TheInfoList

OR:

In
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of th ...
, the Gibbs free energy (or Gibbs energy; symbol $G$) is a thermodynamic potential that can be used to calculate the
maximum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
amount of
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal ...
that may be performed by a thermodynamically closed system at constant
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied ...
and
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
. It also provides a necessary condition for processes such as
chemical reactions A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breaki ...
that may occur under these conditions. The Gibbs free energy change , measured in
joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...
s in SI) is the ''maximum'' amount of non-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 completely reversible 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 the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
forces. The Gibbs energy is the thermodynamic potential that is minimized when a system reaches
chemical equilibrium In a chemical reaction, chemical equilibrium is the state in which both the reactants and products are present in concentrations which have no further tendency to change with time, so that there is no observable change in the properties of th ...
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 $G$ is necessary for a reaction to be
spontaneous Spontaneous may refer to: * Spontaneous abortion * Spontaneous bacterial peritonitis * Spontaneous combustion * Spontaneous declaration * Spontaneous emission * Spontaneous fission * Spontaneous generation * Spontaneous human combustion * ...
under these conditions. The concept of Gibbs free energy, originally called ''available energy'', was developed in the 1870s by the American scientist
Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
. In 1873, Gibbs described this "available energy" as 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 of dissipated energy by reversible processes". In his 1876
magnum opus A masterpiece, ''magnum opus'' (), or ''chef-d’œuvre'' (; ; ) in modern use is a creation that has been given much critical praise, especially one that is considered the greatest work of a person's career or a work of outstanding creativity, ...
'' On 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 thermodynamic
standard state In chemistry, the standard state of a material (pure substance, mixture or solution) is a reference point used to calculate its properties under different conditions. A superscript circle ° (degree symbol) or a Plimsoll (⦵) character is use ...
s, then the defining equation is written as , where ''$H$'' is
enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...
, $T$ is
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynami ...
, and ''$S$'' is
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
.

# Overview According to the
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unless ...
, for systems reacting at fixed temperature and pressure without input of non-''Pressure Volume'' (PV)
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal ...
, there is a general natural tendency to achieve a minimum of the Gibbs free energy. A quantitative measure of the favorability of a given reaction under these conditions is the change Δ''G'' (sometimes written "delta ''G''" or "d''G''") 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, Δ''G'' must be smaller than the non-pressure-volume (non-''PV'', e.g. electrical) work, which is often equal to zero (then Δ''G'' must be negative). Δ''G'' 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 Δ''G'' 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 Δ''G'' to be smaller than the non-''PV'' work and make it possible for the reaction to occur. 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 the
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
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 Δ''G'', and the reaction is called an
exergonic process An exergonic process is one which there is a positive flow of energy from the system to the surroundings. This is in contrast with an endergonic process. Constant pressure, constant temperature reactions are exergonic if and only if the Gibbs ...
. If two chemical reactions are coupled, then an otherwise endergonic reaction (one with positive Δ''G'') can be made to happen. The input of heat into an inherently endergonic reaction, such as the elimination of
cyclohexanol Cyclohexanol is the organic compound with the formula HOCH(CH2)5. The molecule is related to cyclohexane by replacement of one hydrogen atom by a hydroxyl group. This compound exists as a deliquescent colorless solid with a camphor-like odor, whic ...
to
cyclohexene Cyclohexene is a hydrocarbon with the formula C6H10. This cycloalkene is a colorless liquid with a sharp smell. It is an intermediate in various industrial processes. Cyclohexene is not very stable upon long term storage with exposure to light a ...
, can be seen as coupling an unfavourable reaction (elimination) to a favourable 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 the ''total'' Gibbs free energy change of the coupled reactions negative. In traditional use, the term "free" was included in "Gibbs free energy" to mean "available in the form of useful work". The characterization becomes more precise if we add the qualification that it is the energy available for non-pressure-volume work. (An analogous, but slightly different, meaning of "free" applies in conjunction with the
Helmholtz free energy In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz ene ...
, for systems at constant temperature). However, an increasing number of books and journal articles do not include the attachment "free", referring to ''G'' as simply "Gibbs energy". This is the result of a 1988
IUPAC The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
meeting to set unified terminologies for the international scientific community, in which the removal of the adjective "free" was recommended. This standard, however, has not yet been universally adopted. The name "free
enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...
" was also used for ''G'' in the past.

