4.7 – Gibbs Phase Rule

4.7.0 – Learning Objectives

By the end of this section you should be able to:

  1. Understand the concept of Gibbs Phase Rule.
  2. Calculate the degree of freedom of a system.

4.7.1 – Introduction

The Gibbs Phase Rule is used when trying to find the degrees of freedom of a system at equilibrium. Degrees of freedom is the number of intensive variables you need to describe to allow the system to be solved.

4.7.2 – Equation

The Gibbs Phase Rule equation is:

\[DF = 2 + c - \Pi\]

where

\[DF = \text{degrees of freedom}\]
\[c = \text{number of chemical species}\]
\[\Pi = \text{number of phases in a system at equilibrium}\]

4.7.3 – Examples

  1. Pure Liquid water: Pure liquid water is one phase and one chemical species. The degrees of freedom by Gibbs Phase Rule is:

    \[DF = 2 + 1 - 1 = 2\]

    Since pure liquid water exists over a multitude of pressures and temperature we need to specify a specific pressure and temperature at which this equilibrium exists. This is what the degrees of freedom represent. The degrees of freedom can also be thought of as how many dimensions the liquid water exists in a phase diagram. In this case it would be two dimensional.

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  2. A mixture of ice, liquid water, and steam: A mixture of ice, liquid water, and steam is three phases and one chemical species. The degrees of freedom by Gibbs Phase Rule is:

    \[DF = 2 + 1 - 3 = 0\]

    Since ice, liquid water, and steam exists at one point of a phase diagram, we don’t need to specify anything because it only exists at one temperature and pressure. In this case it will be zero dimensional.

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  3. A Vapour-Liquid Equilibrium of acetone and methyl ethyl ketone: A Vapour-Liquid Equilibrium of acetone and methyl ethyl ketone has two phases and two chemical species. The degrees of freedom by Gibbs Phase Rule is:

    \[DF = 2 + 2 - 2 = 2\]

    Since a vapour liquid equilibrium can exist over a multitude of pressures and temperature we need to specify a specific pressure and temperature at which this equilibrium exists.

4.7.4 – Problem Statement

Question

An aquatic centre chlorinates its pools to sanitize the water. What is the degrees of freedom of the aquatic centre? Assume that the aquatic centre is a closed system and that the air in the aquatic centre contains only nitrogen and oxygen. ### Answer Since the aquatic centre contains, air, water and chlorine in two phases, the DF is:

\[c = 4\]
\[\Pi = 2\]
\[\therefore \quad DF = 2 + 4 - 2 = 4\]
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