1.12.23: Expansions- The Difference
For a solution,
\[\varepsilon=\alpha_{p}-\alpha_{s}=\kappa_{T} \, \sigma / T \, \alpha_{p} \nonumber \]
In order to simplify the algebra, we omit (aq) and (\(\ell\)) when describing the properties of an aqueous solution and the pure liquid respectively. Superscript '*' identifies the pure solvent.
\[\varepsilon^{*}=\alpha_{\mathrm{p}}^{*}-\alpha_{\mathrm{S}}^{*}=\kappa_{\mathrm{T}}^{*} \, \sigma^{*} / \mathrm{T} \, \alpha_{\mathrm{p}}^{*} \nonumber \]
Hence,
\[\varepsilon-\varepsilon^{*}=\frac{\varepsilon}{\kappa_{\mathrm{T}} \, \sigma} \,\left[\kappa_{\mathrm{T}} \, \sigma-\kappa_{\mathrm{T}}^{*} \, \sigma^{*}\right]-\frac{\varepsilon^{*}}{\alpha_{\mathrm{p}}} \,\left[\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}\right] \nonumber \]
The latter equation is effectively an identity. According to equation (c)
\[\varepsilon-\varepsilon=\varepsilon-\frac{\varepsilon}{\kappa_{\mathrm{T}} \, \sigma} \, \kappa_{\mathrm{T}}^{*} \, \sigma^{*}-\varepsilon+\frac{\varepsilon}{\alpha_{\mathrm{p}}} \, \alpha_{\mathrm{p}}^{*} \nonumber \]
We use equations (a) and (b) in the second and fourth terms on the right hand side of the latter equation.
\[\varepsilon-\varepsilon^{*}=\varepsilon-\frac{\varepsilon}{\kappa_{\mathrm{T}} \, \sigma} \, \varepsilon^{*} \, \mathrm{T} \, \alpha_{\mathrm{p}}^{*}-\varepsilon^{*}+\frac{\varepsilon^{*} \, \alpha_{\mathrm{p}}^{*} \, \mathrm{T} \, \varepsilon}{\kappa_{\mathrm{T}} \, \sigma} \nonumber \]
Or \(\varepsilon-\varepsilon^{*}=\varepsilon-\varepsilon^{*}\) Further, as an identity,
\[\kappa_{\mathrm{T}} \, \sigma-\kappa_{\mathrm{T}}^{*} \, \sigma^{*}=\sigma \,\left(\kappa_{\mathrm{T}}-\kappa_{\mathrm{T}}^{*}\right)+\kappa_{\mathrm{T}}^{*} \,\left(\sigma-\sigma^{*}\right) \nonumber \]
From equation (c),
\[\varepsilon-\varepsilon^{*}=\frac{\varepsilon}{\kappa_{\mathrm{T}}} \,\left[\kappa_{\mathrm{T}}-\kappa_{\mathrm{T}}^{*}\right]+\frac{\varepsilon \, \kappa_{\mathrm{T}}^{*}}{\kappa_{\mathrm{T}} \, \sigma} \,\left(\sigma-\sigma^{*}\right)-\frac{\varepsilon^{*}}{\alpha_{\mathrm{p}}} \,\left[\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}\right] \nonumber \]
But
\[\phi\left(\mathrm{E}_{\mathrm{sj}} ; \mathrm{def}\right)=\frac{\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}}{\mathrm{c}_{\mathrm{j}}}+\alpha_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
The analogue for \(\phi\left(E_{p j}\right)\) is the following equation. \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\frac{\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}}{\mathrm{c}_{\mathrm{j}}}+\alpha_{\mathrm{p}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\) Hence
\[\phi\left(E_{p j}\right)-\phi\left(E_{S_{j}} ; \operatorname{def}\right)=\frac{\varepsilon-\varepsilon^{*}}{c_{j}}+\varepsilon^{*} \, \phi\left(V_{j}\right) \nonumber \]
From equation (g), dividing by \(\mathrm{c}_{j}\),
\[\begin{gathered}
\frac{\varepsilon-\varepsilon}{c_{j}}=\frac{\varepsilon}{\kappa_{T}} \, \frac{1}{c_{j}} \,\left[\kappa_{T}-\kappa_{T}^{*}\right]+\frac{\varepsilon \, \kappa_{T}^{*}}{\kappa_{T} \, \sigma} \, \frac{1}{c_{j}} \,\left[\sigma-\sigma^{*}\right] \\
-\frac{\varepsilon^{*}}{\alpha_{p}} \, \frac{1}{c_{j}} \,\left[\alpha_{p}-\alpha_{p}^{*}\right]
\end{gathered} \nonumber \]
But from equation (i)
\[\frac{\varepsilon-\varepsilon^{*}}{c_{j}}=\phi\left(E_{p j}\right)-\phi\left(E_{S j} ; \operatorname{def}\right)-\varepsilon^{*} \, \phi\left(V_{j}\right) \nonumber \]
Equations having similar form for \(\left(\kappa_{\mathrm{T}}-\kappa_{\mathrm{T}}^{*}\right),\left(\sigma-\sigma^{*}\right)\) and \(\left(\alpha_{p}-\alpha_{p}^{*}\right)\) are readily generated. Hence
\[\begin{aligned}
\phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right) &=\frac{\varepsilon}{\kappa_{\mathrm{T}}} \,\left[\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)-\kappa_{\mathrm{T}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]+\frac{\varepsilon \, \kappa_{\mathrm{T}}^{*}}{\kappa_{\mathrm{T}} \, \sigma} \,\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\sigma^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \\
&-\frac{\varepsilon^{*}}{\alpha_{\mathrm{p}}} \,\left[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\alpha_{\mathrm{p}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]+\varepsilon^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{aligned} \nonumber \]
Therefore
\[\begin{aligned}
\phi\left(E_{\mathrm{pj}}\right)-\phi\left(E_{\mathrm{Sj}} ; \operatorname{def}\right) &=-\frac{\varepsilon}{\alpha_{\mathrm{p}}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)+\frac{\varepsilon}{\kappa_{\mathrm{T}}} \, \phi\left(\mathrm{K}_{\mathrm{Tj}}\right)+\frac{\varepsilon \, \kappa_{\mathrm{T}}^{*}}{\kappa_{\mathrm{T}} \, \sigma} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \\
&+\left[\varepsilon * \,\left(1+\frac{\alpha_{\mathrm{p}}^{*}}{\alpha_{\mathrm{p}}}\right)-\frac{\varepsilon \, \kappa_{\mathrm{T}}^{*}}{\kappa_{\mathrm{T}}} \,\left(1+\frac{\sigma^{*}}{\sigma}\right)\right] \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \quad(\mathrm{m})
\end{aligned} \nonumber \]
In the limit of infinite dilution,
\[\frac{\phi\left(E_{\mathrm{pj}}\right)^{\infty}-\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}}{\varepsilon_{1}^{*}(\ell)}=-\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\alpha_{\mathrm{p} 1}^{*}(\ell)}+\frac{\phi\left(\mathrm{K}_{\mathrm{T}}\right)^{\infty}}{\kappa_{\mathrm{Tl}}^{*}(\ell)}+\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\sigma_{1}^{*}(\ell)} \nonumber \]