1.10.18: Gibbs Energies- Salt Solutions- Debye-Huckel Equation
The electric potential \(\phi_{j}\) (at ion-\(j\); ion-ion interaction) describes the electric potential at a given reference \(j\)-ion arising from all other \(i\)-ions in solution. The contribution to the chemical potential of one mole of \(j\)-ions is obtained using the Guntleberg charging process. Thus,
\[\Delta \mu_{\mathrm{j}}(\text { ion }-\mathrm{j} \text {; ion }-\text { ion int eractions })=\int_{0}^{\mathrm{z}_{\mathrm{j}} \,{ }_{\mathrm{e}}} \varphi_{\mathrm{j}}(\text { at ion }-\mathrm{j} \text {; ion }-\text { ion }) \, \mathrm{d}\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right) \nonumber \]
Hence[1],
\[\Delta \mu_{\mathrm{j}}(\text { ion }-\mathrm{j} ; \text { ion }-\text { ion int eractions })=-\frac{\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} \, \mathrm{N}_{\mathrm{A}}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}}} \, \frac{\kappa}{1+\kappa \, \mathrm{a}} \nonumber \]
The charge-charge interactions are the only source of deviations in the properties of a given solution from ideal. Then for ion-\(j\),
\[\ln \left(\gamma_{\mathrm{j}}\right)=\Delta \mu_{j}(\text { ion }-\mathrm{j}<-\longrightarrow>\text { ion atmos. }) / \mathrm{R} \, \mathrm{T} \nonumber \]
Hence for the ionic activity coefficient \(\gamma_{j}\),
\[\ln \left(\gamma_{\mathrm{j}}\right)=-\frac{\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} \, \mathrm{N}_{\mathrm{A}}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}} \, \frac{\kappa}{1+\kappa \, \mathrm{a}} \nonumber \]
Ionic activity coefficients have no practical significance. We require an equation for the mean ionic activity coefficient for a salt in solution. For one of mole of salt in solution,
\[v \, \ln \left(\gamma_{\pm}\right)=v_{+} \, \ln \left(\gamma_{+}\right)+v_{-} \, \ln \left(\gamma_{-}\right) \nonumber \]
or,
\[\ln \left(\gamma_{\pm}\right)=\frac{1}{\left(v_{+}+v_{-}\right)}\left[v_{+} \, \ln \left(\gamma_{+}\right)+v_{-} \, \ln \left(\gamma_{-}\right)\right] \nonumber \]
Hence
\[\ln \left(\gamma_{\pm}\right)=-\frac{1}{\left(v_{+}+v_{-}\right)} \,\left[v_{+} \, z_{+}^{2}+v_{-} \, z_{-}^{2}\right] \,\left[\frac{e^{2} \, N_{A}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, R \, T} \, \frac{\kappa}{(1+\kappa \, a)}\right] \nonumber \]
Or[2,3],
\[\ln \left(\gamma_{\pm}\right)=-\left[\frac{\left|z_{+} \, z_{-}\right| \, e^{2} \, N_{A}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{t} \, R \, T} \, \frac{K}{(1+K \, a)}\right] \nonumber \]
In this connection the distance ‘a’ characterises both cations and anions in the salt. In the Debye-Huckel Limiting Law (DHLL) the term \((1+\kappa \, a)\) is approximated to unity thereby assuming that \((1>>K \, a)\). Then using the term \(\mathrm{S}_{\gamma}\), for a 1:1 salt equation (h) is rewritten as follows.
\[\ln \left(\gamma_{\pm}\right)=-\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{S}_{\gamma} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2} /\left[1+\mathrm{b} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\right] \nonumber \]
where
\[\mathrm{b}=\beta \, \mathrm{a} \nonumber \]
and [4]
\[\beta=\left[\frac{2 \, \mathrm{e}^{2} \, \mathrm{N}_{\mathrm{A}}^{2} \, \rho_{1}^{*}(\ell) \, \mathrm{m}^{0}}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{1 / 2} \nonumber \]
For aqueous solutions at ambient pressure and \(298.15 \mathrm{~K}\), \(\beta=3.285 \times 10^{9} \mathrm{~m}^{-1}\) [5].
