Gibbs free energy

In thermodynamics, the Gibbs free energy (or Gibbs energy as the recommended name; symbol is a thermodynamic potential that can be used to calculate the maximum amount of work, other than pressure–volume work, that may be performed by a thermodynamically closed system at constant temperature and pressure. It also provides a necessary condition for processes such as chemical reactions that may occur under these conditions. The Gibbs free energy is expressed as

where:

<math>U</math> is the internal energy of the system
<math>H</math> is the enthalpy of the system
<math>S</math> is the entropy of the system
<math>T</math> is the temperature of the system
<math>V</math> is the volume of the system
<math>p</math> is the pressure of the system (which must be equal to that of the surroundings for mechanical equilibrium).

The Gibbs free energy change (<math>\Delta G = \Delta H - T\,\Delta S</math>, measured in joules in SI) is the maximum 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 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 forces.

The Gibbs energy is the thermodynamic potential that is minimized when a system reaches chemical 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 <math>G</math> is necessary for a reaction to be spontaneous 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. In 1873, Gibbs described this "available energy" as

One can think of <math>\Delta G</math> as the amount of "free" or "useful" energy available to do non-<math>pV</math> 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 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 <math>\Delta G</math>, and the reaction is called an exergonic process.

If two chemical reactions are coupled, then an otherwise endergonic reaction (one with positive <math>\Delta G</math>) can be made to happen. The input of heat into an inherently endergonic reaction, such as the elimination of cyclohexanol to cyclohexene, 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 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". (An analogous, 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 <math>G</math> as simply "Gibbs energy". This is the result of a 1988 IUPAC 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" was also used for <math>G</math> in the past.

In this description, as used by Gibbs, <math>\epsilon</math> refers to the internal energy of the body, <math>\eta</math> refers to the entropy of the body, and <math>v</math> is the volume of the body.

Thereafter, in 1882, the German scientist Hermann von Helmholtz 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 <math>G</math> at constant <math>T</math> and <math>P</math> or Helmholtz free energy <math>F</math> at constant <math>T</math> and <math>V</math>), whilst the heat given out is usually a measure of the diminution of the total energy of the system (internal energy). Thus, <math>G</math> or <math>F</math> 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

class=skin-invert-image|thumb|[[Willard Gibbs' 1873 available energy (free energy) graph, which shows a plane perpendicular to the axis of <math>v</math> (volume) and passing through point <math>\mathrm{A}</math>, which represents the initial state of the body. <math>\mathrm{MN}</math> is the section of the surface of dissipated energy. <math>\mathrm{Q}\epsilon</math> and <math>\mathrm{Q}\eta</math> are sections of the planes <math>\eta = 0</math> and <math>\epsilon = 0</math>, and therefore parallel to the axes of <math>\epsilon</math> (internal energy) and <math>\eta</math> (entropy), respectively. <math>\mathrm{AD}</math> and <math>\mathrm{AE}</math> are the energy and entropy of the body in its initial state, <math>\mathrm{AB}</math> and <math>\mathrm{AC}</math> its available energy (Gibbs free energy) and its capacity 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

which is the same as

where:

<math>U</math> is the internal energy (SI unit: joule),
<math>p</math> is pressure (SI unit: pascal),
<math>V</math> is volume (SI unit: m3),
<math>T</math> is the temperature (SI unit: kelvin),
<math>S</math> is the entropy (SI unit: joule per kelvin),
<math>H</math> is the enthalpy (SI unit: joule).

class=skin-invert-image|frameless|364x364px

The expression for the infinitesimal reversible change in the Gibbs free energy as a function of its "natural variables" <math>p</math> and <math>T</math>, for an open system, subjected to the operation of external forces (for instance, electrical or magnetic) <math>X_i</math>, which cause the external parameters of the system <math>a_i</math> to change by an amount <math>\mathrm{d}a_i</math>, can be derived as follows from the first law for reversible processes:

\begin{align}

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{align}

</math>

where:

<math>\mu_i</math> is the chemical potential of the <math>i</math>-th chemical component. (SI unit: joules per particle or joules per mole 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 <math>N_i</math> 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, 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 <math>-\mathrm{d}l</math> under a force <math>f</math> would result in a term <math>f\,\mathrm{d}l</math> being added. If a quantity of charge <math>-\mathrm{d}e</math> is acquired by a system at an electrical potential <math>\Psi</math>, the electrical work associated with this is <math>-\Psi\,\mathrm{d}e</math>, 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 of an electrochemical cell, and the equilibrium constant for a reversible 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 and entropic driving forces involved in a thermodynamic process.

class=skin-invert-image|thumb|upright=1.9|Relation to other relevant parameters

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

or more conveniently as its chemical potential:

In non-ideal systems, fugacity 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.

