View source for Thermodynamic potential

{{Short description|scalar physical quantities representing system states}}
A '''thermodynamic potential''' (or more accurately, a thermodynamic potential energy)<ref name="ISO 80000-5 20.4">ISO/IEC 80000-5, Quantities an units, Part 5 - Thermodynamics, item 5-20.4 Helmholtz energy, Helmholtz function</ref><ref name="ISO 80000-5 20.5">ISO/IEC 80000-5, Quantities an units, Part 5 - Thermodynamics, item 5-20.5, Gibbs energy, Gibbs function</ref> is a [[Physics:Scalar|scalar]] quantity used to represent the [[Physics:Thermodynamic state|thermodynamic state]] of a [[Physics:Physical system|system]]. The concept of thermodynamic potentials was introduced by [[Biography:Pierre Duhem|Pierre Duhem]] in 1886. [[Biography:Josiah Willard Gibbs|Josiah Willard Gibbs]] in his papers used the term ''fundamental [[Function (mathematics)|functions]]''. One main thermodynamic potential that has a physical interpretation is the [[Physics:Internal energy|internal energy]] {{mvar|U}}. It is the energy of configuration of a given system of [[Physics:Conservative force|conservative force]]s (that is why it is called potential) and only has meaning with respect to a defined set of references (or data). Expressions for all other thermodynamic energy potentials are derivable via [[Legendre transformation|Legendre transform]]s from an expression for {{mvar|U}}. In [[Physics:Thermodynamics|thermodynamics]], external forces, such as [[Physics:Gravity|gravity]], are typically disregarded when formulating expressions for potentials. For example, while all the [[Physics:Working fluid|working fluid]] in a [[Physics:Steam engine|steam engine]] may have higher energy due to gravity while sitting on top of [[Earth:Mount Everest|Mount Everest]] than it would at the bottom of the Mariana Trench, the gravitational potential energy term in the formula for the internal energy would usually be ignored because ''changes'' in gravitational potential within the engine during operation would be negligible. In a large system under even homogeneous external force, like the earth atmosphere under gravity, the intensive parameters (<math>p, T, \rho</math>) should be studied locally having even in equilibrium different values in different places far from each other (see thermodynamic models of [[Earth:Troposphere|troposphere]]].

{{Thermodynamics|cTopic=Potentials}}

== Description and interpretation ==
Five common thermodynamic potentials are:<ref name="Alberty 2001 p1353">Alberty (2001) p. 1353</ref>
{{table of thermodynamic potentials}}
where {{mvar|T}} = [[Physics:Temperature|temperature]], {{mvar|S}} = [[Physics:Entropy|entropy]], {{mvar|p}} = [[Physics:Pressure|pressure]], {{mvar|V}} = [[Physics:Volume (thermodynamics)|volume]]. The Helmholtz free energy is in ISO/IEC standard called Helmholtz energy<ref name="ISO 80000-5 20.4"/> or Helmholtz function. It is often denoted by the symbol {{mvar|F}}, but the use of {{mvar|A}} is preferred by IUPAC,<ref>Alberty (2001) p. 1376</ref> ISO and [[Organization:International Electrotechnical Commission|IEC]].<ref>ISO/IEC 80000-5:2007, item 5-20.4</ref> {{mvar|N<sub>i</sub>}} is the number of particles of type {{mvar|i}} in the system and {{mvar|&mu;<sub>i</sub>}} is the [[Chemistry:Chemical potential|chemical potential]] for an {{mvar|i}}-type particle. The set of all {{mvar|N<sub>i</sub>}} are also included as natural variables but may be ignored when no chemical reactions are occurring which cause them to change.

These five common potentials are all potential energies, but there are also [[Free entropy|entropy potentials]]. The [[Physics:Thermodynamic square|thermodynamic square]] can be used as a tool to recall and derive some of the potentials.

