Adsorption of polyprotic acid at the water/air interface

Abstract

A rigorous thermodynamic treatment appropriate for surface adsorption from mixed aqueous solution of alkali and polyprotic acid was derived. Those equations were applied to mixed aqueous solution/air systems of alkali metal hydroxide and Fe^III complex with ethylenediamine- N, N, N′,N′-tetraacetate (Fe-EDTA). Surface density of each species arising from Fe-EDTA was separately evaluated, and thus, surface activity of Fe-EDTA was studied, especially its dependence on pH and how it is influenced by the counter cations. Fe-EDTA was positively adsorbed at the water/air interface at very low pHs and negatively at high pHs. The pH range of positive adsorption of Fe-EDTA with potassium ion, as a counter ion, was wider than that with sodium ion. Thus, potassium ion, a structure breaker, tended to smooth surface adsorption of Fe-EDTA at the water/air interface, whereas sodium ion, a structure maker, tended to withdraw Fe-EDTA from the interfacial region.

Appendix

Partial differentiations of α ₁, α ₂, δ and $m_{\text {OH}^{-}}$ with respect to m _i

It would be easier if the partial derivatives (∂ α ₁/∂ m _i ) _{T, p, m j ≠ i} , (∂ α ₂/∂ m _i ) _{T, p, m j ≠ i} , (∂ δ/∂ m _i ) _{T, p, m j ≠ i} , and $(\partial m_{\text {OH}^{-}} / \partial m_{i})_{T,p, m_{j\not = i}}$ can be calculated numerically. Partial differentiation of p K _a1, p K _a2, and p K _d and Eq. 22 combined with Eqs. 9 ∼14, with respect to m _i (i = 1 or 2) give the following relations. Namely, $(\partial \mathrm {p}K_{\mathrm {a}_{1}} / \partial m_{i})_{T,p,m_{j\not = i}} = -(\partial \log a_{\mathrm {c}1-} / \partial m_{i})_{T,p,m_{j\not = i}} + (\partial \text {pH} / \partial m_{i})_{T,p,m_{j\not = i}} + (\partial \log a_{\mathrm {c}0} / \partial m_{i})_{T,p,m_{j\not = i}}$ gives

$$\begin{array}{@{}rcl@{}} &&{\kern-.7pc}\frac{1}{\ln{\kern-1.5pt} 1{\kern-.5pt}0}{}\left\{ {}-{}\frac{1}{m_{{\kern-.5pt}\text{OH}^{-}}}{\kern-2.7pt}\left({\kern-.5pt}\frac{\partial m_{\text{OH}^{-}}}{\partial m_{i}}{\kern-.5pt}\right)_{{}T{\kern-.2pt},{\kern-.2pt}p{\kern-.2pt},{\kern-.2pt}m_{j\not= i}}{\kern-3.3pt}+{}\frac{1}{\alpha_{{\kern-.5pt}1}{}({\kern-.5pt}1{}-{}\alpha_{{\kern-.5pt}1}{\kern-.7pt})}{\kern-2.5pt}\left({\kern-.6pt}\frac{\partial \alpha_{1}}{\partial {\kern-.5pt}m_{{\kern-.5pt}i}}{\kern-.6pt}\right)_{{\kern-1.5pt}T{\kern-.5pt},{\kern-.5pt}p{\kern-.5pt},{\kern-.5pt}m_{{\kern-.5pt}j{\kern-.5pt}\not= i}}\right.\\ &&\left.{\kern1.5pc}-\frac{1}{1-\alpha_{2}}\left(\frac{\partial \alpha_{2}}{\partial m_{i}}\right)_{T,p,m_{j\not= i}}\right\} \\ &&{\kern10pt}={}\left({\kern-.5pt}\frac{\partial{} \log {}\gamma_{\text{OH}^{-}}}{\partial m_{i}}{\kern-.5pt}\right)_{{}T,p,m_{j\not=i}}{}-{}\left(\frac{\partial \log \gamma_{\mathrm{c}1-}}{\partial m_{i}}\right)_{{}T,p,m_{j\not= i}} {\kern-.5pt}; \end{array} $$

(50)

