# Chemical Potential

In thermodynamics , chemical potential is the energy of a species that can be absorbed or released due to a change of particle number in a chemical reaction or of a given species, for example phase transition . The rate of change of the chemical potential of a species in a mixture is defined as the free energy of a thermodynamic body with respect to the change in the number of atoms or molecules of the species joining the system. Thus, it is a partial derivative of the free energy with respect to the amount of species, keeping the concentrations of all other species in the mixture constant. molar tooth The chemical potential is also known as the partial molar free energy . [1] When both temperature and pressure are held constant, the chemical potential is the partial molar Gibbs free energy . At chemical equilibrium or in phase balanced form the sum total of the product of the chemical potential and the stoichiometric coefficient is zero, as the free energy is minimal.

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

## Overview

The particles move from high chemical potential to low chemical potential. As such, the chemicals potential is a generalization of “potential” in physics such as the gravitational potential . When a ball rolls down a hill, it is moving from a higher gravitational potential (higher internal energy thus higher potential to do work) to a lower gravitational potential (lower internal energy). In the same way, as molecules move, react, dissolve, melt, etc., they always naturally move from a higher chemicals potential to a lower one, changing the particle number , which is the conjugate variable for chemicals potential. .

A simple example is a system of dilute molecules dispersed in a homogeneous environment. In this system, molecules move from regions of high concentration to regions of low concentration, eventually, the concentration is the same everywhere.

The subtle explanation for this is based on kinetic theory and the random motion of molecules. However, it is easier to describe the process in terms of chemicals potential: for a given temperature, a molecule has a higher chemicals potential in a region of higher concentration and a lower chemicals potential at a region of lower concentration. The movement of molecules from a higher chemicals potential to a lower chemicals potential is accompanied by the release of free energy. Therefore, it is a spontaneous process .

Another example, based not on concentration but on phase, is a glass of liquid water with ice cubes. Above 0 °C, an H2O molecule in the liquid phase (liquid water) has a lower chemicals potential than a water molecule in the solid phase (ice). When some ice melts, the H2O molecules convert from solid to liquid where their chemicals potential is lower, so the ice cubes shrink. Below 0 °C, the molecules in the ice phase have less chemicals potential, so the ice cubes move. Melting point at a temperature of 0 °C, water and ice have the same chemicasl potential; Snowflakes neither grow nor shrink, and the system is in equilibrium .

A third example is illustrated by the chemical reaction of dissociation of a weak acid HA ( such as acetic acid , A = CH3 COO – ) : h a h+ + a

Vinegar contains acetic acid. When the acid molecules dissociate, the concentration of the dissociated acid molecules (HA) decreases and the concentration of the product ions (H + and A− ) increases. Thus the chemicals potentials of HA decreases and the sum of the chemical potentials of H is + and a – increases. When the sum of the chemical potentials of the reactants and products is equal, the system is at equilibrium and there is no tendency for the reaction to proceed in the forward or reverse direction. This explains why vinegar is acidic, as the acetic acid dissociates somewhat, releasing hydrogen ions into the solution.

Chemical potential balances are important in many aspects of chemistry , including melting , boiling , evaporation , solubility , osmosis , partition coefficient , liquid–liquid extraction and chromatography . Each case has a characteristic constant that is a function of the chemicals potential of the species at equilibrium.

In electrochemistry , ions do not always tend to go from higher to lower chemicals potential, but they do always go from higher to lower electrical potential . The electrochemical potential fully characterizes all of the effects on the motion of an ion, while the chemicals potential includes everything except the electric force. (See below for more on this terminology.)

## Thermodynamic definition

The chemical potential μ of species I ( atomic, molecular or atomic), defined as all intensive quantities, are expressed by the phenomenon- fundamental equation of thermodynamics, which holds for both reversible and irreversible Processes:

\mathrm{d}U = T\,\mathrm{d}S - P\,\mathrm{d}V\,{\displaystyle +\sum _{i=1}^{n}\mu _{i}\,\mathrm {d} N_{i},}

where d is the finite change of u internal energy U , ds is the finite change of entropy s , and d is the finite change of V quantity V for a thermodynamic system in solar equilibrium, and d i is the finite change of particle number i species Of the i as the particles are added or subtracted. T is absolute temperature, S is entropy, P is pressure, and V is volume. Other working conditions, such as those involving electric, magnetic or gravitational fields, may be added.

