Chemical potential

The chemical potential of a species in a mixture is defined as the rate of change of free energy of a thermodynamic system with respect to the change in the number of atoms or molecules of the species that are added to the system.

When both temperature and pressure are held constant, and the number of particles is expressed in moles, the chemical potential is the partial molar Gibbs free energy.

[6] In semiconductor physics, the chemical potential of a system of electrons is known as the Fermi level.

In the same way, as molecules move, react, dissolve, melt, etc., they will always tend naturally to go from a higher chemical potential to a lower one, changing the particle number, which is the conjugate variable to chemical potential.

The microscopic explanation for this is based on kinetic theory and the random motion of molecules.

When the sums of chemical potential of reactants and products are equal the system is at equilibrium and there is no tendency for the reaction to proceed in either the forward or backward direction.

Chemical potentials are important in many aspects of multi-phase equilibrium chemistry, including melting, boiling, evaporation, solubility, osmosis, partition coefficient, liquid-liquid extraction and chromatography.

The chemical potential μi of species i (atomic, molecular or nuclear) is defined, as all intensive quantities are, by the phenomenological fundamental equation of thermodynamics.

Other work terms, such as those involving electric, magnetic or gravitational fields may be added.

This expression of the chemical potential as a partial derivative of U with respect to the corresponding species particle number is inconvenient for condensed-matter systems, such as chemical solutions, as it is hard to control the volume and entropy to be constant while particles are added.

A more convenient expression may be obtained by making a Legendre transformation to another thermodynamic potential: the Gibbs free energy

results: and the change in Gibbs free energy of a system that is held at constant temperature and pressure is simply In thermodynamic equilibrium, when the system concerned is at constant temperature and pressure but can exchange particles with its external environment, the Gibbs free energy is at its minimum for the system, that is

It follows that Use of this equality provides the means to establish the equilibrium constant for a chemical reaction.

, expressions for the chemical potential may be obtained in terms of these: These different forms for the chemical potential are all equivalent, meaning that they have the same physical content, and may be useful in different physical situations.

For example, in a binary mixture, at constant temperature and pressure, the chemical potentials of the two participants A and B are related by where

[9] They are used to explain colligative properties such as melting-point depression by the application of pressure.

[11][12] Chemical potential was first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs.

He defined it as follows: If to any homogeneous mass in a state of hydrostatic stress we suppose an infinitesimal quantity of any substance to be added, the mass remaining homogeneous and its entropy and volume remaining unchanged, the increase of the energy of the mass divided by the quantity of the substance added is the potential for that substance in the mass considered.Gibbs later noted[citation needed] also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a substance, whether capable or not of existing by itself as a homogeneous body.

In his 1873 paper A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces, Gibbs introduced the preliminary outline of the principles of his new equation able to predict or estimate the tendencies of various natural processes to ensue when bodies or systems are brought into contact.

In his own words from the aforementioned paper, Gibbs states: If we wish to express in a single equation the necessary and sufficient condition of thermodynamic equilibrium for a substance when surrounded by a medium of constant pressure P and temperature T, this equation may be written:

(where q and m are the charge and mass of the species, Vele and h are the electric potential[15] and height of the container, respectively, and g is the acceleration due to gravity).

Therefore, at thermodynamic equilibrium, the chemical potential of photons is in most physical situations always and everywhere zero.

Since this process occurs extremely rapidly - at least, it occurs rapidly in the presence of dense charged matter or also in the walls of the textbook example for a photon gas of blackbody radiation - it is safe to assume that the photon chemical potential here is never different from zero.

A physical situation where the chemical potential for photons can differ from zero are material-filled optical microcavities, with spacings between cavity mirrors in the wavelength regime.

In fact, each conserved quantity is associated with a chemical potential and a corresponding tendency to diffuse to equalize it out.

At low temperatures, with no positrons present, electrons cannot be created or destroyed.

Generally the chemical potential is given as a sum of an ideal contribution and an excess contribution: In an ideal solution, the chemical potential of species i (μi) is dependent on temperature and pressure.

Given this definition, the chemical potential of species i in an ideal solution is where R is the gas constant, and

This can be corrected for by factoring in the coefficient of activity of species i, defined as γi.

This correction yields The plots above give a very rough picture of the ideal and non-ideal situation.