1.12.10: Expansions- Solutions Apparent Molar Isobaric Expansions- Determination

The volume of an aqueous solution \(\mathrm{V}(\mathrm{aq})\) is related to the amounts of solvent and solute through the molar volume of water \(\mathrm{V}_{1}^{*}(\ell)\) and the apparent molar volume of solute \(\phi \left(\mathrm{V}_{j}\right) at the same temperature and pressure; equation (a).

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

The isobaric temperature dependence of the apparent molar volume of solute \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) yields the apparent molar (isobaric) expansion of solute \(j\), \(\phi\left(\mathrm{E}_{j}\right)\).

\[\phi\left(E_{p j}\right)=\left(\frac{\partial \phi\left(V_{j}\right)}{\partial T}\right)_{p}\]

Equation (a) (as in most treatments of volumetric properties) is the starting equation for the development of equations which relate apparent molar isobaric expansions of a solute \(j\) to the measured isobaric expansibilities of solvent and solution. The following four equivalent equations are frequently quoted [1-8]. A method is also available for direct determination of \(\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})\) from density data determined as functions of \(\mathrm{T}\) and \(\mathrm{m}_{j}\) [9].

Molality Scale [1-3]

\[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{pl}}^{*}(\ell)\right]+\alpha_{\mathrm{p}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]

\[\begin{gathered}
\phi\left(E_{p j}\right)=\left[m_{j} \, \rho(a q) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{p}(a q) \, \rho_{1}^{*}(\ell)-\alpha_{p 1}^{*}(\ell) \, \rho(a q)\right] \\
+\alpha_{p}(a q) \, M_{j} \,[\rho(a q)]^{-1}
\end{gathered}\]

Concentration Scale [4 - 7]

\[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\frac{1}{\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}\right] \,\left[\alpha_{\mathrm{p}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\rho(\mathrm{aq}) \, \alpha_{\mathrm{pl}}^{*}(\ell)\right]+\alpha_{\mathrm{p} 1}^{*}(\ell) \, \mathrm{M}_{\mathrm{j}} / \rho_{1}^{*}(\ell)\]

\[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{pl}}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{p} 1}^{*}(\ell)\]

The four equations (c) - (f) are thermodynamically correct, no assumptions being made in their derivation.

The partial molar isobaric expansion \(\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})\) is obtained using equation (g) [8].

\[E_{p j}(a q)=\phi\left(E_{p j}\right)+m_{j} \,\left(\frac{\partial \phi\left(E_{p j}\right)}{\partial m_{j}}\right)_{p}\]

Footnotes

[1] From equation (a) with respect to the dependence of \(\mathrm{V}(\mathrm{aq}\) on temperature at constant \(\mathrm{p}\) and at “\(\mathrm{A} = 0\)”.

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

Using equation (b), \((\partial \mathrm{V}(\mathrm{aq}) / \partial \mathrm{T})_{\mathrm{p}}=\mathrm{n}_{1} \,\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{p}}+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) Hence,

\[\left(\frac{1}{V(a q)}\right) \,\left(\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\mathrm{n}_{1} \,\left(\frac{\mathrm{V}_{1}^{*}(\ell)}{\mathrm{V}_{1}^{*}(\ell)}\right) \, \frac{1}{\mathrm{~V}(\mathrm{aq})} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left(\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{V}(\mathrm{aq})}\right) \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)\]

Thus, \(\alpha_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \,\left(\frac{\mathrm{V}_{1}^{*}(\ell)}{\mathrm{V}(\mathrm{aq})}\right) \, \alpha_{\mathrm{p} 1}^{*}(\ell)+\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{V}(\mathrm{aq})} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) or, \(\mathrm{V}(\mathrm{aq}) \, \alpha_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \alpha_{\mathrm{p} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)\)

We again use equation (a) for \(\mathrm{V}(\mathrm{aq}\),

\[\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \, \alpha_{\mathrm{p}}(\mathrm{aq})-\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \alpha_{\mathrm{pl}}^{*}(\ell)\]