From the above equation, the chemicals potential is given by

{\displaystyle \mu _{i}=\left({\frac {\partial U}{\partial N_{i}}}\right)_{S,V,N_{j\neq i}}.}

This is an inconvenient expression for condensed-matter systems, such as chemical solutions, because it is difficult to keep volume and entropy constant when adding particles. A more convenient expression can be obtained by performing the Legendre transformation for another thermodynamic potential: the Gibbs free energy . To use the above expression from and to the difference, to obtain an interrelationship relation :

{\displaystyle G=U+PV-TS}{\displaystyle \mathrm {d} G=\mathrm {d} U+P\,\mathrm {d} V+V\,\mathrm {d} P-T\,\mathrm {d} S-S\,\mathrm {d} T}\mathrm {d} U{\displaystyle \mathrm {d} G}
{\displaystyle \mathrm {d} G=-S\,\mathrm {d} T+V\,\mathrm {d} P+\sum _{i=1}^{n}\mu _{i}\,\mathrm {d} N_{i}.}

As a consequence, another expression results for:

{\displaystyle \mu _{i}=\left({\frac {\partial G}{\partial N_{i}}}\right)_{T,P,N_{j\neq i}},}

and the change in Gibbs free energy of a system held at constant temperature and pressure is simple

{\displaystyle \mathrm {d} G=\sum _{i=1}^{n}\mu _{i}\,\mathrm {d} N_{i}.}

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 for the system is minimum, i.e. . it follows that dG = 0

{\displaystyle \mu _{1}\,dN_{1}+\mu _{2}d\,N_{2}+\dots =0.}

The use of this analogy provides a means to establish the equilibrium constant for a chemical reaction.

By making further Legandre transformations from U to other thermodynamic potentials such as enthalpy and Helmholtz free energy , expressions for chemical potentials can be obtained in terms of:

H = U + PV{\displaystyle F=U-TS}

{\displaystyle \mu _{i}=\left({\frac {\partial H}{\partial N_{i}}}\right)_{S,P,N_{j\neq i}},}
{\displaystyle \mu _{i}=\left({\frac {\partial F}{\partial N_{i}}}\right)_{T,V,N_{j\neq i}}.}

These different forms for chemical potential are similar, meaning they have similar physical content and can be useful in different physical conditions.

## Application

The Gibbs–Duhem equation is useful because it relates to individual chemical potentials. For example, in a binary mixture, at constant temperature and pressure, the chemical potentials of the two participants are related.

{\displaystyle d\mu _{\text{B}}=-{\frac {n_{\text{A}}}{n_{\text{B}}}}\,d\mu _{\text{A}}.}

Each instance of a phase or chemical equilibrium is characterized by a constant. For example, the melting of ice is characterized by a temperature, known as the melting point, at which the solid and liquid phases are in equilibrium with each other. The chemicals potential can be used to explain the slopes of lines on a phase diagram using the Clapeyron equation, which in turn can be derived from the Gibbs–Duheim equation. [7] They are used to explain covalent properties such as melting point depression by the application of pressure. [8] Both Raoult’s law and Henry’s law can be derived simply by using chemical potentials. [9]

## History

The chemical potential was first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. He defined it as:

If for any homogeneous mass under hydrostatic stress conditions we assume that an infinite amount of any substance is to be added, the mass remaining homogeneous and its entropy and volume unchanged, the increase in energy of the mass divided by the amount of matter Added mass is the capacity of that substance .

