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{{Short description|Change in energies of a thermodynamic system with respect to particle number}}
In [[Physics:Thermodynamics|thermodynamics]], the '''chemical potential''' of a species is the [[Physics:Energy|energy]] that can be absorbed or released due to a change of the [[Physics:Particle number|particle number]] of the given species, e.g. in a chemical reaction or [[Physics:Phase transition|phase transition]]. The chemical potential of a species in a mixture is defined as the rate of change of [[Physics:Thermodynamic free energy|free energy]] of a [[Physics:Thermodynamic system|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|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 [[Physics:Molar concentration|molar]] [[Physics:Gibbs free energy|Gibbs free energy]]'''.<ref>
{{cite book |last1=Atkins |first1=Peter |last2=de Paula |first2=Julio |title=Atkins' Physical Chemistry|edition=8th |year=2006 |publisher= Oxford University Press|isbn=978-0-19-870072-2}} Page references in this article refer specifically to the 7th or 8th edition of this book.</ref><ref>Opacity, Walter F. Huebner, W. David Barfield, {{ISBN|1461487978}}, [https://books.google.com/books?id=ib-8BAAAQBAJ&dq=partial+molar+free+energy+%22molar+chemical+potential%22&pg=PA105 p. 105].</ref> At [[Chemistry:Chemical equilibrium|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.<ref>Atkins (7th ed.), Section 9.2, p. 227.</ref><ref>{{cite journal | last = Baierlein| first = Ralph |date=April 2001 | title = The elusive chemical potential| journal = American Journal of Physics | volume = 69| issue = 4| pages = 423–434| doi = 10.1119/1.1336839 | url = http://www.physics.udel.edu/~bnikolic/teaching/phys624/PDF/chemical_potential.pdf|bibcode = 2001AmJPh..69..423B }}</ref><ref>{{cite journal| last = Job| first = G.| author2 = Herrmann, F.| date = February 2006| title = Chemical potential–a quantity in search of recognition| journal = European Journal of Physics| volume = 27| pages = 353–371| doi = 10.1088/0143-0807/27/2/018| url = http://www.physikdidaktik.uni-karlsruhe.de/publication/ejp/chem_pot_ejp.pdf| bibcode = 2006EJPh...27..353J| issue = 2| citeseerx = 10.1.1.568.9205| s2cid = 16146320| access-date = 2009-02-12| archive-url = https://web.archive.org/web/20150924072915/http://www.physikdidaktik.uni-karlsruhe.de/publication/ejp/chem_pot_ejp.pdf| archive-date = 2015-09-24| url-status = dead}}</ref> In a system in diffusion equilibrium, the chemical potential of any chemical species is uniformly the same everywhere throughout the system.<ref name="Atkins p. 141">Atkins (7th ed.), Section 6.4, p. 141.</ref>

In [[Physics:Semiconductor|semiconductor]] physics, the chemical potential of a system of electrons at zero absolute temperature is known as the [[Physics:Fermi level|Fermi level]].<ref name=Kittel1980>{{cite book | last = Kittel | first = Charles | author2=Herbert Kroemer  | title = Thermal Physics | publisher = W. H. Freeman | date = 1980-01-15 | page = 357 |  edition = 2nd }}</ref>

==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 [[Physics:Potential energy|"potentials" in physics]] such as [[Physics:Gravitational potential|gravitational potential]]. When a ball rolls down a hill, it is moving from a higher gravitational potential (higher internal energy thus higher potential for work) to a lower gravitational potential (lower internal energy). 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 [[Physics:Particle number|particle number]], which is the [[Physics:Conjugate variables (thermodynamics)|conjugate variable]] to chemical potential.

A simple example is a system of dilute molecules [[Physics:Molecular diffusion|diffusing]] in a homogeneous environment. In this system, the molecules tend to move from areas with high [[Chemistry:Concentration|concentration]] to low concentration, until eventually, the concentration is the same everywhere. The microscopic explanation for this is based on [[Physics:Kinetic theory of gases|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 [[Physics:Spontaneous process|spontaneous process]].

Another example, not based on concentration but on phase, is an ice cube on a plate above 0&nbsp;°C. An H<sub>2</sub>O molecule that is in the solid phase (ice) has a higher chemical potential than a water molecule that is in the liquid phase (water) above 0&nbsp;°C. When some of the ice melts, H<sub>2</sub>O molecules convert from solid to the warmer liquid where their chemical potential is lower, so the ice cube shrinks. At the temperature of the [[Physics:Melting point|melting point]], 0&nbsp;°C, the chemical potentials in water and ice are the same; the ice cube neither grows nor shrinks, and the system is in [[Physics:Thermodynamic equilibrium|equilibrium]].

