1.7.15: Compression- Isentropic and Isothermal- Solutions- Limiting Estimates
The density of an aqueous solutions at defined \(\mathrm{T}\) and \(\mathrm{p}\) and solute molality \(\mathrm{m}_{j}\) yields the apparent molar volume of solute \(j\), \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\). The dependence of \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) on \(\mathrm{m}_{j}\) can be extrapolated to yield the limiting (infinite dilution) property \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}\). The isothermal dependence of densities on pressure can be expressed in terms of an analogous infinite dilution apparent molar isothermal compression, \(\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}\). Similarly the isentropic compressibilities of solutions are characterised by \(\phi\left(\mathrm{K}_{\mathrm{S} j} ; \operatorname{def}\right)^{\infty}\) which is accessible via the density of a solution and the speed of sound in the solution. Nevertheless the isothermal property \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{J}}}\right)^{\infty}\) presents fewer conceptual problems in terms of understanding the properties of solutes and solvents which control volumetric properties. The challenge is to use \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}\) in order to obtain \(\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}\). The linking relationship is the Desnoyers-Philip equation [1]. The apparent molar isothermal compression for solute \(j \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) is related to the concentration \(\mathrm{c}_{j}\) of solute using equation (a) where \(\phi\left(V_{j}\right)\) is the apparent molar volume of the solute.
\[\phi\left(K_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{Tl}}^{*}(\ell)\right] \,\left(\mathrm{c}_{\mathrm{j}}\right)^{-1}+\kappa_{\mathrm{Tl}}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
The corresponding isentropic compression for solute \(j\), \(\phi\left(K_{\mathrm{Sj}} ; \text { def }\right)\) is related to the concentration \(\mathrm{c}_{j}\) using equation (b).
\[\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right) \equiv\left[\kappa_{\mathrm{S}}(\mathrm{aq})-\kappa_{\mathrm{S} 1}^{*}(\ell)\right] \,\left(\mathrm{c}_{\mathrm{j}}\right)^{-1}+\kappa_{\mathrm{S} 1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
[We replace the symbol ≡ by the symbol = in the following account.]
\[\text { By definition } \delta(\mathrm{aq})=\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{S}}(\mathrm{aq}) \nonumber \]
\[\text { And } \delta_{1}^{*}(\ell)=\kappa_{\mathrm{T} 1}^{*}(\ell)-\kappa_{\mathrm{S} 1}^{*}(\ell) \nonumber \]
Hence \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) and \(\phi\left(K_{\mathrm{Sj}} ; \operatorname{def}\right)\) are related by equation (e).
\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)=\left(\mathrm{c}_{\mathrm{j}}\right)^{-1} \,\left[\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)\right]+\delta_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
The difference \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{S}_{\mathrm{j}}} ; \mathrm{def}\right)\) depends on the concentration of the solute \(\mathrm{c}_{j}\). Further \(\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)\) is not zero. In fact,
\[\begin{aligned}
\delta(\mathrm{aq})-& \delta_{1}^{*}(\ell)=\\
&\left\{\mathrm{T} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2} / \sigma(\mathrm{aq})-\left\{\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma_{1}^{*}(\ell)\right\}\right.
\end{aligned} \nonumber \]
Using the technique of adding and subtracting the same quantity, equation (f) is re-expressed as follows.
\[\begin{aligned}
&\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)= \\
&\begin{aligned}
\left\{\delta(\mathrm{aq}) /\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2}\right\} \,\left[\, \alpha_{\mathrm{p}}(\mathrm{aq})+\alpha_{\mathrm{p} 1}^{*}(\ell)\right] \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right] \\
&-\left[\delta_{1}^{*}(\ell) / \sigma(\mathrm{aq})\right] \,\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right]
\end{aligned}
\end{aligned} \nonumber \]
The difference \(\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right]\) is related to \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) using equation (h).
\[\phi\left(E_{p j}\right)=\left[\alpha_{p}(a q)-\alpha_{p 1}^{*}(\ell)\right] \,\left(c_{j}\right)^{-1}+\alpha_{p 1}^{*}(\ell) \, \phi\left(V_{j}\right) \nonumber \]
Similarly, \(\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right]\) is related to \(\phi\left(C_{\mathrm{p} j}\right)\) using equation (i).