The quantity \(\mathrm{b}\) depends on a distance parameter [6] ‘a’ which characterises salt \(j\) and reflects the role of repulsion between ions in determining the chemical potentials of a salt in solution. Hence with increase in distance ‘a’ so the denominator increases and \(\ln \left(\gamma_{j}\right)\) is not so strongly negative as predicted by the DHLL. For large ‘a’ and high ionic strengths, the salts are not stabilised to the extent required by the DHLL. The integrated form of the Gibbs-Duhem equation yields an equation for \((\phi-1)\) in terms of molality \(\mathrm{m}_{j}\) [7]. Thus,
\[\mathrm{m}_{\mathrm{j}} \,(1-\phi)=-\int_{0}^{\mathrm{m}(\mathrm{j})} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\pm}\right) \nonumber \]
Hence, [6]
\[(1-\phi)=\left[\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{S}_{\gamma} / 3\right] \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{-1} \, \sigma(\mathrm{x}) \nonumber \]
and
\[\sigma(x)=\left(3 / x^{3}\right) \,\left[(1+x)-(1+x)^{-1}-2 \, \ln (1+x)\right] \nonumber \]
Then the excess molar Gibbs energy for a solution containing a 1:1 salt is given by equation (o).
\[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{S}_{\gamma} \, \mathrm{m}_{\mathrm{j}}^{3 / 2} \,\left(\mathrm{m}^{0}\right)^{-1} \,\left[\sigma(\mathrm{x}) / 3-(1+\mathrm{x})^{-1}\right] \nonumber \]
Footnotes
[1]
\[\begin{aligned}
\Delta \mu_{j}(---) &=\frac{[\mathrm{A} \mathrm{s}]^{2} \,\left[\mathrm{mol}^{-1}\right]}{[1] \,\left[\mathrm{F} \mathrm{m}^{-1}\right] \,[1]} \, \frac{\left[\mathrm{m}^{-1}\right]}{\left\{[1]+\left[\mathrm{m}^{-1}\right] \,[\mathrm{m}]\right\}} \\
&=\frac{\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right] \,\left[\mathrm{mol}^{-1}\right]}{\left[\mathrm{As} \mathrm{} \mathrm{V}^{-1}\right]}=\frac{\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right] \,\left[\mathrm{mol}^{-1}\right]}{\left[\mathrm{As} \mathrm{As} \mathrm{} \mathrm{J}^{-1}\right]}=\left[\mathrm{J} \mathrm{mol}^{-1}\right]
\end{aligned} \nonumber \]
[2] Condition of electric neutrality; \(v_{+} \, z_{+}=-v_{-} \, z_{-}\) or, \(v_{-}=-v_{+} \, z_{+} / z_{-}\)
Then,
\[\begin{aligned}
& \frac{1}{\left(v_{+}+v_{-}\right)} \,\left[v_{+} \, z_{+}^{2}+v_{-} \, z_{-}^{2}\right] \\
=& {\left[\frac{1}{v_{+}-v_{-} \, z_{+} / z_{-}}\right] \,\left[v_{+} \, z_{+}^{2}-v_{+} \, z_{+} \, z_{-}\right]=\frac{z_{-}}{\left(z_{-}-z_{+}\right)} \,\left[z_{+}^{2}-z_{+} \, z_{-}\right] } \\
=&-z_{+} \, z_{-}=\left|z_{+} \, z_{-}\right|
\end{aligned} \nonumber \]
[3] \(\ln \left(\gamma_{\pm}\right)=\frac{[1] \,[\mathrm{A} \mathrm{s}]^{2} \,[\mathrm{mol}]}{[1] \,[1] \,\left[\mathrm{As} \mathrm{} \mathrm{J}^{-1} \mathrm{As}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]} \, \frac{\left[\mathrm{m}^{-1}\right]}{\left\{1+\left[\mathrm{m}^{-1}\right] \,[\mathrm{m}]\right\}}=[1]\)
[4]
\[\begin{aligned}
\beta=\left\{[1] \,[\mathrm{C}]^{2} \,\left[\mathrm{mol}^{-1}\right]^{2} \,\left[\mathrm{kg} \mathrm{m}^{-3}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]\right\}^{1 / 2} \\
/\left\{\left[\mathrm{J}^{-1} \mathrm{C}^{2} \mathrm{~m}^{-1}\right] \,[1] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]\right\}^{1 / 2}=\left[\mathrm{m}^{-1}\right]
\end{aligned} \nonumber \]
[5] For DH parameters for aqueous solutions to high \(\mathrm{T}\) and \(\mathrm{p}\), see D. J. Bradley and K. S. Pitzer,J. Phys.Chem.,1979, 83 ,1599;1983; 87 ,3798.
