Chemical potential

In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. 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. Thus, it is the partial derivative of the free energy with respect to the amount of the species, all other species' concentrations in the mixture remaining constant. 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. At chemical equilibrium or in phase equilibrium, the total sum of the product of chemical potentials and stoichiometric coefficients is zero, as the free energy is at a minimum. In a system in diffusion equilibrium, the chemical potential of any chemical species is uniformly the same everywhere throughout the system.
In semiconductor physics, the chemical potential of a system of electrons is known as the Fermi level.

Overview

Particles tend to move from higher chemical potential to lower chemical potential because this reduces the free energy. In this way, chemical potential is a generalization of "potentials" in physics such as gravitational potential. When a ball rolls down a hill, it is moving from a higher gravitational potential to a lower gravitational potential. 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.
A simple example is a system of dilute molecules diffusing in a homogeneous environment. In this system, the molecules tend to move from areas with high concentration to low concentration, until eventually, the concentration is the same everywhere. The microscopic explanation for this is based on kinetic theory and the random motion of molecules. However, it is simpler to describe the process in terms of chemical potentials: For a given temperature, a molecule has a higher chemical potential in a higher-concentration area and a lower chemical potential in a low concentration area. Movement of molecules from higher chemical potential to lower chemical potential is accompanied by a release of free energy. Therefore, it is a spontaneous process.
Another example, not based on concentration but on phase, is an ice cube on a plate above 0 °C. An H₂O molecule that is in the solid phase has a higher chemical potential than a water molecule that is in the liquid phase above 0 °C. When some of the ice melts, H₂O molecules convert from solid to the warmer liquid where their chemical potential is lower, so the ice cube shrinks. At the temperature of the melting point, 0 °C, the chemical potentials in water and ice are the same; the ice cube neither grows nor shrinks, and the system is in equilibrium.
A third example is illustrated by the chemical reaction of dissociation of a weak acid HA :
Vinegar contains acetic acid. When acid molecules dissociate, the concentration of the undissociated acid molecules decreases and the concentrations of the product ions increase. Thus the chemical potential of HA decreases and the sum of the chemical potentials of H⁺ and A⁻ increases. 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. This explains why vinegar is acidic, because acetic acid dissociates to some extent, releasing hydrogen ions into the solution.
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. In each case the chemical potential of a given species at equilibrium is the same in all phases of the system.
In electrochemistry, ions do not always tend to go from higher to lower chemical potential, but they do always go from higher to lower electrochemical potential. The electrochemical potential completely characterizes all of the influences on an ion's motion, while the chemical potential includes everything except the electric force.

Thermodynamic definition

The chemical potential μ_i of species i is defined, as all intensive quantities are, by the phenomenological fundamental equation of thermodynamics. This holds for both reversible and irreversible infinitesimal processes:
where dU is the infinitesimal change of internal energy U, dS the infinitesimal change of entropy S, dV is the infinitesimal change of volume V for a thermodynamic system in thermal equilibrium, and dN_i is the infinitesimal change of particle number N_i of species i as particles are added or subtracted. T is absolute temperature, S is entropy, P is pressure, and V is volume. Other work terms, such as those involving electric, magnetic or gravitational fields may be added.
From the above equation, the chemical potential is given by
This is because the internal energy U is a state function, so if its differential exists, then the differential is an exact differential such as
for independent variables x₁, x₂,..., x_N of U.
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. From the differential and using the above expression for, a differential relation for is obtained:
As a consequence, another expression for 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.
By making further Legendre transformations from U to other thermodynamic potentials like the enthalpy and Helmholtz free energy, 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.

Applications

The Gibbs–Duhem equation is useful because it relates individual chemical potentials. 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 is the number of moles of A and is the number of moles of B. Every instance of phase or chemical equilibrium is characterized by a constant. For instance, the melting of ice is characterized by a temperature, known as the melting point at which solid and liquid phases are in equilibrium with each other. Chemical potentials can be used to explain the slopes of lines on a phase diagram by using the Clapeyron equation, which in turn can be derived from the Gibbs–Duhem equation. They are used to explain colligative properties such as melting-point depression by the application of pressure. Henry's law for the solute can be derived from Raoult's law for the solvent using chemical potentials.

History

Chemical potential was first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. He defined it as follows:
Gibbs later noted 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. This freedom to choose the boundary of the system allows the chemical potential to be applied to a huge range of systems. The term can be used in thermodynamics and physics for any system undergoing change. Chemical potential is also referred to as partial molar Gibbs energy. Chemical potential is measured in units of energy/particle or, equivalently, energy/mole.
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. By studying the interactions of homogeneous substances in contact, i.e. bodies, being in composition part solid, part liquid, and part vapor, and by using a three-dimensional volume–entropy–internal energy graph, Gibbs was able to determine three states of equilibrium, i.e. "necessarily stable", "neutral", and "unstable", and whether or not changes will ensue. In 1876, Gibbs built on this framework by introducing the concept of chemical potential so to take into account chemical reactions and states of bodies that are chemically different from each other. In his own words from the aforementioned paper, Gibbs states:
In this description, as used by Gibbs, ε refers to the internal energy of the body, η refers to the entropy of the body, and ν is the volume of the body.