# Thermodynamic potential

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Thermodynamic potential

A thermodynamic potential is a scalar potential function used to represent the thermodynamic state of a system. One main thermodynamic potential which has a physical interpretation is the internal energy, U. It is the energy of configuration of a given system of conservative forces (that is why it is a 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 transforms from an expression for U. In thermodynamics, certain forces, such as gravity, are typically disregarded when formulating expressions for potentials. For example, while all the working fluid in a steam engine may have higher energy due to gravity while sitting on top of Mt. 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. Five common thermodynamic energy potentials are [Alberty (2001) p1353] :

Note that 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 $2^D$ fundamental equations.

The equations of state

We can use the above equations to derive some differential definitions of some thermodynamic parameters. If we define Φ to stand for any of the thermodynamic potentials, then the above equations are of the form:

:$mathrm\left\{d\right\}Phi=sum_i x_i,mathrm\left\{d\right\}y_i,$

where $x_i$ and $y_i$ are conjugate pairs, and the $y_i$ are the natural variables of the potential $Phi$. From the chain rule it follows that:

:$x_j=left\left(frac\left\{partial Phi\right\}\left\{partial y_j\right\} ight\right)_\left\{\left\{y_\left\{i e j\right\}$

Where $\left\{y_\left\{i e j\right\}\right\}$ is the set of all natural variables of $Phi$ except $y_j$. 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 equations of state since they specify parameters of the thermodynamic state [Callen (1985) p37] . If we restrict ourselves to the potentials U,F,H and G, then we have:

:$+T=left\left(frac\left\{partial U\right\}\left\{partial S\right\} ight\right)_\left\{V,\left\{N_i =left\left(frac\left\{partial H\right\}\left\{partial S\right\} ight\right)_\left\{p,\left\{N_i$

:$-p=left\left(frac\left\{partial U\right\}\left\{partial V\right\} ight\right)_\left\{S,\left\{N_i =left\left(frac\left\{partial F\right\}\left\{partial V\right\} ight\right)_\left\{T,\left\{N_i$

:$+V=left\left(frac\left\{partial H\right\}\left\{partial p\right\} ight\right)_\left\{S,\left\{N_i =left\left(frac\left\{partial G\right\}\left\{partial p\right\} ight\right)_\left\{T,\left\{N_i$

:$-S=left\left(frac\left\{partial G\right\}\left\{partial T\right\} ight\right)_\left\{p,\left\{N_i =left\left(frac\left\{partial F\right\}\left\{partial T\right\} ight\right)_\left\{V,\left\{N_i$

:$~mu_j=left\left(frac\left\{partial phi\right\}\left\{partial N_j\right\} ight\right)_\left\{X,Y,\left\{N_\left\{i e j\right\}$

where, in the last equation, $phi$ is any of the thermodynamic potentials U, F, H, G and $\left\{X,Y,\left\{N_\left\{j e i\right\}$ are the set of natural variables for that potential, excluding $N_i$. If we use all potentials, then we will have more equations of state such as

:$-N_j=left\left(frac\left\{partial U \left[mu_j\right] \right\}\left\{partial mu_j\right\} ight\right)_\left\{S,V,\left\{N_\left\{i e j\right\}$

and so on. In all, there will be "D" equations for each potential resulting in a total of "D" 2"D" equations of state. If the "D" 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.

The Maxwell relations

Again, define $x_i$ and $y_i$ to be conjugate pairs, and the $y_i$ to be the natural variables of some potential $Phi$. We may take the "cross differentials" of the state equations, which obey the following relationship:

:$left\left(frac\left\{partial\right\}\left\{partial y_j\right\}left\left(frac\left\{partial Phi\right\}\left\{partial y_k\right\} ight\right)_\left\{\left\{y_\left\{i e k\right\} ight\right)_\left\{\left\{y_\left\{i e j\right\}=left\left(frac\left\{partial\right\}\left\{partial y_k\right\}left\left(frac\left\{partial Phi\right\}\left\{partial y_j\right\} ight\right)_\left\{\left\{y_\left\{i e j\right\} ight\right)_\left\{\left\{y_\left\{i e k\right\}$

From these we get the Maxwell relations [Alberty (2001) p1353] [Callen (1985) p181] . There will be "(D-1)/2" of them for each potential giving a total of "D(D-1)/2" equations in all. If we restrict ourselves the U, F, H, G

:$left\left(frac\left\{partial T\right\}\left\{partial V\right\} ight\right)_\left\{S,\left\{N_i =-left\left(frac\left\{partial p\right\}\left\{partial S\right\} ight\right)_\left\{V,\left\{N_i$

:$left\left(frac\left\{partial T\right\}\left\{partial p\right\} ight\right)_\left\{S,\left\{N_i =+left\left(frac\left\{partial V\right\}\left\{partial S\right\} ight\right)_\left\{p,\left\{N_i$

