1.17.5: Isochoric Thermal Pressure Coefficient
- Page ID
- 394360
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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 equilibrium volume of a given closed system is defined by the following set of independent variables where \(\xi\) is the general composition variable.
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \xi^{\mathrm{eq}} ; \mathrm{A}=0\right] \nonumber \]
We have rather over-defined the system. The aim is to identify the composition variable at equilibrium and under the condition that the affinity for spontaneous change is zero. The system is perturbed by a change in temperature but we require that the system travels a path where the volume remains constant (and at equilibrium). The pressure must be changed in order to satisfy these conditions. By definition the isochoric differential dependence of pressure on temperature defines the isochoric thermal pressure coefficient.
\[\beta_{V}=\left(\frac{\partial p}{\partial T}\right)_{v} \nonumber \]
Three interesting equations follow [1-3].
\[\beta_{\mathrm{V}}=\alpha_{\mathrm{p}} / \kappa_{\mathrm{T}} \nonumber \]
\[\beta_{\mathrm{V}}=-\mathrm{C}_{\mathrm{V}} / \mathrm{T} \, \mathrm{V} \, \alpha_{\mathrm{s}} \nonumber \]
\[\alpha_{\mathrm{p}} / \kappa_{\mathrm{T}}=-\mathrm{C}_{\mathrm{V}} / \mathrm{T} \, \mathrm{V} \, \alpha_{\mathrm{s}} \nonumber \]
Footnotes
[1] From equation (a)
\[\beta_{\mathrm{V}}=-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{V}}\right)_{\mathrm{T}} \nonumber \]
Or,
\[\beta_{\mathrm{v}}=\mathrm{E}_{\mathrm{p}} / \mathrm{K}_{\mathrm{T}}=\alpha_{\mathrm{p}} / \kappa_{\mathrm{T}} \nonumber \]
[2] Using a Maxwell relationship
\[\beta_{V}=\left(\frac{\partial S}{\partial V}\right)_{T}=-\left(\frac{\partial S}{\partial T}\right)_{V} \,\left(\frac{\partial T}{\partial V}\right)_{S} \nonumber \]
But \(\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\mathrm{C}_{\mathrm{V}} / \mathrm{T}\) Then
\[\beta_{\mathrm{v}}=-\mathrm{C}_{\mathrm{v}} / \mathrm{T} \, \mathrm{E}_{\mathrm{s}}=-\mathrm{C}_{\mathrm{v}} / \mathrm{T} \, \mathrm{V} \, \alpha_{\mathrm{s}} \nonumber \]
[3] From [1] and [2],
\[\mathrm{E}_{\mathrm{p}} / \mathrm{K}_{\mathrm{T}}=-\mathrm{C}_{\mathrm{V}} / \mathrm{T} \, \mathrm{E}_{\mathrm{S}} \nonumber \]
Or,
\[\alpha_{\mathrm{p}} / \kappa_{\mathrm{T}}=-\mathrm{C}_{\mathrm{V}} / \mathrm{T} \, \mathrm{V} \, \alpha_{\mathrm{S}} \nonumber \]