## 3 Results
### Diffusion-Based Models
#### 3.3.2 Evaluation of Pore and Surface Diffusion Coefficients
 \tag{19}\]  
where \(\varepsilon_{bc}\) is the internal porosity of the brick composite; \(\rho_{bc}\) is the density of the porous support; \(C\) is the liquid-phase concentration of the cation per unit volume of the liquid phase; and \(Q\) represents the solid-phase concentration per unit mass of brick. As suggested by [PERSON] and his co-workers [100], under favourable (and ideal) adsorption conditions, it could be assumed that: (i) the concentration of adsorbates on the surface of the adsorbent (\(C_{s}\)) was close to the product \"\(Q\).\(\rho_{bc}\)\"; and (ii) as the examined diffusion region tended to attain thermodynamic equilibrium, the \(\partial Q/\partial C\) derivative could be assimilated to the derivative at the equilibrium state, (\(\partial Q/\partial C\))eq. By taking into consideration these hypotheses and according to some authors [98,100], Equation (19) could be rearranged, and the effective diffusion coefficient could be deduced and expressed as a function of both the pore diffusion coefficient (D\({}_{p}\)) and the surface diffusion coefficient (D\({}_{s}\)) as:  
\[D_{eff}=\frac{\varepsilon_{bc}D_{p}+(1-\varepsilon_{bc})\rho_{bc}D_{s}(\frac{ \partial Q}{\partial C})_{eq}}{\varepsilon_{bc}+(1-\varepsilon_{bc})\rho_{bc} (\frac{\partial Q}{\partial C})_{eq}} \tag{20}\]  
On the other hand, from batch experiments carried out on the adsorption of ions (Cs\({}^{+}\), Rb\({}^{+}\), Sr\({}^{2+}\), and Co\({}^{2+}\)) on to the brick composite, the Langmuir [101] and Freundlich [102] isotherm models were fitted to the experimental data. Both models described the equilibrium data quite well, with relatively good correlation coefficients (R\({}^{2}\) > 0.98). However, the Langmuir model provided lower standard errors with higher R\({}^{2}\) coefficients, evidencing preferential monolayer adsorption of cations on brick surfaces. The derivation of the Langmuir equation permitted us to deduce it (\(\partial Q/\partial C\)) and replace it with the following expression:  
\[(\frac{\partial Q}{\partial C})=\frac{K_{L}Q_{\max}}{(1+K_{L}C)^{2}} \tag{21}\]  
From this equation, and by considering the following calculation as a first approximation, (\(\partial Q_{t}/\partial C_{t}\)) was determined at the reaction time values of the different experimental points representative of the second stage of the Weber-Morris kinetics model. These calculations permitted us to show that, for each element, the (\(\partial Q_{t}/\partial C_{t}\)) values remained close enough to (\(\partial Q/\partial C\))eq, which enabled the application of Equation (20) to the delimited intra-particle diffusion zone (second step) revealed by Weber-Morris kinetics curve. On the other hand, as shown in Section 2.9, the ionic diffusivity of each cation could be attained by using the values of the infinite dilution diffusion coefficients \(D_{1}\)\({}^{\circ}\) for cations (Cs\({}^{+}\), Rb\({}^{+}\), Co\({}^{2+}\), or Sr\({}^{2+}\)) and \(D_{2}\)\({}^{\circ}\) for anions (NO\({}_{3}\)\({}^{-}\)) (D\({}_{1}\)\({}^{\circ}\) and D\({}_{2}\)\({}^{\circ}\) values are given in Table S2) in Equation (8). The values of \(D\) are listed in Table S2. From Equations (6) and (7), it was then possible to calculate the \(D_{p}\) coefficient for each cation studied. The data are listed in Table 3.  
Afterwards, the determination of \(D_{p}\) and \(D_{eff}\) allowed us to estimate the surface diffusion coefficients (\(D_{s}\)) for cation adsorption on brick composites from aqueous solutions at 298 K using Equations (20) and (21) (Table 3).