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Gibbs free energy

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$U(S,V)$
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$A(T,V)=U-TS$
Gibbs free energy
$G(T,p)=H-TS$

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Inthermodynamics, theGibbs free energy (orGibbs energy as the recommended name; symbol $G {\displaystyle G}$ ) is athermodynamic potential that can be used to calculate themaximum amount ofwork, other thanpressure–volume work, that may be performed by athermodynamically closed system at constanttemperature andpressure. It also provides a necessary condition for processes such aschemical reactions that may occur under these conditions. The Gibbs free energy is expressed as $G(p,T)=U+pV-TS=H-TS$ where:

${\textstyle U}$ is theinternal energy of the system
${\textstyle H}$ is theenthalpy of the system
${\textstyle S}$ is theentropy of the system
${\textstyle T}$ is the temperature of the system
${\textstyle V}$ is thevolume of the system
${\textstyle p}$ is the pressure of the system (which must be equal to that of the surroundings for mechanical equilibrium).

The Gibbs free energy change (⁠ $\Delta G=\Delta H-T\Delta S$ ⁠, measured injoules inSI) is themaximum amount of non-volume expansion work that can be extracted from a closed system (one that can exchange heat and work with its surroundings, but not matter) at fixed temperature and pressure. This maximum can be attained only in a completelyreversible process. When a system transforms reversibly from an initial state to a final state under these conditions, the decrease in Gibbs free energy equals the work done by the system to its surroundings, minus the work of thepressure forces.^[1]

The Gibbs energy is the thermodynamic potential that is minimized when a system reacheschemical equilibrium at constant pressure and temperature when not driven by an applied electrolytic voltage. Its derivative with respect to the reaction coordinate of the system then vanishes at the equilibrium point. As such, a reduction in $G {\displaystyle G}$ is necessary for a reaction to bespontaneous under these conditions.

The concept of Gibbs free energy, originally calledavailable energy, was developed in the 1870s by the American scientistJosiah Willard Gibbs. In 1873, Gibbs described this "available energy" as^[2]^: 400

the greatest amount of mechanical work which can be obtained from a given quantity of a certain substance in a given initial state, without increasing its totalvolume or allowing heat to pass to or from external bodies, except such as at the close of the processes are left in their initial condition.

The initial state of the body, according to Gibbs, is supposed to be such that "the body can be made to pass from it to states ofdissipated energy byreversible processes". In his 1876magnum opusOn the Equilibrium of Heterogeneous Substances, a graphical analysis of multi-phase chemical systems, he engaged his thoughts on chemical-free energy in full.

If the reactants and products are all in their thermodynamicstandard states, then the defining equation is written as⁠ $\Delta G^{\circ }=\Delta H^{\circ }-T\Delta S^{\circ }$ ⁠, where $H {\displaystyle H}$ isenthalpy, $T {\displaystyle T}$ isabsolute temperature, and $S {\displaystyle S}$ isentropy.

Overview

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The reaction C_(s)^diamond → C_(s)^graphite has a negative change in Gibbs free energy and is therefore thermodynamically favorable at 25 °C and 1 atm. However, the reaction is too slow to be observed, because of its very highactivation energy. Whether a reaction is thermodynamically favorable does not determine its rate.

According to thesecond law of thermodynamics, for systems reacting at fixed temperature and pressure without input of non-Pressure Volume (pV)work, there is a general natural tendency to achieve a minimum of the Gibbs free energy.^{[citation needed]}

A quantitative measure of the favorability of a given reaction under these conditions is the change ΔG (sometimes written "deltaG" or "dG") in Gibbs free energy that is (or would be) caused by the reaction. As a necessary condition for the reaction to occur at constant temperature and pressure, ΔG must be smaller than the non-pressure-volume (non-pV, e.g. electrical) work, which is often equal to zero (then ΔG must be negative). ΔG equals the maximum amount of non-pV work that can be performed as a result of the chemical reaction for the case of a reversible process. If analysis indicates a positive ΔG for a reaction, then energy — in the form of electrical or other non-pV work — would have to be added to the reacting system for ΔG to be smaller than the non-pV work and make it possible for the reaction to occur.^[3]^{: 298–299}

