# Gibbs Free Energy

In thermodynamics , the Gibbs free energy is a thermodynamic potential that can be used to calculate the maximum reversible work that can be done from a thermodynamic body at a constant temperature and pressure . The Gibbs free energy

 {\displaystyle \Delta G=\Delta H~-~t\Delta S}

measured in joules in SI ) is the maximum amount of non-expansion work that can be achieved in a thermodynamically closed system.(one that can exchange heat and work with its surroundings, but doesn’t matter). This maximum can be achieved only in a completely reversible process . When a system reverses from the initial state to the final state, the reduction in Gibbs free energy is equal to the work done by the system in its surroundings, minus the work of the pressure forces. [1]

Gibbs energy (symbol ) is also the thermodynamic potential that decreases when a system reaches chemical equilibrium at constant pressure and temperature. Its derivative with respect to the reaction coordinate of the system vanishes at the equilibrium point. In such . The reduction is necessary for the reaction to be spontaneous at constant pressure and temperature. GG

Gibbs free energy, originally called available energy , was developed in the 1870s by American scientist Josiah Willard Gibbs . In 1873, Gibbs described this “available energy” as [2] : 400. as described

The greatest amount of mechanical work that can be achieved by a given amount of a certain substance in a given initial state without increasing its total volume or allowing heat to pass to or from external bodies, e.g. Close to that processes left in the early stages.

According to Gibbs, the initial state of the body is believed to be such that “the body can be made to go into states of energy dissipated by reversible processes “. In his 1876 magnum opus On the Equilibrium of Heterogeneous Substances , a graphical analysis of multi-phase chemical systems, he covered his views on chemical-free energy more fully.

If the reactants and products are all in their thermodynamic standard states , then the defined equation is written as:

## Overview

According to the second law of thermodynamics , for systems reacting at standard conditions for temperature and pressure (or any other fixed temperature and pressure), there is a general natural tendency to attain a minimum of Gibbs free energy.

A quantitative measure of the adaptability of a given reaction at constant temperature and pressure is the change in Gibbs free energy g (sometimes written “delta g ” or “d g “) that is (or will be) caused by the reaction. As a necessary condition for a reaction to occur at constant temperature and pressure, g must be smaller than the non-pressure-volume (non- PV , e.g. electrical) work , which is often equal to zero (hence ). g must be negative ) g equals the maximum sum of non PVWork that can be done as a result of a chemical reaction for the case of a reversible process. If the analysis indicates a positive , G in the form of electricity or other non – PV work for a reaction, then the energy involved in the reaction system would have for G smaller than the non PV work and it Creating a reaction is possible.

One can think of G as the amount of “free” or “useful” energy available to do work. The equation can also be viewed from the point of view of a system in conjunction with its surroundings (the rest of the universe). First, one assumes that a given reaction at constant temperature and pressure is the only one that is happening. Then the entropy released or absorbed by the system equals the entropy that the environment must absorb or release, respectively. The reaction would be allowed only if the total entropy change of the universe is zero or positive. This is reflected in a negative G , and the reaction is called an extrinsic process .

If two chemical reactions are coupled, an otherwise endogenous reaction (with a positive G ) can occur. A natural reaction can be seen as coupling (elimination) of an unfavorable reaction to a favorable one, such that the input of heat into the endergonic, eg elimination of cyclohexanol to cyclohexane , (burning of coal or other provision of heat) is The change is greater than or equal to zero of the universe, making the total Gibbs free energy difference of the coupled reactions negative.

In traditional usage, the term “free” was included in “Gibbs free energy” meaning “available in the form of useful work”. [1] If we add the ability that this is the energy available for non-pressure-volume work, the characterization becomes more accurate. [4] (A similar, but slightly different, meaning of “free” applies in conjunction with the Helmholtz free energy for systems at constant temperature). However, an increasing number of books and journal articles do not include the attachment “free” , referring to G simply as “Gibbs Energy”. It is the result of a 1988 IUPAC meeting to determine a unified terminology for the international scientific community , including the adjective “free”. was recommended to be removed. [5][6] [7] However, this standard has not yet been universally adopted.

The name “free enthalpy” has also been used for G in the past.

## History

The quantity called “free energy” is a more advanced and precise replacement for the older term affinity , which was used by chemists in the earlier years of physical chemistry to describe the force caused by chemical reactions .

In 1873, Josiah Willard Gibbs published A method of geometric representation of the thermodynamic properties of substances by means of surfaces , in which he sketched the principles of his new equation that were able to predict or predict the tendencies of various natural processes when The body or system is 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 using three-dimensional volume-entropy-internal energy graphs, Gibbs was able to determine three states of equilibrium. were, that is, “necessarily stable”, “neutral”, and “unstable”, and whether or not changes would occur. Furthermore, Gibbs said:

If we wish to express in an equation the necessary and sufficient condition of thermodynamic equilibrium for a substance when it is bounded by means of constant pressure p and temperature T , then this equation can be written as:δ ( ε –  +  ) = 0 When refers to the variation produced by any change in the state of parts of the body, and (when different parts of the body are in different states) in the ratio which is divided between the different states in the body. The condition of stable equilibrium is that the value of the expression in parentheses must be minimum.

