# 17.1: The Maxwell Relations

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

$dw = - PdV$

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

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

Moreover, 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!

## Contributors

• Patrick E. Fleming (Department of Chemistry and Biochemistry; California State University, East Bay)