1.8.11: Enthalpies- Salt Solutions- Dilution

One mole of salt in solution can, with complete dissociation, produce \(ν\) moles of ions. Hence for a given solution prepared using \(\mathrm{n}_{1}\) moles of water(\(\ell\)) and \(\mathrm{n}_{j}\) moles of salt, the enthalpy \(\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)\) is given by equation (a).

\[\begin{aligned}
&\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)= \\
&\begin{aligned}
\mathrm{n}_{1} \, & {\left[\mathrm{H}_{1}^{*}(\ell)+\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\right] } \\
&+\mathrm{n}_{\mathrm{j}} \,\left[\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\partial \ln \gamma_{\pm} / \partial \mathrm{T}\right)_{\mathrm{p}}\right]
\end{aligned}
\end{aligned}\]

With a little re-arrangement,

\[\begin{aligned}
&\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)= \\
&\quad \mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell) \\
&\quad+\mathrm{n}_{\mathrm{j}} \,\left[\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left(\partial \ln \gamma_{\pm} / \partial \mathrm{T}\right)_{\mathrm{p}}+\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\right]
\end{aligned}\]

The terms within the brackets [….] define the apparent molar enthalpy of salt \(j\) in aqueous solution, \(\phi\left(\mathrm{H}_{j}\right)\).

\[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)\]

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

By definition

\[\mathrm{L}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)-\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg} ; \mathrm{id}\right)\]

\[\mathrm{L}_{1}(\mathrm{aq})=\mathrm{H}_{1}(\mathrm{aq})-\mathrm{H}_{1}^{*}(\ell)\]

\[\mathrm{L}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}(\mathrm{aq})-\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]

\[\mathrm{L}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{n}_{1} \, \mathrm{L}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{L}_{\mathrm{j}}(\mathrm{aq})=\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{L}_{\mathrm{j}}\right)\]

Thus,

\[\mathrm{L}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg} ; \mathrm{id}\right)=0\]

Equation (e) forms the basis of comments on changes in enthalpy when a salt solution is diluted by adding \(\Delta \mathrm{n}_{1}\) moles of water(\(\ell\)). Hence

\[\begin{aligned}
\Delta_{\text {dil }} \mathrm{H}=\left[\left(\mathrm{n}_{1}\right.\right.&\left.\left.+\Delta \mathrm{n}_{1}\right) \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}-\text { final }\right)\right] \\
&-\left[\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}-\text { initial }\right)\right]-\Delta \mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)
\end{aligned}\]

Or,

\[\Delta_{\text {dil }} \mathrm{H}=\mathrm{n}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}}-\text { final }\right)-\phi\left(\mathrm{H}_{\mathrm{j}}-\text { initial }\right)\right]\]

If in a given experiment where ‘\(\mathrm{n}_{j} = 1 \mathrm{~mol}\)’ and \(\Delta \mathrm{n}_{1}\) is large such that \(\phi\left(\mathrm{H}_{\mathrm{j}}-\text { final }\right)\) equals \(\phi\left(\mathrm{H}_{\mathrm{j}}-\text { final }\right)^{\infty}\), equation (k) is re-written as shown in equation (l). Then,

\[\Delta_{\text {dil }} \mathrm{H}\left(\mathrm{n}_{\mathrm{j}}=1 \mathrm{~mol}\right)=-\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]\]

Or,

\[\Delta_{\mathrm{dil}} \mathrm{H}\left(\mathrm{n}_{\mathrm{j}}=1 \mathrm{~mol}\right)=-\phi\left(\mathrm{L}_{\mathrm{j}}\right)\]

If for such a dilution, heat passes from the surroundings into the system , \(\Delta_{\text {dil }} \mathrm{H}\left(\mathrm{n}_{\mathrm{j}}=1 \mathrm{~mol}\right)\) is positive and \(\phi\left(L_{j}\right)\) is negative. Thus direct calorimetric measurement of \(\Delta_{\mathrm{dil}} \mathrm{H}\left(\mathrm{n}_{\mathrm{j}}=1 \mathrm{~mol}\right)\) yields the relative apparent molar enthalpy of the salt in solution at molality \(\mathrm{m}_{j}\).