One can think of ∆G as the amount of "free" or "useful" energy available to do non-pV work at constant temperature and pressure. The equation can be also seen from the perspective of the system taken together with its surroundings (the rest of the universe). First, one assumes that the given reaction at constant temperature and pressure is the only one that is occurring. Then theentropy released or absorbed by the system equals the entropy that the environment must absorb or release, respectively. The reaction will only be allowed if the total entropy change of the universe is zero or positive. This is reflected in a negative ΔG, and the reaction is called anexergonic process.^{[citation needed]}

If two chemical reactions are coupled, then an otherwiseendergonic reaction (one with positive ΔG) can be made to happen. The input of heat into an inherently endergonic reaction, such as theelimination ofcyclohexanol tocyclohexene, can be seen as coupling an unfavorable reaction (elimination) to a favorable one (burning of coal or other provision of heat) such that the total entropy change of the universe is greater than or equal to zero, making thetotal Gibbs free energy change of the coupled reactions negative.^{[citation needed]}

In traditional use, the term "free" was included in "Gibbs free energy" to mean "available in the form of useful work".^[1] The characterization becomes more precise if we add the qualification that it is the energy available for non-pressure-volume work.^[4] (An analogous, but slightly different, meaning of "free" applies in conjunction with theHelmholtz free energy, for systems at constant temperature). However, an increasing number of books and journal articles do not include the attachment "free", referring toG as simply "Gibbs energy". This is the result of a 1988IUPAC meeting to set unified terminologies for the international scientific community, in which the removal of the adjective "free" was recommended.^[5]^[6]^[7] This standard, however, has not yet been universally adopted.

The name "freeenthalpy" was also used forG in the past.^[6]

History

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Definitions

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Willard Gibbs' 1873available energy (free energy) graph, which shows a plane perpendicular to the axis ofv (volume) and passing through point A, which represents the initial state of the body. MN is the section of the surface ofdissipated energy. Qε and Qη are sections of the planesη = 0 andε = 0, and therefore parallel to the axes ofε (internal energy) andη (entropy), respectively. AD and AE are the energy and entropy of the body in its initial state, AB and AC itsavailable energy (Gibbs free energy) and itscapacity for entropy (the amount by which the entropy of the body can be increased without changing the energy of the body or increasing its volume), respectively.

The Gibbs free energy is defined as $G(p,T)=U+pV-TS,$ which is the same as $G(p,T)=H-TS,$ where:

U is theinternal energy (SI unit:joule),
p ispressure (SI unit:pascal),
V isvolume (SI unit: m³),
T is thetemperature (SI unit:kelvin),
S is theentropy (SI unit: joule per kelvin),
H is theenthalpy (SI unit: joule).

The expression for the infinitesimal reversible change in the Gibbs free energy as a function of its"natural variables"p andT, for anopen system, subjected to the operation of external forces (for instance, electrical or magnetic)X_i, which cause the external parameters of the systema_i to change by an amount da_i, can be derived as follows from the first law for reversible processes: ${\begin{aligned}T\,\mathrm {d} S&=\mathrm {d} U+p\,\mathrm {d} V-\sum _{i=1}^{k}\mu _{i}\,\mathrm {d} N_{i}+\sum _{i=1}^{n}X_{i}\,\mathrm {d} a_{i}+\cdots \\\mathrm {d} (TS)-S\,\mathrm {d} T&=\mathrm {d} U+\mathrm {d} (pV)-V\,\mathrm {d} p-\sum _{i=1}^{k}\mu _{i}\,\mathrm {d} N_{i}+\sum _{i=1}^{n}X_{i}\,\mathrm {d} a_{i}+\cdots \\\mathrm {d} (U-TS+pV)&=V\,\mathrm {d} p-S\,\mathrm {d} T+\sum _{i=1}^{k}\mu _{i}\,\mathrm {d} N_{i}-\sum _{i=1}^{n}X_{i}\,\mathrm {d} a_{i}+\cdots \\\mathrm {d} G&=V\,\mathrm {d} p-S\,\mathrm {d} T+\sum _{i=1}^{k}\mu _{i}\,\mathrm {d} N_{i}-\sum _{i=1}^{n}X_{i}\,\mathrm {d} a_{i}+\cdots \end{aligned}}$ where:

μ_i is thechemical potential of theithchemical component. (SI unit: joules per particle^[9] or joules per mole^[1])
N_i is thenumber of particles (or number of moles) composing theith chemical component.

