6.1: Maxwell Relations
- Page ID
- 32030
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
\[\biggl( \frac{\partial Z}{\partial Y} \biggr)_X \;\;=\;\; \frac{1}{\bigl( \frac{dY}{dZ} \bigr)_X} \\ \biggl( \frac{\partial X}{\partial Z} \biggr)_Y \biggl( \frac{\partial Z}{\partial Y} \biggr)_X \biggl( \frac{\partial Y}{\partial X} \biggr)_Z \;\;=\;\;-1 \label{6.1.9}\]