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Sunday, July 5, 2015

Helmholtz free energy



From Wikipedia, the free encyclopedia

In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature. The negative of the difference in the Helmholtz energy is equal to the maximum amount of work that the system can perform in a thermodynamic process in which temperature is held constant. If the volume is not held constant, part of this work will be performed as boundary work. The Helmholtz energy is commonly used for systems held at constant volume. Since in this case no work is performed on the environment, the drop in the Helmholtz energy is equal to the maximum amount of useful work that can be extracted from the system. For a system at constant temperature and volume, the Helmholtz energy is minimized at equilibrium.

The Helmholtz free energy was developed by Hermann von Helmholtz, a German physicist, and is usually denoted by the letter A  (from the German “Arbeit” or work), or the letter F . The IUPAC recommends the letter A  as well as the use of name Helmholtz energy.[1] In physics, the letter F can also be used to denote the Helmholtz energy, as Helmholtz energy is sometimes referred to as the Helmholtz function, Helmholtz free energy, or simply free energy (not to be confused with Gibbs free energy).

While Gibbs free energy is most commonly used as a measure of thermodynamic potential, especially in the field of chemistry, it is inconvenient for some applications that do not occur at constant pressure. For example, in explosives research, Helmholtz free energy is often used since explosive reactions by their nature induce pressure changes. It is also frequently used to define fundamental equations of state of pure substances.

Definition

The Helmholtz energy is defined as:[2]
A \equiv U-TS\,
where
  • A  is the Helmholtz free energy (SI: joules, CGS: ergs),
  • U  is the internal energy of the system (SI: joules, CGS: ergs),
  • T  is the absolute temperature (kelvins),
  • S  is the entropy (SI: joules per kelvin, CGS: ergs per kelvin).
The Helmholtz energy is the Legendre transform of the internal energy, U, in which temperature replaces entropy as the independent variable.

Mathematical development

From the first law of thermodynamics (with a constant number of particles) we have
\mathrm{d}U = \delta Q\ - \delta W\,,
where U is the internal energy, \delta Q is the energy added by heating and \delta W is the work done by the system. From the second law of thermodynamics, for a reversible process we may say that \delta Q=T\mathrm{d}S. Also, in case of a reversible change, the work done can be expressed as \delta W = p \mathrm{d}V
\mathrm{d}U = T\mathrm{d}S - p\mathrm{d}V\,
Applying the product rule for differentiation to d(TS) = TdS + SdT, we have:
\mathrm{d}U = d(TS) - S\mathrm{d}T- p\mathrm{d}V\,,
and:
\mathrm{d}(U-TS) = - S\mathrm{d}T - p\mathrm{d}V\,
The definition of A = U - TS enables to rewrite this as
\mathrm{d}A = - S\mathrm{d}T - p\mathrm{d}V\,
This relation is also valid for a process that is not reversible because A is a thermodynamic function of state.

Work in an Isothermal Process and Equilibrium Conditions

The fundamental thermodynamic relation is
\mathrm{d}U = \delta Q\ - \delta W\,
We can make the substitution
\delta Q\leq T\mathrm{d}S\,
Where equality holds for a reversible process

The expression for the internal energy becomes
\mathrm{d}U \leq T\mathrm{d}S - \delta W\,
If we isolate the work term
\mathrm{d}U - T\mathrm{d}S \leq - \delta W\,
And note that, for an isothermal process,
\mathrm{d}A = \mathrm{d}U - T\mathrm{d}S\,
Then
\delta W\leq - \mathrm{d}A, (isothermal process)
Again, equality holds for a reversible process (in which \delta W\, becomes dW) . dW includes all reversible work, mechanical (pressure-volume) work and non-mechanical work (e. g. electrical work).

The maximum energy that can be freed for work is the negative of the change in A. The process is nominally isothermal, but it is only required that the system has the same initial and final temperature, and not that the temperature stays constant along the process.

Now, imagine that our system is also kept at constant volume to prevent work from being done. If temperature and volume are kept constant, then dA = 0; this is a necessary, but not sufficient condition for equilibrium. For any spontaneous process, the change in Helmholtz free energy must be negative, that is A_{f}\leq A_{i},. Therefore, to prevent a spontaneous change, we must also require that A be at a minimum.

