Derivation of Gibb's Phase Rule
When a hetrogeneous 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 of phases provided the equilibrium is not influenced by external factors such as gravity electrical or magnetic forces, surface tension, etc.
Mathematically, the rule is written as
F = C - P + 2
Where,
F = Number of degrees of freedom,
C = Number of components,
P = Number of phases of the system
Degree of Freedom = The smallest number of independently variable factors such as temperature, pressure and concentration which must be required in order to define the system completely. It is also known as variance.
Hetrogeneous System = The system containing two or more phases in equilibrium.
Component and phase are already defined in previous posts, so in case you don't know about these terms please refer previous posts.
Derivation of Phase Rule Equation
The Gibb's phase rule can be derived on the basis of thermodynamic principle as follows.
Let's consider a hetrogeneous system consisting of P number of phases ( i.e. P1, P2, P3,..., P ) and C number of components ( i.e. C1, C2, C3,..., C ) in equilibrium. When the system is in equilibrium state it can be explained completely by specifying the following variables :
i ) Pressure
ii) Temperature
iii) Composition of each phase
Let us assume that the system is non-reacting i.e. the passage of a component from one phase to another does not involve any chemical reaction.
Total number of variable in system
= Composition variables + External variables
= PC + 2 (temperature and pressure)
Total variables = PC + 2
now we only have to define concentration (%x) of (C-1) components since the remaining will be obvious.
Eg. In a 3 component system if we know composition of first two components then concentration of third will be (100 - %x of first two).
thus, our first equation will become
Total variables = P(C-1) + 2
Chemical potential : In thermodynamics, is a form of energy that can be absorbed or released during the chemical reaction or phase transition.
as per total equilibrium condition, the various phases present in system can remain in equilibrium only when the chemical potential ( μ ) of each component is the same in each phase. i.e.
C1 : μ1,P1 = μ1,P2 = ... = μ1,P
C2 : μ2,P1 = μ2,P2 = ... = μ2,P
.
.
.
CC : μC,P1 = μC,P2 = ... = μC,P
where,
number of phases are P ( i.e. P1, P2,..., P )
number of components are C ( i.e. 1, 2, ..., C )
that's why for 1 component if we know the value of chemical potential of 1 phase then value of ( P-1 ) phases gets fixed because it is same.
Eg. If we know the value of μ1,P1 then ( μ1,P2 ), ( μ1,P3 ), upto ( μ1,P ) will be fixed.
Since we have C number of components, number of fixed values will be ( P-1 )C.
Fixed/Dependent variables = (P-1)C
Thus we can find the number of independent variables
Independent variables [ F ]
= Total variables - Fixed variables
= [ P(C-1) + 2 ] - [ (P-1)C ]
= [ PC - P + 2 ] - [ PC - C ]
= [ C - P + 2]
therefore, F = C - P + 2
This equation is known as Generalized form of Gibb's phase rule.
But in reality we perform all the operations at constant pressure i.e. atmospheric pressure.
That is why we only consider temperature as external variable & hence we get a new form of phase rule.
F = C - P + 1
This equation is known as Condensed Phase Rule.
Important Conclusions
- For a system having a specified number of components, the greater the number of phases, the lesser is the number of degrees of freedom.
- A system having a given number of components and the maximum possible number of phases in equilibrium is non-variant ( i.e. degree of freedom will be 0 ).
- 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.