A third example is illustrated by the [[Chemistry:Chemical reaction|chemical reaction]] of [[Chemistry:Dissociation|dissociation]] of a [[Chemistry:Weak acid|weak acid]] H'''A''' (such as [[Chemistry:Acetic acid|acetic acid]], '''A''' = CH<sub>3</sub>COO<sup>−</sup>):
:H'''A''' {{eqm}} H<sup>+</sup> + '''A'''<sup>−</sup>
[[Chemistry:Vinegar|Vinegar]] contains acetic acid. When acid molecules dissociate, the concentration of the undissociated acid molecules (HA) decreases and the concentrations of the product ions (H<sup>+</sup> and A<sup>−</sup>) increase. Thus the chemical potential of HA decreases and the sum of the chemical potentials of H<sup>+</sup> and A<sup>−</sup> 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 [[Physics:Hydrogen ion|hydrogen ion]]s into the solution.

Chemical potentials are important in many aspects of multi-phase [[Chemistry:Equilibrium chemistry|equilibrium chemistry]], including [[Physics:Melting point|melting]], [[Physics:Boiling point|boiling]], [[Physics:Evaporation|evaporation]], [[Chemistry:Solubility|solubility]], [[Physics:Osmosis|osmosis]], [[Physics:Partition coefficient|partition coefficient]], liquid-liquid extraction and [[Physics:Chromatography|chromatography]]. In each case the chemical potential of a given species at equilibrium is the same in all phases of the system.<ref name="Atkins p. 141"/>

In [[Chemistry:Electrochemistry|electrochemistry]], [[Physics:Ion|ion]]s do ''not'' always tend to go from higher to lower chemical potential, but they ''do'' always go from higher to lower ''[[Chemistry:Electrochemical potential|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. (See below for more on this terminology.)

==Thermodynamic definition==
The chemical potential ''μ''<sub>''i''</sub> of species ''i'' (atomic, molecular or nuclear) is defined, as all [[Physics:Intensive and extensive properties|intensive]] quantities are, by the [[Philosophy:Empirical relationship|phenomenological]] [[Physics:Fundamental thermodynamic relation|fundamental equation of thermodynamics]]. This holds for both [[Physics:Reversible process (thermodynamics)|reversible]] and [[Physics:Reversible process (thermodynamics)|irreversible]] infinitesimal processes:<ref>''Statistical Physics'', F Mandl, (Wiley, London, 11971) {{ISBN|0 471 56658 6}}, page 88.</ref>

: <math>\mathrm{d}U = T\,\mathrm{d}S - P\,\mathrm{d}V + \sum_{i=1}^n \mu_i\,\mathrm{d}N_i,</math>

where d''U'' is the infinitesimal change of [[Physics:Internal energy|internal energy]] ''U'', d''S'' the infinitesimal change of [[Physics:Entropy|entropy]] ''S'', d''V'' is the infinitesimal change of [[Physics:Volume (thermodynamics)|volume]] ''V'' for a [[Physics:Thermodynamic system|thermodynamic system]] in thermal equilibrium, and d''N''<sub>''i''</sub> is the infinitesimal change of particle number ''N''<sub>''i''</sub> of species ''i'' as particles are added or subtracted. ''T'' is [[Physics:Absolute temperature|absolute temperature]], ''S'' is [[Physics:Entropy|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

: <math>\mu_i = \left(\frac{\partial U}{\partial N_i} \right)_{S,V, N_{j \ne i}}.</math>

This is because the internal energy ''U'' is a [[Physics:State function|state function]], so if its differential exists, then the differential is an [[Exact differential|exact differential]] such as <math>dU = \sum_{i = 1}^N \left(\frac{\partial U}{\partial x_i} \right)dx_i</math> for independent variables ''x''<sub>1</sub>, ''x''<sub>2</sub>, … , ''x<sub>N</sub>'' 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 [[Physics:Condensed matter physics|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|thermodynamic potential]]: the [[Physics:Gibbs free energy|Gibbs free energy]] <math>G = U + PV - TS</math>. From the differential <math>\mathrm{d}G = \mathrm{d}U + P\,\mathrm{d}V + V\,\mathrm{d}P - T\,\mathrm{d}S - S\,\mathrm{d}T</math> (for <math>PV</math> and <math>TS</math>, the [[Product rule|product rule]] is applied to) and using the above expression for <math>\mathrm{d}U</math>, a differential relation for <math>\mathrm{d}G</math> is obtained:

: <math>\mathrm{d}G = -S\,\mathrm{d}T + V\,\mathrm{d}P + \sum_{i=1}^n \mu_i\,\mathrm{d}N_i.</math>

As a consequence, another expression for <math>\mu_i</math> results:

: <math>\mu_i = \left(\frac{\partial G}{\partial N_i} \right)_{T,P, N_{j \ne i}},</math>

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

: <math>\mathrm{d}G = \sum_{i=1}^n \mu_i\,\mathrm{d}N_i.</math>

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 <math>\mathrm{d}G = 0</math>. It follows that

: <math>\mu_1\,\mathrm{d}N_1 + \mu_2\,\mathrm{d}N_2 + \dots = 0.</math>

Use of this equality provides the means to establish the [[Chemistry:Equilibrium constant|equilibrium constant]] for a chemical reaction.

By making further Legendre transformations from ''U'' to other thermodynamic potentials like the [[Physics:Enthalpy|enthalpy]] <math>H = U + PV</math> and [[Physics:Helmholtz free energy|Helmholtz free energy]] <math>F = U - TS</math>, expressions for the chemical potential may be obtained in terms of these:

: <math>\mu_i = \left(\frac{\partial H}{\partial N_i}\right)_{S,P, N_{j \ne i}},</math>
: <math>\mu_i = \left(\frac{\partial F}{\partial N_i}\right)_{T,V, N_{j \ne i}}.</math>

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 [[Physics:Gibbs–Duhem equation|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
: <math>d\mu_\text{B} = -\frac{n_\text{A}}{n_\text{B}}\,d\mu_\text{A}</math>

where <math>n_\text{A}</math> is the number of moles of A and <math>n_\text{B}</math> 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 [[Physics:Melting point|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|phase diagram]] by using the Clapeyron equation, which in turn can be derived from the Gibbs–Duhem equation.<ref>Atkins (8th ed.), Section 4.1, p. 126.</ref> They are used to explain [[Chemistry:Colligative properties|colligative properties]] such as [[Physics:Melting-point depression|melting-point depression]] by the application of pressure.<ref>Atkins (8th ed.), Section 5.5, pp. 150–155.</ref> [[Physics:Henry's law|Henry's law]] for the solute can be derived from [[Physics:Raoult's law|Raoult's law]] for the solvent using chemical potentials.<ref name="McQuarrie&Simon">McQuarrie, D. A.; Simon, J. D. ''Physical Chemistry – A Molecular Approach'', p. 968, University Science Books, 1997.</ref><ref>Atkins (8th ed.), Section 5.3, pp. 143–145.</ref>

==History==
Chemical potential was first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. He defined it as follows:

{{Blockquote|If to any homogeneous mass in a [[Physics:Thermodynamic state|state]] of [[Hydrostatic stress|hydrostatic stress]] we suppose an infinitesimal quantity of any substance to be added, the mass remaining homogeneous and its [[Physics:Entropy|entropy]] and [[Physics:Volume|volume]] remaining unchanged, the increase of the [[Physics:Internal energy|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 also that for the purposes of this definition, any [[Chemistry:Chemical element|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 [[Physics:Thermodynamics|thermodynamics]] and [[Physics:Physics|physics]] for any system undergoing change. Chemical potential is also referred to as '''partial molar Gibbs energy''' (see also [[Physics:Partial molar property|partial molar property]]). Chemical potential is measured in units of energy/particle or, equivalently, energy/[[Mole (unit)|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 [[Physics:Volume|volume]]–[[Physics:Entropy|entropy]]–[[Physics:Internal energy|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:
{{Blockquote|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:

:<math>\displaystyle \delta (\epsilon - T\eta + P\nu) = 0</math>

Where ''δ'' refers to the variation produced by any variations in the state of the parts of the body, and (when different parts of the body are in different states) in the proportion in which the body is divided between the different states.  The condition of stable equilibrium is that the value of the expression in the parenthesis shall be a minimum.}}

In this description, as used by Gibbs, ''ε'' refers to the [[Physics:Internal energy|internal energy]] of the body, ''η'' refers to the [[Physics:Entropy|entropy]] of the body, and ''ν'' is the [[Physics:Volume|volume]] of the body.