But, \(\mathrm{V}_{1}^{*}(\ell)=\mathrm{M}_{1} / \rho_{1}^{*}(\ell)\) where \(\mathrm{M}_{1}\) is the molar mass of the solvent water.

\[\phi\left(E_{p j}\right)=\frac{n_{1} \, M_{1}}{n_{j} \, \rho_{1}^{*}(\ell)} \, \alpha_{p}(\mathrm{aq})-\frac{n_{1} \, M_{1}}{n_{j} \, \rho_{1}^{*}(\ell)} \, \alpha_{1}^{*}(\ell)+\alpha_{p}(\mathrm{aq}) \, \phi\left(V_{j}\right)\]

But \(\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{\mathrm{l}} \, \mathrm{M}_{\mathrm{l}}\).

\[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right]+\alpha_{\mathrm{p}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]

Hence we obtain equation (c).

[2] With reference to equation (c),

\[\begin{aligned}
&{\left[\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}\right] \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right]=\left[\frac{\mathrm{kg}}{\mathrm{mol}}\right] \,\left[\frac{\mathrm{m}^{3}}{\mathrm{~kg}}\right] \,\left[\mathrm{K}^{-1}\right]=\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]} \\
&\alpha_{\mathrm{p}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left[\mathrm{K}^{-1}\right] \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]
\end{aligned}\]

[3] From \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right]+\alpha_{\mathrm{p}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\) and, \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\frac{\mathrm{V}(\mathrm{aq})-\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)}{\mathrm{n}_{\mathrm{j}}}\)

\[\phi\left(E_{p j}\right)=\left[m_{j} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{p}(a q)-\alpha_{p l}^{*}(\ell)\right]+\alpha_{p}(a q) \,\left[\frac{\mathrm{V}(\mathrm{aq})-\mathrm{n}_{1} \, V_{1}^{*}(\ell)}{\mathrm{n}_{\mathrm{j}}}\right]\]

\[\phi\left(E_{p j}\right)=\left[m_{j} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{p}(\mathrm{aq})-\alpha_{p 1}^{*}(\ell)\right]+\alpha_{p}(\mathrm{aq}) \,\left[\frac{1}{\mathrm{c}_{j}}-\frac{\mathrm{n}_{1} \, \mathrm{M}_{1}}{\rho_{1}^{*}(\ell) \, \mathrm{n}_{\mathrm{j}}}\right]\]

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

\[\begin{aligned}
&\phi\left(E_{\mathrm{pj}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{pl}}^{*}(\ell)\right]+\left[\frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq})}+\frac{\alpha_{\mathrm{p}}(\mathrm{aq}) \, \mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}-\frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}}}\right] \\
&\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}-\frac{\alpha_{\mathrm{pl}}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}+\frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq})}+\frac{\alpha_{\mathrm{p}}(\mathrm{aq}) \, \mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}-\frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}}}
\end{aligned}\]

Hence we obtain equation (d).

\[\begin{gathered}
\phi\left(E_{p j}\right)=\left[m_{j} \, \rho(a q) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{p}(a q) \, \rho_{1}^{*}(\ell)-\alpha_{p 1}^{*}(\ell) \, \rho(a q)\right] \\
+\alpha_{p}(a q) \, M_{j} \,[\rho(a q)]^{-1}
\end{gathered}\]

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

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

Or, \(\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\mathrm{V}(\mathrm{aq}) \, \alpha_{\mathrm{p}}(\mathrm{aq})-\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \alpha_{\mathrm{p} 1}^{*}(\ell)\)