[6] Parameter ‘a’ is sometimes called ‘ion size’ . But as S. Glasstone [Introduction to Electrochemistry, D.van Nostrand, New Jersey, 1943, page 145, footnote] points out ‘the exact physical significance cannot be expressed precisely’. Nevertheless an important consideration is the relative sizes of ions and solvent molecules; B. E. Conway and R. E. Verrall, J.Phys.Chem.,1966, 70 ,1473.
[7] From equation (i) when by definition \(\mathrm{x}=\mathrm{b} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\) and \(\mathrm{k}=\left|\mathrm{Z}_{+} \, \mathrm{Z}_{-}\right| \, \mathrm{S}_{\gamma} / \mathrm{b}\)
Hence,
\[\begin{aligned}
&\ln \left(\gamma_{\pm}\right)=-(b \, k) \,(x / b) /(1+x)=-k \, x /(1+x) \\
&\mathrm{d} \ln \left(\gamma_{\pm}\right)=-\mathrm{k} \,\left\{[1 /(1+\mathrm{x})]-\left[\mathrm{x} /(1+\mathrm{x})^{2}\right]\right\} \, \mathrm{dx}=-\mathrm{k} \,\left\{[1+\mathrm{x}-\mathrm{x}] /[1+\mathrm{x}]^{2}\right\} \, \mathrm{dx} \\
&\mathrm{d} \ln \left(\gamma_{\pm}\right)=-\mathrm{k} \, \mathrm{dx} /[1+\mathrm{x}]^{2}
\end{aligned} \nonumber \]
Therefore,
\[(1-\phi) \, m_{j}=-\int_{0}^{x} m_{j} \,\left\{-k /(1+x)^{2}\right\} \, d x \nonumber \]
Or, \((1-\phi)=\left(k / x^{2}\right) \, \int_{0}^{x}\left\{x^{2} /(1+x)^{2}\right\} \, d x\)
Standard integral:
\[\begin{aligned}
&\int_{0}^{x}\left\{x^{2} /(a \, x+b)^{2}\right\} \, d x= \\
&\left\{(a \, x+b) / a^{3}\right\}-\left\{b^{2} / a^{3} \,(a \, x+b)\right\}-\left(2 \, b / a^{3}\right) \, \ln (a \, x+b)
\end{aligned} \nonumber \]
With \(a=b=1\),
\[\int_{0}^{x}\left\{x^{2} /(1+x)^{2}\right\} \, d x=(1+x)-[1 /(1+x)]-2 \, \ln (1+x) \nonumber \]
Thus,
\[(1-\phi)=(k \, x / 3) \,\left\{\left(3 / x^{3}\right) \,\left[(1+x)-(1+x)^{-1}-2 \, \ln (1+x)\right]\right. \nonumber \]