Gibbs later citation needed ] also noted that for the purposes of this definition, any chemicals element or combination of elements in a given proportion can be considered a substance, whether it exists as a homogeneous body or Don’t be This freedom to choose the range of the system allows the chemicals potential to be applied to a vast range of systems. The term may be used in thermodynamics and physics for any system undergoing change. The chemicals potential is also known as the partial molar Gibbs energy (see also partial molar property). Chemicals 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 of Means of SurfaceIn this article, Gibbs introduced an initial outline of the principles of his new equation, which is able to predict or estimate the tendencies of various natural processes when bodies or systems occur. 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 using three-dimensional quantity-entropy-internal energy graphs, the three states of Gibbs equilibrium can be determined. was able to. , i.e. “necessarily stable”, “neutral”, and “volatile”, and whether changes will occur. In 1876, Gibbs built on this framework by introducing the concept of chemical potentials to take into account chemical reactions and states of bodies that are chemically different from each other. in your own words,citation needed ]

If we want to express in an equation the necessary and sufficient condition of thermodynamic equilibrium for a substance when it is bounded by means of constant pressure P and temperature T , then this equation can be written as:

\displaystyle \delta (\epsilon -T\eta +P\nu )=0

Where refers to the variation produced by any change in the state of parts of the body, and (when different parts of the body are in different states) in the ratio which is divided between different states in the body. The condition of stable equilibrium is that the value of the expression in parentheses must be minimum.

In this description, as used by Gibbs, refers to the internal energy of the body, refers to the entropy of the body, and is the quantity of the body.

## Electrochemical, intrinsic, extrinsic and total chemical potentials

The abstract definition of chemical potential given above—the total change in free energy per additional mole of substance—is specifically called the total chemical potential . [10] [11] If two locations have different total chemical potentials for a species, some of them may be due to potentials associated with “external” force fields (electric potential energy difference, gravitational potential energy difference, etc.). , while the rest will be due to “internal” factors (density, temperature, etc.) [10] Therefore, the total chemical potential can be divided into internal chemical potential and external chemical potential :

{\displaystyle \mu _{\text{tot}}=\mu _{\text{int}}+\mu _{\text{ext}},}

Where from

{\displaystyle \mu _{\text{ext}}=qV+mgh+\cdots ,}

i.e., the external potential is the sum of the electric potential, the gravitational potential, etc. ( q and m are the mass of the charge and species, v and h are the voltage and height of the container, and g is the acceleration due to gravity) internal chemical Capacity includes everything other than external potential, such as density, temperature, and enthalpy. This formalism can be understood by assuming that the total energy, , of a system is the sum of two parts: an internal energy, due to the interaction of each particle with the external field, and an external energy, . The definition of chemical potential applied produces the expression above .

U{\displaystyle U_{\text{int}}}{\displaystyle U_{\text{ext}}=N(qV+mgh+\cdots )}{\displaystyle U_{\text{int}}+U_{\text{ext}}}{\displaystyle \mu _{\text{tot}}}

The phrase “chemical potential” is sometimes used to mean “total chemical potential”, but it is not universal. [10] In some fields, notably electrochemistry, semiconductor physics, and solid state physics, the term “chemical potential” refers to the internal chemical potential, while the term electrochemical potential is used for the total chemical potential. [12] [13] [14] [15] [16]

## System of Particles

### Electrons in Solid

Electrons in solids have a chemical potential, defined in the same way as the chemical potential of a chemical species: the change in free energy when electrons are added or removed from the system. In the case of electrons, chemical potential is usually expressed in energy per particle rather than energy per particle, and energy per particle is traditionally given in units of electronvolt (eV).

The chemical potential plays a particularly important role in solid-state physics and is closely related to the concepts of work function, Fermi energy and Fermi level. For example, n-type silicon has a higher intrinsic chemical potential of electrons than p-type silicon. In a diode at p-n junction equilibrium the chemical potential ( internal chemical potential) varies from p-type to n-type side, while the total chemical potential (electric potential, or, Fermi level) across the diode is constant.

When describing a chemical potential, as described above, one has to say “relative to what”. In the case of electrons in semiconductors, the intrinsic chemical potential is often specified relative to some vantage point in the band structure, for example, to the bottom of the conduction band. It can also be specified “relative to vacuum”, which is known as the work function, however, the work function varies from surface to surface even on perfectly homogeneous materials. On the other hand, the total chemical potential is usually specified relative to the electric ground.

