6.1: Maxwell Relations

Thermodynamics and Statistical Mechanics · LibreTexts

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    From LibreTexts, under CC BY-NC-SA 4.0, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    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

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

    This page titled 6.1: Maxwell Relations was last modified on Mon, 15 Mar 2021 19:17:22 GMT and is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by V. Parameswaran Nair via source content that was edited to the style and standards of the LibreTexts platform.