6.1: Maxwell Relations

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For a system with one constituent with fixed number of particles, from the first and second laws, and from Equation 5.1.10, we have the basic relations

\[dU\;=\;TdS\;-\;pdV \\ dH\;=\;TdS\;-\;Vdp \\ dF\;=\;-SdT\;-\;pdV \\ dG\;=\;-SdT\;-\;Vdp \label{6.1.1}\]

The quantities on the left are all perfect differentials. For a general differential \(dR\) of the form

\[dR = Xdx + Y dy\]

to be a perfect differential, the necessary and sufficient condition is

\[\biggl( \frac{\partial X}{\partial y} \biggr)_x \;=\; \biggl( \frac{\partial Y}{\partial x} \biggr)_y \label{6.1.3}\]

Applying this to the four differentials in \ref{6.1.1}, we get

\[ \biggl( \frac{\partial T}{\partial V} \biggr)_S \;=\; -\biggl( \frac{\partial p}{\partial S} \biggr)_V \\ \biggl( \frac{\partial T}{\partial p} \biggr)_S \;=\; \biggl( \frac{\partial V}{\partial S} \biggr)_p \\ \biggl( \frac{\partial S}{\partial V} \biggr)_T \;=\; \biggl( \frac{\partial p}{\partial T} \biggr)_V \\ \biggl( \frac{\partial S}{\partial p} \biggr)_T \;=\; -\biggl( \frac{\partial V}{\partial T} \biggr)_p \]

These four relations are called the Maxwell relations.

A Mathematical Result

Let \(X\), \(Y\), \(Z\) be three variables, of which only two are independent. Taking \(Z\) to be a function of \(X\) and \(Y\), we can write

\[dZ\;=\; \biggl( \frac{\partial Z}{\partial X} \biggr)_Y dX \;+\; \biggl( \frac{\partial Z}{\partial Y} \biggr)_X dY \label{6.1.5}\]

If now we take \(X\) and \(Z\) as the independent variables, we can write

\[dY\;=\; \biggl( \frac{\partial Y}{\partial X} \biggr)_Y dX \;+\; \biggl( \frac{\partial Y}{\partial Z} \biggr)_X dZ \]

Upon substituting this result into \ref{6.1.5}, we get

\[dZ\;=\; \biggl[ \biggl( \frac{\partial Z}{\partial X} \biggr)_Y \;+\; \biggl( \frac{\partial Z}{\partial Y} \biggr)_X \biggl( \frac{\partial Y}{\partial X} \biggr)_Z \biggr]dX \;+\; \biggl( \frac{\partial Z}{\partial Y} \biggr)_X \biggl( \frac{\partial Y}{\partial Z} \biggr)_X dZ\]

Since we are considering \(X\) and \(Z\) as independent variables now, this equation immediately yields the relations

\[\biggl( \frac{\partial Z}{\partial Y} \biggr)_X \biggl( \frac{\partial Y}{\partial Z} \biggr)_X dZ \;\;=\;\;1 \\ \biggl( \frac{\partial Z}{\partial X} \biggr)_Y \;+\; \biggl( \frac{\partial Z}{\partial Y} \biggr)_X \biggl( \frac{\partial Y}{\partial X} \biggr)_Z \;\;=\;\;0 \]

These relations can be rewritten as

\[\biggl( \frac{\partial Z}{\partial Y} \biggr)_X \;\;=\;\; \frac{1}{\bigl( \frac{dY}{dZ} \bigr)_X} \\ \biggl( \frac{\partial X}{\partial Z} \biggr)_Y \biggl( \frac{\partial Z}{\partial Y} \biggr)_X \biggl( \frac{\partial Y}{\partial X} \biggr)_Z \;\;=\;\;-1 \label{6.1.9}\]