1.17.7: Isopiestic: Aqueous Salt Solutions

An extensive literature reports applications of the isopiestic technique to the determination of osmotic coefficients and ionic activity coefficients for salt solutions [1-11]. In effect the technique probes the role of ion-ion interactions in determining the properties of real salt solutions.

Several approaches have been reported for analyzing isopiestic results. A common method starts with the isopiestic ratio \(\mathrm{R}_{\text{iso}}\). For solutions in dishes \(\mathrm{A}\) and \(\mathrm{B}\) at equilibrium, the isopiestic equilibrium conditions is given by equation (a).

\[\left(\phi_{j} \, V_{j} \, m_{j}\right)_{A}=\left(\phi_{i} \, V_{i} \, m_{i}\right)_{B}\]

The isopiestic ratio,

\[\mathrm{R}_{\text {iso }}=\left(\mathrm{v}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right)_{\mathrm{B}} /\left(\mathrm{v}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right)_{\mathrm{A}}\]

An important task formulates an equation relating the osmotic coefficient for a given salt solution and the mean ionic coefficient \(\gamma_{\pm}\)

If the salt solution contains a single salt, then according to the Gibbs-Duhem Equation,

\[\left(1 / M_{1}\right) \, d \mu_{1}(a q)=-m_{j} \, d \mu_{j}(a q)\]

Hence (where pressure \(\mathrm{p}\) is close to the standard pressure)

\[\begin{aligned}
\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d}\left[\mu_{1}^{*}(\mathrm{l})\right.&\left.-\left(\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{v} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right)\right]=\\
&-\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\left(\mathrm{v} \, \mathrm{Q} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\right]\right.
\end{aligned}\]

Then,

\[\left.\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right)\right]=-\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\left(\ln \left(\mathrm{m}_{\mathrm{j}}\right)+\ln \left(\gamma_{\pm}\right)\right]\right.\]

Or,

\[\left.-\phi \, \mathrm{dm}_{\mathrm{j}}-\mathrm{m}_{\mathrm{j}} \, \mathrm{d}[\phi)\right]=-\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{d}\left(\mathrm{m}_{\mathrm{j}}\right) / \mathrm{m}_{\mathrm{j}}+\mathrm{d} \ln \left(\gamma_{\pm}\right)\right]\]

Equation (f) is integrated between the limits ‘\(\mathrm{m}_{\mathrm{j}} = 0\)’ and \(\mathrm{m}_{\mathrm{j}}\) [3,4]. Then,

\[\ln \left(\gamma_{\pm}\right)=(\phi-1)+\int_{0}^{m(j)}(\phi-1) \, d \ln \left(m_{j}\right)\]

And,

\[\phi=1+\frac{1}{m_{j}} \, \int_{0}^{m(j)} m_{j} \, d \ln \left(\gamma_{\pm}\right)\]

Hence the dependences of both \(\gamma_{\pm}\) and \(\phi\) are obtained [1] for salt solutions and of both \(\gamma_{\mathrm{j}} and \(\phi\) for solutions containing neutral solutes [5].

An important challenge at this stage is to express the experimentally determined dependence of \(\phi\) on \(\mathrm{m}_{\mathrm{j}}\). Having expressed this dependence quantitively, the dependence of \(\gamma_{\pm}\) on \(\mathrm{m}_{\mathrm{j}}\) is obtained using equation (g). The integration can be done graphically [6] or numerically using a computer- based analysis. The Debye-Huckle Limiting Law plus extended form can be used to express the dependence of \(\phi\) on \(\mathrm{m}_{\mathrm{j}}\).

\[\phi=1-\left(\mathrm{S}_{\gamma} / 3\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\sum_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{j}} \mathrm{A}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{\mathrm{r}(\mathrm{i})}\]

The parameter \(\mathrm{r}(\mathrm{i})\) increases in quarter powers. Then [7,8],

\[\ln \left(\gamma_{\pm}\right)=-\mathrm{S}_{\mathrm{\gamma}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}+\sum_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{j}} \mathrm{A}_{\mathrm{i}} \,\left(\frac{\mathrm{r}_{\mathrm{i}}+1}{\mathrm{r}_{\mathrm{i}}}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{\mathrm{r}(\mathrm{i})}\]

In more recent accounts, Pitzer’s equations have been used to represent the dependence of \(\phi\) on ionic strength [9,10].

If the isopiestic experiments are repeated at several temperatures, the relative partial molar enthalpy of the solvent \(\mathrm{L}_{1}(\mathrm{aq})\) is obtained [10].

In summary a large scientific literature reports thermodynamic data for aqueous solutions containing salts [11] and mixed salt [12] systems.

Footnotes

[1] G. Scatchard, W. J. Hamer and S. E. Wood, J.Am.Chem.Soc.,1938,60,3061.

[2] For reviews and further data compilations see

1. R. N. Goldberg and R. L. Nuttall, J.Phys.Chem.Ref.Data, 1978,7,263.
2. E. C. W. Clarke, J.Phys.Chem.Ref.Data, 1985,14,489.

[3] R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd edn. (revised), 1965, p. 34.

[4] A. K. Covington and R. A. Matheson, J. Solution Chem., 1977, 6, 263; \(\mathrm{NH}_{4} \mathrm{~CNS}(\mathrm{aq})\).