# History

The quantity called "free energy" is a more advanced and accurate replacement for the outdated term ''affinity'', which was used by chemists in the earlier years of physical chemistry to describe the ''force'' that caused
chemical reaction A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breaki ...
s. In 1873,
Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
published ''A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces'', in which he sketched the principles of his new equation that was 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 composed of part solid, part liquid, and part vapor, and by using a three-dimensional
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
-
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
-
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
graph, Gibbs was able to determine three states of equilibrium, i.e., "necessarily stable", "neutral", and "unstable", and whether or not changes would ensue. Further, Gibbs stated: In this description, as used by Gibbs, ''ε'' refers to the
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
of the body, ''η'' refers to the
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
of the body, and ''ν'' is the
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
of the body... Thereafter, in 1882, the German scientist
Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associat ...
characterized the affinity as the largest quantity of work which can be gained when the reaction is carried out in a reversible manner, e.g., electrical work in a reversible cell. The maximum work is thus regarded as the diminution of the free, or available, energy of the system (''Gibbs free energy'' ''G'' at ''T'' = constant, ''P'' = constant or ''Helmholtz free energy'' ''F'' at ''T'' = constant, ''V'' = constant), whilst the heat given out is usually a measure of the diminution of the total energy of the system (
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
). Thus, ''G'' or ''F'' is the amount of energy "free" for work under the given conditions. Until this point, the general view had been such that: "all chemical reactions drive the system to a state of equilibrium in which the affinities of the reactions vanish". Over the next 60 years, the term affinity came to be replaced with the term free energy. According to chemistry historian Henry Leicester, the influential 1923 textbook ''Thermodynamics and the Free Energy of Chemical Substances'' by Gilbert N. Lewis and Merle Randall led to the replacement of the term "affinity" by the term "free energy" in much of the English-speaking world.

# Definitions 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 the
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
(SI unit:
joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...
), * ''p'' is
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
(SI unit: pascal), * ''V'' is
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
(SI unit: m3), * ''T'' is the
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied ...
(SI unit:
kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ...
), * ''S'' is the
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
(SI unit: joule per kelvin), * ''H'' is the
enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...
(SI unit: joule). The expression for the infinitesimal reversible change in the Gibbs free energy as a function of its "natural variables" ''p'' and ''T'', for an open system, subjected to the operation of external forces (for instance, electrical or magnetic) ''Xi'', which cause the external parameters of the system ''ai'' to change by an amount d''ai'', can be derived as follows from the first law for reversible processes: $\begin T\,\mathrmS &= \mathrmU + p\,\mathrmV - \sum_^k \mu_i \,\mathrmN_i + \sum_^n X_i \,\mathrma_i + \cdots \\ \mathrm(TS) - S\,\mathrmT &= \mathrmU + \mathrm(pV) - V\,\mathrmp - \sum_^k \mu_i \,\mathrmN_i + \sum_^n X_i \,\mathrma_i + \cdots \\ \mathrm(U - TS + pV) &= V\,\mathrmp - S\,\mathrmT + \sum_^k \mu_i \,\mathrmN_i - \sum_^n X_i \,\mathrma_i + \cdots \\ \mathrmG &= V\,\mathrmp - S\,\mathrmT + \sum_^k \mu_i \,\mathrmN_i - \sum_^n X_i \,\mathrma_i + \cdots \end$ where: *''μ''''i'' is the
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 ...
of the ''i''-th chemical component. (SI unit: joules per particle or joules per mole) *''N''''i'' is the
number of particles The particle number (or number of particles) of a thermodynamic system, conventionally indicated with the letter ''N'', is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which is ...
(or number of moles) composing the ''i''-th chemical component. This is one form of the Gibbs fundamental equation. 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 an open system or for a closed, chemically reacting system where the ''Ni'' 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 from
mechanical work In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force stre ...
, 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 −d''l'' under a force ''f'' would result in a term ''f'' d''l'' being added. If a quantity of charge −d''e'' is acquired by a system at an electrical potential Ψ, the electrical work associated with this is −Ψ d''e'', which would be included in the infinitesimal expression. Other work terms are added on per system requirements. Each quantity in the equations above can be divided by the amount of substance, measured in moles, to form ''molar 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 the
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
of an
electrochemical cell An electrochemical cell is a device capable of either generating electrical energy from chemical reactions or using electrical energy to cause chemical reactions. The electrochemical cells which generate an electric current are called voltaic o ...
, and the
equilibrium constant The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency ...
for a
reversible reaction A reversible reaction is a reaction in which the conversion of reactants to products and the conversion of products to reactants occur simultaneously. : \mathit aA + \mathit bB \mathit cC + \mathit dD A and B can react to form C and D or, in the ...
. 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 and entropic driving forces involved in a thermodynamic process. The temperature dependence of the Gibbs energy for an
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is ...
is given by the Gibbs–Helmholtz equation, and its pressure dependence is given by $\frac = \frac + kT\ln \frac.$ or more conveniently as its
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 ...
: $\frac = \mu = \mu^\circ + kT\ln \frac.$ In non-ideal systems,
fugacity In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. It is equal to the pressure of an ideal gas whi ...
comes into play.

