1.12.24: Expansions- Equations
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, \(\frac{\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}}{\alpha_{\mathrm{S} 1}^{*}(\ell)}=-\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\alpha_{\mathrm{p} 1}^{*}(\ell)}+\frac{\phi\left(\mathrm{K}_{\mathrm{s} j} ; \mathrm{def}\right)^{\infty}}{\kappa_{\mathrm{s} 1}^{*}(\ell)}+\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\sigma_{1}^{*}(\ell)}\)