1.7.16: Compressions- Isentropic and Isothermal- Apparent Molar Volume

A given solution is perturbed by a change in pressure to a neighbouring state at constant affinity, \(\mathrm{A}\).

\[(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{s}}=(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}}-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}} \,(\partial \mathrm{T} / \partial \mathrm{S})_{\mathrm{p}} \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}\]

But for the pure solvent (at constant affinity \(\mathrm{A}\)) S * 1 * S1 K (l) = −(∂V (l)/ ∂p) and T * 1 * T1 K (l) = −(∂V (l)/ ∂p) (b)

We confine attention to perturbation at ‘\(\mathrm{A} = 0\)’; i.e. an equilibrium process. [Note the change in sign.]

\[\mathrm{K}_{\mathrm{s} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\left(\partial \mathrm{S}_{1}^{*}(\ell) / \partial \mathrm{p}\right)_{\mathrm{T}} \,\left(\partial \mathrm{T} / \partial \mathrm{S}_{1}^{*}(\ell)\right)_{\mathrm{p}} \,\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{p}}\]

From a Maxwell Equation,

\[\left(\partial \mathrm{S}_{1}^{*}(\ell) / \partial \mathrm{p}\right)_{\mathrm{T}}=-\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{p}}\]

From the Gibbs-Helmholtz Equation,

\[\left(\partial \mathrm{S}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{p}}=\mathrm{C}_{\mathrm{p} 1}^{*}(\ell) / \mathrm{T}\]

\(C_{p 1}^{*}(\ell)\) is the molar ( equilibrium) isobaric heat capacity of the solvent at defined \(\mathrm{T}\) and \(\mathrm{p}\). From equation (c), [Note change of sign.]

\[\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)-\left[\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{p}}\right]^{2} \, \mathrm{T} / \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)\]

But

\[\alpha_{1}^{*}(\ell) \, V_{1}^{*}(\ell)=\left(\partial V_{1}^{*}(\ell) / \partial T\right)_{p}\]

\[\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)-\left[\alpha_{1}^{*}(\ell) \, \mathrm{V}_{1}^{*}(\ell)\right]^{2} \, \mathrm{T} / \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)\]

But the ratio of isobaric heat capacity of the solvent to its molar volume,

\[\mathrm{C}_{\mathrm{p} 1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)=\sigma_{1}^{*}(\ell) .\]

\[\left.\mathrm{K}_{\mathrm{s} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)-\left[\alpha_{1}^{*}(\ell)\right]^{2} \, \mathrm{V}_{1}^{*}(\ell)\right] \, \mathrm{T} / \sigma_{1}^{*}(\ell)\]

But

\[\kappa_{\mathrm{s} 1}^{*}(\ell)=\left[\mathrm{V}_{1}^{*}(\ell)\right]^{-1} \, \mathrm{K}_{\mathrm{S} 1}^{*}(\ell) \text { and } \kappa_{\mathrm{T} 1}^{*}(\mathrm{l})=\left[\mathrm{V}_{1}^{*}(\ell)\right]^{-1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)\]

\[\kappa_{\mathrm{S} 1}^{*}(\ell)=\kappa_{\mathrm{T} 1}^{*}(\ell)-\left[\alpha_{1}^{*}(\ell)\right]^{2} \, \mathrm{T} / \sigma_{1}^{*}(\ell)\]

Similarly for an aqueous solution, molality \(\mathrm{m}_{j}\),

\[\kappa_{\mathrm{S}}(\mathrm{aq})=\kappa_{\mathrm{T}}(\mathrm{aq})-[\alpha(\mathrm{aq})]^{2} \, \mathrm{T} / \sigma(\mathrm{aq})\]

Also

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

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

We convert from compressions to compressibilities.

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

But we know \(\kappa_{\mathrm{T}}(\mathrm{aq})\) in terms of ) \(\kappa_{\mathrm{S}}(\mathrm{aq})\) (aq [see equation (m)] and \(\kappa_{\mathrm{T} 1}^{*}(\ell)\) in terms of \(\kappa_{\mathrm{s} 1}^{*}(\ell)\). Then [NB change of sign]

\[\begin{aligned}
\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\frac{\mathrm{V}(\mathrm{aq})}{\mathrm{n}_{\mathrm{j}}}\right] \,\left[\mathrm{K}_{\mathrm{S}}(\mathrm{aq})+\frac{\{\alpha(\mathrm{aq})\}^{2} \, \mathrm{T}}{\sigma(\mathrm{aq})}\right] \\
&-\left[\frac{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)}{\mathrm{n}_{\mathrm{j}}}\right] \,\left[\kappa_{\mathrm{Sl}}^{*}(\ell)+\frac{\left\{\alpha_{1}^{*}(\ell)\right\}^{2} \, \mathrm{T}}{\sigma_{1}^{*}(\ell)}\right]
\end{aligned}\]