This is one form of theGibbs fundamental equation.^[10] In the infinitesimal expression, the term involving the chemical potential accounts for changes in Gibbs free energy resulting from an influx or outflux of particles. In other words, it holds for anopen system or for aclosed, chemically reacting system where theN_i are changing. For a closed, non-reacting system, this term may be dropped.

Any number of extra terms may be added, depending on the particular system being considered. Aside frommechanical work, a system may, in addition, perform numerous other types of work. For example, in the infinitesimal expression, the contractile work energy associated with a thermodynamic system that is a contractile fiber that shortens by an amount −dl under a forcef would result in a termf dl being added. If a quantity of charge −de is acquired by a system at an electrical potential Ψ, the electrical work associated with this is −Ψ de, which would be included in the infinitesimal expression. Other work terms are added on per system requirements.^[11]

Each quantity in the equations above can be divided by the amount of substance, measured inmoles, to formmolar Gibbs free energy. The Gibbs free energy is one of the most important thermodynamic functions for the characterization of a system. It is a factor in determining outcomes such as thevoltage of anelectrochemical cell, and theequilibrium constant for areversible reaction. In isothermal, isobaric systems, Gibbs free energy can be thought of as a "dynamic" quantity, in that it is a representative measure of the competing effects of the enthalpic^{[clarification needed]} and entropic driving forces involved in a thermodynamic process.

The temperature dependence of the Gibbs energy for anideal gas is given by theGibbs–Helmholtz equation, and its pressure dependence is given by^[12] ${\frac {G}{N}}={\frac {G^{\circ }}{N}}+kT\ln {\frac {p}{p^{\circ }}}.$ or more conveniently as itschemical potential: ${\frac {G}{N}}=\mu =\mu ^{\circ }+kT\ln {\frac {p}{p^{\circ }}}.$

In non-ideal systems,fugacity comes into play.

Derivation

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The Gibbs free energytotal differential with respect tonatural variables may be derived byLegendre transforms of theinternal energy.

\mathrm {d} U=T\,\mathrm {d} S-p\,\mathrm {d} V+\sum _{i}\mu _{i}\,\mathrm {d} N_{i}.

The definition ofG from above is

G=U+pV-TS

Taking the total differential, we have

\mathrm {d} G=\mathrm {d} U+p\,\mathrm {d} V+V\,\mathrm {d} p-T\,\mathrm {d} S-S\,\mathrm {d} T.

Replacing dU with the result from the first law gives^[13]

{\begin{aligned}\mathrm {d} G&=T\,\mathrm {d} S-p\,\mathrm {d} V+\sum _{i}\mu _{i}\,\mathrm {d} N_{i}+p\,\mathrm {d} V+V\,\mathrm {d} p-T\,\mathrm {d} S-S\,\mathrm {d} T\\&=V\,\mathrm {d} p-S\,\mathrm {d} T+\sum _{i}\mu _{i}\,\mathrm {d} N_{i}.\end{aligned}}

The natural variables ofG are thenp,T, and {N_i}.

Homogeneous systems

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BecauseS,V, andN_i areextensive variables, anEuler relation allows easy integration of dU:^[13]

U=TS-pV+\sum _{i}\mu _{i}N_{i}.

Because some of the natural variables ofG are intensive, dG may not be integrated using Euler relations as is the case with internal energy. However, simply substituting the above integrated result forU into the definition ofG gives a standard expression forG:^[13]

{\begin{aligned}G&=U+pV-TS\\&=\left(TS-pV+\sum _{i}\mu _{i}N_{i}\right)+pV-TS\\&=\sum _{i}\mu _{i}N_{i}.\end{aligned}}

This result shows that the chemical potential of a substance $i {\displaystyle i}$ is its (partial) molecular Gibbs free energy. It applies to homogeneous, macroscopic systems, but not to all thermodynamic systems.^[14]

Gibbs free energy of reactions

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The system under consideration is held at constant temperature and pressure, and is closed (no matter can come in or out). The Gibbs energy of any system is⁠ $G=U+pV-TS$ ⁠ and an infinitesimal change inG, at constant temperature and pressure, yields

dG=dU+pdV-TdS.