Minimum free energy and maximum work principles

The laws of thermodynamics are only directly applicable to systems in thermal equilibrium. If we wish to describe phenomena like chemical reactions, then the best we can do is to consider suitably chosen initial and final states in which the system is in (metastable) thermal equilibrium. If the system is kept at fixed volume and is in contact with a heat bath at some constant temperature, then we can reason as follows.

Since the thermodynamical variables of the system are well defined in the initial state and the final state, the internal energy increase, \Delta U, the entropy increase \Delta S, and the total amount of work that can be extracted, performed by the system, W, are well-defined quantities. Conservation of energy implies:
\Delta U_{\text{bath}} + \Delta U + W = 0\,
The volume of the system is kept constant. This means that the volume of the heat bath does not change either and we can conclude that the heat bath does not perform any work. This implies that the amount of heat that flows into the heat bath is given by:
Q_{\text{bath}} = \Delta U_{\text{bath}} =-\left(\Delta U + W\right) \,
The heat bath remains in thermal equilibrium at temperature T no matter what the system does. Therefore the entropy change of the heat bath is:
\Delta S_{\text{bath}} = \frac{Q_{\text{bath}}}{T}=-\frac{\Delta U + W}{T} \,
The total entropy change is thus given by:
\Delta S_{\text{bath}} +\Delta S= -\frac{\Delta U -T\Delta S+ W}{T} \,
Since the system is in thermal equilibrium with the heat bath in the initial and the final states, T is also the temperature of the system in these states. The fact that the system's temperature does not change allows us to express the numerator as the free energy change of the system:
\Delta S_{\text{bath}} +\Delta S=-\frac{\Delta A+ W}{T} \,
Since the total change in entropy must always be larger or equal to zero, we obtain the inequality:
W\leq -\Delta A\,
If no work is extracted from the system then
\Delta A\leq 0\,
We see that for a system kept at constant temperature and volume, the total free energy during a spontaneous change can only decrease, that the total amount of work that can be extracted is limited by the free energy decrease, and that increasing the free energy requires work to be done on the system.

This result seems to contradict the equation dA = -S dT - P dV, as keeping T and V constant seems to imply dA = 0 and hence A = \text{ constant}. In reality there is no contradiction. After the spontaneous change, the system, as described by thermodynamics, is a different system with a different free energy function than it was before the spontaneous change. Thus, we can say that \Delta A= A_{2} - A_{1}\leq 0 where the A_{i} are different thermodynamic functions of state.

One can imagine that the spontaneous change is carried out in a sequence of infinitesimally small steps. To describe such a system thermodynamically, one needs to enlarge the thermodynamical state space of the system. In case of a chemical reaction, one must specify the number of particles of each type. The differential of the free energy then generalizes to:
dA = -S dT - p dV + \sum_{j}\mu_{j}dN_{j}\,
where the N_{j} are the numbers of particles of type j and the \mu_{j} are the corresponding chemical potentials. This equation is then again valid for both reversible and non-reversible changes. In case of a spontaneous change at constant T and V, the last term will thus be negative.

In case there are other external parameters the above equation generalizes to:
dA = -S dT - \sum_{i}X_{i}dx_{i} +\sum_{j}\mu_{j}dN_{j}\,
Here the x_{i} are the external variables and the X_{i} the corresponding generalized forces.

Relation to the canonical partition function

A system kept at constant volume, temperature, and particle number is described by the canonical ensemble. The probability to find the system in some energy eigenstate r is given by:
P_{r}= \frac{e^{-\beta E_r}}{Z}\,
where
\beta\equiv\frac{1}{k T}\,
E_{r}=\text{ energy of eigenstate }r\,
Z = \sum_{r} e^{-\beta E_{r}}
Z is called the partition function of the system. The fact that the system does not have a unique energy means that the various thermodynamical quantities must be defined as expectation values. In the thermodynamical limit of infinite system size, the relative fluctuations in these averages will go to zero.