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

\[\begin{aligned}
&n_{j} \, \phi\left(E_{p j}\right)=\left[V(a q) \, \alpha_{p}(a q)\right]-\left[V(a q) \, \rho(a q)-n_{j} \, M_{j}\right] \, V_{1}^{*}(\ell) \, \alpha_{p l}^{*}(\ell) / M_{1} \\
&\phi\left(E_{p j}\right)=\left[\frac{V(a q) \, \alpha_{p}(a q)}{n_{j}}\right]-\left[\frac{V(a q) \, \rho(a q) \, V_{1}^{*}(\ell) \, \alpha_{1}^{*}(\ell)}{n_{j} \, M_{1}}\right]+\left[\frac{V_{1}^{*}(\ell) \, \alpha_{p 1}^{*}(\ell) \, M_{j}}{M_{1}}\right]
\end{aligned}\]

But concentration \(\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}(\mathrm{aq})\) and \(\rho_{1}^{*}=\mathrm{M}_{1} / \mathrm{V}_{1}^{*}(\ell)\).

\[\phi\left(\mathrm{E}_{\mathrm{j}}\right)=\left[\frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\mathrm{c}_{\mathrm{j}}}\right]-\left[\frac{\rho(\mathrm{aq}) \, \alpha_{\mathrm{pl}}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}\right]+\left[\frac{\alpha_{\mathrm{p} 1}^{*}(\ell) \, \mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\right]\]

Hence we obtain equation (e).

\[\phi\left(E_{p j}\right)=\left[\frac{1}{c_{j} \, \rho_{1}^{*}(\ell)}\right] \,\left[\alpha_{p}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\rho(a q) \, \alpha_{p 1}^{*}(\ell)\right]+\alpha_{p 1}^{*}(\ell) \, M_{j} / \rho_{1}^{*}(\ell)\]

[5] With reference to equation (e),

\[\begin{aligned}
&{\left[\frac{1}{\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}\right] \,\left[\alpha_{\mathrm{p}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\rho(\mathrm{aq}) \, \alpha_{\mathrm{pl} 1}^{*}(\ell)\right]=\left[\frac{\mathrm{m}^{3}}{\mathrm{~mol}}\right] \,\left[\frac{\mathrm{m}^{3}}{\mathrm{~kg}}\right] \,\left[\mathrm{K}^{-1}\right] \,\left[\mathrm{kg} \mathrm{m}^{-3}\right]} \\
&=\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]
\end{aligned}\]

[6] The volume of a solution, \(\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{1} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\)

Concentration \(\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}(\mathrm{aq})\) or, \(\mathrm{c}_{\mathrm{j}}=\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)}\)

But molality \(\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{\mathrm{l}} \, \mathrm{M}_{\mathrm{l}}\) \(\mathrm{c}_{\mathrm{j}}=\frac{\mathrm{m}_{\mathrm{j}} \, \mathrm{n}_{1} \, \mathrm{M}_{1}}{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)}\) or, \(\frac{1}{\mathrm{c}_{\mathrm{j}}}=\frac{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \mathrm{n}_{1} \, \mathrm{M}_{1}}+\frac{\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\mathrm{n}_{1} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}}\) or, \(\frac{1}{c_{j}}=\frac{1}{m_{j} \, \rho_{1}^{*}(\ell)}+\phi\left(V_{j}\right)\) or, \(\frac{1}{\mathrm{~m}_{\mathrm{j}}}=\frac{\rho_{1}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}}}-\rho_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\)

From equation (c).

\[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\frac{1}{\rho_{1}^{*}(\ell)}\right] \,\left[\frac{\rho_{1}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}}}-\rho_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right]+\alpha_{\mathrm{p}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]

Or,

\[\begin{aligned}
\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\frac{1}{\mathrm{c}_{\mathrm{j}}}\right] \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right.&\left.-\alpha_{\mathrm{pl}}^{*}(\ell)\right]-\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{p}}(\mathrm{aq})+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{pl}}^{*}(\ell) \\
&+\alpha_{\mathrm{p}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{aligned}\]

We obtain equation (f)

\[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{pl}}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{pl}}^{*}(\ell)\]

