6.1: Maxwell Relations

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Mar 15, 2021
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- 6: Thermodynamic Relations and Processes
- 6.2: Other 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

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

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