1.7.19: Compressions- Isothermal- Solutions- Apparent Molar- Determination

A given solution (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) is prepared using \(1 \mathrm{~kg}\) of solvent water and \(\mathrm{m}_{j}\) moles of solute \(j\). The compression of this solution \(\mathrm{K}_{\mathrm{T}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\) is given by equations (a) and (b).

\[\mathrm{K}_{\mathrm{T}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\]

\[\mathrm{K}_{\mathrm{T}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{K}_{\mathrm{T} 1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})\]

where,

\[\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}(\mathrm{aq})=-\left(\frac{\partial \mathrm{V}_{\mathrm{j}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]

and

\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=-\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]

Both \(\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})\) and \(\phi\left(\mathrm{K}_{\mathrm{Tj}^{\mathrm{j}}}\right)\) are Lewisian variables. With reference to partial molar volumes,

\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]

Hence

\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})\]

\(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}\) is the limiting (infinite dilution) apparent molar compression of solute--\(j\). For a given solution \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) is calculated using one of the following equations together with the isothermal compressions of solution and solvent [1-3].

Molality Scale

\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]+\kappa_{\mathrm{T}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]

\[\begin{gathered}
\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})\right] \\
+\kappa_{\mathrm{T}}(\mathrm{aq}) \, \mathrm{M}_{\mathrm{j}} \,[\rho(\mathrm{aq})]^{-1}
\end{gathered}\]

Concentration Scale

\[\begin{gathered}
\phi\left(\mathrm{K}_{\mathrm{TJ}}\right)=\left[\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})\right] \\
+\kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{M}_{\mathrm{j}} / \rho_{1}^{*}(\ell)
\end{gathered}\]

Also

\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{Tl}}^{*}(\ell)\right]+\kappa_{\mathrm{T} 1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]

The latter four equations are thermodynamically correct, no assumption being made in their derivation.

In 1933, Gucker reviewed the direct determination of compressibilities of solutions leading to apparent molar compressions of solutes in aqueous solution calculated using equation (k) [4-6].

\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\frac{\kappa_{\mathrm{T}}(\mathrm{aq})}{\rho(\mathrm{aq})} \,\left[\frac{1}{\mathrm{~m}_{\mathrm{j}}}+\mathrm{M}_{\mathrm{j}}\right]-\frac{\kappa_{1}^{*}(\ell)}{\rho_{1}^{*}(\ell)} \, \frac{1}{\mathrm{~m}_{\mathrm{j}}}\]

Compressibilties of solutions were directly determined by measuring the sensitivity of \(\mathrm{V}(\mathrm{aq})\) to an increase in pressure. Gucker showed that for aqueous salt solutions, \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) is negative and a linear function of \(\left(\mathrm{c}_{\mathrm{j}}\right)^{1 / 2}\). Moreover the limiting value, \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}\) is an additive property of \(\phi\left(\mathrm{K}_{\mathrm{T}}-1 \mathrm{ion}\right)^{\infty}\)

A useful approximation is that for dilute solutions at constant \(\mathrm{T}\) and \(\mathrm{p}\) containing a neutral solute \(j\), \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) is linear function of molaity \(\mathrm{m}_{j}\).

\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{e} \, \mathrm{m}_{\mathrm{j}}+\mathrm{f}\]

Hence,

\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{K}_{\mathrm{T}_{\mathrm{j}}}(\mathrm{aq})=\mathrm{K}_{\mathrm{Tj}}^{\infty}(\mathrm{aq})=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}\]

We identify ‘f’ in equation (l) as the limiting isothermal apparent molar isothermal compression of solute \(j\) in solution (at equilibrium).