The average internal energy of the system is the expectation value of the energy and can be expressed in terms of Z as follows:
U\equiv\left\langle E \right\rangle = \sum_{r}P_{r}E_{r}= -\frac{\partial \log Z}{\partial \beta}\,
If the system is in state r, then the generalized force corresponding to an external variable x is given by
X_{r} = -\frac{\partial E_{r}}{\partial x}\,
The thermal average of this can be written as:
X = \sum_{r}P_{r}X_{r}=\frac{1}{\beta}\frac{\partial \log Z}{\partial x}\,
Suppose the system has one external variable x. Then changing the system's temperature parameter by d\beta and the external variable by dx will lead to a change in \log Z:
d\left(\log Z\right)= \frac{\partial\log Z}{\partial\beta}d\beta + \frac{\partial\log Z}{\partial x}dx = -U\,d\beta + \beta X\,dx\,
If we write U\,d\beta as:
U\,d\beta = d\left(\beta U\right) - \beta\, dU\,
we get:
d\left(\log Z\right)=-d\left(\beta U\right) + \beta\, dU+ \beta X \,dx\,
This means that the change in the internal energy is given by:
dU =\frac{1}{\beta}d\left(\log Z+\beta U\right) - X\,dx \,
In the thermodynamic limit, the fundamental thermodynamic relation should hold:
dU = T\, dS - X\, dx\,
This then implies that the entropy of the system is given by:
S = k\log Z + \frac{U}{T} + c\,
where c is some constant. The value of c can be determined by considering the limit T → 0. In this limit the entropy becomes S = k \log \Omega_{0} where \Omega_{0} is the ground state degeneracy. The partition function in this limit is \Omega_{0}e^{-\beta U_{0}} where U_{0} is the ground state energy. Thus, we see that c = 0 and that:
A = -kT\log\left(Z\right)\,

Bogoliubov inequality

Computing the free energy is an intractable problem for all but the simplest models in statistical physics. A powerful approximation method is mean field theory, which is a variational method based on the Bogoliubov inequality. This inequality can be formulated as follows.

Suppose we replace the real Hamiltonian H of the model by a trial Hamiltonian \tilde{H}, which has different interactions and may depend on extra parameters that are not present in the original model. If we choose this trial Hamiltonian such that
\left\langle\tilde{H}\right\rangle =\left\langle H\right\rangle\,
where both averages are taken with respect to the canonical distribution defined by the trial Hamiltonian \tilde{H}, then
A\leq \tilde{A}\,
where A is the free energy of the original Hamiltonian and \tilde{A} is the free energy of the trial Hamiltonian. By including a large number of parameters in the trial Hamiltonian and minimizing the free energy we can expect to get a close approximation to the exact free energy.

The Bogoliubov inequality is often formulated in a slightly different but equivalent way. If we write the Hamiltonian as:
H = H_{0} + \Delta H\,
where H_{0} is exactly solvable, then we can apply the above inequality by defining
\tilde{H} = H_{0} + \left\langle\Delta H\right\rangle_{0}\,
Here we have defined \left\langle X\right\rangle_{0} to be the average of X over the canonical ensemble defined by H_{0}. Since \tilde{H} defined this way differs from H_{0} by a constant, we have in general
\left\langle X\right\rangle_{0} =\left\langle X\right\rangle\,
Therefore
\left\langle\tilde{H}\right\rangle = \left\langle H_{0} + \left\langle\Delta H\right\rangle\right\rangle =\left\langle H\right\rangle\,
And thus the inequality
A\leq \tilde{A}\,
holds. The free energy \tilde{A} is the free energy of the model defined by H_{0} plus \left\langle\Delta H\right\rangle. This means that
\tilde{A}=\left\langle H_{0}\right\rangle_{0} - T S_{0} + \left\langle\Delta H\right\rangle_{0}=\left\langle H\right\rangle_{0} - T S_{0}\,
and thus:
A\leq \left\langle H\right\rangle_{0} - T S_{0} \,

Proof

For a classical model we can prove the Bogoliubov inequality as follows. We denote the canonical probability distributions for the Hamiltonian and the trial Hamiltonian by P_{r} and \tilde{P}_{r}, respectively. The inequality:
\sum_{r} \tilde{P}_{r}\log\left(\tilde{P}_{r}\right)\geq \sum_{r} \tilde{P}_{r}\log\left(P_{r}\right) \,
then holds. To see this, consider the difference between the left hand side and the right hand side. We can write this as:
\sum_{r} \tilde{P}_{r}\log\left(\frac{\tilde{P}_{r}}{P_{r}}\right) \,
Since
\log\left(x\right)\geq 1 - \frac{1}{x}\,
it follows that:
\sum_{r} \tilde{P}_{r}\log\left(\frac{\tilde{P}_{r}}{P_{r}}\right)\geq \sum_{r}\left(\tilde{P}_{r} - P_{r}\right) = 0 \,
where in the last step we have used that both probability distributions are normalized to 1.

