduced phase rule, 

F = C – P + 1 

instead of F = C – P + 2.

If the two components ( 2 chemical species) are miscible with each other in the liquid state then we come across the following cases:

1.The two components are not miscible in the solid state and form a eutectic mixture. 

A eutectic mixture is a homogeneous solid mixture (not a compound) that melts or solidifies at a temperature lower than any of the individual ingredient’s (component’s) meting point. Example: Pb – Ag system, Tin – Lead system, Sodium chloride – water system.

2.The two components form a stable compounds with congruent melting point. 

A compound formed (from the chemical combination of two components) is said to be having congruent melting point if it is formed from the cooling of a liquid with same composition without the formation of other solids. The entire liquid turns into a solid compound with same composition. Example: Al-Mg, Zn-Mg, Au-Sn, FeCl₃ - H₂O System.

  Symbolically,

Liquid --------> Solid compound

During congruent melting no other solid is formed, only the liquid phase is obtained.

3. The two components form a compound with incongruent melting point. 

A compound formed (from the chemical combination of two components) is said to be having incongruent melting point if it is formed from the cooling of a liquid along with another solids. The entire liquid turns into two different solids. Symbolically,

Liquid --------> Solid compound + Another Solid 2

During in concongruent melting a new solid with different composition with thwe melt is obtained. For example, on melting, Orthoclase (KAlSi₃O₈) produces another solid Leucite (KAlSi₂O₆) along with the melt. 

Also in low pressure condition, Enstatite (MgSiO₃) on melting produces another solid (Mg₂SiO₄) along with the melt.

Simple Eutectic System:

Silver – Lead System:

Silver and lead are completely miscible in molten state and do not form any chemical compound. They constitute four possible phases:

Solution of silver and lead,

Solid silver,

Solid lead and

Vapour.

The vapour phase is practically negligible due to very high boiling points of metals.

As shown in the figure the diagram consists of two curves AC and BC. Pure lead melts at 327 ⁰c. Curve AC is the freezing or melting point curve of lead and represents the effect of adding silver on the melting point of lead. Clearly addition of silver lowers the melting point of lead. A similar explanation can be done for curve BC.