# Derivation

The Gibbs free energy total differential with respect to natural variables may be derived by Legendre transforms of the
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
. :$\mathrmU = T\,\mathrmS - p \,\mathrmV + \sum_i \mu_i \,\mathrm N_i.$ The definition of ''G'' from above is :$G = U + p V - T S$. Taking the total differential, we have :$\mathrmG = \mathrmU + p\,\mathrmV + V\,\mathrmp - T\,\mathrmS - S\,\mathrmT.$ Replacing d''U'' with the result from the first law gives :$\begin \mathrmG &= T\,\mathrmS - p\,\mathrmV + \sum_i \mu_i \,\mathrmN_i + p \,\mathrmV + V\,\mathrmp - T\,\mathrmS - S\,\mathrmT\\ &= V\,\mathrmp - S\,\mathrmT + \sum_i \mu_i \,\mathrm N_i. \end$ The natural variables of ''G'' are then ''p'', ''T'', and .

## Homogeneous systems

Because ''S'', ''V'', and ''N''''i'' are extensive variables, an Euler relation allows easy integration of d''U'': :$U = T S - p V + \sum_i \mu_i N_i.$ Because some of the natural variables of ''G'' are intensive, d''G'' may not be integrated using Euler relations as is the case with internal energy. However, simply substituting the above integrated result for ''U'' into the definition of ''G'' gives a standard expression for ''G'': :$\begin G &= U + p V - TS\\ &= \left\left(T S - p V + \sum_i \mu_i N_i \right\right) + p V - T S\\ &= \sum_i \mu_i N_i. \end$ This result shows that the chemical potential of a substance ''$i$'' is its (partial) mol(ecul)ar Gibbs free energy. It applies to homogeneous, macroscopic systems, but not to all thermodynamic systems.

# Gibbs free energy of reactions

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 and an infinitesimal change in ''G'', at constant temperature and pressure, yields :$dG=dU+pdV-TdS$. By the
first law of thermodynamics The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant amo ...
, a change in the internal energy ''U'' is given by :$dU=\delta Q+\delta W$ where is energy added as heat, and is energy added as work. The work done on the system may be written as , where is the mechanical work of compression/expansion done on or by the system and is all other forms of work, which may include electrical, magnetic, etc. Then :$dU=\delta Q-pdV+\delta W_x$ and the infinitesimal change in ''G'' is :$dG=\delta Q-TdS+\delta W_x$. The
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unless ...
states that for a closed system at constant temperature (in a heat bath), and so it follows that :$dG \le \delta W_x$ Assuming that only mechanical work is done, this simplifies to :$dG \le 0$ This means that for such a system when not in equilibrium, the Gibbs energy will always be decreasing, and in equilibrium, the infinitesimal change ''dG'' 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