[7] With reference to equation (f)

\[\left[\frac{1}{\mathrm{c}_{\mathrm{j}}}\right] \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{pl}}^{*}(\ell)\right]=\left[\frac{\mathrm{m}^{3}}{\mathrm{~mol}}\right] \,\left[\mathrm{K}^{-1}\right]=\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]\]

[8] From \(\mathrm{E}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{p} j}\right)\) Then, \(\left(\frac{\partial E_{p}}{\partial n_{j}}\right)_{T, p, n_{j}}=n_{j} \,\left[\frac{\partial \phi\left(E_{p j}\right)}{\partial n_{j}}\right]+\phi\left(E_{p j}\right)\)

Or, \(\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{m}_{\mathrm{j}} \,\left[\frac{\partial \phi\left(\mathrm{E}_{\mathrm{pj}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right]+\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\)

[9] M. J. Blandamer and H. Hoiland, Phys.Chem.Chem.Phys.,1999,1,1873.
This method starts out with the measured dependence of the density \(\rho(\mathrm{aq})\) on temperature and molality at fixed pressure about density \(\rho_{1}^{*}(\ell, \theta)\), at temperature \(\theta\) at same pressure. For example the data might be fitted to an equation having the following form yielding the b-coefficients.

\[\begin{aligned}
&\rho\left(\mathrm{m}_{\mathrm{j}}, \mathrm{T}\right)=\rho_{1}^{*}(\ell, \theta)+\mathrm{b}_{2} \,(\mathrm{T}-\theta) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) / \theta+\mathrm{b}_{3} \,(\mathrm{T}-\theta) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} / \theta \\
&+\mathrm{b}_{4} \,(\mathrm{T}-\theta)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) / \theta^{2}+\mathrm{b}_{5} \,(\mathrm{T}-\theta)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} / \theta^{2} \\
&\left(\frac{\partial \rho\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T}\right)}{\partial \mathrm{T}}\right)_{\mathrm{m}_{\mathrm{j}}}=\mathrm{b}_{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) / \theta+\mathrm{b}_{3} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} / \theta
\end{aligned}\]

\[+2 \, b_{4} \,(\mathrm{T}-\theta) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) / \theta^{2}+2 \, \mathrm{b}_{5} \,(\mathrm{T}-\theta) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) / \theta^{2}\]

and,

\[\begin{aligned}
&\left(\frac{\partial \rho\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T}\right)}{\partial\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)}\right)_{\mathrm{T}}=\mathrm{b}_{2} \,(\mathrm{T}-\theta) / \theta+2 \, \mathrm{b}_{3} \,(\mathrm{T}-\theta) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) / \theta \\
&+\mathrm{b}_{4} \,(\mathrm{T}-\theta)^{2} / \theta^{2}+2 \, \mathrm{b}_{5} \,(\mathrm{T}-\theta)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) / \theta^{2}
\end{aligned}\]

The density \(\rho(\mathrm{aq})\) of an aqueous solution molality \(\mathrm{m}_{j}\) prepared using \(1 \mathrm{~kg}\) of water is given by the following equation.

\[\rho(\mathrm{aq})=\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right] / \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\]

\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right] / \rho(\mathrm{aq})\]

Also,

\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\]

\[\left(\frac{\partial \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}=\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}-\frac{1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}}{[\rho(\mathrm{aq})]^{2}} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}\]

But,

\[\mathrm{V}_{\mathrm{j}}(\mathrm{aq})=\left(\frac{\partial \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}\]

And,

\[E_{p j}(a q)=\left(\frac{\partial V_{j}(a q)}{\partial T}\right)_{p, m_{j}}\]

\[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\frac{\partial}{\partial \mathrm{T}} \,\left\{\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}-\left(\frac{1}{\rho(\mathrm{aq})}\right)^{2} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right) \,\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right]\right\}\]

\[\begin{aligned}
&\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=-\frac{\mathrm{M}_{\mathrm{j}}}{(\rho(\mathrm{aq}))^{2}} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{m}_{\mathrm{j}}}+\frac{2}{\left(\rho(\mathrm{aq})^{3}\right)} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{m}_{\mathrm{j}}} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right) \,\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right] \\
&-\frac{1}{(\rho(\mathrm{aq}))^{2}} \, \frac{\partial}{\partial \mathrm{T}} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right) \,\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right]
\end{aligned}\]