We can write the inequality as:
\left\langle\log\left(\tilde{P}_{r}\right)\right\rangle\geq \left\langle\log\left(P_{r}\right)\right\rangle\,
where the averages are taken with respect to \tilde{P}_{r}. If we now substitute in here the expressions for the probability distributions:
P_{r}=\frac{\exp\left[-\beta H\left(r\right)\right]}{Z}\,
and
\tilde{P}_{r}=\frac{\exp\left[-\beta\tilde{H}\left(r\right)\right]}{\tilde{Z}}\,
we get:
\left\langle -\beta \tilde{H} - \log\left(\tilde{Z}\right)\right\rangle\geq \left\langle -\beta H - \log\left(Z\right)\right\rangle
Since the averages of H and \tilde{H} are, by assumption, identical we have:
A\leq\tilde{A}
Here we have used that the partition functions are constants with respect to taking averages and that the free energy is proportional to minus the logarithm of the partition function.

We can easily generalize this proof to the case of quantum mechanical models. We denote the eigenstates of \tilde{H} by \left|r\right\rangle. We denote the diagonal components of the density matrices for the canonical distributions for H and \tilde{H} in this basis as:
P_{r}=\left\langle r\left|\frac{\exp\left[-\beta H\right]}{Z}\right|r\right\rangle\,
and
\tilde{P}_{r}=\left\langle r\left|\frac{\exp\left[-\beta\tilde{H}\right]}{\tilde{Z}}\right|r\right\rangle=\frac{\exp\left(-\beta\tilde{E}_{r}\right)}{\tilde{Z}}\,
where the \tilde{E}_{r} are the eigenvalues of \tilde{H}

We assume again that the averages of H and \tilde{H} in the canonical ensemble defined by \tilde{H} are the same:
\left\langle\tilde{H}\right\rangle = \left\langle H\right\rangle \,
where
\left\langle H\right\rangle = \sum_{r}\tilde{P}_{r}\left\langle r\left|H\right|r\right\rangle\,
The inequality
\sum_{r} \tilde{P}_{r}\log\left(\tilde{P}_{r}\right)\geq \sum_{r} \tilde{P}_{r}\log\left(P_{r}\right) \,
still holds as both the P_{r} and the \tilde{P}_{r} sum to 1. On the l.h.s. we can replace:
\log\left(\tilde{P}_{r}\right)= -\beta \tilde{E}_{r} - \log\left(\tilde{Z}\right)\,
On the right hand side we can use the inequality
\left\langle\exp\left(X\right)\right\rangle_{r}\geq\exp\left(\left\langle X\right\rangle_{r}\right)\,
where we have introduced the notation
\left\langle Y\right\rangle_{r}\equiv\left\langle r\left|Y\right|r\right\rangle\,
for the expectation value of the operator Y in the state r. See here for a proof. Taking the logarithm of this inequality gives:
\log\left[\left\langle\exp\left(X\right)\right\rangle_{r}\right]\geq\left\langle X\right\rangle_{r}\,
This allows us to write:
\log\left(P_{r}\right)=\log\left[\left\langle\exp\left(-\beta H - \log\left(Z\right)\right)\right\rangle_{r}\right]\geq\left\langle -\beta H - \log\left(Z\right)\right\rangle_{r}\,
The fact that the averages of H and \tilde{H} are the same then leads to the same conclusion as in the classical case:
A\leq\tilde{A}