[5]

1. G . Barone, E. Rizzo and V. Volpe, J. Chem. Eng Data. 1976,21,59; alkyureas(aq)
2. O. D. Bonner, C. F. Jordan, R. K. Arisman and J. Bednarek, J. Chem. Thermodyn., 1976,8,1173; thioureas(aq)
3. O. D. Bonner and W. H. Breazeale, J. Chem. Eng. Data,1965,10,325; dextrose(aq); dimethylurea(aq).
4. H. D. Ellerton and P. J. Dunlop, J. Phys.Chem.,1966,70,1831; sucrose(aq).

[6] J. A. Rard and D. J. Miller, J. Chem.Eng. Data 1982,27,169; CsCl(aq) and \(\mathrm{SrCl}_{2}(\mathrm{aq})\).

[7] J. A. Rard, J.Chem.Eng.Data,1987,32,92. \(\mathrm{La}\left(\mathrm{NO}_{3}\right)_{3}(\mathrm{aq}) \text { and } \mathrm{Eu}\left(\mathrm{NO}_{3}\right)_{3}(\mathrm{aq})\).

[8] J. B. Maskill and R. G. Bates, J.Solution Chem.,1986,15,418 Tris(aq).

[9] L.M. Mukherjee and R. G. Bates, J.Solution Chem.,1985,14,255; \(\mathrm{R}_{4}\mathrm{N}^{+} \mathrm{Br}^{-} \left(\mathrm{D}_{2}\mathrm{O}\right)\).

[10] S. Lindenbaum, L. Leifer, G. E. Boyd and J. W. Chase, J. Phys. Chem., 1970,74, 761; \(\mathrm{R}_{4}\mathrm{NX}(\mathrm{aq})\)

[11]

1. KCl(aq) at 45 Celsius; T.M.Davis, L.M.Duckett, J.F.Owen, C.S.Patterson and R.Saleeby, J.Chem.Eng. Data, 1985, 30,432.
2. \(\mathrm{NH}_{4}\mathrm{Br}(\mathrm{aq})\); A.K.Covington and D.Irish, J.Chem.Eng.Data, 1972,17,175.
3. Sodium benzoate and hydroxybenzoates; J.E.Desnoyers, R.Page, G. Perron, J.-L.Fortier, P.-A.Leduc and R.F.Platford, Can. J.Chem.,1973,51,2129.
4. \(\mathrm{CaCl}_{2}(\mathrm{aq})\); L. M. Duckett, J. M. Hollifield and C. S. Patterson, J.Chem. Eng. Data, 1986, 31, 213.
5. \(\mathrm{CaCl}_{2}(\mathrm{aq})\); J.A.Rard and F.H.Spedding, J.Chem.Eng.Data, 1977,22,56.
6. Borates(aq); R. F. Platford, Can. J.Chem.,1969,47,2271.
7. \(\mathrm{Pr}\left(\mathrm{NO}_{3}\right)_{3}(\mathrm{aq}) \text { and } \mathrm{Lu}\left(\mathrm{NO}_{3}\right)_{3}\); J.A.Rard, J. Chem..Eng.Data, 1987,32,334.
8. Alkali metal trifluoroethanoates(aq); O.D.Bonner, J.Chem.Thermodyn.,1982,14,275.

[12]

1. J.A Rard and D.G.Miller, J.Chem.Eng. Data, 1987,32,85; and references therein.
2. G. E. Boyd, J. Solution Chem.,1977,6,95; NaCl+Na p-ethylbenzenesulfonate.
3. A. K.Covington, T. H. Lilley and R. A. Robinson, J.Phys.Chem.,1968,72,2579; \(\mathrm{M}^{+}\mathrm{X}^{-}\text { pairs}(\mathrm{aq})\).
4. C. C. Briggs, R. Charlton and T. H. Lilley, J. Chem.Thermodyn., 1973, 5, 445; \(\mathrm{HClO}_{4} + \mathrm{~NaClO}_{4} + \mathrm{~LiClO}_{4}(\mathrm{aq})\).
5. C. P. Bezboruah, A. K. Covington and R. A. Robinson, J. Chem.Thermodyn., 1970, 2, 431; \(\mathrm{KCL} + \mathrm{~NaNO}_{3}(\mathrm{aq})\).
6. S. Lindenbaum, R. M. Rush and R. A. Robinson, J. Chem.Thermodyn., 1972,4,381.
7. D. Rosenzweig, J. Padova and Y. Marcus, J.Phys.Chem.,1976,80,601; \(\mathrm{NaBr}+ \mathrm{~R}_{4}\mathrm{NBr}(\mathrm{aq})\).
8. I.R.Lantzke, A.K.Covington and R.A.Robinson, J.Chem. Eng. Data, 1973,18,421; \(\mathrm{Na}_{2}\mathrm{S}_{2}\mathrm{O}_{6}(\mathrm{aq}), \mathrm{~Na}_{2}\mathrm{SO}_{3}(\mathrm{aq})\).
9. W.-Y. Wen, S.Saito and C-m. Lee, J. Phys. Chem.,1966,70,1244; \(\mathrm{R}_{4}\mathrm{NF}(\mathrm{aq})\).
10. A.K.Covington, R.A.Robinson and R.Thomson, J.Chem.Eng. Data, 1973,18,422; methane sulfonic acid(aq).