$(\partial \mathrm {p}K_{\mathrm {a}_{2}} / \partial m_{i})_{T,p,m_{j\not = i}} = -(\partial \log a_{\mathrm {c}2-} / \partial m_{i})_{T,p,m_{j\not = i}} \\+ (\partial \text {pH} / \partial m_{i})_{T,p,m_{j\not = i}} + (\partial \log a_{\mathrm {c}1-} / \partial m_{i})_{T,p,m_{j\not = i}}$ gives

$$\begin{array}{@{}rcl@{}} &&\frac{1}{\ln10}\left\{ -\frac{1}{m_{\text{OH}^{-}}}\left(\frac{\partial m_{\text{OH}^{-}}}{\partial m_{i}}\right)_{T,p,m_{j\not= i}}+\frac{1}{\alpha_{2}(1-\alpha_{2})}\right.\notag\\ &&{\kern2.2pc}\left.\times\left(\frac{\partial \alpha_{2}}{\partial m_{i}}\right)_{T,p,m_{j\not= i}}-\frac{2}{1-2\delta}\left(\frac{\partial \delta}{\partial m_{i}}\right)_{T,p,m_{j\not= i}}\right\}\\ &&{\kern2pc}=\left(\frac{\partial \log \gamma_{\text{OH}^{-}}}{\partial m_{i}}\right)_{T,p,m_{j\not=i}}+\left(\frac{\partial \log \gamma_{\mathrm{c}1-}}{\partial m_{i}}\right)_{T,p,m_{j\not=i}}\notag\\ &&{\kern2.2pc}-\left(\frac{\partial \log \gamma_{\mathrm{c}2-}}{\partial m_{i}}\right)_{T,p,m_{j\not=i}} ; \end{array} $$

(51)

$(\partial \mathrm {p}K_{\mathrm {d}} / \partial m_{i})_{T,p,m_{j\not = i}} = -(\partial \log a_{\text {cd}} / \partial m_{i})_{T,p,m_{j\not = i}} - (\partial \text {pH} / \partial m_{i})_{T,p,m_{j\not = i}} + 2(\partial \log a_{\text {cd}} / \partial m_{i})_{T,p,m_{j\not = i}}$ gives

$$\begin{array}{@{}rcl@{}} &&{}\frac{1}{\ln10}\left\{ \frac{1}{\alpha_{1}}\left(\frac{\partial \alpha_{1}}{\partial m_{i}}\right)_{T,p,m_{j\not= i}}+\frac{1}{\alpha_{2}}\left(\frac{\partial \alpha_{2}}{\partial m_{i}}\right)_{T,p,m_{j\not= i}}\right.\notag\\[6pt] &&{\kern1.6pc}\left.-\frac{1+2\delta}{\delta(1-2\delta)}\left(\frac{\partial \delta}{\partial m_{i}}\right)_{T,p,m_{j\not= i}}\right\}\nonumber\\[6pt] &&{\kern.5pc}=-\frac{1}{(\ln 10) m_{2}}\chi(i) -2\left(\frac{\partial \log \gamma_{\mathrm{c}2-}}{\partial m_{i}}\right)_{T,p,m_{j\not= i}}\notag\\[6pt] &&{\kern1.5pc}+\left(\frac{\partial \log \gamma_{\mathrm{cd}}}{\partial m_{i}}\right)_{T,p,m_{j\not= i}} , \end{array} $$

(52)

with χ(i) being

$$\begin{array}{@{}rcl@{}} \chi(i) &=&0, (i=1)\\ &=&1, (i= 2); \end{array} $$

(53)

and $\left \{\partial \left (m_{\text {OH}^{-}} -\frac {K_{\mathrm {w}}}{\gamma _{\mathrm {H}^{+}}\gamma _{\text {OH}^{-}}m_{\text {OH}^{-}}}\right )/ \partial m_{i}\right \}_{T,p,m_{j\not = i}} = \left [ \partial \{m_{1} -m_{2} \alpha _{1} (1+\alpha _{2})\} / \partial m_{i} \right ]_{T,p,m_{j\not = i}}$ gives