==Electrochemical, internal, external, and total chemical potential==
The abstract definition of chemical potential given above—total change in free energy per extra mole of substance—is more specifically called '''total chemical potential'''.<ref name=Kittel>''[https://books.google.com/books?id=c0R79nyOoNMC&pg=PA124 Thermal Physics]'' by Kittel and Kroemer, second edition, page 124.</ref><ref>''Thermodynamics in Earth and Planetary Sciences'' by Jibamitra Ganguly, [https://books.google.com/books?id=aD6TJAuCTVsC&pg=PA240 p. 240]. This text uses "internal", "external", and "total chemical potential" as in this article.</ref> If two locations have different total chemical potentials for a species, some of it may be due to potentials associated with "external" force fields ([[Physics:Electric potential energy|electric potential energy]], gravitational potential energy, etc.), while the rest would be due to "internal" factors (density, temperature, etc.)<ref name=Kittel/> Therefore, the total chemical potential can be split into '''internal chemical potential''' and '''external chemical potential''':
: <math>\mu_\text{tot} = \mu_\text{int} + \mu_\text{ext},</math>
where
: <math>\mu_\text{ext} = qV_\text{ele} + mgh + \cdots,</math>
i.e., the external potential is the sum of electric potential, gravitational potential, etc. (where ''q'' and ''m'' are the charge and mass of the species, ''V''<sub>ele</sub> and ''h'' are the [[Physics:Electric potential|electric potential]]<ref name="MortimerBook2">Mortimer, R. G. ''Physical Chemistry'', 3rd ed., p. 352, Academic Press, 2008.</ref> and height of the container, respectively, and ''g'' is the [[Physics:Standard gravity|acceleration due to gravity]]). The internal chemical potential includes everything else besides the external potentials, such as density, temperature, and enthalpy.  This formalism can be understood by assuming that the total energy of a system, <math>U</math>, is the sum of two parts: an internal energy, <math>U_\text{int}</math>, and an external energy due to the interaction of each particle with an external field, <math>U_\text{ext} = N (qV_\text{ele} + mgh + \cdots)</math>.  The definition of chemical potential applied to <math>U_\text{int} + U_\text{ext}</math> yields the above expression for <math>\mu_\text{tot}</math>.

The phrase "chemical potential" sometimes means "total chemical potential", but that is not universal.<ref name=Kittel/> In some fields, in particular [[Chemistry:Electrochemistry|electrochemistry]], semiconductor physics, and [[Physics:Solid-state physics|solid-state physics]], the term "chemical potential" means ''internal'' chemical potential, while the term [[Chemistry:Electrochemical potential|electrochemical potential]] is used to mean ''total'' chemical potential.<ref>''Electrochemical Methods'' by Bard and Faulkner, 2nd edition, Section 2.2.4(a), 4–5.</ref><ref>''Electrochemistry at Metal and Semiconductor Electrodes'', by Norio Sato, [https://books.google.com/books?id=olQzaXNgM74C&pg=PA4 pages 4–5].</ref><ref>''Physics Of Transition Metal Oxides'', by Sadamichi Maekawa, [https://books.google.com/books?id=iyNzfufnkBgC&pg=PA323 p. 323].</ref><ref>''The Physics of Solids: Essentials and Beyond'', by Eleftherios N. Economou, [https://books.google.com/books?id=SyGsHxH071MC&pg=PA140 page 140]. In this text, total chemical potential is usually called "electrochemical potential", but sometimes just "chemical potential". The internal chemical potential is referred to by the unwieldy phrase "chemical potential in the absence of the [electric] field".</ref><ref>''Solid State Physics'' by Ashcroft and Mermin, page 257 note 36. Page 593 of the same book uses, instead, an unusual "flipped" definition, where "chemical potential" is the total chemical potential, which is constant in equilibrium, and "electrochemical potential" is the internal chemical potential; presumably this unusual terminology was an unintentional mistake.</ref>

==Systems of particles==

===Electrons in solids===
{{Main|Physics:Fermi level}}
Electrons in solids have a chemical potential, defined 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, the chemical potential is usually expressed in energy per particle rather than energy per mole, and the energy per particle is conventionally given in units of [[Physics:Electronvolt|electronvolt]] (eV).