When electric charge ''dQ''ele is passed between the electrodes of an electrochemical cell generating an emf $\mathcal$, a electrical work term appears in the expression for the change in Gibbs energy: $dG = -SdT + Vdp + \mathcal dQ_ ,$ where ''S'' is the
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
, ''V'' is the system volume, ''p'' is its pressure and ''T'' is its
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynami ...
. The combination ($\mathcal$, ''Qele'') is an example of a conjugate pair of variables. At constant pressure the above equation produces a Maxwell relation that links the change in open cell voltage with temperature ''T'' (a measurable quantity) to the change in entropy ''S'' when charge is passed
isothermally In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and ...
and
isobarically In thermodynamics, an isobaric process is a type of thermodynamic process in which the pressure of the system stays constant: Δ''P'' = 0. The heat transferred to the system does work, but also changes the internal energy (''U'') of t ...
. The latter is closely related to the reaction
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
of the electrochemical reaction that lends the battery its power. This Maxwell relation is: :$\left\left(\frac\right\right)_ = -\left\left( \frac \right\right)_$ If a mole of ions goes into solution (for example, in a Daniell cell, as discussed below) the charge through the external circuit is :$\Delta Q_ = -n_0 F_0 \, ,$ where ''n''0 is the number of electrons/ion, and ''F''0 is the
Faraday constant In physical chemistry, the Faraday constant, denoted by the symbol and sometimes stylized as ℱ, is the electric charge per mole of elementary charges. It is named after the English scientist Michael Faraday. Since the 2019 redefinition of ...
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 :$\Delta H = -n_0 F_0 \left\left( \mathcal - T \frac \right\right) ,$ where Δ''H'' is the
enthalpy of reaction The standard enthalpy of reaction (denoted \Delta_ H^\ominus or \Delta H_^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. This can i ...
. The quantities on the right are all directly measurable.

# Useful identities to derive the Nernst equation

During a reversible electrochemical reaction at constant temperature and pressure, the following equations involving the Gibbs free energy hold: *$\Delta_\text G = \Delta_\text G^\circ + R T \ln Q_\text$ (see
chemical equilibrium In a chemical reaction, chemical equilibrium is the state in which both the reactants and products are present in concentrations which have no further tendency to change with time, so that there is no observable change in the properties of th ...
), *$\Delta_\text G^\circ = -R T \ln K_\text$ (for a system at chemical equilibrium), *$\Delta_\text G = w_\text = -nF\mathcal$ (for a reversible electrochemical process at constant temperature and pressure), *$\Delta_\text G^\circ = -nF\mathcal^\circ$ (definition of $\mathcal^\circ$), and rearranging gives $\begin nF\mathcal^\circ &= RT \ln K_\text, \\ nF\mathcal &= nF\mathcal^\circ - R T \ln Q_\text, \\ \mathcal &= \mathcal^\circ - \frac \ln Q_\text, \end$ which relates the cell potential resulting from the reaction to the equilibrium constant and
reaction quotient In chemical thermodynamics, the reaction quotient (''Q''r or just ''Q'') is a dimensionless quantity that provides a measurement of the relative amounts of products and reactants present in a reaction mixture for a reaction with well-defined overall ...
for that reaction (
Nernst equation In electrochemistry, the Nernst equation is a chemical thermodynamical relationship that permits the calculation of the reduction potential of a reaction ( half-cell or full cell reaction) from the standard electrode potential, absolute tem ...
), where * , Gibbs free energy change per mole of reaction, * , Gibbs free energy change per mole of reaction for unmixed reactants and products at standard conditions (i.e. 298K, 100kPa, 1M of each reactant and product), * ,
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
, * , absolute
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied ...
, * ,
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, ...
, * ,
reaction quotient In chemical thermodynamics, the reaction quotient (''Q''r or just ''Q'') is a dimensionless quantity that provides a measurement of the relative amounts of products and reactants present in a reaction mixture for a reaction with well-defined overall ...
(unitless), * ,
equilibrium constant The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency ...
(unitless), * ,
electrical work Electric field work is the work performed by an electric field on a charged particle in its vicinity. The particle located experiences an interaction with the electric field. The work per unit of charge is defined by moving a negligible test cha ...
in a reversible process (chemistry sign convention), * , number of moles of
electrons The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
transferred in the reaction, * ,
Faraday constant In physical chemistry, the Faraday constant, denoted by the symbol and sometimes stylized as ℱ, is the electric charge per mole of elementary charges. It is named after the English scientist Michael Faraday. Since the 2019 redefinition of ...
(charge per
mole Mole (or Molé) may refer to: Animals * Mole (animal) or "true mole", mammals in the family Talpidae, found in Eurasia and North America * Golden moles, southern African mammals in the family Chrysochloridae, similar to but unrelated to Talpida ...
of electrons), * $\mathcal$, cell potential, * $\mathcal^\circ$, standard cell potential. Moreover, we also have $\begin K_\text &= e^, \\ \Delta_\text G^\circ &= -RT\left(\ln K_\text\right) = -2.303\,RT\left(\log_ K_\text\right), \end$ which relates the equilibrium constant with Gibbs free energy. This implies that at equilibrium $Q_\text = K_\text$ and $\Delta_\text G = 0.$