$$\begin{array}{@{}rcl@{}} &&{}\left(\frac{\partial m_{\text{OH}^{-}}}{\partial m_{1}}\right)_{T,p,m_{2}}= A \left[1-m_{2}(1+\alpha_{2})\left(\frac{\partial \alpha_{1}}{\partial m_{1}}\right)_{T,p,m_{2}}\right.\notag\\ &&{\kern8pc}-m_{2}\alpha_{1}\left. \left(\frac{\partial \alpha_{2}}{\partial m_{1}}\right)_{T,p,m_{2}}\right.\\ &&{\kern8pc}-\frac{a_{\mathrm{H}^{+}}}{\gamma_{\mathrm{H}^{+}}}\left\{\left(\frac{\partial \log \gamma_{\mathrm{H}^{+}}}{\partial m_{1}}\right)_{T,p,m_{2}}\right.\notag\\ &&{\kern8pc}\left.\left.+\left(\frac{\partial \log \gamma_{\text{OH}^{-}}}{\partial m_{1}}\right)_{T,p,m_{2}} \right\} \right]\\ \end{array} $$

(54)

and

$$\begin{array}{@{}rcl@{}} &&\left(\frac{\partial m_{\text{OH}^{-}}}{\partial m_{2}}\right)_{T,p,m_{1}}\!\!= A \left[{\vphantom{\frac{\partial \alpha_{2}}{\partial m_{2}}}}-\alpha_{1}(1+\alpha_{2})-m_{2}(1+\alpha_{2})\right.\notag\\ &&{\kern6.5pc}\times \left(\frac{\partial \alpha_{1}}{\partial m_{2}}\right)_{T,p,m_{1}}\!\!-m_{2}\alpha_{1} \left(\frac{\partial \alpha_{2}}{\partial m_{2}}\right)_{T,p,m_{1}}\notag\\ &&{\kern6.5pc}-\frac{a_{\mathrm{H}^{+}}}{\gamma_{\mathrm{H}^{+}}}\left\{\left(\frac{\partial \log \gamma_{\mathrm{H}^{+}}}{\partial m_{2}}\right)_{T,p,m_{1}}\right.\notag\\ &&{\kern6.5pc}\left.\left.+\left(\frac{\partial \log \gamma_{\text{OH}^{-}}}{\partial m_{2}}\right)_{T,p,m_{1}} \right\}\right]. \end{array} $$

(55)

Thus, partial derivatives of the molality of hydroxide ion $m_{\text {OH}^{-}}$, the degrees of dissociation of the complex α ₁ and α ₂, and the degree of dimerization of the complex δ with respect to the molality of sodium hydroxide, m ₁ constitutes the following simultaneous linear equations.

$$ M\left(\begin{array}{c} \left(\frac{\partial m_{\text{OH}^{-}}}{\partial m_{1}}\right)_{T,p,m_{2}}\\ \\\left(\frac{\partial \alpha_{1}}{\partial m_{1}}\right)_{T,p,m_{2}}\\ \\\left(\frac{\partial \alpha_{2}}{\partial m_{1}}\right)_{T,p,m_{2}}\\ \\\left(\frac{\partial \delta}{\partial m_{1}}\right)_{T,p,m_{2}} \end{array} \right) =\left(\begin{array}{c} 1\\ \\ 0\\ \\ 0\\ \\ 0 \end{array} \right) , $$

(56)