By thefirst law of thermodynamics, a change in the internal energyU is given by

dU=\delta Q+\delta W

whereδQ is energy added as heat, andδW is energy added as work. The work done on the system may be written asδW = −pdV +δW_x, where−pdV is the mechanical work of compression/expansion done on or by the system andδW_x is all other forms of work, which may include electrical, magnetic, etc. Then

dU=\delta Q-pdV+\delta W_{x}

and the infinitesimal change inG is

dG=\delta Q-TdS+\delta W_{x}.

Thesecond law of thermodynamics states that for a closed system at constant temperature (in a heat bath), $TdS\geq \delta Q$ , and so it follows that

dG\leq \delta W_{x}

Assuming that only mechanical work is done, this simplifies to

dG\leq 0

This means that for such a system when not in equilibrium, the Gibbs energy will always be decreasing, and in equilibrium, the infinitesimal changedG will be zero. In particular, this will be true if the system is experiencing any number of internal chemical reactions on its path to equilibrium.

In electrochemical thermodynamics

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When electric chargedQ_ele is passed between the electrodes of an electrochemical cell generating anemf ${\mathcal {E}}$ , an electrical work term appears in the expression for the change in Gibbs energy: $dG=-SdT+Vdp+{\mathcal {E}}dQ_{ele},$ whereS is theentropy,V is the system volume,p is its pressure andT is itsabsolute temperature.

The combination ( ${\mathcal {E}}$ ,Q_ele) is an example of aconjugate pair of variables. At constant pressure the above equation produces aMaxwell relation that links the change in open cell voltage with temperatureT (a measurable quantity) to the change in entropyS when charge is passedisothermally andisobarically. The latter is closely related to the reactionentropy of the electrochemical reaction that lends the battery its power. This Maxwell relation is:^[15]

\left({\frac {\partial {\mathcal {E}}}{\partial T}}\right)_{Q_{ele},p}=-\left({\frac {\partial S}{\partial Q_{ele}}}\right)_{T,p}

If a mole of ions goes into solution (for example, in a Daniell cell, as discussed below) the charge through the external circuit is

\Delta Q_{ele}=-n_{0}F_{0}\,,

wheren₀ is the number of electrons/ion, andF₀ is theFaraday constant and the minus sign indicates discharge of the cell. Assuming constant pressure and volume, the thermodynamic properties of the cell are related strictly to the behavior of its emf by

\Delta H=-n_{0}F_{0}\left({\mathcal {E}}-T{\frac {d{\mathcal {E}}}{dT}}\right),

where ΔH is theenthalpy of reaction. The quantities on the right are all directly measurable.

Useful identities to derive the Nernst equation

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This sectionmay beconfusing or unclear to readers. In particular, the physical situation is not explained. Also, the circle notation is not well explained (even in the one case where it is attempted). It's just bare equations. Please helpclarify the section. There might be a discussion about this onthe talk page.(March 2015) (Learn how and when to remove this message)

During a reversible electrochemical reaction at constant temperature and pressure, the following equations involving the Gibbs free energy hold:

$\Delta _{\text{r}}G=\Delta _{\text{r}}G^{\circ }+RT\ln Q_{\text{r}}$ (seechemical equilibrium),
$\Delta _{\text{r}}G^{\circ }=-RT\ln K_{\text{eq}}$ (for a system at chemical equilibrium),
$\Delta _{\text{r}}G=w_{\text{elec,rev}}=-nF{\mathcal {E}}$ (for a reversible electrochemical process at constant temperature and pressure),
$\Delta _{\text{r}}G^{\circ }=-nF{\mathcal {E}}^{\circ }$ (definition of ${\mathcal {E}}^{\circ }$ ),

and rearranging gives ${\begin{aligned}nF{\mathcal {E}}^{\circ }&=RT\ln K_{\text{eq}},\\nF{\mathcal {E}}&=nF{\mathcal {E}}^{\circ }-RT\ln Q_{\text{r}},\\{\mathcal {E}}&={\mathcal {E}}^{\circ }-{\frac {RT}{nF}}\ln Q_{\text{r}},\end{aligned}}$ which relates the cell potential resulting from the reaction to the equilibrium constant andreaction quotient for that reaction (Nernst equation),where

