1.12.12: Expansions- Isobaric- Salt Solutions- Apparent Molar
In general terms the dependence of apparent molar isobaric expansions for salt \(j\) on the composition of a given solution can be described using the following empirical equation.
\[\phi\left(E_{p j}\right)=a_{1}+a_{2} \,\left(m_{j} / m^{0}\right)^{1 / 2}+a_{3} \,\left(m_{j} / m^{0}\right) \nonumber \]
The presence of a term in \(\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\) is not unexpected in the case of salt solutions. Moreover for dilute solutions the term in \(\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\) is dominant. Hence
\[\left[\partial \phi\left(E_{p j}\right) / \partial m_{j}\right]=(1 / 2) \, a_{2} \,\left(m_{j} \, m^{0}\right)^{-1 / 2}+a_{3} \,\left(1 / m^{0}\right) \nonumber \]
Then,
\[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{a}_{1}+(3 / 2) \, \mathrm{a}_{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}+2 \, \mathrm{a}_{3} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
Therefore parameter \(\mathrm{a}_{1}\) is the limiting partial molar and apparent molar isobaric expansion of solute \(j\) in solution. An explanation of the term in \(\left(m_{j} / m^{0}\right)^{1 / 2}\) based on the Debye-Huckel Limiting Law (DHLL).
In general terms the chemical potential of salt \(j\) in aqueous solution at fixed \(\mathrm{T}\) and \(\mathrm{p}\) is related to molality \(\mathrm{m}_{j}\) using equation (d).
\[\mu_{j}(a q)=\mu_{j}^{0}(a q)+v \, R \, T \, \ln \left(Q \, m_{j} \, \gamma_{\pm} / m^{0}\right)+\int_{p^{0}}^{p} V_{j}^{\infty}(a q) \, d p \nonumber \]
\[\mathrm{V}_{\mathrm{j}}(\mathrm{aq})=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \,\left(\partial \ln \gamma_{\pm} / \partial \mathrm{p}\right)_{\mathrm{T}} \nonumber \]
Therefore
\[\left.\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \,\left\{\left[\mathrm{d} \ln \left(\gamma_{\pm}\right) / \mathrm{dp}\right)\right]_{\mathrm{T}}+\mathrm{T} \,\left[\mathrm{d}^{2} \ln \left(\gamma_{\pm}\right) / \mathrm{dp} \, \mathrm{dT}\right]\right\} \nonumber \]
According to the DHLL,
\[\ln \left(\gamma_{\pm}\right)=-S_{\gamma} \,\left(m_{j} / m^{0}\right)^{1 / 2} \nonumber \]
By definition
\[\mathrm{S}_{\mathrm{V}}=\left\lfloor\partial \mathrm{S}_{\gamma} / \partial \mathrm{p}\right\rfloor_{\mathrm{T}} \nonumber \]
Then [1],
\[\mathrm{T} \,\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}=-\mathrm{T} \, \mathrm{S}_{\mathrm{v}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2} \nonumber \]
Hence we write [2]
\[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})-\mathrm{v}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{S}_{\mathrm{Ep}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2} \nonumber \]
where [3],
\[\mathrm{S}_{\mathrm{Ep}}=\mathrm{S}_{\mathrm{V}}+\mathrm{T} \,\left[\partial \mathrm{S}_{\mathrm{V}} / \partial \mathrm{T}\right)_{\mathrm{p}} \nonumber \]
Therefore a linear dependence of \(\mathrm{E}_{\mathrm{p j}}(\mathrm{aq})\) on \(\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\) for dilute solutions is predicted by the DHLL. Hence for dilute solutions
\[\phi\left(E_{p j}\right)=\phi\left(E_{p j}\right)^{\infty}-(2 / 3) \, v \, R \, S_{E p} \,\left(m_{j} / m^{0}\right)^{1 / 2} \nonumber \]
For tetra-alkylammonium iodides(aq) \(\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} / \partial \mathrm{T}\right]_{\mathrm{p}}\) is positive, the magnitude increasing on going from \(\mathrm{Me}_{4}\mathrm{N}^{+}\) to \(\mathrm{Bu}_{4}\mathrm{N}^{+}\) [4,5,].
Apparent molar isobaric expansions for divalent metal chlorides(aq) lead to estimates of ionic molar isobaric expansions based on \(\mathrm{E}_{p}^{\infty}\left(\mathrm{Cl}^{-} ; \mathrm{aq}\right)\) set at \(+0.046 \mathrm{~cm}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\) [6,7]. The ionic estimates show a linear dependence on \(\left(\mathrm{r}_{\mathrm{j}}\right)^{-1}\), a pattern predicted by the Born equation. \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}\) for \(\mathrm{NaBPh}_{4}\) decreases gradually over the range 0 to 60 Celsius [8].
\(\phi\left(E_{p j}\right)^{\infty}\) for \(\mathrm{NaF}(\mathrm{aq})\), \(\mathrm{Na}_{2}\mathrm{SO}_{4}(\mathrm{aq})\) and \(\mathrm{KCl}(\mathrm{aq})\) is positive [9].
Footnotes
[1]
\[\begin{aligned}
&\mathrm{S}_{\gamma}=[1] \quad \mathrm{S}_{\mathrm{V}}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1} \quad \mathrm{~T} \, \mathrm{S}_{\mathrm{V}}=[\mathrm{K}] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1} \\
&\left\{\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}+\mathrm{T} \,\left[\partial^{2} \ln \left(\gamma_{\pm}\right) / \partial \mathrm{p} \, \partial \mathrm{T}\right]\right\} \\
&=\left\{\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}+[\mathrm{K}] \,[1] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}[\mathrm{~K}]^{-1}\right\}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}
\end{aligned} \nonumber \]
[2]
\[\begin{aligned}
&\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]-[1] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1} \\
&=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]-\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]
\end{aligned} \nonumber \]
[3] \(\mathrm{S}_{\mathrm{Ep}}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}+[\mathrm{K}] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1} \,\left[\mathrm{K}^{-1}\right]=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}\)
[4] R. Gopal and M. A. Siddiqi, J.Phys.Chem.,1968, 72 ,1814.
[5] F. Franks and H. T. Smith, Trans. Faraday Soc.,1967, 63 ,2586.
[6] F. J. Millero and W. Drost –Hansen, J. Phys.Chem.,1968, 72 ,1758.
[7] F. J. Millero, J. Phys. Chem., 1968, 72 , 4589.
[8] F. J. Millero, J. Chem. Eng. Data, 1970, 15 ,562.
[9] F. J. Millero and J. H. Knox, J. Chem. Eng. Data, 1973, 18 ,407.