Footnotes

[1] \(\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right] ; \mathrm{K}_{\mathrm{Tl}}^{*}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right]\); \(\mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\left[\mathrm{m}^{3} \mathrm{~Pa}^{-1}\right]\)

For the solution, \(\kappa_{\mathrm{T}}=\mathrm{K}_{\mathrm{T}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})=\left[\mathrm{Pa}^{-1}\right]\)

For the solvent, \(\kappa_{\mathrm{T} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)=\left[\mathrm{Pa}^{-1}\right]\)

[2] Isothermal compressions have units ‘volume per unit of pressure’ whereas compressibilities have units of ‘reciprocal pressure’. \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) is an apparent molar isothermal compression on the grounds that the units of this property are \(\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right]\). Some reports use the term ‘apparent molar isothermal compressibility’.

[3] For an aqueous solution at fixed temperature and pressure prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of solute \(j\),

\[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]

Hence with respect to an equilibrium displacement (i.e. at \(\mathrm{A} = 0\)) at defined temperature,

\[(\partial \mathrm{V}(\mathrm{aq}) / \partial \mathrm{p})_{\mathrm{T}}=\mathrm{n}_{1} \,\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{p}\right)_{\mathrm{T}}+\mathrm{n}_{\mathrm{j}} \,\left(\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right)_{\mathrm{T}}\]

Hence

\[(\partial \mathrm{V}(\mathrm{aq}) / \partial \rho)_{\mathrm{T}}=\mathrm{n}_{1} \,\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{p}\right)_{\mathrm{T}}-\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\]

For the solution the (equilibrium) isothermal compressibility,

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

Similarly for the pure solvent,

\[\kappa_{\mathrm{T} 1}^{*}(\ell)=-\frac{1}{\mathrm{~V}_{1}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]

Hence from equation (c),

\[\mathrm{V}(\mathrm{aq}) \, \mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{Tj}}\right)\]

We use equation (as) for \(\mathrm{V}(\mathrm{aq})\) in conjunction with equation (f).

\[\begin{aligned}
\kappa_{\mathrm{T}}(\mathrm{aq}) \,\left[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]=\\
\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)
\end{aligned}\]

\[\begin{aligned}
&\phi\left(\mathrm{K}_{\mathrm{T} \mathrm{j}}\right)= \\
&{\left[\kappa_{\mathrm{T}}(\mathrm{aq}) \, \mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)\right] / \mathrm{n}_{\mathrm{j}}-\left[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell) / \mathrm{n}_{\mathrm{j}}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T}}(\mathrm{aq})}
\end{aligned}\]

But, \(\mathrm{V}_{1}^{*}(\ell)=\mathrm{M}_{1} / \rho_{1}^{*}(\ell)\) Then,

\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\kappa_{\mathrm{T}}(\mathrm{aq}) \, \frac{\mathrm{n}_{1} \, \mathrm{M}_{1}}{\rho_{1}^{*}(\ell) \, \mathrm{n}_{\mathrm{j}}}-\frac{\mathrm{n}_{1} \, \mathrm{M}_{1}}{\rho_{1}^{*}(\ell) \, \mathrm{n}_{\mathrm{j}}} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \mathrm{K}_{\mathrm{T}}(\mathrm{aq})\]

Molality \(\begin{aligned}
&\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{\mathrm{l}} \, \mathrm{M}_{\mathrm{l}} \\
&\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\frac{\kappa_{\mathrm{T}}(\mathrm{aq})}{\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}-\frac{\kappa_{\mathrm{T} 1}^{*}(\ell)}{\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}}}+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T}}(\mathrm{aq})
\end{aligned}\)

Or,

\[\phi\left(\mathrm{K}_{\mathrm{TJ}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T}}(\mathrm{aq})\]

Using again equation (a) to substitute for \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\),

\[\begin{aligned}
\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \, & {\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right] } \\
&+\kappa_{\mathrm{T}}(\mathrm{aq}) \,\left[\frac{\mathrm{V}(\mathrm{aq})-\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)}{\mathrm{n}_{\mathrm{j}}}\right]
\end{aligned}\]

