1.14.3: Absorption Isotherms - Two Absorbates
We consider the case where in addition to the adsorbent there are two adsorbates, chemical substances \(i\) and \(j\) in aqueous solution [1]. In the simplest case the thermodynamic properties of the system are ideal. In other words for both solutes and adsorbates there are no \(i - i\), \(j - j\) and \(i - j\) interactions. Analysis of the adsorption using the Langmuir adsorption isotherm leads to two terms describing the surface coverage, \(\theta_{i}\) and \(\theta_{j}\) plus the total surface coverage, \(\theta_{i} + \theta_{j}\). The chemical potentials of the solutes in solution are described in terms of their concentrations (assuming pressure \(p\) is close to the standard pressure); \mathrm{c}_{\mathrm{r}}=1 \mathrm{~mol dm}^{-3}.
\[\mu_{j}(a q)=\mu_{j}^{0}(a q)+R \, T \, \ln \left(c_{j} / c_{r}\right) \nonumber \]
\[\mu_{i}(a q)=\mu_{i}^{0}(a q)+R \, T \, \ln \left(c_{i} / c_{r}\right) \nonumber \]
The upper limit of the total surface occupancy is unity and so we expect as \(\left(\theta_{i} + \theta_{j} \right)\) approaches unity the sum of the chemical potentials \(\mu_{j}(\mathrm{ad})\) and \(\mu_{i}(\mathrm{ad})\) approaches \(+\infty\), thereby opposing any tendency for further solute to be adsorbed. If we assert that there are no substrate-substrate interactions on the surface, the chemical potential of adsorbate \(j\) can be formulated as follows.
\[\mu_{j}(\mathrm{ad})=\mu_{j}^{0}(\mathrm{ad})+\mathrm{R} \, \mathrm{T} \, \ln \left[\theta_{\mathrm{j}} /\left(1-\theta_{\mathrm{i}}-\theta_{\mathrm{j}}\right)\right] \nonumber \]
The denominator \(\left(1-\theta_{i}-\theta_{j}\right)\) takes account of the fact that adsorbate \(i\) also occupies the surface. Thus as \(\left(\theta_{i} + \theta_{j} \right)\) approaches unity there are no more sites on the surface for adsorbate \(j\) (and adsorbate \(i\) ) to occupy.
\[\text { Thus } \operatorname{limit}\left[\left(1-\theta_{\mathrm{i}}-\theta_{\mathrm{j}}\right) \rightarrow 0\right] \mu_{\mathrm{j}}(\mathrm{ad})=+\infty \nonumber \]
The equilibrium between chemical substance \(j\) as solute and adsorbate is described as follows.
\[\text { Solute }-\mathrm{j}+\text { Polymer Surface } \Leftrightarrow \text { Adsorbate } \mathrm{j} \nonumber \]
\[\text { Prepared } \mathrm{n}_{\mathrm{j}}^{0} \quad \quad \quad 0 \mathrm{~mol} \nonumber \]
\[\text { Equilib. } \quad \mathrm{n}_{\mathrm{j}}^{0}-\xi_{\mathrm{j}} \quad \xi_{\mathrm{j}}+\xi_{\mathrm{i}} \quad \xi_{\mathrm{j}} \mathrm{mol} \nonumber \]
We express the fraction of surface coverage as proportional to the extent of adsorption via a proportionality constant \(\pi\) which is a function of the sizes of the solutes and geometric parameters describing the surface.
\[\text { Then, } \quad \theta_{i}=\pi_{i} \, \xi_{i} \quad \text { and } \theta_{j}=\pi_{j} \, \xi_{j} \nonumber \]
\[\text { At equilibrium } \mu_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq})=\mu_{\mathrm{j}}^{\mathrm{eq}} \text { (ad) and } \mu_{\mathrm{i}}^{\mathrm{eq}}(\mathrm{aq})=\mu_{\mathrm{i}}^{\mathrm{eq}}(\mathrm{ad}) \nonumber \]
Hence for a system having volume \(\mathrm{V}\),
\[\begin{aligned}
\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left[\left(\mathrm{n}_{\mathrm{j}}^{0}-\xi_{\mathrm{j}}\right) / \mathrm{V} \, \mathrm{c}_{\mathrm{r}}\right] \\
&=\mu_{\mathrm{j}}^{0}(\mathrm{ad})+\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\pi_{\mathrm{j}} \, \xi_{\mathrm{j}}}{1-\pi_{\mathrm{j}} \, \xi_{\mathrm{j}}-\pi_{\mathrm{i}} \, \xi_{\mathrm{i}}}\right]
\end{aligned} \nonumber \]
\[\text { By definition, } \Delta_{\mathrm{ad}} \mathrm{G}_{\mathrm{j}}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \mathrm{K}_{\mathrm{j}}=\mu_{\mathrm{j}}^{0}(\mathrm{ad})-\mu_{\mathrm{j}}^{0}(\mathrm{aq}) \nonumber \]
\[\text { Hence, } \mathrm{K}_{\mathrm{j}}=\left[\frac{\pi_{\mathrm{j}} \, \xi_{\mathrm{j}}}{1-\pi_{\mathrm{j}} \, \xi_{\mathrm{j}}-\pi_{\mathrm{i}} \, \xi_{\mathrm{i}}}\right] \, \frac{\mathrm{V} \, \mathrm{c}_{\mathrm{r}}}{\left(\mathrm{n}_{\mathrm{j}}^{0}-\xi_{\mathrm{j}}\right)} \nonumber \]
A similar equation is obtained for equilibrium constant \(\mathrm{K}_{i}\). Both equations are quadratics in the extent of adsorption
Footnotes
[1] M. J. Blandamer, B. Briggs, P. M. Cullis, K. D. Irlam, J. B. F. N. Engberts and J. Kevelam, J. Chem. Soc. Faraday Trans.,1998, 94 , 259.