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

Subsequently, in 1882, the German scientist Hermann von Helmholtz described affinity as the greatest amount of work that could be achieved when the reaction was carried out in a reversible manner, e.g., electrical work in a reversible cell. . Thus the maximum work is assumed to be the loss 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), while the heat given is usually But is a measure of the loss of the total energy (internal energy) of the system. Thus, g or f is the amount of “free” energy for work under the given conditions.

Up to this point, the general view was such that: “all chemical reactions lead the system to a state of equilibrium in which the equivalences of the reactions disappear”. Over the next 60 years, the term affinity was replaced by 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 most parts of English-speaking. of World.

### Definitions

Gibbs free energy is defined as

{\displaystyle G(p,T)=U+pV-TS,}

which is similar

{\displaystyle G(p,T)=H-TS,}

Where from:U is internal energy : (SI unit joule,)p is pressure (SI unit: pascal),V is Quantity : (m SI unit 3 ,)T is temperature : (SI unit Kelvin,)s is entropy (SI unit: joule per kelvin),H is enthalpy (SI unit: joule).

The expression for the infinitesimally reversible change in Gibbs free energy as a function of its “natural variables” P and T , for an open system, subject to the operation of external forces (for example, electric or magnetic) , which is Reason to replace the external parameter i by a sum d i of the system , from the first law for reversible processes can be obtained as follows:

{\displaystyle {\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}}}

μ is the chemical potential of the i – th chemical component. (SI unit: joule per particle [9] or joule per mole  )i is the number of particles (or number of moles) of the chemical constituents of the composition i may.

It is a form of Gibbs’ fundamental equation . [10] In infinitesimal expressions, the chemical potential term refers to a change in Gibbs free energy as a result of an inflow or outflow of particles. In other words, it is for an open system or for a closed, chemically reacting system where is changing. For a closed, non-reactive system, this term can be dropped.

Any number of additional conditions may be added, depending on the particular system being considered. In addition to mechanical work, a system can, in addition, perform a variety of other functions. For example, in the infinitesimal expression, the contractile work energy associated with a thermodynamic system that is a contractile fiber that shrinks under a force f by an amount −d l will result in the addition of a term f d l . If the amount of charge -d e is gained by a system at an electric potential , then the electrical work associated with it is −Ψ d e , which would be included in the infinitesimal expression. Other working conditions are added per system requirements. [11 1]

Each of the quantities in the above equations can be divided by the amount of matter measured in moles, making the molar Gibbs free energy . Gibbs free energy is one of the most important thermodynamic functions for the characterization of a system. It is a factor in determining results such as the voltage of an electrochemical cell, and the equilibrium constant for a reversible reaction. In isothermal, isobaric systems, the Gibbs free energy can be thought of as a “kinetic” quantity, in that it is a representative measure of the competing effects of the enthalpy and the entropic driving forces involved in the thermodynamic process.

The temperature dependence of the Gibbs energy for an ideal gas is given by the Gibbs–Helmholtz equation, and its pressure dependence is given by.

{\displaystyle {\frac {G}{N}}={\frac {G^{\circ }}{N}}+kT\ln {\frac {p}{p^{\circ }}}.}

or more simply as its chemical potential:

{\displaystyle {\frac {G}{N}}=\mu =\mu ^{\circ }+kT\ln {\frac {p}{p^{\circ }}}.}

In non-ideal systems, runaway comes in handy.

## Etymology

The total difference of Gibbs free energy with respect to the natural variable can be obtained by the internal energy of the Legandre transform.

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

The definition of G from above is

{\displaystyle G=U+pV-TS}.

Taking the total difference, we have

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

Substituting dU with the result of the first rule, gives

{\displaystyle {\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 of G are then p , T , and { i } .

### Homogeneous system

Because s , v , and n are broad variables, a Euler integral d allows easy integration of U

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

Because some of the natural variables of G are intensive, dG cannot be integrated using Euler integrals as is the case with internal energy. However , simply substituting the integrated result for the above in the definition of u gives a standard expression for g :

{\displaystyle {\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 applies to homogeneous, macroscopic systems, but not all thermodynamic systems.

## Gibbs Free Energy of Reactions

The system in question is kept at a constant temperature and pressure, and is closed (no matter can come in or out). Any system has Gibbs energy and an infinitesimal change in G at constant temperature and pressure yields :

G=U+PV-TS
{\displaystyle dG=dU+PdV-TdS}

By the first law of thermodynamics, the change in internal energy U is given by

dU = \delta Q + \delta W

Where Q is the energy added in the form of heat, and W is the energy added in the form of work. The work done on the system can be written as W = – PdV + x , where -PdV is the mechanical work of compression/expansion done on the system and x is all other types of work, including electric, magnetic, e.t.c. Assuming that only mechanical work is done,

{\displaystyle dU=\delta Q-PdV}

And the subtle change in G is:

{\displaystyle dG=\delta Q-TdS}

The second law of thermodynamics states that for a closed system, , and so it follows:

{\displaystyle TdS\geq \delta Q}
{\displaystyle dG\leq 0}
{\displaystyle dG=-SdT+VdP+{\mathcal {E}}dQ,}

where G is Gibbs’ free energy, S is entropy, V is system volume, P is pressure and T is its absolute temperature.