With \(\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}(\mathrm{aq})\),

\[\begin{aligned}
\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right] \\
&+\kappa_{\mathrm{T}}(\mathrm{aq}) \,\left[\frac{1}{\mathrm{c}_{\mathrm{j}}}-\frac{\mathrm{n}_{1} \, \mathrm{M}_{1}}{\rho_{1}^{*}(\ell) \, \mathrm{n}_{\mathrm{j}}}\right]
\end{aligned}\]

But \(\frac{1}{c_{j}}=\frac{M_{j}}{\rho(a q)}+\frac{1}{m_{j} \, \rho(a q)}\) Hence,

\[\begin{aligned}
\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{Tl}}^{*}(\ell)\right] \\
&+\kappa_{\mathrm{T}}(\mathrm{aq}) \,\left[\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}+\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho(\mathrm{aq})}-\frac{\mathrm{n}_{1} \, \mathrm{M}_{\mathrm{1}}}{\rho_{1}^{*}(\ell) \, \mathrm{n}_{\mathrm{j}}}\right]
\end{aligned}\]

\[\begin{gathered}
\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})\right] \\
+\frac{\kappa_{\mathrm{T}}(\mathrm{aq}) \, \mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}
\end{gathered}\]

We start with equation (f).

\[\mathrm{V}(\mathrm{aq}) \, \mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \kappa_{\mathrm{T} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\]

Mass of solution,

\[V(a q) \, \rho(a q)=n_{1} \, M_{1}+n_{j} \, M_{j}\]

Or,

\[\mathrm{n}_{1}=\left[\mathrm{V}(\mathrm{aq}) \, \rho(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}\right] / \mathrm{M}_{1}\]

We combine equations (f) and (q).

\[\begin{aligned}
&\mathrm{V}(\mathrm{aq}) \, \kappa_{\mathrm{T}}(\mathrm{aq})= \\
&\quad \mathrm{V}_{1}^{*}(\ell) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \,\left[\frac{\mathrm{V}(\mathrm{aq}) \, \rho(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}{\mathrm{M}_{1}}\right]+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)
\end{aligned}\]

With \(c_{j}=n_{j} / V(a q)\)

\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\frac{\kappa_{\mathrm{T}}(\mathrm{aq})}{\mathrm{c}_{\mathrm{j}}}-\frac{\mathrm{V}_{1}^{*}(\ell) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})}{\mathrm{c}_{\mathrm{j}} \, \mathrm{M}_{1}}+\frac{\mathrm{V}_{1}^{*}(\ell) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{M}_{\mathrm{j}}}{\mathrm{M}_{1}}\]

Hence,

\[\begin{array}{r}
\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\left[\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})\right] \\
+\kappa_{\mathrm{Tl}}^{*}(\ell) \, \mathrm{M}_{\mathrm{j}} / \rho_{1}^{*}(\ell)
\end{array}\]

Again from equation (j)

\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\mathrm{K}_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T}}(\mathrm{aq})\]

Hence,

\[\begin{aligned}
&\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)= \\
&{\left[\frac{\rho_{1}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}}}-\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \rho_{1}^{*}(\ell)\right] \,\left[\rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]} \\
&\quad+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T}}(\mathrm{aq})
\end{aligned}\]

Or

\[\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\mathrm{K}_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T} 1}^{*}(\ell)\]

[4] F. T. Gucker, J.Am.Chem.Soc.,1933,55,2709.

[5] F. T. Gucker, Chem. Rev.,1933,13,111.

[6] From equation (h),

\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\frac{\kappa_{\mathrm{T}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)}-\frac{\kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)}+\frac{\kappa_{\mathrm{T}}(\mathrm{aq}) \, \mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}\]

\[\phi\left(\mathrm{K}_{\mathrm{T}}\right)=\frac{\kappa_{\mathrm{T}}(\mathrm{aq})}{\rho(\mathrm{aq})} \,\left[\frac{1}{\mathrm{~m}_{\mathrm{j}}}+\mathrm{M}_{\mathrm{j}}\right]-\frac{\kappa_{\mathrm{Tl}}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{\star}(\ell)}\]