<math display="block">\mathrm{d}U = T\,\mathrm{d}S - p\,\mathrm{d}V + \sum_i \mu_i \,\mathrm{d} N_i\, .</math>

The definition of <math>G</math> from above is

Taking the total differential, we have

<math display="block">\mathrm{d}G = \mathrm{d}U + p\,\mathrm{d}V + V\,\mathrm{d}p - T\,\mathrm{d}S - S\,\mathrm{d}T\, .</math>

Replacing <math>\mathrm{d}U</math> with the result from the first law gives

Because some of the natural variables of <math>G</math> are intensive, <math>\mathrm{d}G</math> may not be integrated using Euler relations as is the case with internal energy. However, simply substituting the above integrated result for <math>U</math> into the definition of <math>G</math> gives a standard expression for <math>G</math>:

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 <math>1 = G = U + pV - TS</math> and an infinitesimal change in <math>G</math>, at constant temperature and pressure, yields

<math display="block">\mathrm{d}G = \mathrm{d}U + p\,\mathrm{d}V - T\,\mathrm{d}S\, .</math>

By the first law of thermodynamics, a change in the internal energy <math>U</math> is given by

<math display="block">\mathrm{d}U = \delta Q + \delta W</math>

where <math>\delta Q</math> is energy added as heat, and <math>\delta W</math> is energy added as work. The work done on the system may be written as <math>\delta W = -p\,\mathrm{d}V + \delta W_x</math>, where <math>-p\,\mathrm{d}V</math> is the mechanical work of compression/expansion done on or by the system and <math>\delta W_x</math> is all other forms of work, which may include electrical, magnetic, etc. Then

<math display="block>\mathrm{d}U = \delta Q - p\,\mathrm{d}V + \delta W_x</math>

and the infinitesimal change in <math>G</math> is

<math display="block">\mathrm{d}G = \delta Q - T\,\mathrm{d}S + \delta W_x\, .</math>

The second law of thermodynamics states that for a closed system at constant temperature (in a heat bath), <math>T\,\mathrm{d}S \ge \delta Q</math>, and so it follows that

<math display="block">\mathrm{d}G \le \delta W_x\, .</math>

Assuming that only mechanical work is done, this simplifies to

<math display="block">\mathrm{d}G \le 0\, .</math>

This means that for such a system when not in equilibrium, the Gibbs energy will always be decreasing, and in equilibrium, the infinitesimal change <math>\mathrm{d}G</math> 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 <math>\mathrm{d}Q_\mathrm{ele}</math> is passed between the electrodes of an electrochemical cell generating an emf <math>\mathcal{E}</math>, an electrical work term appears in the expression for the change in Gibbs energy:

<math display="block">\mathrm{d}G = -S\,\mathrm{d}T + V\,\mathrm{d}p + \mathcal{E} \,\mathrm{d}Q_\mathrm{ele}\, ,</math>

where <math>S</math> is the entropy, <math>V</math> is the system volume, <math>p</math> is its pressure and <math>T</math> is its absolute temperature.

The combination (<math>\mathcal{E}</math>, <math>Q_\mathrm{ele}</math>) 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 <math>T</math> (a measurable quantity) to the change in entropy <math>S</math> when charge is passed isothermally and isobarically. The latter is closely related to the reaction entropy of the electrochemical reaction that lends the battery its power. This Maxwell relation is:

<math display="block">\left(\frac{\partial \mathcal{E{\partial T}\right)_{Q_\mathrm{ele},p} = -\left( \frac{\partial S}{\partial Q_\mathrm{ele \right)_{T,p}</math>

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

<math display="block">\Delta Q_\mathrm{ele} = -n_0 F_0 \, ,</math>

where <math>n_0</math> is the number of electrons/ion, and <math>F_0</math> is the Faraday 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

<math display="block">\Delta H = -n_0 F_0 \left(\mathcal{E} - T\frac{\mathrm{d}\mathcal{E{\mathrm{d}T}\right) \, ,</math>

where <math>\Delta H</math> is the enthalpy of reaction. 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:

<math>\Delta_\mathrm{r} G = \Delta_\text{r} G^\circ + R T \ln Q_\mathrm{r}</math> (see chemical equilibrium),
<math>\Delta_\mathrm{r} G^\circ = -R T \ln K_\mathrm{eq}</math> (for a system at chemical equilibrium),
<math>\Delta_\mathrm{r} G = w_{\mathrm{elec},\mathrm{rev = -nF\mathcal{E}</math> (for a reversible electrochemical process at constant temperature and pressure),
<math>\Delta_\mathrm{r} G^\circ = -nF\mathcal{E}^\circ</math> (definition of <math>\mathcal{E}^\circ</math>),

and rearranging gives

\begin{align}

nF\mathcal{E}^\circ &= RT \ln K_\text{eq} \\

nF\mathcal{E} &= nF\mathcal{E}^\circ - R T \ln Q_\text{r} \\

\mathcal{E} &= \mathcal{E}^\circ - \frac{R T}{n F} \ln Q_\text{r}\, ,

\end{align}

</math>

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

<math>\Delta_\mathrm{r}G</math>, Gibbs free energy change per mole of reaction,
<math>\Delta_\mathrm{r}G^\circ</math>, 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),

<math>R</math>, gas constant,
<math>T</math>, absolute temperature,
<math>Q_\mathrm{r}</math>, reaction quotient (unitless),
<math>K_\mathrm{eq}</math>, equilibrium constant (unitless),
<math>w_{\mathrm{elec},\mathrm{rev</math>, electrical work in a reversible process (chemistry sign convention),
<math>n</math>, number of moles of electrons transferred in the reaction,
<math>F = N_\mathrm{A}e \approx 96485\,\mathrm{C/mol}</math>, Faraday constant (charge per mole of electrons),
<math>\mathcal{E}</math>, cell potential,
<math>\mathcal{E}^\circ</math>, standard cell potential.

Moreover, we also have

\begin{align}

K_\mathrm{eq} &= e^{-\frac{\Delta_\mathrm{r} G^\circ}{RT\, , \\

\Delta_\mathrm{r} G^\circ &= -RT\left(\ln K_\mathrm{eq}\right) = -2.303\,RT\left(\log_{10} K_\mathrm{eq}\right)\, ,

\end{align}

</math>

which relates the equilibrium constant with Gibbs free energy. This implies that at equilibrium, <math>Q_\mathrm{r} = K_\mathrm{eq}</math> and <math>\Delta_\mathrm{r}G = 0</math>.

Standard Gibbs energy change of formation

{| class="wikitable sortable floatright" style="text-align:right"

|+ Table of selected substances

! rowspan=2 | Substance (state)

! colspan=2 | <math>\Delta_\mathrm{f}G^\circ</math>

! (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

The standard Gibbs free energy of formation of a compound is the change of Gibbs free energy that accompanies the formation of 1 mol of that substance from its component elements, in their standard states (the most stable form of the element at 25 °C and 100 kPa). Its symbol is <math>\Delta_\mathrm{f}G^\circ</math>.

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

<math display="block">\Delta_\mathrm{f}G = \Delta_\mathrm{f}G^\circ + RT \ln Q_\mathrm{f}\, ,</math>

where <math>Q_\mathrm{f}</math> is the reaction quotient.

At equilibrium, <math>\Delta_\mathrm{f}G = 0</math> and <math>Q_\mathrm{f} = K</math>, so the equation becomes

<math display="block">\Delta_\mathrm{f}G^\circ = -RT \ln K\, ,</math>

where <math>K</math> is the equilibrium constant 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 used Gibbs' figures to make a 3D energy-entropy-volume thermodynamic surface of a fictitious water-like substance. 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.

class=skin-invert-image|750px|center|thumb|American scientist [[Willard Gibbs' 1873 figures two and three (above left and middle) used by Scottish physicist James Clerk Maxwell in 1874 to create a three-dimensional entropy, volume, energy thermodynamic 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-dimensional Cartesian coordinates; the region <math>\mathrm{AB}</math> being the first-ever three-dimensional representation of Gibbs free energy, or what Gibbs called "available energy"; the region <math>\mathrm{AC}</math> being its capacity for entropy, 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

External links

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

Definitions

Derivation

Gibbs free energy of reactions

In electrochemical thermodynamics

Useful identities to derive the Nernst equation

Standard Gibbs energy change of formation

Graphical interpretation by Gibbs

See also

Notes and references

External links