We introduce densities into equation (q). For a solution having mass \(\mathrm{w}\),

\[\begin{aligned}
\mathrm{V}(\mathrm{aq}) / \mathrm{n}_{\mathrm{j}} &=\left[1 / \mathrm{n}_{\mathrm{j}}\right] \,[\mathrm{w} / \rho(\mathrm{aq})]=\left[1 / \mathrm{n}_{\mathrm{j}} \, \rho(\mathrm{aq})\right] \,\left[\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}+\mathrm{n}_{1} \, \mathrm{M}_{1}\right] \\
&=\left[1 / \mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq})\right]+\left[\mathrm{M}_{\mathrm{j}} / \rho(\mathrm{aq})\right]
\end{aligned}\]

Also,

\[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) / \mathrm{n}_{\mathrm{j}}=\left[\mathrm{n}_{1} / \mathrm{n}_{\mathrm{j}}\right] \,\left[\mathrm{M}_{1} / \rho_{1}^{*}(\ell)\right]=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}\]

From equations (q), (r) and (s),

\[\begin{aligned}
\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\left[\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho(\mathrm{aq})}+\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}\right] \,\left[\kappa_{\mathrm{s}}(\mathrm{aq})+\frac{\{\alpha(\mathrm{aq})\}^{2} \, \mathrm{T}}{\sigma(\mathrm{aq})}\right] \\
&-\left[\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}\right] \,\left[\kappa_{\mathrm{s} 1}^{*}(\ell)+\frac{\left\{\alpha_{1}^{*}(\ell)\right\}^{2} \, \mathrm{T}}{\sigma_{1}^{*}(\ell)}\right]
\end{aligned}\]

We factor out the six terms . The order in which we write these terms anticipates the next but one step.

\[\begin{aligned}
\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\frac{\kappa_{\mathrm{s}}(\mathrm{aq})}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq})}\right]-\left[\frac{\kappa_{\mathrm{Sl}}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}\right]+\left[\frac{\mathrm{M}_{\mathrm{j}} \, \kappa_{\mathrm{s}}(\mathrm{aq})}{\rho(\mathrm{aq})}\right] \\
&+\left[\frac{\{\alpha(\mathrm{aq})\}^{2} \, \mathrm{T}}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \sigma(\mathrm{aq})}\right] \\
&-\left[\frac{\left\{\alpha_{1}^{*}(\ell)\right\}^{2} \, \mathrm{T}}{\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell)}\right]+\left[\frac{\mathrm{M}_{\mathrm{j}} \,\{\alpha(\mathrm{aq})\}^{2} \, \mathrm{T}}{\rho(\mathrm{aq}) \, \sigma(\mathrm{aq})}\right]
\end{aligned}\]

Hence,

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

where,

\[\mathrm{A}=\left[\frac{\mathrm{T}}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)}\right] \,\left[\left(\frac{\{\alpha(\mathrm{aq})\}^{2} \, \rho_{1}^{*}(\ell)}{\sigma(\mathrm{aq})}\right)-\left(\frac{\left\{\alpha_{1}^{*}(\ell) \, \rho(\mathrm{aq})\right.}{\sigma_{1}^{*}(\ell)}\right)\right]\]

and

\[\mathrm{B}=\mathrm{M}_{\mathrm{j}} \,\{\alpha(\mathrm{aq})\}^{2} \, \mathrm{T} / \rho(\mathrm{aq}) \, \sigma(\mathrm{aq})\]

With reference to solutions we compare the isentropic and isothermal dependences of \(\phi\left(V_{j}\right)\) on pressure.