Just as in [[Physics:Mechanics|mechanics]], where [[Physics:Potential energy|potential energy]] is defined as capacity to do work, similarly different potentials have different meanings:
* [[Physics:Internal energy|Internal energy]] ({{mvar|U}}) is the capacity to do work plus the capacity to release heat.
* [[Physics:Gibbs free energy|Gibbs energy]]<ref name="ISO 80000-5 20.5"/> ({{mvar|G}}) is the capacity to do non-mechanical work.
* [[Physics:Enthalpy|Enthalpy]] ({{mvar|H}}) is the capacity to do non-mechanical work plus the capacity to release heat.
* [[Physics:Helmholtz free energy|Helmholtz energy]]<ref name="ISO 80000-5 20.4"/> ({{mvar|F}}) is the capacity to do mechanical plus non-mechanical work.
From these meanings (which actually apply in specific conditions, e.g. constant pressure, temperature, etc), we can say that {{math|Δ''U''}} is the energy added to the system, {{math|Δ''F''}} is the total work done on it, {{math|Δ''G''}} is the non-mechanical work done on it, and {{math|Δ''H''}} is the sum of non-mechanical work done on the system and the heat given to it.
Thermodynamic potentials are very useful when calculating the [[Chemistry:Chemical equilibrium|equilibrium results of a chemical reaction]], or when measuring the properties of materials in a chemical reaction. The chemical reactions usually take place under some constraints such as constant pressure and temperature, or constant entropy and volume, and when this is true, there is a corresponding thermodynamic potential that comes into play. Just as in mechanics, the system will tend towards lower values of potential and at equilibrium, under these constraints, the potential will take on an unchanging minimum value. The thermodynamic potentials can also be used to estimate the total amount of energy available from a thermodynamic system under the appropriate constraint.

In particular: (see [[Physics:Principle of minimum energy|principle of minimum energy]] for a derivation)<ref>Callen (1985) p. 153</ref>

* When the entropy {{mvar|S}} and "external parameters" (e.g. volume) of a [[Closed system|closed system]] are held constant, the internal energy {{mvar|U}} decreases and reaches a minimum value at equilibrium. This follows from the first and second laws of thermodynamics and is called the principle of minimum energy. The following three statements are directly derivable from this principle.
* When the temperature {{mvar|T}} and external parameters of a closed system are held constant, the Helmholtz free energy {{mvar|F}} decreases and reaches a minimum value at equilibrium.
* When the pressure {{mvar|p}} and external parameters of a closed system are held constant, the enthalpy {{mvar|H}} decreases and reaches a minimum value at equilibrium.
* When the temperature {{mvar|T}}, pressure {{mvar|p}} and external parameters of a closed system are held constant, the Gibbs free energy {{mvar|G}} decreases and reaches a minimum value at equilibrium.

== Natural variables ==
The variables that are held constant in this process are termed the '''natural variables''' of that potential.<ref name="Alberty 2001 p1352">Alberty (2001) p. 1352</ref> The natural variables are important not only for the above-mentioned reason, but also because if a thermodynamic potential can be determined as a function of its natural variables, all of the thermodynamic properties of the system can be found by taking partial derivatives of that potential with respect to its natural variables and this is true for no other combination of variables. On the converse, if a thermodynamic potential is not given as a function of its natural variables, it will not, in general, yield all of the thermodynamic properties of the system.

The set of natural variables for the above four potentials are formed from every combination of the {{mvar|T}}-{{mvar|S}} and {{mvar|P}}-{{mvar|V}} variables, excluding any pairs of [[Physics:Conjugate variables (thermodynamics)|conjugate variables]]. There is no reason to ignore the {{math|''N<sub>i</sub>''}} − {{math|''&mu;<sub>i</sub>''}} conjugate pairs, and in fact we may define four additional potentials for each species.<ref>Alberty (2001) p. 1355</ref> Using IUPAC notation in which the brackets contain the natural variables (other than the main four), we have:

{| class="wikitable" style="background: white"
|- 
|'''Formula'''
|'''Natural variables'''
|-
|<math>U[\mu_j]=U-\mu_jN_j</math>
|align="center"|<math>~~~~~S,V,\{N_{i\ne j}\},\mu_j</math>
|- 
|<math>F[\mu_j]=U-TS-\mu_jN_j</math>
|align="center"|<math>~~~~~T,V,\{N_{i\ne j}\},\mu_j</math>
|- 
|<math>H[\mu_j]=U+pV-\mu_jN_j</math>
|align="center"|<math>~~~~~S,p,\{N_{i\ne j}\},\mu_j</math>
|- 
|<math>G[\mu_j]=U+pV-TS-\mu_jN_j</math>
|align="center"|<math>~~~~~T,p,\{N_{i\ne j}\},\mu_j</math>
|}

If there is only one species, then we are done.  But, if there are, say, two species, then there will be additional potentials such as <math>U[\mu_1,\mu_2] = U-\mu_1 N_1-\mu_2 N_2</math> and so on. If there are {{mvar|D}} dimensions to the thermodynamic space, then there are {{math|[[Power of two|2<sup>''D''</sup>]]}} unique thermodynamic potentials. For the most simple case, a single phase ideal gas, there will be three dimensions, yielding eight thermodynamic potentials.

== The fundamental equations ==
{{main|Physics:Fundamental thermodynamic relation}}
The definitions of the thermodynamic potentials may be differentiated and, along with the first and second laws of thermodynamics, a set of differential equations known as the ''fundamental equations'' follow.<ref name="Alberty 2001 p1354">Alberty (2001) p. 1354</ref> (Actually they are all expressions of the same fundamental thermodynamic relation, but are expressed in different variables.) By the [[Physics:First law of thermodynamics|first law of thermodynamics]], any differential change in the internal energy {{mvar|U}} of a system can be written as the sum of heat flowing into the system and work done by the system on the environment, along with any change due to the addition of new particles to the system:

:<math>\mathrm{d}U = \delta Q - \delta W+\sum_i \mu_i\,\mathrm{d}N_i</math>

where {{math|''δQ''}} is the [[Infinitesimal|infinitesimal]] heat flow into the system, and {{math|''δW''}} is the infinitesimal work done by the system, {{mvar|&mu;<sub>i</sub>}} is the [[Chemistry:Chemical potential|chemical potential]] of particle type {{mvar|i}} and {{mvar|N<sub>i</sub>}} is the number of type {{mvar|i}} particles. (Neither {{math|''δQ''}} nor {{math|''δW''}} are [[Exact differential|exact differential]]s. Small changes in these variables are, therefore, represented with {{math|<VAR>&delta;</VAR>}} rather than {{math|d}}.)

By the [[Physics:Second law of thermodynamics|second law of thermodynamics]], we can express the internal energy change in terms of state functions and their differentials. In case of reversible changes we have:

:<math>\delta Q = T\,\mathrm{d}S</math>

:<math>\delta W = p\,\mathrm{d}V</math>

where
: {{mvar|T}} is [[Physics:Temperature|temperature]],
: {{mvar|S}} is [[Physics:Entropy|entropy]],
: {{mvar|p}} is [[Physics:Pressure|pressure]],
and {{mvar|V}} is [[Physics:Volume (thermodynamics)|volume]], and the equality holds for reversible processes.

This leads to the standard differential form of the internal energy in case of a quasistatic reversible change:

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

Since {{mvar|U}}, {{mvar|S}} and {{mvar|V}} are thermodynamic functions of state, the above relation holds also for arbitrary non-reversible changes. If the system has more external variables than just the volume that can change, the fundamental thermodynamic relation generalizes to:

:<math>dU = T\,dS - p\,dV + \sum_j \mu_j\,dN_j + \sum_i X_i \, dx_{i} </math>

Here the {{mvar|X<sub>i</sub>}} are the [[Physics:Generalized forces|generalized forces]] corresponding to the external variables {{mvar|x<sub>i</sub>}}.<ref>For example, ionic species ''N<sub>j</sub>'' (measured in [[Mole (unit)|moles]]) held at a certain potential ''V<sub>j</sub>'' will include the term <math>\sum_j V_j dq_j = F\sum_j V_j z_j dN_j</math> where ''F'' is the [[Physics:Faraday constant|Faraday constant]] and ''z<sub>j</sub>'' is the multiple of the elementary charge of the ion.</ref>