In nuclear physics, the chemical potential of electrons in an atom is sometimes [17] referred to as the negative of the atom’s electronegativity. Similarly, the process of chemical potential equalization is sometimes referred to as the process of electronegativity equalization. This connection comes from the Mulliken electronegativity scale. By incorporating the energetic definitions of ionization potential and electron affinity into the Mulliken electronegativity, it is seen that the Mulliken chemical potential is a finite difference approximation of the electronic energy with respect to the number of electrons, that is,

{\displaystyle \mu _{\text{Mulliken}}=-\chi _{\text{Mulliken}}=-{\frac {IP+EA}{2}}=\left[{\frac {\delta E[N]}{\delta N}}\right]_{N=N_{0}}.}

#### Subatomic Particles

In recent years, thermal physics has applied the definition of chemical potential to systems in particle physics and its associated processes. For example, in quark-gluon plasma or other QCD matter, at every point in space there is a chemical potential for photons, a chemical potential for electrons, a chemical potential for baryon numbers, an electric charge, and more.

In the case of photons, photons are bosons and can appear or disappear very easily and rapidly. Therefore, at thermodynamic equilibrium, the chemical potential of a photon is always and everywhere zero. This is because, if the chemical potential was greater than zero, the photons would automatically disappear from the region until the chemical potential returned to zero; Similarly, if somewhere the chemical potential was less than zero, photons would spontaneously appear until the chemical potential returned to zero. Since this process occurs very rapidly (at least, it is rapid in the presence of densely charged matter), it is safe to assume that the photon chemical potential never differs from zero.

Electric charge is different because it is conserved, i.e. it can neither be created nor destroyed. However, it can spread. The “chemical potential of electric charge” governs this diffusion: Electric charge, like anything, will diffuse from regions of high chemical potential to regions of low chemical potential. [18] Other conserved quantities such as the Baryon number are the same. In fact, each conserved quantity is associated with a chemical potential and has a corresponding tendency to expand to equalize it. [19]

In the case of electrons, the behavior depends on temperature and context. At low temperatures, electrons cannot be created or destroyed if positrons are not present. Therefore, an electron is a chemical potential that can vary in space, leading to diffusion. At very high temperatures, however, electrons and positrons can spontaneously appear outside the vacuum (pair production), so the chemical potential of electrons in itself becomes a less useful quantity than the chemical potential of conserved quantities (electrons minus positrons). .

The chemical potentials of bosons and fermions are related to the number and temperature of particles by Bose–Einstein statistics and Fermi–Dirac statistics, respectively.

### Ideal vs. Non-Ideal Solutions

Usually the chemical potential is given as the sum of the ideal contribution and the excess contribution:

{\displaystyle \mu _{i}=\mu _{i}^{\text{ideal}}+\mu _{i}^{\text{excess}},}

In an ideal solution, the chemical potential I (μ I ) of the species is temperature and pressure dependent. μ i 0 ( T ,  P ) is defined as the chemical potential of the pure species i . Given this definition, the chemical potential of species i in an ideal solution is

{\displaystyle \mu _{i}^{\text{ideal}}\approx \mu _{i0}+RT\ln(x_{i}),}

where R is the gas constant, and I is the mole fraction of the species contained in the solution. Note that the approximation is only valid for not close to zero. xi xi

This equation assumes that only the mole fraction ( ) is contained in the solution. This interaction neglects between molecular species I as well as other species [ I – ( J I )]. This can be improved by factoring in the coefficient of activity of the species for I , defined as . it yields improvement

\mu _ {i}x_{i}
{\displaystyle \mu _{i}=\mu _{i0}(T,P)+RT\ln(x_{i})+RT\ln(\gamma _{i})=\mu _{i0}(T,P)+RT\ln(x_{i}\gamma _{i}).}

The above plots give a rough picture of ideal and non-ideal situation.