\[\phi\left(C_{p j}\right)=\left[\sigma(a q)-\sigma_{1}^{*}(\ell)\right] \,\left(c_{j}\right)^{-1}+\sigma_{1}^{*}(\ell) \, \phi\left(V_{j}\right) \nonumber \]
Using equations (g) - (i), we express equation (e) as follows.
\[\begin{aligned}
\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right)=& \\
{\left[\delta(\mathrm{aq}) / \alpha_{\mathrm{p}}(\mathrm{aq})\right] \,\left\{1+\left[\alpha_{\mathrm{p} 1}^{*}(\ell) / \alpha_{\mathrm{p}}(\mathrm{aq})\right]\right\} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right) } \\
&-\left[\delta_{1}^{*}(\ell) / \sigma(\mathrm{aq})\right] \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right)+\left\{\delta_{1}^{*}(\ell)\right.\\
&\left.-\left[\delta(\mathrm{aq}) \, \alpha_{\mathrm{p} 1}^{*}(\ell) / \alpha_{\mathrm{p}}(\mathrm{aq})\right]\right\} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{aligned} \nonumber \]
Equation (j) was obtained by Desnoyers and Philip [1] who showed that if \(\phi\left(K_{T_{j}}\right)^{\infty}\) and \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}\) are the limiting (infinite dilution) apparent molar properties, the difference is given by equation (k).
\[\begin{aligned}
\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}-\phi\left(\mathrm{K}_{\mathrm{Sj}} ;\right.&\operatorname{def})^{\infty}=\\
\delta_{1}^{*}(\ell) \,\left\{\left[2 \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty} / \alpha_{\mathrm{pl} 1}^{*}(\ell)\right]-\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty} / \sigma_{1}^{*}(\ell)\right]\right\}
\end{aligned} \nonumber \]
Using equation (b), \(\phi\left(K_{\mathrm{Sj}} ; \text { def }\right)\) is plotted as a function of cj across a set of different solutions having different entropies. \(\operatorname{Limit}\left(\mathrm{c}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)\) defines \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}\). Granted two limiting quantities, \(\phi\left(E_{p j}\right)^{\infty}\) and \(\phi\left(C_{p j}\right)^{\infty}\) are available for the solution at the same \(\mathrm{T}\) and \(\mathrm{p}\), equation (k) is used to calculate \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{J}}}\right)^{\infty}\) using \(\phi\left(K_{\mathrm{S}_{\mathrm{j}}} ; \text { def }\right)^{\infty}\).
An alternative form of equation (j) refers to a solution, molality mj [2].
\[\begin{aligned}
&\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)= \\
&\quad \delta_{1}^{*}(\ell) \,\left\{\left[2 \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right) / \alpha_{\mathrm{p} 1}^{*}(\ell)\right]-\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right) / \sigma_{1}^{*}(\ell)\right]\right. \\
&\quad+\left[\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\right]^{2} /\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2}\right\} \,\left\{1+\left[\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) / \sigma_{1}^{*}(\ell)\right]\right\}^{-1}
\end{aligned} \nonumber \]
The fact that \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}\) can be obtained from \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)^{\infty}\) indicates the importance of the Desnoyers-Philip equation. Bernal and Van Hook [3] used the Desnoyers-Philip equation to calculate \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}\) for glucose(aq), sucrose(aq) and fructose(aq) at \(348 \mathrm{~K}\). Similarly Hedwig et. al. used the Desnoyers –Philip equation to obtain estimates of \(\phi\left(K_{\mathrm{Tj}}\right)^{\infty}\) for glycyl dipeptides (aq) at \(298 \mathrm{~K}\) [4].
Footnotes
[1] J. E. Desnoyers and P. R. Philip, Can. J.Chem.,1972, 50 ,1095.
[2] J. C. R. Reis, J. Chem. Soc. Faraday Trans.,1998, 94 ,2385.
[3] P. J. Bernal and W. A. Hook, J.Chem.Thermodyn.,1986, 18 ,955.
[4] G. R. Hedwig, J. D. Hastie and H. Hoiland, J. Solution Chem.,1996, 25 , 615.