Applying [[Legendre transformation|Legendre transforms]] repeatedly, the following differential relations hold for the four potentials:

{| border="0" cellpadding="2"  style="margin: 0 0 1em 1em"
|- 
| <math>\mathrm{d}U</math>||<math>\!\!=</math>  ||                ||<math>T\mathrm{d}S</math>
||<math>-</math> ||<math>p\mathrm{d}V</math>||<math>+\sum_i \mu_i \,\mathrm{d}N_i</math>
|- 
| <math>\mathrm{d}F</math>||<math>\!\!=</math>  ||<math>-</math>||<math>S\,\mathrm{d}T</math>
||<math>-</math> ||<math>p\mathrm{d}V</math>||<math>+\sum_i \mu_i \,\mathrm{d}N_i</math>
|-
| <math>\mathrm{d}H</math>||<math>\!\!=</math>  ||                ||<math>T\,\mathrm{d}S</math>
||<math>+</math> ||<math>V\mathrm{d}p</math>||<math>+\sum_i \mu_i \,\mathrm{d}N_i</math>
|- 
| <math>\mathrm{d}G</math>||<math>\!\!=</math>  ||<math>-</math>||<math>S\,\mathrm{d}T</math>
||<math>+</math> ||<math>V\mathrm{d}p</math>||<math>+\sum_i \mu_i \,\mathrm{d}N_i</math>
|}

The infinitesimals on the right-hand side of each of the above equations are of the natural variables of the potential on the left-hand side.
Similar equations can be developed for all of the other thermodynamic potentials of the system. There will be one fundamental equation for each thermodynamic potential, resulting in a total of {{math|2<sup>''D''</sup>}} fundamental equations.

The differences between the four thermodynamic potentials can be summarized as follows:

:<math>\mathrm{d}(pV) = \mathrm{d}H - \mathrm{d}U = \mathrm{d}G - \mathrm{d}F </math>
:<math>\mathrm{d}(TS) = \mathrm{d}U - \mathrm{d}F = \mathrm{d}H - \mathrm{d}G </math>

== The equations of state ==
We can use the above equations to derive some differential definitions of some thermodynamic parameters. If we define {{math|&Phi;}} to stand for any of the thermodynamic potentials, then the above equations are of the form:

:<math>\mathrm{d}\Phi=\sum_i x_i\,\mathrm{d}y_i</math>

where {{mvar|x<sub>i</sub>}} and {{mvar|y<sub>i</sub>}} are conjugate pairs, and the {{mvar|y<sub>i</sub>}} are the natural variables of the potential {{math|&Phi;}}. From the [[Chain rule|chain rule]] it follows that:

:<math>x_j=\left(\frac{\partial \Phi}{\partial y_j}\right)_{\{y_{i\ne j}\}}</math>

Where {{math|''y''<sub>''i'' ≠ ''j''</sub>}} is the set of all natural variables of {{math|&Phi;}} except {{mvar|y<sub>i</sub>}} . This yields expressions for various thermodynamic parameters in terms of the derivatives of the potentials with respect to their natural variables. These equations are known as [[Physics:Equation of state|''equations of state'']] since they specify parameters of the [[Physics:Thermodynamic state|thermodynamic state]].<ref>Callen (1985) p. 37</ref> If we restrict ourselves to the potentials {{mvar|U}}, {{mvar|F}}, {{mvar|H}} and {{mvar|G}}, then we have:

:<math>
+T=\left(\frac{\partial U}{\partial S}\right)_{V,\{N_i\}}
  =\left(\frac{\partial H}{\partial S}\right)_{p,\{N_i\}}
</math>

