Adsorption layer and its characteristic to modulate the electro-oxidation runway of organic species

Abstract

Studies of organic compounds oxidation at constant current conditions with different anodes, indicate that the oxidative action of these anodes is related to their absorption characteristic i.e. the low or high difficulty of the generated hydroxyl radicals in reaching the solution. This paper shows that the adsorption does not concern the anodic surface, but an adjacent thin layer (adsorption layer) of the aqueous solution. An estimation of the fraction of hydroxyl radicals that move from the adsorption layer to the adjacent reactive/diffusive layer is obtained as well as a suitable tool for comparison the result of this approach with the experimental results. This comparison shows how the adsorption layer modulates the runway of the organic species oxidation.

Appendix

The differential equations is of second order:

$$ \frac{d^{2}y}{d\xi^{2}}=y^{2} $$

(20)

While the boundary conditions of this equation are:

$$ \frac{dy}{d\xi}(\infty)=0, \quad y(0)=1 $$

(21)

The following new variable ϕ = ϕ[y(ξ)], is introduced:

$$ \phi=\frac{dy}{d\xi} $$

(22)

This new variable is elaborated in the following way:

$$ \frac{d\phi}{d\xi}=\frac{d\phi}{dy}\frac{dy} {d\xi}=\phi\frac{d\phi}{dy}=\frac{d^{2}y}{d\xi^{2}}.$$

(23)

Using the obtained relationship in the second order differential equation it is reduced to a first order equation that is separable:

$$ \phi d\phi=y^{2}dy.$$

(24)

The integral of this equation is estimated by assuming as initial values a generic value of y at which correspond a generic value of ϕ, while as final values are zero for both variables. In fact when ξ→∞ both variables assumes as value zero:

$$ \int\limits^{0}_{\phi}\phi d\phi=\int\limits^{0}_{y}y^{2}dy $$

(25)

And from the solution of this integral the following equation is obtained:

$$ \phi^{2}=\frac{2}{3}y^{3} $$

(26)

The new variable ϕ is substituted by dy/dξ:

$$ \frac{dy}{d\xi}=\pm\sqrt{\frac{2}{3}}\sqrt{y^{3}}.$$

(27)

And the following integral equation is obtained by considering the initial point (0, 1) and a generic point (ξ, y):

$$ \int\limits^{y}_{1} \frac{dy}{\sqrt{y^{3}}}=\pm\sqrt{\frac{2} {3}}\int\limits^{\xi}_{0}d\xi $$

(28)

The solution of this integral is:

$$ \frac{1}{\sqrt{y}}=\mp\sqrt{\frac{8}{3}}\xi+1.$$

(29)

And the obtained equation relates the dimensionless concentration y, to dimensionless distance ξ:

$$ y=\frac{1}{\left(\mp\sqrt{\frac{8}{3}}\xi+1\right)^{2}}.$$

(30)

The obtained concentration versus dimensionless distance must be constantly decreasing with the dimensionless distance and then the sign plus can be accepted.

$$ y=\frac{1}{\left(\sqrt{\frac{8}{3}}\xi+1\right)^{2}}.$$

(31)

At the end of adsorption layer ξ = δ _a /λ _a , this dimensionless concentration assumes the following value:

$$ y(1)=\frac{1}{\left(\sqrt{\frac{8}{3}}\frac{\delta_{a}} {\lambda_{a}}+1\right)^{2}}.$$

(32)

The derivative of this concentration that is proportional to the hydroxyl radical flow is given by the following relationship:

$$ \frac{dy}{d\xi}=-\frac{\sqrt{\frac{2}{3}}\frac{\delta_{a}} {\lambda_{a}}}{\left(\sqrt{\frac{8}{3}}\frac{\delta_{a}} {\lambda_{a}}\xi+1\right)^{3}}.$$

(33)

The ratio of hydroxyl radical flow that reaches the diffusion layer respect to the produced radicals is indicated as efficiency of the considered anode to produce free hydroxyl radicals layer and its value is given by the following relationship:

$$ \eta_{a}=\frac{1}{\left(\sqrt{\frac{8}{3}}\frac{\delta_{a}} {\lambda_{a}}+1\right)^{3}} $$

(34)