1.12.24: Expansions- Equations
- Page ID
- 377921
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We simplify the algebra by omitting the descriptors (aq) and (\(\ell\)) in the following equations. The starting point is the following equation.
\[\alpha_{S}-\alpha_{S}^{*}=\frac{\alpha_{S}}{\kappa_{S} \, \sigma} \,\left(\kappa_{S} \, \sigma-\kappa_{S}^{*} \, \sigma^{*}\right)-\frac{\alpha_{S}^{*}}{\alpha_{p}} \,\left(\alpha_{p}-\alpha_{p}^{*}\right) \nonumber \]
The latter equation is effectively an identity. Thus from equation (a)
\[\alpha_{S}-\alpha_{S}^{*}=\alpha_{S}-\frac{\alpha_{S}}{\kappa_{S} \, \sigma} \, \kappa_{S}^{*} \, \sigma^{*}-\alpha_{S}^{*}+\frac{\alpha_{S}^{*}}{\alpha_{p}} \, \alpha_{p}^{*} \nonumber \]
But \(\alpha_{\mathrm{s}}=-\kappa_{\mathrm{s}} \, \sigma / \mathrm{T} \, \alpha_{\mathrm{p}}\) and \(\alpha_{\mathrm{p}}^{*} / \alpha_{\mathrm{s}}^{*}=-\kappa_{\mathrm{s}}^{*} \, \sigma^{*} / \mathrm{T}\)
Then from (b), \(\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}+\frac{\kappa_{\mathrm{s}} \, \sigma}{\mathrm{T} \, \alpha_{\mathrm{p}} \, \kappa_{\mathrm{s}} \, \sigma} \, \kappa_{\mathrm{s}}^{*} \, \sigma^{*}-\frac{\kappa_{\mathrm{s}}^{*} \, \sigma^{*}}{\alpha_{\mathrm{p}} \, \mathrm{T}}\) or \(\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}\)
But as an identity,
\[\kappa_{\mathrm{S}} \, \sigma-\kappa_{\mathrm{S}}^{*} \, \sigma^{*}=\sigma \,\left(\kappa_{\mathrm{S}}-\kappa_{\mathrm{S}}^{*}\right)+\kappa_{\mathrm{S}}^{*} \,\left(\sigma-\sigma^{*}\right) \nonumber \]
Then from equations (a) and (c),
\[\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \,\left(\kappa_{\mathrm{s}}-\kappa_{\mathrm{s}}^{*}\right)+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}} \, \sigma} \,\left(\sigma-\sigma^{*}\right)-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \,\left(\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}\right) \nonumber \]
But,
\[\phi\left(E_{\mathrm{S}} ; \operatorname{def}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left(\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}\right)+\alpha_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
Hence
\[\begin{aligned}
\phi\left(E_{\mathrm{s} j} ; \operatorname{def}\right)=& \frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \,\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left(\kappa_{\mathrm{s}}-\kappa_{\mathrm{s}}^{*}\right)+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}} \, \sigma} \,\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left(\sigma-\sigma^{*}\right) \\
&-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \,\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left(\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}\right)+\alpha_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{aligned} \nonumber \]
For isobaric heat capacities,
\[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left(\sigma-\sigma^{*}\right)+\sigma^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
Also
\[\phi\left(K_{\mathrm{S}_{j}} ; \operatorname{def}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left(\kappa_{\mathrm{s}}-\kappa_{\mathrm{s}}^{*}\right)+\kappa_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
Hence
\[\begin{aligned}
\phi\left(\mathrm{E}_{\mathrm{s} j} ; \operatorname{def}\right)=& \frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \,\left[\phi\left(\mathrm{K}_{\mathrm{s} j} ; \operatorname{def}\right)-\kappa_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}} \, \sigma} \,\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\sigma^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \\
&-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \,\left[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\alpha_{\mathrm{p}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]+\alpha_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{aligned} \nonumber \]
Then with a little reorganisation,
\[\begin{aligned}
\phi\left(\mathrm{E}_{\mathrm{s} j} ; \operatorname{def}\right)=&-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \, \phi\left(\mathrm{E}_{\mathrm{p} j}\right)+\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \, \phi\left(\mathrm{K}_{\mathrm{s} j} ; \operatorname{def}\right)+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}} \, \sigma} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \\
&+\left[\alpha_{\mathrm{s}}^{*} \,\left(1+\frac{\alpha_{\mathrm{p}}^{*}}{\alpha_{\mathrm{p}}}\right)-\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}}} \,\left(1+\frac{\sigma^{*}}{\sigma}\right)\right] \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{aligned} \nonumber \]
Hence, in the limit of infinite dilution, \(\frac{\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}}{\alpha_{\mathrm{S} 1}^{*}(\ell)}=-\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\alpha_{\mathrm{p} 1}^{*}(\ell)}+\frac{\phi\left(\mathrm{K}_{\mathrm{s} j} ; \mathrm{def}\right)^{\infty}}{\kappa_{\mathrm{s} 1}^{*}(\ell)}+\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\sigma_{1}^{*}(\ell)}\)