Generalized Helmholtz energy

In the more general case, the mechanical term (p{\rm d}V) must be replaced by the product of volume, stress, and an infinitesimal strain:[3]
{\rm d}A = V\sum_{ij}\sigma_{ij}\,{\rm d}\varepsilon_{ij} - S{\rm d}T + \sum_i \mu_i \,{\rm d}N_i\,
where \sigma_{ij} is the stress tensor, and \varepsilon_{ij} is the strain tensor. In the case of linear elastic materials that obey Hooke's Law, the stress is related to the strain by:
\sigma_{ij}=C_{ijkl}\epsilon_{kl}
where we are now using Einstein notation for the tensors, in which repeated indices in a product are summed. We may integrate the expression for {\rm d}A to obtain the Helmholtz energy:
A = \frac{1}{2}VC_{ijkl}\epsilon_{ij}\epsilon_{kl} - ST + \sum_i \mu_i N_i\,
= \frac{1}{2}V\sigma_{ij}\epsilon_{ij} - ST + \sum_i \mu_i N_i\,

Application to fundamental equations of state

The Helmholtz free energy function for a pure substance (together with its partial derivatives) can be used to determine all other thermodynamic properties for the substance. See, for example, the equations of state for water, as given by the IAPWS in their IAPWS-95 release.
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Gibbs free energy



From Wikipedia, the free encyclopedia

In thermodynamics, the Gibbs free energy (IUPAC recommended name: Gibbs energy or Gibbs function; also known as free enthalpy[1] to distinguish it from Helmholtz free energy) is a thermodynamic potential that measures the "usefulness" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure (isothermal, isobaric). Just as in mechanics, where potential energy is defined as capacity to do work, similarly different potentials have different meanings. The Gibbs free energy (SI units kJ/mol) is the maximum amount of non-expansion work that can be extracted from a thermodynamically closed system (one that can exchange heat and work with its surroundings, but not matter); this maximum can be attained only in a completely reversible process. When a system changes from a well-defined initial state to a well-defined final state, the Gibbs free energy change ΔP equals the work exchanged by the system with its surroundings, minus the work of the pressure forces, during a reversible transformation of the system from the initial state to the final state.[2]
Gibbs energy (also referred to as ∆P) is also the chemical potential that is minimized when a system reaches equilibrium at constant pressure and temperature. Its derivative with respect to the reaction coordinate of the system vanishes at the equilibrium point. As such, it is a convenient criterion for the spontaneity of processes with constant pressure and temperature.

The Gibbs free energy, originally called available energy, was developed in the 1870s by the American mathematician Josiah Willard Gibbs. In 1873, Gibbs described this "available energy" as
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 total volume 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.[3]
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 On the Equilibrium of Heterogeneous Substances, a graphical analysis of multi-phase chemical systems, he engaged his thoughts on chemical free energy in full.

Overview


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, even though favorable, it is so slow that it is not observed. Whether a reaction is thermodynamically favorable does not determine its rate.

For systems reacting at STP (or any other fixed temperature and pressure), there is a general natural tendency to achieve a minimum of the free energy.

A quantitative measure of the favorability of a given reaction is the difference ΔG in Gibbs free energy that is (or would be) effected by proceeding with the reaction. When the calculated energetics of the process indicate that ΔG is negative, it means that the reaction will be favoured and will release energy. The energy released equals the maximum amount of work that can be performed as a result of the chemical reaction. In contrast, if conditions indicated a positive ΔG, then energy—in the form of work—would have to be added to the reacting system for the reaction to occur.

The equation can be also seen from the perspective of the system taken together with its surroundings (the rest of the universe). First assume that the given reaction 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 ΔG, and the reaction is called exergonic.

If we allow other reactions to occur on the side, then an otherwise endergonic chemical reaction (one with positive ΔG) 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 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 difference 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."[2] The characterization becomes more precise if we add the qualification that it is the energy available for non-volume work.[4] (A 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 G 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 adjective ‘free’ was supposedly banished.[5][6][7] This standard, however, has not yet been universally adopted.

History

The quantity called "free energy" is a more advanced and accurate replacement for the outdated term affinity, which was used by chemists in previous years[when?] to describe the force that caused chemical reactions.
In 1873, Willard Gibbs 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-entropy-internal energy graph, Gibbs was able to determine three states of equilibrium, i.e., "necessarily stable", "neutral", and "unstable", and whether or not changes would ensue.