:<math>
-p=\left(\frac{\partial U}{\partial V}\right)_{S,\{N_i\}}
  =\left(\frac{\partial F}{\partial V}\right)_{T,\{N_i\}}
</math>

:<math>
+V=\left(\frac{\partial H}{\partial p}\right)_{S,\{N_i\}}
  =\left(\frac{\partial G}{\partial p}\right)_{T,\{N_i\}}
</math>

:<math>
-S=\left(\frac{\partial G}{\partial T}\right)_{p,\{N_i\}}
  =\left(\frac{\partial F}{\partial T}\right)_{V,\{N_i\}}
</math>

:<math>
~\mu_j=
\left(\frac{\partial \phi}{\partial N_j}\right)_{X,Y,\{N_{i\ne j}\}}
</math>

where, in the last equation, {{mvar|ϕ}} is any of the thermodynamic potentials {{mvar|U}}, {{mvar|F}}, {{mvar|H}}, {{mvar|G}} and <math>{X,Y,\{N_{j\ne i}\}}</math> are the set of natural variables for that potential, excluding {{mvar|N<sub>i</sub>}} . If we use all potentials, then we will have more equations of state such as

:<math>
-N_j=\left(\frac{\partial U[\mu_j]}{\partial \mu_j}\right)_{S,V,\{N_{i\ne j}\}}
</math>

and so on. In all, there will be {{mvar|D}}&nbsp;equations for each potential, resulting in a total of {{math|''D'' 2<sup>''D''</sup>}} equations of state. If the {{mvar|D}}&nbsp;equations of state for a particular potential are known, then the fundamental equation for that potential can be determined. This means that all thermodynamic information about the system will be known, and that the fundamental equations for any other potential can be found, along with the corresponding equations of state.

== Measurement of thermodynamic potentials ==

The above equations of state suggest methods to experimentally measure changes in the thermodynamic potentials using physically measureable parameters. For example the free energy expressions

:<math>
+V=\left(\frac{\partial G}{\partial p}\right)_{T,\{N_i\}}
</math>

and

:<math>
-p=\left(\frac{\partial F}{\partial V}\right)_{T,\{N_i\}}
</math>

can be integrated at constant temperature and quantities to obtain:

:<math>
\Delta G = \int_{P1}^{P2}V\,dp\,\,\,\,
</math>(at constant ''T'', {N<sub>j</sub>} )

:<math>
\Delta F = -\int_{V1}^{V2}p\,dV\,\,\,\,
</math>(at constant ''T'', {N<sub>j</sub>} )

which can be measured by monitoring the measureable variables of pressure, temperature and volume. Changes in the enthalpy and internal energy can be measured by [[Physics:Calorimetry|calorimetry]] (which measures the amount of heat ''&Delta;Q'' released or absorbed by a system). The expressions

:<math>
+T=\left(\frac{\partial U}{\partial S}\right)_{V,\{N_i\}}
  =\left(\frac{\partial H}{\partial S}\right)_{p,\{N_i\}}
</math>

can be integrated:

:<math>
\Delta H = \int_{S1}^{S2}T\,dS = \Delta Q\,\,\,\,
</math>(at constant ''P'', {N<sub>j</sub>} )

:<math>
\Delta U = \int_{S1}^{S2}T\,dS = \Delta Q\,\,\,\,
</math>(at constant ''V'', {N<sub>j</sub>} )

Note that these measurements are made at constant {''N<sub>j</sub>'' } and are therefore not applicable to situations in which chemical reactions take place.

