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| {{Thermodynamics|cTopic=[[Thermodynamic equations|Equations]]}}
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| {{For|electromagnetic equations|Maxwell's equations}}
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| '''Maxwell's relations''' are a set of equations in [[thermodynamics]] which are derivable from the definitions of the [[thermodynamic potentials]]. These relations are named for the nineteenth-century physicist [[James Clerk Maxwell]]. | |
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| ==Equation==
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| The Maxwell relations are statements of equality among the second derivatives of the thermodynamic potentials. They follow directly from the fact that the order of differentiation of an [[analytic function]] of two variables is irrelevant. If Φ is a thermodynamic potential and ''x<sub>i</sub>'' and ''x<sub>j</sub>'' are two different [[Thermodynamic potential#Natural variables|natural variables]] for that potential, then the Maxwell relation for that potential and those variables is:
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| {{Equation box 1
| | '''MathML''' |
| |title = '''Maxwell's relations''' ''(general)''
| | :<math forcemathmode="mathml">E=mc^2</math> |
| |indent =:
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| |equation = <math>\frac{\partial }{\partial x_j}\left(\frac{\partial \Phi}{\partial x_i}\right)=
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| \frac{\partial }{\partial x_i}\left(\frac{\partial \Phi}{\partial x_j}\right)
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| </math> | |
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| where the [[partial derivatives]] are taken with all other natural variables held constant. It is seen that for every thermodynamic potential there are ''n''(''n'' − 1)/2 possible Maxwell relations where ''n'' is the number of natural variables for that potential.
| | <!--'''PNG''' (currently default in production) |
| | :<math forcemathmode="png">E=mc^2</math> |
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| == The four most common Maxwell relations == | | '''source''' |
| | :<math forcemathmode="source">E=mc^2</math> --> |
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| The four most common Maxwell relations are the equalities of the second derivatives of each of the four thermodynamic potentials, with respect to their thermal natural variable ([[temperature]] ''T''; or [[entropy]] ''S'') and their ''mechanical'' natural variable ([[pressure]] ''P''; or [[volume]] ''V''):
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| {{Equation box 1
| | ==Demos== |
| |title = '''Maxwell's relations''' ''(common)''
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| |indent =:
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| |equation =
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| <math> \begin{align}
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
| +\left(\frac{\partial T}{\partial V}\right)_S &=& -\left(\frac{\partial P}{\partial S}\right)_V &=& \frac{\partial^2 U }{\partial S \partial V}\\
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| +\left(\frac{\partial T}{\partial P}\right)_S &=& +\left(\frac{\partial V}{\partial S}\right)_P &=& \frac{\partial^2 H }{\partial S \partial P}\\
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| +\left(\frac{\partial S}{\partial V}\right)_T &=& +\left(\frac{\partial P}{\partial T}\right)_V &=& -\frac{\partial^2 A }{\partial T \partial V}\\
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| -\left(\frac{\partial S}{\partial P}\right)_T &=& +\left(\frac{\partial V}{\partial T}\right)_P &=& \frac{\partial^2 G }{\partial T \partial P} | | * accessibility: |
| \end{align}\,\!</math>
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| |border colour = #0073CF
| | ==Test pages == |
| |background colour=#F5FFFA}}
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| where the potentials as functions of their natural thermal and mechanical variables are the [[internal energy]] ''U''(''S, V''), [[Enthalpy]] ''H''(''S, P''), [[Helmholtz free energy]] ''A''(''T, V'') and [[Gibbs free energy]] ''G''(''T, P''). The [[thermodynamic square]] can be used as a [[mnemonic]] to recall and derive these relations.
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| === Derivation ===
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
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| Maxwell relations are based on simple partial differentiation rules, in particular the [[Total derivative|total]] [[differential of a function]] and the symmetry of evaluating second order partial derivatives.