Δ_rG, Gibbs free energy change per mole of reaction,
Δ_rG°, Gibbs free energy change per mole of reaction for unmixed reactants and products at standard conditions (i.e. 298 K, 100 kPa, 1 M of each reactant and product),
R,gas constant,
T, absolutetemperature,
ln,natural logarithm,
Q_r,reaction quotient (unitless),
K_eq,equilibrium constant (unitless),
w_elec,rev,electrical work in a reversible process (chemistry sign convention),
n, number ofmoles ofelectrons transferred in the reaction,
F =N_Ae ≈ 96485 C/mol,Faraday constant (charge permole of electrons),
${\mathcal {E}}$ ,cell potential,
${\mathcal {E}}^{\circ }$ ,standard cell potential.

Moreover, we also have ${\begin{aligned}K_{\text{eq}}&=e^{-{\frac {\Delta _{\text{r}}G^{\circ }}{RT}}},\\\Delta _{\text{r}}G^{\circ }&=-RT\left(\ln K_{\text{eq}}\right)=-2.303\,RT\left(\log _{10}K_{\text{eq}}\right),\end{aligned}}$ which relates the equilibrium constant with Gibbs free energy. This implies that at equilibrium $Q_{\text{r}}=K_{\text{eq}}$ and $\Delta _{\text{r}}G=0.$

Standard Gibbs energy change of formation

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Table of selected substances^[16]
Substance (state)	Δ_fG°
Substance (state)	(kJ/mol)	(kcal/mol)
NO(g)	87.6	20.9
NO₂(g)	51.3	12.3
N₂O(g)	103.7	24.78
H₂O(g)	−228.6	−54.64
H₂O(l)	−237.1	−56.67
CO₂(g)	−394.4	−94.26
CO(g)	−137.2	−32.79
CH₄(g)	−50.5	−12.1
C₂H₆(g)	−32.0	−7.65
C₃H₈(g)	−23.4	−5.59
C₆H₆(g)	129.7	29.76
C₆H₆(l)	124.5	31.00

Thestandard Gibbs free energy of formation of a compound is the change of Gibbs free energy that accompanies the formation of 1mole of that substance from its component elements, in theirstandard states (the most stable form of the element at 25 °C and 100 kPa). Its symbol is Δ_fG˚.

All elements in their standard states (diatomicoxygen gas,graphite, etc.) have standard Gibbs free energy change of formation equal to zero, as there is no change involved.

Δ_fG = Δ_fG˚ +RT lnQ_f,

whereQ_f is thereaction quotient.

At equilibrium, Δ_fG = 0, andQ_f =K, so the equation becomes

Δ_fG˚ = −RT lnK,

whereK is theequilibrium constant of the formation reaction of the substance from the elements in their standard states.

Graphical interpretation by Gibbs

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Gibbs free energy was originally defined graphically. In 1873, American scientistWillard Gibbs published his first thermodynamics paper, "Graphical Methods in the Thermodynamics of Fluids", in which Gibbs used the two coordinates of the entropy and volume to represent the state of the body. In his second follow-up paper, "A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces", published later that year, Gibbs added in the third coordinate of the energy of the body, defined on three figures. In 1874, Scottish physicistJames Clerk Maxwell used Gibbs' figures to make a 3D energy-entropy-volumethermodynamic surface of a fictitious water-like substance.^[17] Thus, in order to understand the concept of Gibbs free energy, it may help to understand its interpretation by Gibbs as section AB on his figure 3, and as Maxwell sculpted that section on his3D surface figure.

American scientistWillard Gibbs' 1873 figures two and three (above left and middle) used by Scottish physicistJames Clerk Maxwell in 1874 to create a three-dimensionalentropy,volume,energythermodynamic surface diagram for a fictitious water-like substance, transposed the two figures of Gibbs (above right) onto the volume-entropy coordinates (transposed to bottom of cube) and energy-entropy coordinates (flipped upside down and transposed to back of cube), respectively, of a three-dimensionalCartesian coordinates; the region AB being the first-ever three-dimensional representation of Gibbs free energy, or what Gibbs called "available energy"; the region AC being its capacity forentropy, what Gibbs defined as "the amount by which the entropy of the body can be increased without changing the energy of the body or increasing its volume.