Chemical potential plays an especially important role in [[Physics:Solid-state physics|solid-state physics]] and is closely related to the concepts of [[Physics:Work function|work function]], [[Physics:Fermi energy|Fermi energy]], and [[Physics:Fermi level|Fermi level]]. For example, [[Physics:Doping (semiconductor)|n-type]] silicon has a higher internal chemical potential of electrons than [[Physics:Doping (semiconductor)|p-type]] silicon. In a [[Physics:P–n junction|p–n junction]] diode at equilibrium the chemical potential (''internal'' chemical potential) varies from the p-type to the n-type side, while the total chemical potential (electrochemical potential, or, [[Physics:Fermi level|Fermi level]]) is constant throughout the diode.

As described above, when describing chemical potential, one has to say "relative to what". In the case of electrons in semiconductors, internal chemical potential is often specified relative to some convenient point in the band structure, e.g., to the bottom of the conduction band. It may also be specified "relative to vacuum", to yield a quantity known as [[Physics:Work function|work function]], however, work function varies from surface to surface even on a completely homogeneous material. Total chemical potential, on the other hand, is usually specified relative to electrical ground.

In atomic physics, the chemical potential of the electrons in an atom is sometimes<ref>Morell, Christophe, [http://inac.cea.fr/Phocea/file.php?class=pisp&reload=1261486766&file=christophe.morell/files/98/introduction_to_Density_Functional_Theory_of_Chemical_Reactivity.pdf Introduction to Density Functional Theory of Chemical Reactivity: The so-called Conceptual DFT] {{Webarchive|url=https://web.archive.org/web/20170828231922/http://inac.cea.fr/Phocea/file.php?class=pisp&reload=1261486766&file=christophe.morell%2Ffiles%2F98%2Fintroduction_to_Density_Functional_Theory_of_Chemical_Reactivity.pdf |date=2017-08-28 }}, retrieved May 2016.</ref> said to be the negative of the atom's [[Chemistry:Electronegativity|electronegativity]].  Likewise, 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 inserting the energetic definitions of the ionization potential and [[Physics:Electron affinity|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., i.e.,
: <math>\mu_\text{Mulliken} = -\chi_\text{Mulliken} = -\frac{IP + EA}{2} = \left[\frac{\delta E[N]}{\delta N}\right]_{N=N_0}.</math>

===Sub-nuclear particles===
In recent years,{{When|date=November 2021}} [[Physics:Thermal physics|thermal physics]] has applied the definition of chemical potential to systems in [[Physics:Particle physics|particle physics]] and its associated processes. For example, in a [[Physics:Quark–gluon plasma|quark–gluon plasma]] or other [[Physics:QCD matter|QCD matter]], at every point in space there is a chemical potential for [[Physics:Photon|photon]]s, a chemical potential for electrons, a chemical potential for [[Physics:Baryon number|baryon number]], [[Physics:Electric charge|electric charge]], and so forth.

In the case of photons, photons are [[Physics:Boson|boson]]s and can very easily and rapidly appear or disappear. Therefore, at thermodynamic equilibrium, the chemical potential of photons is always and everywhere zero. The reason is, if the chemical potential somewhere was higher than zero, photons would spontaneously disappear from that area until the chemical potential went back to zero; likewise, if the chemical potential somewhere was less than zero, photons would spontaneously appear until the chemical potential went back to zero. Since this process occurs extremely rapidly (at least, it occurs rapidly in the presence of dense charged matter), it is safe to assume that the photon chemical potential is never different from zero.

Electric charge is different because it is conserved, i.e. it can be neither created nor destroyed. It can, however, diffuse. The "chemical potential of electric charge" controls this diffusion: Electric charge, like anything else, will tend to diffuse from areas of higher chemical potential to areas of lower chemical potential.<ref>{{cite book |author=Baierlein, Ralph |title=Thermal Physics |publisher=Cambridge University Press |year=2003 |isbn=978-0-521-65838-6 |oclc=39633743 |url-access=registration |url=https://archive.org/details/thermalphysics00ralp}}</ref> Other conserved quantities like [[Physics:Baryon number|baryon number]] are the same. In fact, each conserved quantity is associated with a chemical potential and a corresponding tendency to diffuse to equalize it out.<ref>[https://books.google.com/books?id=SAlbKkdor1gC&pg=PA91 Hadrons and Quark-Gluon Plasma], by Jean Letessier, Johann Rafelski, p. 91.</ref>

In the case of electrons, the behaviour depends on temperature and context. At low temperatures, with no [[Physics:Positron|positron]]s present, electrons cannot be created or destroyed. Therefore, there is an electron chemical potential that might vary in space, causing diffusion. At very high temperatures, however, electrons and positrons can spontaneously appear out of the vacuum ([[Physics:Pair production|pair production]]), so the chemical potential of electrons by themselves becomes a less useful quantity than the chemical potential of the conserved quantities like (electrons minus positrons).