:$left\left(frac\left\{partial S\right\}\left\{partial V\right\} ight\right)_\left\{T,\left\{N_i =+left\left(frac\left\{partial p\right\}\left\{partial T\right\} ight\right)_\left\{V,\left\{N_i$

:$left\left(frac\left\{partial S\right\}\left\{partial p\right\} ight\right)_\left\{T,\left\{N_i =-left\left(frac\left\{partial V\right\}\left\{partial T\right\} ight\right)_\left\{p,\left\{N_i$

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

:$left\left(frac\left\{partial T\right\}\left\{partial N_j\right\} ight\right)_\left\{V,S,\left\{N_\left\{i e j\right\} =left\left(frac\left\{partial mu_j\right\}\left\{partial S\right\} ight\right)_\left\{V,\left\{N_i$

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

:$left\left(frac\left\{partial N_j\right\}\left\{partial V\right\} ight\right)_\left\{S,mu_j,\left\{N_\left\{i e j\right\} =-left\left(frac\left\{partial p\right\}\left\{partial mu_j\right\} ight\right)_\left\{S,V\left\{N_\left\{i e j\right\}$

:$left\left(frac\left\{partial N_j\right\}\left\{partial N_k\right\} ight\right)_\left\{S,V,mu_j,\left\{N_\left\{i e j,k\right\} =-left\left(frac\left\{partial mu_k\right\}\left\{partial mu_j\right\} ight\right)_\left\{S,V\left\{N_\left\{i e j\right\}$

Euler integrals

Again, define $x_i$ and $y_i$ to be conjugate pairs, and the $y_i$ to be the natural variables of the internal energy.Since all of the natural variables of the internal energy "U" are extensive quantities

:$U\left(\left\{alpha y_i\right\}\right)=alpha U\left(\left\{y_i\right\}\right),$

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

:$U\left(\left\{y_i\right\}\right)=sum_j y_jleft\left(frac\left\{partial U\right\}\left\{partial y_j\right\} ight\right)_\left\{\left\{y_\left\{i e j\right\}$

From the equations of state, we then have:

:$U=TS-pV+sum_i mu_i N_i,$

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

:$F= -pV+sum_i mu_i N_i,$

:$H=TS +sum_i mu_i N_i,$

:$G= sum_i mu_i N_i,$

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

The Gibbs-Duhem relation

Deriving the Gibbs-Duhem equation from basic thermodynamic state equations is straightforward [Moran & Shapiro, p538] [Alberty (2001) p1354] [Callen (1985) p60] . The Gibbs free energy $G,$ can be expanded locally at equilibrium in terms of the thermodynamic state as:

:$mathrm\left\{d\right\}G=left. frac\left\{partial G\right\}\left\{partial p\right\} ight | _\left\{T,N\right\}mathrm\left\{d\right\}p+left. frac\left\{partial G\right\}\left\{partial T\right\} ight | _\left\{p,N\right\}mathrm\left\{d\right\}T+sum_\left\{i=1\right\}^I left. frac\left\{partial G\right\}\left\{partial N_i\right\} ight | _\left\{p,N_\left\{j eq imathrm\left\{d\right\}N_i ,$

With the substitution of two of the Maxwell relations and the definition of chemical potential, this is transformed into:

:$mathrm\left\{d\right\}G=V mathrm\left\{d\right\}p-S mathrm\left\{d\right\}T+sum_\left\{i=1\right\}^I mu_i mathrm\left\{d\right\}N_i ,$

The chemical potential is just another name for the partial molar Gibbs free energy, and as such::$G = sum_\left\{i=1\right\}^I mu_i N_i ,$:$mathrm\left\{d\right\}G = sum_\left\{i=1\right\}^I mu_i mathrm\left\{d\right\}N_i + sum_\left\{i=1\right\}^I N_i mathrm\left\{d\right\}mu_i ,$

Subtracting yields the Gibbs-Duhem relation:

:$sum_\left\{i=1\right\}^I N_imathrm\left\{d\right\}mu_i = - Smathrm\left\{d\right\}T + Vmathrm\left\{d\right\}p ,$

The Gibbs-Duhem is a relationship among the intensive parameters of the system. It follows that for a simple system with $I$ components, there will be $I+1$ 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 Josiah Willard Gibbs and 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. "Δ" denotes the change in the potential and at equilibrium the change will be zero.

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

Notes

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=Vol. 73 | issue=8 | pages=1349–1380
doi=10.1351/pac200173081349
format=PDF

*cite book
first = Herbert B. | last = Callen | authorlink = Herbert Callen | year = 1985
title = Thermodynamics and an Introduction to Themostatistics | edition = 2nd Ed.
publisher = John Wiley & Sons | location = New York | id = ISBN 0-471-86256-8

*cite book
first = Herbert B. | last =Moran | first=Michael J. |coauthors=Shapiro, Howard N.
title = Fundamentals of Engineering Thermodynamics | edition = 3rd Ed.| id = ISBN 0-471-07681-3

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

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