Chemical Potential

Chemical potential, or more specifically the total chemical potential of a species is the total amount of energy that is absorbed or released during a change in the number of particles of that species (e.g. due to a reaction or phase change).

It was first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs.

Chemical potential is also referred to as partial molar Gibbs energy (\(\text{E/mol}\)).

Formula

The chemical potential, \({\displaystyle \mu _{\mathrm {B}}}\), of a substance \(\text{B}\) in an ideal mixture of liquids or an ideal solution is given by:

\({\displaystyle \mu _{\mathrm {B} }=\mu _{\mathrm {B} }^{\ominus }+\text{RT}\chi_{\mathrm {B}}}\),

where \(\chi\) is the chemical potential of a pure substance \({\displaystyle \mathrm {B} }\), and \({\displaystyle \chi_{\mathrm {B} }}\) is the mole fraction of the substance in the mixture.

This is generalized to include non-ideal behavior by writing:

\({\displaystyle \mu = \mu_{\mathrm{ideal}}+\mu_{\mathrm{excess}}}\),
\({\displaystyle \Rightarrow \mu_\text{ideal} = \mu^\ominus + \text{RT}\ln\chi}\), and
\(\displaystyle \Rightarrow \mu_\text{excess}=\text{RT}\ln\gamma\)
\(\displaystyle \Rightarrow \mu=\mu^\ominus + \text{RT}\ln\chi + \text{RT}\ln\gamma\)
\(\displaystyle \Rightarrow \mu=\mu^\ominus + \text{RT}\ln\chi\gamma\)

\({\displaystyle \therefore \mu _{\mathrm {B} }=\mu _{\mathrm {B} }^{\ominus }+\text {RT} \ln a_{\mathrm {B}}}\),

when \({\displaystyle a_{\mathrm {B} }}\) is the activity of the substance in the mixture,

\({\displaystyle a_{\mathrm {B} }=\chi_{\mathrm {B} }\gamma _{\mathrm {B} }}\),

\({\displaystyle \gamma _{\mathrm {B} }}\) will be the activity coefficient, which may itself depend on \({\displaystyle \chi_{\mathrm {B} }}\). As \({\displaystyle \gamma _{\mathrm {B} }}\) approaches 1, the substance behaves as if it were ideal.

For instance, if \({\displaystyle \gamma _{\mathrm {B} }}\) ≈ 1, then Raoult's Law is accurate. For \({\displaystyle \gamma _{\mathrm {B} }}\) > 1 and \({\displaystyle \gamma _{\mathrm {B} }}\) < 1, substance B shows positive and negative deviation from Raoult's law, respectively. A positive deviation implies that substance B is more volatile.

In many cases, as \({\displaystyle \chi_{\mathrm {B} }}\) goes to zero, the activity coefficient of substance B approaches a constant; this relationship is Henry's law for the solvent. These relationships are related to each other through the Gibbs–Duhem equation.

Notes

The total chemical potential can be divided into the internal and external chemical potentials.

In fields such as electrochemistry, semiconductor physics and solid state physics, the term chemical potential means just the internal chemical potential.

Electrochemical Potential is an example of total chemical potential.

In semiconductor physics, the chemical potential of a system of electrons at zero absolute temperature is known as the Fermi level.