However we need to comment in more detail on the analysis of heats of dilution for salt solutions. We envisage a situation where a calorimeter records the heat associated with dilution of a given salt solution from an initial molality \(\mathrm{m}_{\mathrm{i}}\) to a final molality \(\mathrm{m}_{\mathrm{f}}\). A data set often includes pairs of \(\mathrm{m}_{\mathrm{i}}-\mathrm{m}_{\mathrm{f}}\) values together with the accompanying enthalpy change, \(\Delta \mathrm{H}(\text { old } \rightarrow \text { new })\) which yields the difference in apparent molar enthalpies of the two salt solutions, cf. equation (k). Thus

\[\Delta \mathrm{H}(\text { old } \rightarrow \text { new })=\mathrm{n}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)\right]\]

Or,

\[\left[\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)\right]=\Delta \mathrm{H}(\text { old } \rightarrow \text { new }) / \mathrm{n}_{\mathrm{j}}\]

We note that the molalities of the ‘new’ and ‘old’ solutions differ and therefore the contributions of ion-ion interactions to the apparent molar enthalpies differ. In the event that sufficient solvent is added that \(\mathrm{m}_{\mathrm{f}}\) is effectively zero, then \(\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)\) is the infinitely dilute property \(\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\).

The excess enthalpy \(\mathrm{H}^{\mathrm{E}}\) is given by equation (p).

\[\mathrm{H}^{\mathrm{E}}=\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{L}_{\mathrm{j}}\right)=\mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]\]

For salt solutions \(\mathrm{H}^{\mathrm{E}}\) is not negligible as a consequence of intense ion-ion interaction. However in order to calculate \(\mathrm{H}^{\mathrm{E}}\) and hence obtain an indication of the strength of these interactions we return to equation (m) and note that experiment yields the difference between \(\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)\) and \(\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)\). Since there are no ion-ion interactions at infinite dilution, the difference \(\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]\left\{\text { i.e. } \phi\left(\mathrm{L}_{\mathrm{j}}\right)\right\}\) is obtained as a function of \(\mathrm{m}_{j}\)(old).

A key component of the difference \(\left[\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)\right]\) is charge-charge interaction in the real solutions which is calculated using, for example, the Debye-Huckel equations. These equations start out with a relation between \(\ln \left(\gamma_{\pm}\right)\) where \(\gamma_{\pm}\) is the mean ionic activity coefficient and I the ionic strength (or, in a simple solution, molality \(\mathrm{m}_{j}\)). These equations are differentiated with respect to temperature (at fixed pressure) requiring therefore the corresponding dependences of molar volume \(\mathrm{V}_{1}^{*}(\ell)\) and relative permittivity \(\varepsilon_{\mathrm{r}}^{*}(\ell)\) of the solvent. Not surprisingly a large chemical literature describes a range of procedures for analysing the calorimetric results. In most cases the starting point is the Debye-Huckel Limiting Law.

For \(\mathrm{Bu}_{4}\mathrm{N}^{+}\mathrm{Br}^{-}(\mathrm{aq})\), the dependence of \(\phi\left(\mathrm{L}_{\mathrm{j}}\right)\) on \(\mathrm{m}_{j}\) was expressed [1] using equation (q). \(\mathrm{S}_{\mathrm{H}}\) was taken from the compilation published by Helgeson and Kirkham [2].

\[\phi\left(L_{j}\right)=S_{H} \,\left(m_{j} / m^{0}\right)^{1 / 2}+\sum B_{i} \,\left(m_{j} / m^{0}\right)^{(i+1) / 2}\]

For \(\left(\mathrm{HOC}_{2}\mathrm{H}_{4}\right)_{4}\mathrm{N}^{+}\mathrm{Br}^{-}(\mathrm{aq})\), an extended Debye –Huckel equation was used having the following form [3].

