It was the American mathematical physicist Josiah Willard Gibbs who introduced the concepts of phase and chemical potential in his milestone monograph *On the Equilibrium of Heterogeneous Substances (1876-1878)* with which he almost single-handedly laid the theoretical foundations of chemical thermodynamics.

In a paragraph under the heading *“On Coexistent Phases of Matter”* Gibbs mentions – in passing – that for a system of coexistent phases in equilibrium at constant temperature and pressure, the chemical potential μ of any component must have the same value in every phase.

This simple statement turns out to have considerable practical value as we shall see. But first, let’s go through the formal proof of Gibbs’ assertion.

**An important result**

Consider a system of two phases, each containing the same components, in equilibrium at constant temperature and pressure. Suppose a small quantity *dn _{i}* moles of any component

*i*is transferred from phase A in which its chemical potential is

*μ’*to phase B in which its chemical potential is

_{i}*μ”*. The Gibbs free energy of phase A changes by –

_{i}*μ’*while that of phase B changes by +

_{i}dn_{i}*μ”*. Since the system is in equilibrium at constant temperature and pressure, the net change in Gibbs free energy for this process is zero and we can write

_{i}dn_{i}hence

This result can be generalized for any number of phases: *for a system in equilibrium at constant temperature and pressure, the chemical potential of any given component has the same value in every phase*.

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**Visualizing variance**

The equality of pressure P, temperature T and component chemical potentials μ_{n} between coexistent phases in equilibrium provides a convenient way to visualize variance, or the number of degrees of freedom a system possesses. For example, the triple point of a single component system can be visualized as the array

where the solid, liquid and vapor phases are indicated by one, two and three primes respectively.

Each row represents a single variable, so the number of rows equates to the total number of variables. Each column lists the variables in a single phase. All but one of these may be independently varied; the last is determined by the Gibbs-Duhem relation

There are one of these for each phase, so the number of columns equates to the number of relations (=constraints) to which the system variables are subject. The variance, or number of degrees of freedom (f) of the system is defined

For arrays of the kind presented above, this transposes into

For the triple point of a single component system, there are three rows and three columns, so f =0. With zero degrees of freedom, the triple point is not subject to independent variation and is represented by a fixed point in the PT plane.

The above rule implies that a system of coexistent phases in equilibrium cannot have more phases than intensive system variables.

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**Generating useful equations**

For a component present in any pair of coexistent phases in equilibrium at constant temperature and pressure, the chemical potential of that component has the same value in both phases

From this general relation, equations may be deduced for computing various properties of thermodynamic systems such as ideal solutions, for example the elevation of boiling point, the depression of freezing point, and the variation of the solubility of a solute with temperature.

The key point to grasp is that *μ _{i}* is the chemical potential of component

*i*in an

*arbitrary*state, i.e. in a mixture of components. In order to compute this potential we need to know two things: the chemical potential of the pure substance

*μ*at a pressure p (such as that of the atmosphere), and the mole fraction (

_{i}^{0}*x*) of the component in the mixture. Assuming an ideal solution, use can then be made of the textbook formula

_{i}where for a given phase, *μ _{i}* is the arbitrary chemical potential of

*i*in the mixture,

*μ°*is the chemical potential of the pure substance, and

_{i}*x*is the mole fraction of the component.

_{i}As an example, let us take the relation

where the chemical potential of the solid solvent is necessarily the standard potential because the mole fraction x is unity. The above relation will generate an equation for the depression of the solvent freezing point in a solution at a fixed pressure (p).

Substituting (1) for the liquid phase in (2) gives

where by convention the subscript 1 refers to the solvent. Differentiating with respect to T at constant pressure

using the quotient rule for ΔG/T gives

Now since

equation (3) simplifies to

Integrating from the pure solvent state, where the mole fraction x_{1}=1 and T^{0}_{fus} is the freezing point of the pure solvent, to the solution state where the mole fraction x_{1}= x_{1} and T_{fus} is the freezing point of the solvent in the solution

yields the equation for the depression of the solvent freezing point in a solution at a fixed pressure (p)

Since x_{1}<1 in a solution, the logarithm is negative and therefore the freezing point of the solvent in the solution must be lower than the freezing point of the pure solvent.

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Ok, so maybe that wasn’t the simplest procedure for generating a useful thermodynamic equation. But the point to be made here is that the same procedure applies in the other cases, so you only need to understand the principle once.

For example, the equation for elevation of solvent boiling point in solution with a non-volatile solute at a fixed pressure (p) is

The similarity to the previous equation is evident.

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