# Standard Gibbs energy change of formation

The
standard Gibbs free energy of formation The standard Gibbs free energy of formation (''G''f°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most s ...
of a compound is the change of Gibbs free energy that accompanies the formation of 1
mole Mole (or Molé) may refer to: Animals * Mole (animal) or "true mole", mammals in the family Talpidae, found in Eurasia and North America * Golden moles, southern African mammals in the family Chrysochloridae, similar to but unrelated to Talpida ...
of that substance from its component elements, in their
standard state In chemistry, the standard state of a material (pure substance, mixture or solution) is a reference point used to calculate its properties under different conditions. A superscript circle ° (degree symbol) or a Plimsoll (⦵) character is use ...
s (the most stable form of the element at 25 °C and 100
kPa KPA may refer to: * Keele Postgraduate Association, Keele University, UK, formerly Keele Research Association (KRA) * Kensington (Olympia) station, London, England, National Rail station code * Kenya Ports Authority * ''Kiln phosphoric acid'', ...
). Its symbol is Δ''f''''G''˚. All elements in their standard states (diatomic
oxygen Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements a ...
gas,
graphite Graphite () is a crystalline form of the element carbon. It consists of stacked layers of graphene. Graphite occurs naturally and is the most stable form of carbon under standard conditions. Synthetic and natural graphite are consumed on la ...
, etc.) have standard Gibbs free energy change of formation equal to zero, as there is no change involved. : Δf''G'' = Δ''f''''G''˚ + ''RT'' ln ''Qf'', where ''Qf'' is the
reaction quotient In chemical thermodynamics, the reaction quotient (''Q''r or just ''Q'') is a dimensionless quantity that provides a measurement of the relative amounts of products and reactants present in a reaction mixture for a reaction with well-defined overall ...
. At equilibrium, Δ''f''''G'' = 0, and ''Qf'' = ''K'', so the equation becomes : Δ''f''''G''˚ = −''RT'' ln ''K'', where ''K'' is the
equilibrium constant The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency ...
of the formation reaction of the substance from the elements in their standard states.

# Graphical interpretation by Gibbs

Gibbs free energy was originally defined graphically. In 1873, American scientist Willard 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 physicist
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
used Gibbs' figures to make a 3D energy-entropy-volume thermodynamic surface of a fictitious water-like substance.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, , p. 248.
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 his 3D surface figure. *
Bioenergetics Bioenergetics is a field in biochemistry and cell biology that concerns energy flow through living systems. This is an active area of biological research that includes the study of the transformation of energy in living organisms and the study of ...
* Calphad (CALculation of PHAse Diagrams) *
Critical point (thermodynamics) In thermodynamics, a critical point (or critical state) is the end point of a phase equilibrium curve. The most prominent example is the liquid–vapor critical point, the end point of the pressure–temperature curve that designates conditions ...
* Electron equivalent * Enthalpy-entropy compensation * Free entropy * Gibbs–Helmholtz equation * Grand potential * Non-random two-liquid model (NRTL model) – Gibbs energy of excess and mixing calculation and activity coefficients *
Spinodal In thermodynamics, the limit of local stability with respect to small fluctuations is clearly defined by the condition that the second derivative of Gibbs free energy is zero. : 0 The locus of these points (the inflection point within a G-x or G- ...
– Spinodal Curves (Hessian matrix) * Standard molar entropy *
Thermodynamic free energy The thermodynamic free energy is a concept useful in the thermodynamics of chemical or thermal processes in engineering and science. The change in the free energy is the maximum amount of work that a thermodynamic system can perform in a process ...
* UNIQUAC model – Gibbs energy of excess and mixing calculation and activity coefficients