\[\begin{gathered}
\phi\left(\mathrm{L}_{\mathrm{j}}\right)=\mathrm{S}_{\mathrm{H}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2} \,\left[\frac{1}{1+\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}}-\frac{\sigma \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}}{3}\right] \\
+\mathrm{B} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{C} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3 / 2}
\end{gathered}\]

The dependence of \(\phi\left(\mathrm{L}_{\mathrm{j}}\right)\) on \(\mathrm{m}_{j}\) for 1,1’-dimethyl-4,4’-dipyridinium dichloride(aq; \(298 \mathrm{~K}\)) was expressed [3] using a simple polynomial in \(\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\).

The Pitzer equations describing the properties of salt solutions also provide a basis for examining the enthalpies of dilution of, for example [4], \(\mathrm{NaCl}(\mathrm{aq})\). An interesting group of papers [5] compares relative apparent molar enthalpies of salts in \(\mathrm{D}_{2}\mathrm{O}\) and \(\mathrm{H}_{2}\mathrm{O}\); i.e. \(\phi\left(\mathrm{L}_{\mathrm{j}} ; \mathrm{D}_{2} \mathrm{O}\right)-\phi\left(\mathrm{L}_{\mathrm{j}} ; \mathrm{H}_{2} \mathrm{O}\right)\). The compositions of the salt solutions are expressed in aquamolalities; i.e. \(\mathrm{m}_{j}\) moles of salt in \(55.1\) moles of solvent. The difference is expressed as a quadratic in aqueous molality using Kerwin’s equation.

\[\phi\left(\mathrm{L}_{\mathrm{j}} ; \mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{D}_{2} \mathrm{O}\right)=\mathrm{k}_{1} \, \mathrm{m}_{\mathrm{j}}+\mathrm{k}_{2} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2}\]

Further examples are listed in reference [6].

Footnotes

[1] J. E. Mayrath and R. H. Wood, J. Chem. Thermodyn., 1983,15,625; and references therein.

[2] H. C. Helgeson and D. H Kirkham, Am. J. Sci.,1974,274,1199.

[3] G. Perron and J. E. Desnoyers, J. Solution Chem.,1972,1,537.

[4] R. H. Busey, H. F. Holmes and R. E. Mesmer, J.Chem.Thermodyn., 1984,16, 343.

[5]

1. A. S. Levine and R. H. Wood, J.Phys.Chem.,1973,77,2390.
2. Y.-C. Wu and H.L.Friedman, J.Phys.Chem.,1966,70,166.
3. J. E. Desnoyers, R. Francescon, P. Picker and C. Jolicoeur, Can. J. Chem., 1971, 49,3460

[6]

1. S. Lindenbaum, J.Chem.Thermodyn., 1971, 3,625; J. Phys. Chem., 1971,75,3733; \(\mathrm{Na}^{+}\) and \(\mathrm{Bu}_{4}\mathrm{N}^{+}\) salts of carboxylic acids(aq).
2. R. H. Wood and F. Belkin, J. Chem. Eng. Data, 1973, 18,184; \(\left(\mathrm{HOC}_{2}\mathrm{H}_{4})_{4}\mathrm{N}^{+}\mathrm{Br}(\mathrm{aq})\).
3. D. D. Ensor, H. L. Anderson and T. G. Conally, J. Phys. Chem.,1974,78,77.
4. D. D. Ensor and H. L. Anderson, J. Chem. Eng. Data, 1973, 18,205; \(\mathrm{NaCl}(\mathrm{aq})\).
5. G. E. Boyd, J. W. Chase and F. Vaslow, J. Phys. Chem., 1967, 71, 573; \(\mathrm{R}_{4} \mathrm{~N}^{+} \mathrm{X}(\mathrm{aq})\).
6. S. Lindenbaum, J. Phys.Chem.,1969,73,4734; \(\left[\mathrm{Bu}_{3} \mathrm{~N}-\left(\mathrm{CH}_{2}\right)_{8}-\mathrm{NBu}_{3}\right] \mathrm{X}_{2}\)