# The Diffusion Role in Adsorption of Hexavalent Chromium on Solid Olive Mill Waste
## 3 Diffusion-Binding Model  
Considering a mass \(m\) of the adsorbent, which is suspended in K\({}_{2}\)CrO\({}_{4}\) solution of concentration \(C_{0}\), the adsorbate diffuses into the solid and is adsorbed on the interior surface area. The rate accumulation in the solid results [15]:  
\[\frac{\partial C_{A}}{\partial t}+(-r_{A}(C_{A}))=D_{A}\bigtriangledown^{2}C_{A} \tag{2}\]  
where \(D_{A}\) is the diffusion coefficient of Cr(VI) ion in solution, \(t\) is the time, and \(\bigtriangledown^{2}\) is the Laplacian operator. The reaction term, \(r_{A}(C_{A})\) is an empirical expression. In general, the Laplacian operator involves all three spatial coordinates, however, we will mainly be concerned with systems where the symmetry allows this diffusion term to be written as a function of a single spatial coordinate. Cr(VI) ions penetrate the pores of the composite, diffuse into the solid and bind to specific sites. We assume that the binding process is described by the following reaction [16]:  
\[A+S\xrightleftharpoons{<}{\Longleftrightarrow}AS \tag{3}\]  
where \(A\), \(S\) and \(AS\) represent Cr(VI), adsorption site, and complex Cr(VI)-solid, respectively. Therefore, one can write:  
\[-r_{A}=\frac{dC_{AS}}{dt}=k_{1}C_{A}C_{S}-k_{-1}C_{AS} \tag{4}\]  
where \(C_{A}(z,t)\), i.e. the concentration at position \(z\) and time \(t\) in the solid, assumes the form:  
\[\frac{\partial C_{A}}{\partial t}+\frac{dC_{AS}}{dt}=D_{A}\frac{\partial^{2} C_{A}}{\partial z^{2}} \tag{5}\]  
In order to solve this differential equation, a relationship between \(C_{AS}\) and \(C_{A}\) must exist. Let:  
\[C_{AS}=f(C_{A}) \tag{6}\]then, Equation (5) becomes:  
\[\frac{\partial C_{A}}{\partial t}=\frac{D_{A}}{1+\frac{df}{dC_{A}}}\frac{ \partial^{2}C_{A}}{\partial z^{2}} \tag{7}\]  
Actual solid concentration is related to \(C_{AS}\) by the matter balance, thus:  
\[C_{S}^{0}=C_{S}+C_{AS} \tag{8}\]  
Since the binding reaction proceeds very rapidly compared with the pore diffusion, we made the assumption that a local steady state  
\[\frac{dC_{AS}}{dt}\simeq 0\]  
and consequently, we have:  
\[f(C_{A})=\frac{K_{1}C_{1}C_{S}^{0}}{K_{-1}+K_{1}C_{A}}=\frac{KC_{1}C_{S}^{0}}{ 1+KC_{A}} \tag{9}\]  
where \(K=\frac{k_{1}}{K_{-1}}\) has been used. Substitution of Equation (9) into Equation (7), yields:  
\[\frac{\partial C_{A}}{\partial t}=\left[\frac{D_{A}}{1+\frac{KC_{S}^{0}}{(1+KC _{A})^{2}}}\right]\frac{\partial^{2}C_{A}}{\partial z^{2}} \tag{10}\]  
At low penetrant concentration \(KC_{A}\ll 1\) so that Equation (10) can be rewritten as  
\[A+S\xrightleftharpoons{<}{\Longleftrightarrow}AS \tag{3}\]0  
Equation (11) describes the rate of change in Cr(VI) concentration in terms of an effective diffusion coefficient, defined as  
\[A+S\xrightleftharpoons{<}{\Longleftrightarrow}AS \tag{3}\]1  
where it is evident that the binding to specific sites hinders the molecular diffusivity by the factor \(1+KC_{S}^{0}\). Equation (11) has to be solved in some region such as \(0<z<\ell\) with a given initial value of \(C_{A}(z,t)\) at the instant \(t=0\), and with boundary conditions at the ends of the region. We seek a solution of Equation (11) in a solid sheet which satisfies the boundary and initial conditions:  
\[A+S\xrightleftharpoons{<}{\Longleftrightarrow}AS \tag{3}\]2 \[A+S\xrightleftharpoons{<}{\Longleftrightarrow}AS \tag{3}\]3 \[A+S\xrightleftharpoons{<}{\Longleftrightarrow}AS \tag{3}\]4  
For the sake of simplicity, we assume that the concentration at one end of the region reaches the equilibrium value, i.e., \(C_{A}=C_{A}^{e}\). By applying the well-known method of variable separation [17], one finds the following solution:  
\[A+S\xrightleftharpoons{<}{\Longleftrightarrow}AS \tag{3}\]5  
with \(\alpha_{j}=(2j+1)\frac{\pi}{2}\).