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 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). 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.

Graphical interpretation

Gibbs free energy was originally defined graphically. In 1873, American engineer 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.[8] Thus, in order to understand the very difficult concept of Gibbs free energy one must be able to understand its interpretation as Gibbs defined originally by section AB on his figure 3 and as Maxwell sculpted that section on his 3D surface figure.

American engineer 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 (x), volume (y), energy (z) 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 AB being the first-ever three-dimensional representation of Gibbs free energy, or what Gibbs called "available energy"; the region AC 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.

Definitions


Willard Gibbs’ 1873 available energy (free energy) graph, which shows a plane perpendicular to the axis of v (volume) and passing through point A, which represents the initial state of the body. MN is the section of the surface of dissipated energy. Qε and Qη are sections of the planes η = 0 and ε = 0, and therefore parallel to the axes of ε (internal energy) and η (entropy), respectively. AD and AE are the energy and entropy of the body in its initial state, AB and AC 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:
G(p,T) = U + pV - TS
which is the same as:
G(p,T) = H - TS
where:
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 magnetical) Xi, which cause the external parameters of the system ai to change by an amount dai, can be derived as follows from the First Law for reversible processes:
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
where:
This is one form of Gibbs 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 an open system. For a closed 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 −dl under a force f would result in a term f 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 −Ψde, 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 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[clarification needed] and entropic driving forces involved in a 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:
\frac{G}{N} = \frac{G}{N}^\circ + kT\ln \frac{p}{{p^\circ }}
if the volume is known rather than pressure then it becomes:
\frac{G}{N} = \frac{G}{N}^\circ + kT\ln \frac{V^\circ}{{V }}
or more conveniently as its chemical potential:
\frac{G}{N} = \mu = \mu^\circ + kT\ln \frac{p}{{p^\circ }}.
In non-ideal systems, fugacity comes into play.

Derivation

The Gibbs free energy total differential natural variables may be derived via Legendre transforms of the internal energy.
\mathrm{d}U = T\mathrm{d}S - p \,\mathrm{d}V + \sum_i \mu_i \,\mathrm{d} N_i\,.
Because S, V, and Ni are extensive variables, Euler's homogeneous function theorem allows easy integration of dU:[12]
U = T S - p V + \sum_i \mu_i N_i\,.
The definition of G from above is
G = U + p V - T S\,.
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[12]
\begin{align} \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{align}.
The natural variables of G are then p, T, and {Ni}.

Homogeneous systems

Because some of the natural variables are intensive, dG may not be integrated using Euler integrals as is the case with internal energy. However, simply substituting the Gibbs-Duhem relation result for U into the definition of G gives a standard expression for G:[12]
\begin{align} G &= T S - p V + \sum_i \mu_i N_i + p V - T S\\ &= \sum_i \mu_i N_i \end{align}.
This result applies to homogeneous, macroscopic systems, but not to all thermodynamic systems.[13]

Gibbs free energy of reactions

To derive the Gibbs free energy equation for an isolated system, let Stot be the total entropy of the isolated system, that is, a system that cannot exchange energy(heat and work) or mass with its surroundings. According to the second law of thermodynamics:
\Delta S_{tot} \ge 0 \,
and if ΔStot = 0 then the process is reversible. The heat transfer Q vanishes for an adiabatic system. Any adiabatic process that is also reversible is called an isentropic \left( {Q\over T} = \Delta S = 0 \right) \, process.

Now consider a subsystem having internal entropy Sint. Such a system is thermally connected to its surroundings, which have entropy Sext. The entropy form of the second law applies only to the closed system formed by both the system and its surroundings. Therefore a process is possible only if
\Delta S_{int} + \Delta S_{ext} \ge 0 \,.
If Q is the heat transferred to the system from the surroundings, then −Q is the heat lost by the surroundings, so that \Delta S_{ext} = - {Q \over T}, corresponds to the entropy change of the surroundings.