where

$$\begin{array}{@{}rcl@{}} M \equiv\left( \begin{array}{@{\,} cccc@{\,}} \dfrac{1}{A}+(\ln 10)\dfrac{a_{\mathrm{H}^{+}}}{\gamma_{\mathrm{H}^{+}}}\mathcal{D} & m_{2}\left\{(1+\alpha_{2})+{\vphantom{\left.\dfrac{a_{\mathrm{H}^{+}}}{\gamma_{\mathrm{H}^{+}}}\mathcal{D}(1+2\delta)\right\}}}(\ln 10)\times\right. &m_{2}\alpha_{1}\left\{1 +{\vphantom{\left.\dfrac{a_{\mathrm{H}^{+}}}{\gamma_{\mathrm{H}^{+}}}\mathcal{D}(1+2\delta)\right\}}}2(\ln 10)\times \right.&4m_{2}(\ln 10)\dfrac{a_{\mathrm{H}^{+}}}{\gamma_{\mathrm{H}^{+}}}\mathcal{D} \alpha_{2}\\ &\left.\dfrac{a_{\mathrm{H}^{+}}}{\gamma_{\mathrm{H}^{+}}}\mathcal{D}(1+2\alpha_{2} +4\alpha_{2} \delta)\right\}&\left.\dfrac{a_{\mathrm{H}^{+}}}{\gamma_{\mathrm{H}^{+}}}\mathcal{D}(1+2\delta)\right\}&\\ &&&\\ -\dfrac{1}{m_{\mathrm{OH}^{-}}} & \dfrac{1}{\alpha_{1}(1-\alpha_{1})} & -\dfrac{1}{1-\alpha_{2}} & 0 \\ &&&\\ -\dfrac{1}{m_{\mathrm{OH}^{-}}}+(\ln 10)\mathcal{D} & m_{2} (\ln 10)\mathcal{D}\times & \dfrac{1}{\alpha_{2}(1-\alpha_{2})} +2m_{2}\times & -\dfrac{2}{1-2\delta}+\\ &(1+2\alpha_{2} +4\alpha_{2} \delta)&(\ln 10)\mathcal{D} \alpha_{1} (1+2\delta) & 4m_{2} (\ln 10)\mathcal{D}\alpha_{1} \alpha_{2}\\ &&&\\ -4(\ln 10) \mathcal{D} & \dfrac{1}{\alpha_{1}}-4m_{2}(\ln 10)\times & \dfrac{1}{\alpha_{2}}-8m_{2}(\ln 10) \times & -\dfrac{1+2\delta}{\delta(1-2\delta)}-\\ &\mathcal{D}(1+2\alpha_{2}+4\alpha_{2} \delta) & \mathcal{D}\alpha_{1}(1+2\delta) &16m_{2}(\ln 10)\mathcal{D}\alpha_{1} \alpha_{2}\\\end{array}\right) \end{array} $$

(57)

where 𝒟 is defined by

$$ \mathcal{D} \equiv \left\{ \frac{-\mathcal{A}}{\sqrt{I}(1+\sqrt{I})^{2}}+0.2 \right\} $$

(58)

Similarly, partial derivatives of $m_{\text {OH}^{-}}$, α ₁, α ₂, and δ with respect to m ₂ constitute the following simultaneous equations.

$$ M\left(\begin{array}{c} \left(\frac{\partial m_{\text{OH}^{-}}}{\partial m_{2}}\right)_{T,p,m_{1}}\\ \\ \left(\frac{\partial \alpha_{1}}{\partial m_{2}}\right)_{T,p,m_{1}}\\ \\ \left(\frac{\partial \alpha_{2}}{\partial m_{2}}\right)_{T,p,m_{1}}\\ \\ \left(\frac{\partial \delta}{\partial m_{2}}\right)_{T,p,m_{1}} \end{array} \right) = \left(\begin{array}{c} -\alpha_{1} (1+\alpha_{2})-(\ln 10)\frac{a_{\mathrm{H}^{+}}}{\gamma_{\mathrm{H}^{+}}}\mathcal{D}\alpha_{1} (1+2\alpha_{2} +4\alpha_{2} \delta)\\ \\ 0\\ \\ -(\ln 10)\mathcal{D}\alpha_{1} (1+2\alpha_{2} +4\alpha_{2} \delta)\\ \\ -\frac{1}{m_{2}} +4(\ln 10)\mathcal{D}\alpha_{1} (1+2\alpha_{2} +4\alpha_{2} \delta) \end{array} \right) $$

(59)

The partial derivatives $(\partial m_{\text {OH}^{-}} /\partial m_{i})_{T,p,m_{j\neq i}}$, (∂ α ₁/∂ m _i ) _{T, p, m j ≠ i} , (∂ α ₂/∂ m _i ) _{T, p, m j ≠ i} , and (∂ δ/∂ m _i ) _{T, p, m j ≠ i} (i = 1 or 2) are obtained by solving the above simultaneous equations by using Cramer’s rule.