1.0.s: The Clapeyron Equation
- Page ID
- 206900
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The analysis in the two previous sections can be repeated for any phase change of a pure substance. Let \(\alpha\) and \(\beta\) denote the two phases that are at equilibrium.
\[\alpha \rightleftharpoons \beta\]
Let \(\overline{G}_{\alpha }\), \(\overline{S}_{\alpha }\), and \({\overline{V}}_{\alpha }\) represent the Gibbs free energy, the entropy, and the volume of one mole of pure phase \(\alpha\) at pressure \(P\) and temperature \(T\). Let \(\overline{G}_{\beta }\), \(\overline{S}_{\beta }\), and \(\overline{V}_{\beta }\) represent the corresponding properties of one mole of pure phase \(\beta\). The equations
\[d\overline{G}\left(\alpha \right)=\overline{V}_\alpha dP-\overline{S}_{\alpha }dT\]
and
\[d\overline{G}\left(\beta \right)= \overline{V}_{\beta } dP- \overline{S}_{\beta }dT\]
describe the changes in the Gibbs free energy of a mole of \(\alpha\) and a mole of \(\beta\) when they go from one \(\alpha-\beta\)-equilibrium state at \(P\) and \(T\) to a second \(\alpha-\beta\)-equilibrium state at \(P+dP\) and \(T+dT\). Since these Gibbs free energy changes must be equal, we have
\[ \begin{align*} d\overline{G}\left(\beta \right)-d\overline{G}\left(\alpha \right) &=\left({\overline{V}}_{\beta }-{\overline{V}}_{\alpha }\right)dP-\left({\overline{S}}_{\beta }-\overline{S}_{\alpha }\right)dT \\[4pt] &=\Delta \overline{V}dP-\Delta \overline{S}dT \\[4pt] &=0 \end{align*}\]
and
\[\frac{dP}{dT}=\frac{\Delta \overline{S}}{\Delta \overline{V}}\]
where \(\Delta \overline{S}\) and \(\Delta \overline{V}\) are the entropy and volume changes that occur when one mole of the substance goes from phase \(\alpha\) to phase \(\beta\). Since \(\Delta \overline{S}=\Delta \overline{H}/T\), the condition for equilibrium between phases \(\alpha\) and \(\beta\) becomes
\[\frac{dP}{dT}=\frac{\Delta \overline{H}}{T\ \Delta \overline{V}} \label{Clap1}\]
Equation \ref{Clap1} is known as the Clapeyron equation.