We now have:
\Delta S_{int} - {Q \over T} \ge 0 \,
Multiplying both sides by T:
T \Delta S_{int} - Q \ge 0 \,
Q is the heat transferred to the system; if the process is now assumed to be isobaric, then Q = ΔH:
T \Delta S_{int} - \Delta H \ge 0\,
ΔH is the enthalpy change of reaction (for a chemical reaction at constant pressure). Then:
\Delta H - T \Delta S_{int} \le 0 \,
for a possible process. Let the change ΔG in Gibbs free energy be defined as
\Delta G = \Delta H - T \Delta S_{int} \, (eq.1)
Notice that it is not defined in terms of any external state functions, such as ΔSext or ΔStot. Then the second law, which also tells us about the spontaneity of the reaction, becomes:
\Delta G < 0 \, favoured reaction (Spontaneous)
\Delta G = 0 \, Neither the forward nor the reverse reaction prevails (Equilibrium)
\Delta G > 0 \, disfavoured reaction (Nonspontaneous)
Gibbs free energy G itself is defined as
G = H - T S_{int} \, (eq.2)
but notice that to obtain equation (1) from equation (2) we must assume that T is constant. Thus, Gibbs free energy is most useful for thermochemical processes at constant temperature and pressure: both isothermal and isobaric. Such processes don't move on a P-V diagram, such as phase change of a pure substance, which takes place at the saturation pressure and temperature. Chemical reactions, however, do undergo changes in chemical potential, which is a state function. Thus, thermodynamic processes are not confined to the two dimensional P-V diagram. There is a third dimension for n, the quantity of gas. For the study of explosive chemicals, the processes are not necessarily isothermal and isobaric. For these studies, Helmholtz free energy is used.

If an isolated system (Q = 0) is at constant pressure (Q = ΔH), then
\Delta H = 0 \,
Therefore the Gibbs free energy of an isolated system is
\Delta G = -T \Delta S \,
and if ΔG ≤ 0 then this implies that ΔS ≥ 0, back to where we started the derivation of ΔG.

Useful identities

\Delta G = \Delta H - T \Delta S \, (for constant temperature)
\Delta_r G^\circ = -R T \ln K \, (for a system at equilibrium)
\Delta_r G = \Delta_r G^\circ + R T \ln Q_r \, (see Chemical equilibrium)
\Delta G = -nFE \,
\Delta G^\circ = -nFE^\circ \,
and rearranging gives
nFE^\circ = RT \ln K \,
nFE = nFE^\circ - R T \ln Q_r \, \,
E = E^\circ - \frac{R T}{n F} \ln Q_r \, \,
which relates the electrical potential of a reaction to the equilibrium coefficient for that reaction (Nernst equation).
where
Moreover, we also have:
K_{eq}=e^{- \frac{\Delta_r G^\circ}{RT}}
\Delta_r G^\circ = -RT(\ln K_{eq}) = -2.303\,RT(\log_{10} K_{eq})
which relates the equilibrium constant with Gibbs free energy.

Gibbs free energy, the second law of thermodynamics, and metabolism

A chemical reaction will (or can) proceed spontaneously if the change in the total entropy of the universe that would be caused by the reaction is nonnegative. As discussed in the overview, if the temperature and pressure are held constant, the Gibbs free energy is a (negative) proxy for the change in total entropy of the universe. It's "negative" because S appears with a negative coefficient in the expression for G, so the Gibbs free energy moves in the opposite direction from the total entropy. Thus, a reaction with a positive Gibbs free energy will not proceed spontaneously. However, in biological systems (among others), energy inputs from other energy sources (including the sun and exothermic chemical reactions) are "coupled" with reactions that are not entropically favored (i.e. have a Gibbs free energy above zero). Taking into account the coupled reactions, the total entropy in the universe increases. This coupling allows endergonic reactions, such as photosynthesis and DNA synthesis, to proceed without decreasing the total entropy of the universe. Thus biological systems do not violate the second law of thermodynamics.

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 component elements, at their standard states (the most stable form of the element at 25 degrees Celsius and 101.3 kilopascals). Its symbol is ΔfG˚.

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.
ΔfG = ΔfG˚ + RT ln Qf ; Qf is the reaction quotient.
At equilibrium, ΔfG = 0 and Qf = K so the equation becomes ΔfG˚ = −RT ln K; K is the equilibrium constant.

Table of selected substances[14]

Substance State ΔfG°(kJ/mol) ΔfG°(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
at
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