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

Noting signs [cf. equations (d) and (e)] and the definition of \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\),

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

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

We turn our attention to \(\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\). We recall that

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

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

Hence

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

We combine equations (za) and (zd). Hence

\[\begin{aligned}
&-\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})}= \\
&\phi\left(\mathrm{K}_{\left.\mathrm{T}_{\mathrm{j}}\right)}\right. \\
&-\left[\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}}\right]_{\mathrm{T}} \,\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \,\left[\frac{1}{\mathrm{n}_{\mathrm{j}}} \,\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}}-\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}} \,\left[\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right]_{\mathrm{p}}\right]
\end{aligned}\]

Or,

\[\begin{aligned}
-\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}]_{\mathrm{S}(\mathrm{aq})}}\right]=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right) \\
&-\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \, \frac{1}{\mathrm{n}_{\mathrm{j}}} \,\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}}^{2} \\
&+\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \,\left[\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}} \,\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}}\left[\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right]_{\mathrm{p}}\right]
\end{aligned}\]

But

\[\alpha(\mathrm{aq}) \, \mathrm{V}(\mathrm{aq})=[\partial \mathrm{V}(\mathrm{aq}) / \partial \mathrm{T}]_{\mathrm{p}}\]

And

\[\alpha_{1}^{*}(\ell) \, V_{1}^{*}(\ell)=\left[\partial V_{1}^{*}(\ell) / \partial \mathrm{T}\right]_{\mathrm{p}}\]

We introduce the latter two equations into equation (zf).

\[\begin{aligned}
{\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})} } &=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right) \\
&-\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \, \frac{1}{\mathrm{n}_{\mathrm{j}}} \,[\mathrm{V}(\mathrm{aq}) \, \alpha(\mathrm{aq})]^{2} \\
&+\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \,\left[\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}}\right] \,\left[\alpha(\mathrm{aq}) \, \mathrm{V}(\mathrm{aq}) \, \alpha_{1}^{*}(\ell) \, \mathrm{V}_{1}^{*}(\ell)\right]
\end{aligned}\]

But \(\sigma(\mathrm{aq})=\mathrm{C}_{\mathrm{p}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\) and \(\sigma_{1}^{*}(\ell)=\mathrm{C}_{\mathrm{p} 1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)\). Also \(\mathrm{M}_{1} / \mathrm{V}_{1}^{*}(\ell)=\rho_{1}^{*}(\ell)\). Hence,

\[\begin{aligned}
-\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})} &=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right) \\
-& {\left[\frac{\mathrm{T}}{\sigma(\mathrm{aq})}\right] \, \frac{1}{\mathrm{n}_{\mathrm{j}}} \, \mathrm{V}(\mathrm{aq}) \,[\alpha(\mathrm{aq})]^{2} } \\
&+\left[\frac{\mathrm{T}}{\sigma(\mathrm{aq})}\right] \,\left[\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}}\right] \,\left[\alpha(\mathrm{aq}) \, \alpha_{1}^{*}(\ell) \, \mathrm{M}_{1} / \rho_{1}^{*}(\ell)\right]
\end{aligned}\]

Also, \(\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}=\mathrm{m}_{\mathrm{j}}\). And [2] \(\mathrm{V}(\mathrm{aq}) / \mathrm{n}_{\mathrm{j}}=\left[1 / \mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq})\right]+\left[\mathrm{M}_{\mathrm{j}} / \rho(\mathrm{aq})\right]\)

Hence we obtain a relation between the two compressions of the apparent molar volumes

\[\begin{aligned}
-\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})} &=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right) \\
-\left[\frac{\mathrm{T}}{\sigma(\mathrm{aq})}\right] \,\left[\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho(\mathrm{aq})}+\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}\right] \,[\alpha(\mathrm{aq})]^{2} \\
&+\left[\frac{\mathrm{T}}{\sigma(\mathrm{aq})}\right] \,\left[\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}\right] \, \alpha(\mathrm{aq}) \, \alpha_{1}^{*}(\ell)
\end{aligned}\]

Footnotes

[1] Unit check on equation (l). \(\left.\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}-\left\{\left[\mathrm{K}^{-1}\right]^{2} \,[\mathrm{K}]\right\} /\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{-1}\right\}\)
\(\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}-\left[\mathrm{J} \mathrm{m}^{-3}\right]^{-1}\) But \(\left[\mathrm{N} \mathrm{m}^{-2}\right]=\left[\mathrm{J} \mathrm{m}^{-3}\right]\)

[2] From
\(\begin{aligned}
&V(a q)=\frac{w}{\rho(a q)}=\frac{n_{1} \, M_{1}}{\rho(a q)}+\frac{n_{j} \, M_{j}}{\rho(a q)} \\
&\frac{V(a q)}{n_{j}}=\frac{n_{1} \, M_{1}}{n_{j} \, \rho(a q)}+\frac{M_{j}}{\rho(a q)} \\
&\frac{V(a q)}{n_{j}}=\frac{1}{m_{j} \, \rho(a q)}+\frac{M_{j}}{\rho(a q)}
\end{aligned}\)