1.12.16: Expansions- Solutions- Apparent Molar Isentropic and Isobaric

In the context of the properties of aqueous solutions the concept of apparent molar properties is important with respect to the analysis of experimental results; e.g. apparent molar volume for solute \(j\) \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) calculated from the densities of a given solution and solvent at fixed \(\mathrm{T}\) and \(\mathrm{p}\). Similarly apparent molar isobaric expansions \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) characterise the dependence of \phi\left(\mathrm{V}_{j}\right) on temperature at fixed pressure. Nevertheless problems emerge when we turn attention to comparable isentropic properties. The way ahead involves definition of apparent molar isentropic expansions \(\phi\left(\mathrm{E}_{\mathrm{sj}} ; \text { def }\right)\) and apparent molar isentropic compressions \(\phi\left(\mathrm{K}_{\mathrm{sj}} ; \text { def }\right)\). These two properties are related [1]; equation(a).

\[\begin{aligned}
&\phi\left(\mathrm{E}_{\mathrm{sj}} ; \mathrm{def}\right)= \\
&-\frac{\alpha_{\mathrm{S} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)+\frac{\alpha_{\mathrm{s}}(\mathrm{aq})}{\kappa_{\mathrm{s}}(\mathrm{aq})} \, \phi\left(\mathrm{K}_{\mathrm{s}}\right)+\frac{\alpha_{\mathrm{s}}(\mathrm{aq}) \, \kappa_{\mathrm{S} 1}^{*}(\ell)}{\kappa_{\mathrm{s}}(\mathrm{aq}) \, \sigma(\mathrm{aq})} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \\
&\quad+\left[\alpha_{\mathrm{p} 1}^{*}(\ell) \,\left(1+\frac{\alpha_{\mathrm{pl} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})}\right)-\frac{\alpha_{\mathrm{s}}(\mathrm{aq}) \, \kappa_{\mathrm{S} 1}^{*}(\ell)}{\kappa_{\mathrm{s}}(\mathrm{aq})} \,\left(1+\frac{\sigma_{1}^{*}(\ell)}{\sigma(\mathrm{aq})}\right)\right] \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{aligned}\]

Equation (b) relates the corresponding properties at infinite dilution [1].

\[\frac{\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}}{\alpha_{\mathrm{S} 1}^{*}(\ell)}=-\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\alpha_{\mathrm{p} 1}^{*}(\ell)}+\frac{\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}}{\kappa_{\mathrm{S} 1}^{*}(\ell)}+\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\sigma_{1}^{*}(\ell)}\]

‘Semi’ apparent molar isentropic expansions and compressions are related using equation (c)

\[\frac{1}{\alpha_{\mathrm{S}}(\mathrm{aq})} \,\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right]_{\mathrm{S}(\mathrm{aq})}=-\frac{1}{\kappa_{\mathrm{S}}(\mathrm{aq})} \,\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})}\]

Equations (a) , (b) and (c) illustrate the power of thermodynamics in drawing together and relating the several properties of a solution.

Footnotes

[1] In the following we simplify the algebra by omitting the descriptors (aq) and (\(\ell\)). The starting point is the following equation.

\[\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}} \, \sigma} \,\left(\kappa_{\mathrm{s}} \, \sigma-\kappa_{\mathrm{s}}^{*} \, \sigma^{*}\right)-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \,\left(\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}\right)\]

The latter equation is effectively an identity. From equation (a),

\[\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\alpha_{\mathrm{s}}-\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}} \, \sigma} \, \kappa_{\mathrm{s}}^{*} \, \sigma^{*}-\alpha_{\mathrm{s}}^{*}+\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \, \alpha_{\mathrm{p}}^{*}\]

From equation (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,

\[\kappa_{\mathrm{S}} \, \sigma-\kappa_{\mathrm{S}}^{*} \, \sigma^{*}=\sigma \,\left(\kappa_{\mathrm{S}}-\kappa_{\mathrm{S}}^{*}\right)+\kappa_{\mathrm{S}}^{*} \,\left(\sigma-\sigma^{*}\right)\]

From equations (a) and (c). \(\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \,\left(\kappa_{\mathrm{s}}-\kappa_{\mathrm{s}}^{*}\right)+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}} \, \sigma} \,\left(\sigma-\sigma^{*}\right)-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}}\left(\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}\right)\)

But,

\[\begin{aligned}
&\phi\left(\mathrm{E}_{\mathrm{s} j} ; \text { def }\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s}}^{*} \\
&\phi\left(\mathrm{E}_{\mathrm{S}_{\mathrm{j}}} ; \operatorname{def}\right)=\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \, \frac{1}{\mathrm{c}_{\mathrm{j}}} \,\left(\kappa_{\mathrm{s}}-\kappa_{\mathrm{s}}^{*}\right)+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}} \, \sigma} \, \frac{1}{\mathrm{c}_{\mathrm{j}}} \,\left(\sigma-\sigma^{*}\right) \\
&-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \, \frac{1}{\mathrm{c}_{\mathrm{j}}} \,\left(\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}\right)+\alpha_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{aligned}\]

But for the isobaric heat capacities \(\phi\left(\mathrm{C}_{\mathrm{pj}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\sigma-\sigma^{*}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \sigma^{*}\)

Also,

\[\begin{array}{r}
\phi\left(\mathrm{K}_{\mathrm{s} j} ; \text { def }\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\kappa_{\mathrm{s}}-\kappa_{\mathrm{s}}^{*}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{s}}^{*} \\
\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \text { def }\right)=\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{S}}} \,\left[\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)-\kappa_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]
\end{array}\]

Hence,

\[\begin{aligned}
&+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}} \, \sigma} \,\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\sigma^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \\
&-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \,\left[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\alpha_{\mathrm{p}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]+\alpha_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{aligned}\]

With a little reorganisation,

\[\begin{aligned}
\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right)=-& \frac{\alpha_{\mathrm{S}}^{*}}{\alpha_{\mathrm{p}}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)+\frac{\alpha_{\mathrm{S}}}{\kappa_{\mathrm{S}}} \, \phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{S}} \, \sigma} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \\
&+\left[\alpha_{\mathrm{s}}^{*} \,\left(1+\frac{\alpha_{\mathrm{p}}^{*}}{\alpha_{\mathrm{p}}}\right)-\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{S}}}\left(1+\frac{\sigma^{*}}{\sigma}\right)\right] \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{aligned}\]

Hence in the limit of infinite dilution,

\[\frac{\phi\left(E_{\mathrm{S} j} ; \mathrm{def}\right)^{\infty}}{\alpha_{\mathrm{s}}^{*}}=-\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\alpha_{\mathrm{p}}^{*}}+\frac{\phi\left(\mathrm{K}_{\mathrm{s} j} ; \mathrm{def}\right)^{\infty}}{\kappa_{\mathrm{s}}^{*}}+\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\sigma^{*}}\]

[2]

\[\begin{aligned}
&{\frac{\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}}{\alpha_{\mathrm{s}}^{*}}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]}{\left[\mathrm{K}^{-1}\right]}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]} \\
&\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\alpha_{\mathrm{p}}^{*}}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]}{\left[\mathrm{K}^{-1}\right]}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \\
&\frac{\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}}{\kappa_{\mathrm{S}}^{*}}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} / \mathrm{N} \mathrm{m}^{-2}\right]}{\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \\
&\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\sigma^{*}}=\frac{\left[\mathrm{J} \mathrm{mol}^{-1}\right]}{\left[\mathrm{J} \mathrm{m}^{-3}\right]}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]
\end{aligned}\]