== The Maxwell relations ==
{{Main|Physics:Maxwell relations}}

Again, define {{mvar|x<sub>i</sub>}} and {{mvar|y<sub>i</sub>}} to be conjugate pairs, and the {{mvar|y<sub>i</sub>}} to be the natural variables of some potential {{math|&Phi;}}.  We may take the "cross differentials" of the state equations,  which obey the following relationship:

:<math>
\left(\frac{\partial}{\partial y_j}
\left(\frac{\partial \Phi}{\partial y_k}
\right)_{\{y_{i\ne k}\}}
\right)_{\{y_{i\ne j}\}}
=
\left(\frac{\partial}{\partial y_k}
\left(\frac{\partial \Phi}{\partial y_j}
\right)_{\{y_{i\ne j}\}}
\right)_{\{y_{i\ne k}\}}
</math>

From these we get the [[Physics:Maxwell relations|Maxwell relations]].<ref name="Alberty 2001 p1353"/><ref>Callen (1985) p. 181</ref> There will be {{sfrac|(''D'' − 1)|2}} of them for each potential giving a total of {{sfrac|''D''(''D'' − 1)|2}} equations in all. If we restrict ourselves the {{mvar|U}}, {{mvar|F}}, {{mvar|H}}, {{mvar|G}}

:<math>
\left(\frac{\partial T}{\partial V}\right)_{S,\{N_i\}} =
-\left(\frac{\partial p}{\partial S}\right)_{V,\{N_i\}}
</math>

:<math>
\left(\frac{\partial T}{\partial p}\right)_{S,\{N_i\}} =
+\left(\frac{\partial V}{\partial S}\right)_{p,\{N_i\}}
</math>

:<math>
\left(\frac{\partial S}{\partial V}\right)_{T,\{N_i\}} =
+\left(\frac{\partial p}{\partial T}\right)_{V,\{N_i\}}
</math>

:<math>
\left(\frac{\partial S}{\partial p}\right)_{T,\{N_i\}} =
-\left(\frac{\partial V}{\partial T}\right)_{p,\{N_i\}}
</math>

Using the equations of state involving the chemical potential we get equations such as:

:<math>
\left(\frac{\partial T}{\partial N_j}\right)_{V,S,\{N_{i\ne j}\}} =
\left(\frac{\partial \mu_j}{\partial S}\right)_{V,\{N_i\}}
</math>

and using the other potentials we can get equations such as:

:<math>
\left(\frac{\partial N_j}{\partial V}\right)_{S,\mu_j,\{N_{i\ne j}\}} =
-\left(\frac{\partial p}{\partial \mu_j}\right)_{S,V\{N_{i\ne j}\}}
</math>

:<math>
\left(\frac{\partial N_j}{\partial N_k}\right)_{S,V,\mu_j,\{N_{i\ne j,k}\}} =
-\left(\frac{\partial \mu_k}{\partial \mu_j}\right)_{S,V\{N_{i\ne j}\}}
</math>

== Euler integrals ==
Again, define {{mvar|x<sub>i</sub>}} and {{mvar|y<sub>i</sub>}} to be conjugate pairs, and the {{mvar|y<sub>i</sub>}} to be the natural variables of the internal energy.
Since all of the natural variables of the internal energy {{mvar|U}} are extensive quantities

:<math>U(\{\alpha y_i\})=\alpha U(\{y_i\})</math>

it follows from [[Homogeneous function|Euler's homogeneous function theorem]] that the internal energy can be written as:

:<math>U(\{y_i\})=\sum_j y_j\left(\frac{\partial U}{\partial y_j}\right)_{\{y_{i\ne j}\}}</math>

From the equations of state, we then have:

:<math>U=TS-pV+\sum_i \mu_i N_i</math>

Substituting into the expressions for the other main potentials we have:

:<math>F=  -pV+\sum_i \mu_i N_i</math>

:<math>H=TS   +\sum_i \mu_i N_i</math>

:<math>G=      \sum_i \mu_i N_i</math>

As in the above sections, this process can be carried out on all of the other thermodynamic potentials. The Euler integrals are sometimes also referred to as fundamental equations.