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| :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
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| !Derivation
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| |-
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| |Derivation of the Maxwell relations can be deduced from the differential forms of the [[thermodynamic potentials]]:
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| :<math>\begin{align}
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| dU &=& TdS-PdV \\
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| dH &=& TdS+VdP \\
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| dA &=& -SdT-PdV \\
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| dG &=& -SdT+VdP \\
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| \end{align}\,\!</math>
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| These equations resemble [[Total derivative|total differentials]] of the form
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| :<math>dz = \left(\frac{\partial z}{\partial x}\right)_y\!dx +
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| \left(\frac{\partial z}{\partial y}\right)_x\!dy</math>
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| And indeed, it can be shown for any equation of the form
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| :<math>dz = Mdx + Ndy \,</math>
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| that
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| :<math>M = \left(\frac{\partial z}{\partial x}\right)_y, \quad
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| N = \left(\frac{\partial z}{\partial y}\right)_x</math>
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| Consider, as an example, the equation <math>dH=TdS+VdP\,</math>. We can now immediately see that
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| :<math>T = \left(\frac{\partial H}{\partial S}\right)_P, \quad
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| V = \left(\frac{\partial H}{\partial P}\right)_S</math>
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| Since we also know that for functions with continuous second derivatives, the mixed partial derivatives are identical ([[Symmetry of second derivatives]]), that is, that
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| :<math>\frac{\partial}{\partial y}\left(\frac{\partial z}{\partial x}\right)_y =
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| \frac{\partial}{\partial x}\left(\frac{\partial z}{\partial y}\right)_x =
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| \frac{\partial^2 z}{\partial y \partial x} = \frac{\partial^2 z}{\partial x \partial y}</math>
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| we therefore can see that
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| :<math> \frac{\partial}{\partial P}\left(\frac{\partial H}{\partial S}\right)_P =
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| \frac{\partial}{\partial S}\left(\frac{\partial H}{\partial P}\right)_S </math>
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| and therefore that
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| :<math>\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P</math>
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| Each of the four Maxwell relationships given above follows similarly from one of the [[Gibbs equations]].
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| |}
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| :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
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| !Extended derivation
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| |Combined form first and second law of thermodynamics,
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| :<math>TdS = dU+PdV</math> (Eq.1)
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| U, S, and V are state functions.
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| Let,
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| :<math>U = U(x,y)</math>
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| :<math>S = S(x,y)</math>
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| :<math>V = V(x,y)</math>
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| :<math>dU = \left(\frac{\partial U}{\partial x}\right)_y\!dx +
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| \left(\frac{\partial U}{\partial y}\right)_x\!dy</math>
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| :<math>dS = \left(\frac{\partial S}{\partial x}\right)_y\!dx +
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| \left(\frac{\partial S}{\partial y}\right)_x\!dy</math>
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| :<math>dV = \left(\frac{\partial V}{\partial x}\right)_y\!dx +
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| \left(\frac{\partial V}{\partial y}\right)_x\!dy</math>
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| Substitute them in Eq.1 and one gets,
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| :<math>T\left(\frac{\partial S}{\partial x}\right)_y\!dx +
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| T\left(\frac{\partial S}{\partial y}\right)_x\!dy = \left(\frac{\partial U}{\partial x}\right)_y\!dx +
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| \left(\frac{\partial U}{\partial y}\right)_x\!dy + P\left(\frac{\partial V}{\partial x}\right)_y\!dx +
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| P\left(\frac{\partial V}{\partial y}\right)_x\!dy</math>
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| And also written as,
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| :<math>\left(\frac{\partial U}{\partial x}\right)_y\!dx +
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| \left(\frac{\partial U}{\partial y}\right)_x\!dy = T\left(\frac{\partial S}{\partial x}\right)_y\!dx +
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| T\left(\frac{\partial S}{\partial y}\right)_x\!dy - P\left(\frac{\partial V}{\partial x}\right)_y\!dx -
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| P\left(\frac{\partial V}{\partial y}\right)_x\!dy</math>
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| comparing the coefficient of dx and dy, one gets
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| :<math>\left(\frac{\partial U}{\partial x}\right)_y = T\left(\frac{\partial S}{\partial x}\right)_y - P\left(\frac{\partial V}{\partial x}\right)_y</math> | |
| :<math>\left(\frac{\partial U}{\partial y}\right)_x = T\left(\frac{\partial S}{\partial y}\right)_x - P\left(\frac{\partial V}{\partial y}\right)_x</math>
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| Differentiating above equations by y, x respectively<br />