Using equation (k) in conjunction with equations (a) - (c), partial molar isobaric expansion \(\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})\) is calculated from the density and its dependence on both temperature and molality of solute. In another development \(\mathrm{E}_{\mathrm{pj}}\) is related to \(\alpha_{\mathrm{p}}\) and its dependence on molality of solute. By definition,

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

Or,

\[\alpha_{p}(a q)=-\frac{1}{\rho(a q)} \,\left(\frac{\partial \rho(a q)}{\partial T}\right)_{p, m(j)}\]

At temperature \(\mathrm{T}\) and molality \(\mathrm{m}_{j}\),

\[\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{m}(\mathrm{j})}=-\alpha_{\mathrm{p}}(\mathrm{aq}) \, \rho(\mathrm{aq})\]

Using equation (n)

\[\begin{aligned}
&E_{p j}(a q)=\frac{M_{j} \, \alpha_{p}}{\rho(a q)}-\frac{2}{(\rho(a q))^{2}} \, \alpha_{p} \,\left(\frac{\partial \rho(a q)}{\partial m_{j}}\right) \,\left[1+M_{j} \, m_{j}\right] \\
&-\frac{1}{(\rho(a q))^{2}} \, \frac{\partial}{\partial m_{j}} \,\left(\frac{\partial \rho}{\partial T}\right) \,\left[1+M_{j} \, m_{j}\right]
\end{aligned}\]

But from equation (n)

\[\frac{\partial}{\partial m_{j}} \,\left(\frac{\partial \rho}{\partial T}\right)=-\alpha_{p} \, \frac{\partial \rho(\mathrm{aq})}{\partial m_{j}}-\rho(\mathrm{aq}) \,\left(\frac{\partial \alpha_{p}}{\partial m_{j}}\right)\]

Therefore,

\[\begin{aligned}
&\mathrm{E}_{\mathrm{pj}}=\frac{\mathrm{M}_{\mathrm{j}} \, \alpha_{\mathrm{p}}}{\rho(\mathrm{aq})}-\frac{2}{(\rho(\mathrm{aq}))^{2}} \, \alpha_{\mathrm{p}} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right) \,\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right] \\
&+\frac{1}{(\rho(\mathrm{aq}))^{2}} \, \alpha_{\mathrm{p}} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right) \,\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right]+\frac{1}{\rho(\mathrm{aq})} \,\left(\frac{\partial \alpha_{\mathrm{p}}}{\partial \mathrm{m}_{\mathrm{j}}}\right) \,\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right]
\end{aligned}\]

or, (with reordering of terms)

\[\mathrm{E}_{\mathrm{p} j}=-\frac{\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right]}{(\rho(\mathrm{aq}))^{2}} \, \alpha_{\mathrm{p}} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right)+\frac{1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}}{\rho(\mathrm{aq})} \,\left(\frac{\partial \alpha_{\mathrm{p}}}{\partial \mathrm{m}_{\mathrm{j}}}\right)+\frac{\mathrm{M}_{\mathrm{j}} \, \alpha_{\mathrm{p}}}{\rho(\mathrm{aq})}\]

Using equation (g) for \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq})\)

\[\mathrm{E}_{\mathrm{pj}}=\mathrm{V}_{\mathrm{j}}(\mathrm{aq}) \, \alpha_{\mathrm{p}}+\frac{\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right]}{\rho(\mathrm{aq})} \,\left(\frac{\partial \alpha_{\mathrm{p}}}{\partial \mathrm{m}_{\mathrm{j}}}\right)\]

The partial molar isobaric expansion \(\mathrm{E}_{\mathrm{pj}}\) is calculated from isobaric expansibility and its dependence on molality of solute.