EOS: Maxwell Relations

Fundamental Property Relations and the Maxwell Equations (cont.)

Further manipulating these relations by using our second mathematical fact yields the Maxwell relations as follows:

Take our equations that originated from the $du$ analysis:

$T = \left (\frac{\partial u}{\partial s} \right )_v$
$-P = \left (\frac{\partial u}{\partial v} \right )_s$

and take the partial derivative with respect to $v$ for the top and with respect to $s$ for the bottom yields:

$\left (\frac{\partial T}{\partial v} \right )_s = \left [\frac{\partial}{\partial v} \left (\frac{\partial u}{\partial s} \right )_v \right ]_s$
$-\left (\frac{\partial P}{\partial s} \right )_v = \left [\frac{\partial}{\partial s}\left (\frac{\partial u}{\partial v} \right )_s \right ]_v$

Our second mathematical fact tells us that the right-hand side of both of these expressions is equivalent, so that we can write:

$\left (\frac{\partial T}{\partial v} \right )_s = -\left (\frac{\partial P}{\partial s} \right )_v $

Performing the same analysis on the expressions that originated from the $dh$, $da$, and $dg$ expressions gives the other three maxwell relations:

$\left (\frac{\partial v}{\partial s} \right )_P = \left (\frac{\partial T}{\partial P} \right )_s $

$\left (\frac{\partial P}{\partial T} \right )_v = \left (\frac{\partial s}{\partial v} \right )_T $

$-\left (\frac{\partial v}{\partial T} \right )_P = \left (\frac{\partial s}{\partial P} \right )_T $

Note:

The "beauty" of the maxwell equations, as well as the fundamental property relations derived early, is that they can be used to relate measurable quantities (like $T, P$, and $v$) to fundamental and derived quantities. Also, in many cases, we can substitute equations of state into these expressions to aid in calculating quantities of interest.