The chemical potentials of [[Physics:Boson|boson]]s and [[Physics:Fermion|fermion]]s is related to the number of particles and the temperature by [[Physics:Bose–Einstein statistics|Bose–Einstein statistics]] and [[Physics:Fermi–Dirac statistics|Fermi–Dirac statistics]] respectively.

===Ideal vs. non-ideal solutions===
[[File:Chempotential.jpg|thumb|500px|The chemical potential of component ''i'' in solution for (left) ideal [incorrectly linearized] and (right) real solutions ]]
Generally the chemical potential is given as a sum of an ideal contribution and an excess contribution:
: <math> \mu_i = \mu_i^\text{ideal} + \mu_i^\text{excess},</math>

In an [[Physics:Ideal solution|ideal solution]], the chemical potential of species ''i'' (μ<sub>''i''</sub>) is dependent on temperature and pressure.

μ<sub>''i''0</sub>(''T'', ''P'') is defined as the chemical potential of pure species ''i''. Given this definition, the chemical potential of species ''i'' in an ideal solution is

: <math>\mu_i^\text{ideal} \approx \mu_{i0} + RT \ln(x_i),</math>

where ''R'' is the gas constant, and <math>x_i</math> is the mole fraction of species ''i'' contained in the solution. The chemical potential becomes negative infinity when <math>x_i = 0</math>, but this does not lead to nonphysical results because <math>x_i = 0</math> means that species ''i'' is not present in the system.

This equation assumes that <math>\mu_i</math> only depends on the mole fraction (<math>x_i</math>) contained in the solution. This neglects intermolecular interaction between species ''i'' with itself and other species [''i''–(''j''≠''i'')]. This can be corrected for by factoring in the coefficient of activity of species ''i'', defined as γ<sub>''i''</sub>. This correction yields

: <math>\mu_i = \mu_{i0}(T, P) + RT \ln(x_i) + RT \ln(\gamma_i) = \mu_{i0}(T, P) + RT \ln(x_i \gamma_i).</math>

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

==See also==
* [[Chemistry:Chemical equilibrium|Chemical equilibrium]]
* [[Chemistry:Electrochemical potential|Electrochemical potential]]
* [[Chemistry:Equilibrium chemistry|Equilibrium chemistry]]
* [[Chemistry:Excess chemical potential|Excess chemical potential]]
* [[Physics:Fugacity|Fugacity]]
* [[Physics:Partial molar property|Partial molar property]]
* [[Chemistry:Thermodynamic activity|Thermodynamic activity]]
* [[Physics:Thermodynamic equilibrium|Thermodynamic equilibrium]]

==References==
{{Reflist}}
<!-- * {{cite journal | last = Kaplan| first = T. A. |date=March 2006
 | title = The Chemical Potential| journal = Journal of Statistical Physics
 | volume = 122| issue = 6| pages = 1237–1260| doi = 10.1007/s10955-005-8067-x|bibcode = 2006JSP...122.1237K }} -->

==External links==
* G. Cook and R. H. Dickerson, "Understanding the chemical potential", American Journal of Physics '''63''', pp. 737–742 (1995), {{doi|10.1119/1.17844}}.
* T. A. Kaplan, "The Chemical Potential", Journal of Statistical Physics '''122''', pp. 1237–1260 (2006), {{doi|10.1007/s10955-005-8067-x}}.
* {{cite web |url=http://www.job-stiftung.de/index.php?id=54,0,0,1,0,0 |title=Values of the chemical potential of 1300 substances |website=Eduard-Job-Foundation for Thermo- and Matterdynamics}}


[[Category:Physical chemistry]]
[[Category:Potentials]]
[[Category:Chemical thermodynamics]]
[[Category:Thermodynamic properties]]
[[Category:Chemical engineering thermodynamics]]

{{Sourceattribution|Chemical potential}}