# 9.3: The Maxwell Relations

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Modeling the dependence of the Gibbs and Helmholtz functions behave with varying temperature, pressure, and volume is fundamentally useful. But in order to do that, a little bit more development is necessary. To see the power and utility of these functions, it is useful to combine the First and Second Laws into a single mathematical statement. In order to do that, one notes that since

$dS = \dfrac{dq}{T}$

for a reversible change, it follows that

$dq= TdS$

And since

$dw = TdS - pdV$

for a reversible expansion in which only p-V works is done, it also follows that (since $$dU=dq+dw$$):

$dU = TdS - pdV$

This is an extraordinarily powerful result. This differential for $$dU$$ can be used to simplify the differentials for $$H$$, $$A$$, and $$G$$. But even more useful are the constraints it places on the variables T, S, p, and V due to the mathematics of exact differentials!

## Maxwell Relations

The above result suggests that the natural variables of internal energy are $$S$$ and $$V$$ (or the function can be considered as $$U(S, V)$$). So the total differential ($$dU$$) can be expressed:

$dU = \left( \dfrac{\partial U}{\partial S} \right)_V dS + \left( \dfrac{\partial U}{\partial V} \right)_S dV$

Also, by inspection (comparing the two expressions for $$dU$$) it is apparent that:

$\left( \dfrac{\partial U}{\partial S} \right)_V = T \label{eq5A}$

and

$\left( \dfrac{\partial U}{\partial V} \right)_S = -p \label{eq5B}$

But the value doesn’t stop there! Since $$dU$$ is an exact differential, the Euler relation must hold that

$\left[ \dfrac{\partial}{\partial V} \left( \dfrac{\partial U}{\partial S} \right)_V \right]_S= \left[ \dfrac{\partial}{\partial S} \left( \dfrac{\partial U}{\partial V} \right)_S \right]_V$

By substituting Equations \ref{eq5A} and \ref{eq5B}, we see that

$\left[ \dfrac{\partial}{\partial V} \left( T \right)_V \right]_S= \left[ \dfrac{\partial}{\partial S} \left( -p \right)_S \right]_V$

or

$\left( \dfrac{\partial T}{\partial V} \right)_S = - \left( \dfrac{\partial p}{\partial S} \right)_V$

This is an example of a Maxwell Relation. These are very powerful relationship that allows one to substitute partial derivatives when one is more convenient (perhaps it can be expressed entirely in terms of $$\alpha$$ and/or $$\kappa_T$$ for example.)

A similar result can be derived based on the definition of $$H$$.

$H \equiv U +pV$

Differentiating (and using the chain rule on $$d(pV)$$) yields

$dH = dU +pdV + Vdp$

Making the substitution using the combined first and second laws ($$dU = TdS – pdV$$) for a reversible change involving on expansion (p-V) work

$dH = TdS – \cancel{pdV} + \cancel{pdV} + Vdp$

This expression can be simplified by canceling the $$pdV$$ terms.

$dH = TdS + Vdp \label{eq2A}$

And much as in the case of internal energy, this suggests that the natural variables of $$H$$ are $$S$$ and $$p$$. Or

$dH = \left( \dfrac{\partial H}{\partial S} \right)_p dS + \left( \dfrac{\partial H}{\partial p} \right)_S dV \label{eq2B}$

Comparing Equations \ref{eq2A} and \ref{eq2B} show that

$\left( \dfrac{\partial H}{\partial S} \right)_p= T \label{eq6A}$

and

$\left( \dfrac{\partial H}{\partial p} \right)_S = V \label{eq6B}$

It is worth noting at this point that both (Equation \ref{eq5A})

$\left( \dfrac{\partial U}{\partial S} \right)_V$

and (Equation \ref{eq6A})

$\left( \dfrac{\partial H}{\partial S} \right)_p$

are equation to $$T$$. So they are equation to each other

$\left( \dfrac{\partial U}{\partial S} \right)_V = \left( \dfrac{\partial H}{\partial S} \right)_p$

Morevoer, the Euler Relation must also hold

$\left[ \dfrac{\partial}{\partial p} \left( \dfrac{\partial H}{\partial S} \right)_p \right]_S= \left[ \dfrac{\partial}{\partial S} \left( \dfrac{\partial H}{\partial p} \right)_S \right]_p$

so

$\left( \dfrac{\partial T}{\partial p} \right)_S = \left( \dfrac{\partial V}{\partial S} \right)_p$

This is the Maxwell relation on $$H$$. Maxwell relations can also be developed based on $$A$$ and $$G$$. The results of those derivations are summarized in Table 6.2.1..

Table 6.2.1: Maxwell Relations
Function Differential Natural Variables Maxwell Relation
$$U$$ $$dU = TdS - pdV$$ $$S, \,V$$ $$\left( \dfrac{\partial T}{\partial V} \right)_S = - \left( \dfrac{\partial p}{\partial S} \right)_V$$
$$H$$ $$dH = TdS + Vdp$$ $$S, \,p$$ $$\left( \dfrac{\partial T}{\partial p} \right)_S = \left( \dfrac{\partial V}{\partial S} \right)_p$$
$$A$$ $$dA = -pdV - SdT$$ $$V, \,T$$ $$\left( \dfrac{\partial p}{\partial T} \right)_V = \left( \dfrac{\partial S}{\partial V} \right)_T$$
$$G$$ $$dG = Vdp - SdT$$ $$p, \,T$$ $$\left( \dfrac{\partial V}{\partial T} \right)_p = - \left( \dfrac{\partial S}{\partial p} \right)_T$$

The Maxwell relations are extraordinarily useful in deriving the dependence of thermodynamic variables on the state variables of p, T, and V.

Example $$\PageIndex{1}$$

Show that

$\left( \dfrac{\partial V}{\partial T} \right)_p = T\dfrac{\alpha}{\kappa_T} - p \nonumber$

Solution:

$dU = TdS - pdV \nonumber$

Divide both sides by $$dV$$ and constraint to constant $$T$$:

$\left.\dfrac{dU}{dV}\right|_{T} = \left.\dfrac{TdS}{dV}\right|_{T} - p \left.\dfrac{dV}{dV} \right|_{T} \nonumber$

Noting that

$\left.\dfrac{dU}{dV}\right|_{T} =\left( \dfrac{\partial U}{\partial V} \right)_T$

$\left.\dfrac{TdS}{dV}\right|_{T} = \left( \dfrac{\partial S}{\partial V} \right)_T$

$\left.\dfrac{dV}{dV} \right|_{T} = 1$

The result is

$\left( \dfrac{\partial U}{\partial V} \right)_T = T \left( \dfrac{\partial S}{\partial V} \right)_T -p \nonumber$

Now, employ the Maxwell relation on $$A$$ (Table 6.2.1)

$\left( \dfrac{\partial p}{\partial T} \right)_V = \left( \dfrac{\partial S}{\partial V} \right)_T \nonumber$

to get

$\left( \dfrac{\partial U}{\partial V} \right)_T = T \left( \dfrac{\partial p}{\partial T} \right)_V -p \nonumber$

and since

$\left( \dfrac{\partial p}{\partial T} \right)_V = \dfrac{\alpha}{\kappa_T} \nonumber$

It is apparent that

$\left( \dfrac{\partial V}{\partial T} \right)_p = T\dfrac{\alpha}{\kappa_T} - p \nonumber$

Note: How cool is that? This result was given without proof in Chapter 4, but can now be proven analytically using the Maxwell Relations! 9.3: The Maxwell Relations is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.