== The Gibbs–Duhem relation ==
Deriving the [[Physics:Gibbs–Duhem equation|Gibbs–Duhem equation]] from basic thermodynamic state equations is straightforward.<ref name="Alberty 2001 p1354"/><ref>Moran & Shapiro, p. 538</ref><ref>Callen (1985) p. 60</ref> Equating any thermodynamic potential definition with its Euler integral expression yields:

:<math>U=TS-PV+\sum_i \mu_i N_i</math>

Differentiating, and using the second law:

:<math>dU=TdS-PdV+\sum_i\mu_i\,dN_i</math>

yields:

:<math>0=SdT-VdP+\sum_i N_i d\mu_i</math>

Which is the Gibbs–Duhem relation. The Gibbs–Duhem is a relationship among the intensive parameters of the system. It follows that for a simple system with {{mvar|I}} components, there will be {{math|''I'' + 1}} independent parameters, or degrees of freedom. For example, a simple system with a single component will have two degrees of freedom, and may be specified by only two parameters, such as pressure and volume for example. The law is named after [[Biography:Josiah Willard Gibbs|Josiah Willard Gibbs]] and [[Biography:Pierre Duhem|Pierre Duhem]].

== Chemical reactions ==
Changes in these quantities are useful for assessing the degree to which a chemical reaction will proceed. The relevant quantity depends on the reaction conditions, as shown in the following table. {{math|Δ}} denotes the change in the potential and at equilibrium the change will be zero.

{| class="wikitable"
! !! Constant {{mvar|V}} !! Constant {{mvar|p}}
|-
! Constant {{mvar|S}}
| {{math|Δ''U''}} || {{math|Δ''H''}}
|-
! Constant {{mvar|T}}
| {{math|Δ''F''}} || {{math|Δ''G''}}
|}

Most commonly one considers reactions at constant {{mvar|p}} and {{mvar|T}}, so the Gibbs free energy is the most useful potential in studies of chemical reactions.

==See also==
* [[Physics:Coomber's relationship|Coomber's relationship]]

==Notes==
{{reflist}}

==References==
* {{cite journal
 | author=Alberty, R. A.
 | url = http://www.iupac.org/publications/pac/2001/pdf/7308x1349.pdf
 | title = Use of Legendre transforms in chemical thermodynamics
 | journal=Pure Appl. Chem.
 | year=2001 | volume= 73 | issue=8 | pages=1349–1380
 | doi=10.1351/pac200173081349
 }}
* {{cite book
 | title = Thermodynamics and an Introduction to Thermostatistics | edition = 2nd
 | publisher = John Wiley & Sons | location = New York | isbn = 978-0-471-86256-7
 }}
* {{cite book | first=Michael J. | last=Moran | author2=Shapiro, Howard N. | title=Fundamentals of Engineering Thermodynamics | edition=3rd | isbn=978-0-471-07681-0 | year=1996 | publisher=J. Wiley & Sons | location=New York ; Toronto | url=https://archive.org/details/fundamentalsofen00mora_0 }}

==Further reading==
* ''McGraw Hill Encyclopaedia of Physics'' (2nd Edition), C.B. Parker, 1994, {{ISBN|0-07-051400-3}}
* ''Thermodynamics, From Concepts to Applications'' (2nd Edition), A. Shavit, C. Gutfinger, CRC Press (Taylor and Francis Group, USA), 2009, {{ISBN|9781420073683}}
* ''Chemical Thermodynamics'', D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, {{ISBN|0-356-03736-3}}
* ''Elements of Statistical Thermodynamics'' (2nd Edition), L.K. Nash, Principles of Chemistry, Addison-Wesley, 1974, {{ISBN|0-201-05229-6}}
* ''Statistical Physics'' (2nd Edition), F. Mandl, Manchester Physics, John Wiley & Sons, 2008, {{ISBN|9780471566588}}

==External links==
* [http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/thepot.html Thermodynamic Potentials] – Georgia State University
* [https://arxiv.org/abs/physics/0004055 Chemical Potential Energy: The 'Characteristic' vs the Concentration-Dependent Kind]

{{DEFAULTSORT:Thermodynamic Potential}}
[[Category:Thermodynamics]]
[[Category:Potentials]]
[[Category:Fundamental theorems|*Thermodynamic potential]]
[[Category:Thermodynamic equations]]

{{Sourceattribution|Thermodynamic potential|1}}