Notes and references

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^^a ^b ^cPerrot, Pierre (1998).A to Z of Thermodynamics. Oxford University Press.ISBN 0-19-856552-6.
^^a ^bGibbs, Josiah Willard (December 1873)."A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces"(PDF).Transactions of the Connecticut Academy of Arts and Sciences.2:382–404.
^Peter Atkins; Loretta Jones (1 August 2007).Chemical Principles: The Quest for Insight. W. H. Freeman.ISBN 978-1-4292-0965-6.
^Reiss, Howard (1965).Methods of Thermodynamics. Dover Publications.ISBN 0-486-69445-3.
^Calvert, J. G. (1 January 1990)."Glossary of atmospheric chemistry terms (Recommendations 1990)".Pure and Applied Chemistry.62 (11):2167–2219.doi:10.1351/pac199062112167.
^^a ^b"Gibbs energy (function), G".IUPAC Gold Book (Compendium of Chemical Technology). IUPAC (International Union of Pure and Applied Chemistry). 2008.doi:10.1351/goldbook.G02629. Retrieved24 December 2020.It was formerly called free energy or free enthalpy.
^Lehmann, H. P.; Fuentes-Arderiu, X.; Bertello, L. F. (1 January 1996)."Glossary of terms in quantities and units in Clinical Chemistry (IUPAC-IFCC Recommendations 1996)".Pure and Applied Chemistry.68 (4):957–1000.doi:10.1351/pac199668040957.S2CID 95196393.
^Henry Marshall Leicester (1971).The Historical Background of Chemistry. Courier Corporation.ISBN 978-0-486-61053-5.
^Chemical Potential, IUPAC Gold Book.
^Müller, Ingo (2007).A History of Thermodynamics – the Doctrine of Energy and Entropy. Springer.ISBN 978-3-540-46226-2.
^Katchalsky, A.; Curran, Peter F. (1965).Nonequilibrium Thermodynamics in Biophysics.Harvard University Press. CCN 65-22045.
^Atkins, Peter; de Paula, Julio (2006).Atkins' Physical Chemistry (8th ed.). W. H. Freeman. p. 109.ISBN 0-7167-8759-8.
^^a ^b ^cSalzman, William R. (2001-08-21)."Open Systems".Chemical Thermodynamics.University of Arizona. Archived fromthe original on 2007-07-07. Retrieved2007-10-11.
^Brachman, M. K. (1954). "Fermi Level, Chemical Potential, and Gibbs Free Energy".The Journal of Chemical Physics.22 (6): 1152.Bibcode:1954JChPh..22.1152B.doi:10.1063/1.1740312.
^H. S. Harned, B. B. Owen, The Physical Chemistry of Electrolytic Solutions, third edition, Reinhold Publishing Corporation, N.Y.,1958, p. 2-6
^CRC Handbook of Chemistry and Physics, 2009, pp. 5-4–5-42, 90th ed., Lide.
^James Clerk Maxwell,Elizabeth Garber, Stephen G. Brush, and C. W. Francis Everitt (1995),Maxwell on heat and statistical mechanics: on "avoiding all personal enquiries" of molecules, Lehigh University Press,ISBN 0-934223-34-3, p. 248.

External links

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IUPAC definition (Gibbs energy)
Gibbs Free Energy – Georgia State University

v t e Statistical mechanics
Theory	Principle of maximum entropy ergodic theory
Statistical thermodynamics	Ensembles partition functions equations of state thermodynamic potential: U H F G Maxwell relations
Models	Ferromagnetism models Ising Potts Heisenberg percolation Particles withforce field depletion force Lennard-Jones potential
Mathematical approaches	Boltzmann equation H-theorem Vlasov equation BBGKY hierarchy stochastic process mean-field theory andconformal field theory
Critical phenomena	Phase transition Critical exponents correlation length size scaling
Entropy	Boltzmann Shannon Tsallis Rényi von Neumann
Applications	Statistical field theory elementary particle superfluidity Condensed matter physics Complex system chaos information theory Boltzmann machine