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| :<math>\left(\frac{\partial^2U}{\partial y\partial x}\right) = \left(\frac{\partial T}{\partial y}\right)_x \left(\frac{\partial S}{\partial x}\right)_y + T\left(\frac{\partial^2 S}{\partial y\partial x}\right) - \left(\frac{\partial P}{\partial y}\right)_x \left(\frac{\partial V}{\partial x}\right)_y - P\left(\frac{\partial^2 V}{\partial y\partial x}\right)</math> (Eq.2)
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| :and
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| :<math>\left(\frac{\partial^2U}{\partial x\partial y}\right) = \left(\frac{\partial T}{\partial x}\right)_y \left(\frac{\partial S}{\partial y}\right)_x + T\left(\frac{\partial^2 S}{\partial x\partial y}\right) - \left(\frac{\partial P}{\partial x}\right)_y \left(\frac{\partial V}{\partial y}\right)_x - P\left(\frac{\partial^2 V}{\partial x\partial y}\right)</math> (Eq.3)
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| U, S, and V are exact differentials, therefore,
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| :<math>\left(\frac{\partial^2U}{\partial y\partial x}\right) = \left(\frac{\partial^2U}{\partial x\partial y}\right)</math>
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| :<math>\left(\frac{\partial^2S}{\partial y\partial x}\right) = \left(\frac{\partial^2S}{\partial x\partial y}\right)
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| :\left(\frac{\partial^2V}{\partial y\partial x}\right) = \left(\frac{\partial^2V}{\partial x\partial y}\right)</math>
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| Subtract eqn(2) and (3) and one gets<br />
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| :<math>\left(\frac{\partial T}{\partial y}\right)_x \left(\frac{\partial S}{\partial x}\right)_y - \left(\frac{\partial P}{\partial y}\right)_x \left(\frac{\partial V}{\partial x}\right)_y = \left(\frac{\partial T}{\partial x}\right)_y \left(\frac{\partial S}{\partial y}\right)_x - \left(\frac{\partial P}{\partial x}\right)_y \left(\frac{\partial V}{\partial y}\right)_x</math>
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| :''Note: The above is called the general expression for Maxwell's thermodynamical relation.''
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| ;Maxwell's first relation
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| :Allow x = S and y = V and one gets
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| :<math>\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V</math>
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| ;Maxwell's second relation
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| :Allow x = T and y = V and one gets
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| :<math>\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V</math>
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| ;Maxwell's third relation
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| :Allow x = S and y = P and one gets
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| :<math>\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P</math>
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| ;Maxwell's fourth relation
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| :Allow x = T and y = P and one gets
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| :<math>\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P</math>
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| ;Maxwell's fifth relation
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| :Allow x = P and y = V
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| :<math>\left(\frac{\partial T}{\partial P}\right)_V \left(\frac{\partial S}{\partial V}\right)_P</math><math>-\left(\frac{\partial T}{\partial V}\right)_P \left(\frac{\partial S}{\partial P}\right)_V</math> = 1
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| ;Maxwell's sixth relation
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| :Allow x = T and y = S and one gets
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| :<math>\left(\frac{\partial P}{\partial T}\right)_S \left(\frac{\partial V}{\partial S}\right)_T -\left(\frac{\partial P}{\partial S}\right)_T \left(\frac{\partial V}{\partial T}\right)_S</math> = 1
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| |}
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| == General Maxwell relationships ==
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| The above are not the only Maxwell relationships. When other work terms involving other natural variables besides the volume work are considered or when the [[Particle number|number of particles]] is included as a natural variable, other Maxwell relations become apparent. For example, if we have a single-component gas, then the number of particles ''N'' is also a natural variable of the above four thermodynamic potentials. The Maxwell relationship for the enthalpy with respect to pressure and particle number would then be:
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| :<math>
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| \left(\frac{\partial \mu}{\partial P}\right)_{S, N} =
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| \left(\frac{\partial V}{\partial N}\right)_{S, P}\qquad=
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| \frac{\partial^2 H }{\partial P \partial N}
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| </math>
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| where μ is the [[chemical potential]]. In addition, there are other thermodynamic potentials besides the four that are commonly used, and each of these potentials will yield a set of Maxwell relations.
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| Each equation can be re-expressed using the relationship
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| :<math>\left(\frac{\partial y}{\partial x}\right)_z
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| =
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| 1\left/\left(\frac{\partial x}{\partial y}\right)_z\right.</math>
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| which are sometimes also known as Maxwell relations.
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| == See also ==
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| * [[Table of thermodynamic equations]]
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| * [[Thermodynamic equations]]
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| ==External links==
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| *http://theory.ph.man.ac.uk/~judith/stat_therm/node48.html a partial derivation of Maxwell's relations
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| [[Category:Thermodynamics]]
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| [[Category:Concepts in physics]]
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| [[Category:James Clerk Maxwell]]
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| [[Category:Thermodynamic equations]]
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