Combination (ℰ, Q ) is an example of a conjugate pair of variables. The above relation at constant pressure yields the Maxwell relation which links the change in open cell voltage with temperature T (a measurable quantity) to the change in entropy S when charge is passed isothermally and isobaric. The latter is closely related to the reaction entropy of the electrochemical reaction that gives the battery its power. Maxwell is concerned:

{\displaystyle \left({\frac {\partial {\mathcal {E}}}{\partial T}}\right)_{Q}=-\left({\frac {\partial S}{\partial Q}}\right)_{T}}

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

{\displaystyle \Delta Q=-n_{0}F_{0}\,,}

where n0 is the number of electrons/ion, and f0 is the Faraday constant and the minus sign indicates the cell’s discharge. Assuming constant pressure and volume, the thermodynamic properties of a cell are strictly related to the behavior of its emf:

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

where H is the enthalpy of the reaction. The quantities on the right are all directly measurable.

## Useful Identities for Deriving the Nernst Equation

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

{\displaystyle \Delta _{\text{r}}G=\Delta _{\text{r}}G^{\circ }+RT\ln Q_{\text{r}}}~~(see ~~chemical ~~equilibrium),
{\displaystyle \Delta _{\text{r}}G^{\circ }=-RT\ln K_{\text{eq}}}~(for ~a ~system ~at~ chemical ~equilibrium),
{\displaystyle \Delta _{\text{r}}G=w_{\text{elec,rev}}=-nFE}(~for~ a ~reversible ~electrochemical ~process~ at~ constant~ temperature ~and~ pressure),
{\displaystyle \Delta _{\text{r}}G^{\circ }=-nFE^{\circ }}( Definition~ of~ E ~degree),

and rearrange

{\displaystyle {\begin{aligned}nFE^{\circ }&=RT\ln K_{\text{eq}},\\nFE&=nFE^{\circ }-RT\ln Q_{\text{r}}r,\\E&=E^{\circ }-{\frac {RT}{nF}}\ln Q_{\text{r}},\end{aligned}}}

which is related to the equilibrium constant for that reaction and the cell potential resulting from the reaction quotient (Nernst equation),

Where fromΔ RG Gibbs free energy conversion per mole of reaction,r g ° , Gibbs free energy change per mole of reaction for immiscible reactants and products at standard conditions (ie 298 K , 100 kPa, 1 M of each reactant and product),r , gas constant,T , absolute temperature,ln, natural logarithm,r , reaction quotient (unitless),eq , equilibrium constant (unitless),election, rev , Electrical work in a reversible process (Chemistry Signature Convention),n , moles of electrons transferred in the reaction,F = E 96485  c/mol, Faraday’s constant (charge per mole of electrons),E , cell potential , , standard cell potential.

{\displaystyle {\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 the Gibbs free energy. This means that at equilibrium

{\displaystyle Q_{\text{r}}=K_{\text{eq}}}~And~{\displaystyle \Delta _{\text{r}}G=0}

## Standard Energy Change of Formation

The standard Gibbs free energy of formation of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of that substance from its constituent elements, in their standard states at 25 °C and 100 kPa (the most stable form of the element ). Its symbol is f G.

All elements in their standard states (diatomic oxygen gas, graphite, etc.) have the standard Gibbs free energy change of formation equal to zero, because there is no change involved.f g = g + rt ln f , _

where f is the reaction quotient.

At equilibrium, f G = 0, and f = K , then the equation becomesf g = – t ln k ,

where k is the equilibrium constant.

## Graphical Interpretation by Gibbs

The 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 entropy and volume to represent the position of a body. In his second follow-up paper, “A Method for the Geometrical Representation of the Thermodynamic Properties of Matters by means of Surfaces”, published later that year, Gibbs added in the third coordinate the energy of the body, defined on three figures. In 1874, Scottish physicist James Clerk Maxwell used Gibbs figures to create a 3D energy-entropy-volume thermodynamic surface of a hypothetical water-like substance. [17]Thus, to understand the concept of Gibbs free energy, it may be helpful to understand its interpretation by Gibbs as the segment AB on his figure 3, and Maxwell sculpting that segment on his 3D surface figure.

American scientist Willard Gibbs’ use of two and three (above left and middle) in 1873 by Scottish physicist James Clerk Maxwell in 1874 to create three-dimensional entropy, volume, energy thermodynamic surface diagrams for a hypothetical water-like substance Gibbs figures (above right) on both the volume-entropy coordinates of the shifted three-dimensional Cartesian coordinates (shifted under the cube) and the energy-entropy coordinates (flipped upside down and shifted behind the cube); The field AB is the first three-dimensional representation of the Gibbs free energy, or what Gibbs calls “available energy”; The field is its potential for ac entropy, which is defined by Gibbs as “