Derivation of Phase Rule

Mathematical physicist Josiah Gibbs is responsible for laying down the theoretical groundwork for chemical thermodynamics. His phase rule equation paved the way for many subsequent discoveries and breakthroughs. 

Gibbs Phase Rule is a look at the degrees of freedom for a compound in a closed physical system. The rule states that the freedom degree is always equal to the number of components minus the exact number of phases, plus 2.

While you state Gibbs Phase Rule, also remember that such a system in equilibrium is free from the effects of magnetic, electric and gravitational forces.

The equation is as follows – 

F = C – P + 2

Here, F represents the degree of freedom, while C is the chemical component numbers. ‘P’ denotes the types of phases in a particular system. Another way to look at Gibbs Phase Rule definition is that it is an equation to determine the stability of phases present in any material. Keep in mind that equilibrium conditions are the key for this rule to apply.

Before proceeding to Gibbs Phase Rule derivation, here is what you must know about the variables.

Variables of Phase Rule Equation

1. Phase (P)

Any material that you can physically separate in a system is a phase. Therefore, igneous melts, liquids and vapour are considered phases in such a system. Two or more phases can occur in the same state of matter. 

A phase can either be pure or a mixture of two or more elements. Nevertheless, each element in a phase must share physical and chemical properties.

2. Chemical Components (C)

C defines the minimum number of components necessary to define all the phases in a particular system.

3. Number of Degrees of Freedom (F)

This signifies the number of variables that you can change without altering the system’s state. Variables can include temperature, pressure and other factors too.

Thermodynamic Derivation of Phase Rule

Gibbs rule relies greatly on the Gibbs-Duhem equation, which is a fundamental basis for thermodynamics. The Gibbs-Duhem equation clarifies the relationship between pressure (P), temperature (T) and potential for chemical components (μ). This equation is as follows -  

dG = Vdp – Sdt + ΣNidμi 

However, a simpler way to derive the Phase rule is to understand that the composition of each phase is defined as P (C-1). Thus, the total number of variables is equal to

P (C-1) + 2 (Let us consider this as equation 1)

Number of equilibrium for each component’s each phase is P-1

Now for C number of components, the number of equilibrium = P (C-1)

Therefore, total number of equilibria E = C (P-1) (Let us consider this as equation 2)

Now, considering equation 1 and 2, we arrive at the following – 

F = {P (C-1) + 2} – {C (P-1)}

F = {CP - P + 2 – CP + C}

F = C – P + 2

Therefore, using this technique, one can arrive at Gibbs Phase Rule easily.

Multiple Choice Question

1. What Does ‘P’ Stand for in the Phase Rule?

a. Pressure

b. Pascal

c. Momentum

d. Phase

Ans: (d) Phase

Example of Phase Rule on Water

Example 1

Consider water (H₂O) as the system. At the triple point, i.e. P = 3 (steam, ice and liquid), the C = 1. Therefore, determining its degree of freedom is simple

F = C - P + 2

F = 1 – 3 + 2

F = 0

Example 2

Now, consider water as the system with a liquid-solid curve. In such a case, P = 2, while C is still 1. With derivation of phase rule, we can determine that 

F = 1 – 2 + 2

F= 1

Thus, for a liquid-solid curve system, only one variable (pressure or temperature) can be changed, while maintaining equilibrium.

Phase rule derivation is vital to understand and apply the equation. Vedantu’s online teaching platform can help you learn and assess such complex topics in detail. We employ the finest teaching staff to assist students from across India and now you can access us even through our Vedantu app!