Editing Josiah Willard Gibbs (section)

=== Chemical and electrochemical thermodynamics ===
[[File:Wykres Gibbsa.svg|thumb|250px|alt=Diagram representing the free energy of a substance|Graphical representation of the free energy of a body, from the latter of the papers published by Gibbs in 1873. This shows a plane of constant volume, passing through the point ''A'' that represents the body's initial state. The curve ''MN'' is the section of the "surface of dissipated energy". ''AD'' and ''AE'' are, respectively, the energy (''ε'') and entropy (''η'') of the initial state. ''AB'' is the "available energy" (now called the [[Helmholtz free energy]]) and ''AC'' the "capacity for entropy" (i.e., the amount by which the entropy can be increased without changing the energy or volume).]]

Gibbs's papers from the 1870s introduced the idea of expressing the internal energy&nbsp;''U'' of a system in terms of the [[entropy]]&nbsp;''S'', in addition to the usual [[state variable]]s of volume&nbsp;''V'', pressure&nbsp;''p'', and temperature&nbsp;''T''. He also introduced the concept of the [[chemical potential]]&nbsp;<math>\mu</math> of a given chemical species, defined to be the rate of the increase in ''U'' associated with the increase in the number ''N'' of molecules of that species (at constant entropy and volume). Thus, it was Gibbs who first combined the first and second [[laws of thermodynamics]] by expressing the infinitesimal change in the internal energy, d''U'', of a [[closed system]] in the form<ref name=Klein1990 />
: <math>\mathrm{d}U = T\mathrm{d}S - p \,\mathrm{d}V + \sum_i \mu_i \,\mathrm{d} N_i,</math>
where ''T'' is the [[absolute temperature]], ''p'' is the pressure, d''S'' is an infinitesimal change in entropy and d''V'' is an infinitesimal change of volume. The last term is the sum, over all the chemical species in a chemical reaction, of the chemical potential, ''μ''<sub>i</sub>, of the ''i''-th species, multiplied by the infinitesimal change in the number of moles, d''N''<sub>i</sub> of that species. By taking the [[Legendre transformation|Legendre transform]] of this expression, he defined the concepts of [[enthalpy]] ''H'' and [[Gibbs free energy]] ''G'':
: <math>G_{(p,T)} = H - TS.</math>
This compares to the expression for [[Helmholtz free energy]] ''A'':
: <math>A_{(v,T)} = U - TS.</math>

When the Gibbs free energy for a chemical reaction is negative, the reaction will proceed spontaneously. When a chemical system is at [[equilibrium chemistry|equilibrium]], the change in Gibbs free energy is zero. An [[equilibrium constant]] is simply related to the free energy change when the reactants are in their [[standard state]]s:
: <math>\Delta G^\ominus = -RT \ln K^\ominus.</math>
[[Chemical potential]] is usually defined as partial molar Gibbs free energy:
: <math>\mu_i = \left(\frac{\partial G}{\partial N_i}\right)_{T,P,N_{j\neq i}}.</math>

Gibbs also obtained what later came to be known as the "[[Gibbs–Duhem equation]]".<ref name="Ott" />

In an [[Electrochemistry|electrochemical reaction]] characterized by an [[electromotive force]] ℰ and an amount of transferred charge ''Q'', Gibbs's starting equation becomes 
: <math>\mathrm{d}U = T\mathrm{d}S - p \,\mathrm{d}V + \mathcal{E}\mathrm{d}Q.</math>

[[File:Apparatus for investigating the Phase Rule of an iron-nitrogen system 9p290969x.tif|thumb|right|Apparatus for investigating the phase rule of an iron–nitrogen system, U.S. Fixed Nitrogen Research Laboratory, 1930]]
The publication of the paper "[[On the Equilibrium of Heterogeneous Substances]]" (1874–1878) is now regarded as a landmark in the development of [[chemistry]].<ref name="MacTutor" /> In it, Gibbs developed a rigorous mathematical theory for various [[transport phenomena]], including [[adsorption]], [[electrochemistry]], and the [[Marangoni effect]] in fluid mixtures.<ref name="Wheeler-thermodynamics" /> He also formulated the [[phase rule]]
: <math>F = C - P + 2</math>
for the number ''F'' of [[Intensive and extensive properties|variables]] that may be independently controlled in an equilibrium mixture of ''C'' components existing in ''P'' [[Phase (matter)|phases]]. The phase rule is very useful in diverse areas, such as metallurgy, mineralogy, and petrology. It can also be applied to various research problems in physical chemistry.<ref>Wheeler 1998, p.&nbsp;79.</ref>