## 3 Results and Discussion
### Transport Study of Cr(VI) through Aliquat 336/PolyHIPE Membrane
#### 3.4.3 Modeling and Parameters of Cr(VI) Diffusion
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**[Cr(VI)]\({}_{\rm b}\)** & **Feeding Phase** & **Membrane** & & \(J_{f}\)\({}^{\rm b}\) & **Membrane** & \\  & & **Type/Base** & **Carrier** & & **Thickness** & \\ \hline
**(mol/L)** & **pH** & **Polymer** & & **(mol/m\({}^{2}\) s)** & **(um)** & \\ \hline \(2.0\times 10^{-4}\) & 0.12 & PIM/CTA & Calix[4]arene & 2.253 & 42 & [39] \\ \(2.0\times 10^{-3}\) & 1 & PIM/CTA & Aliquat 336 & 8.84 & 28 & [40] \\ \(2.3\times 10^{-4}\) & 1.2 & PIM/CTA & Aliquat 336 & 3.11 & 80 & [41] \\ \(1.8\times 10^{-6}\) & 8 & PIM/CTA & Aliquat 336 & 0.002 & 62 & [42] \\ \(1.0\times 10^{-3}\) & 1 & ACM & Cyaney 923 & 10.91 & 90 & [14] \\ \(1.0\times 10^{-3}\) & 1 & ACM & Cyaney 923 & 11.94 & 25 & [14] \\ \(3\times 10^{-4}\) & 2 & SLM & CYPHOS IL101 & 5.5 & 125 & [43] \\ \(7.5\times 10^{-4}\) & 0 & SLM & Cyaney 921 & 6.7 & 125 & [44] \\ \(1.0\times 10^{-3}\) & 4 & SLM/PolyHIPE & Aliquat 336 & 14.38 & 120 & this study \\ \hline \hline \end{tabular} Note: polymer inclusion membrane (PIM), cellulose triacetate (CTA), activated composite membrane (ACM).  
\end{table} 
Table 3: Comparison of the Cr(VI) transport efficiency with different membranes. \[[AC]_{f}=\frac{[A]_{o}\times K\times[C]_{f}}{1+K\times[C]_{f}} \tag{18}\]  
Since the formation of Cr(VI) complex is a fast process, based on the quasi-steady state assumption \(J_{m}{}^{0}\approx J_{f}{}^{0}\), the initial flux can be described as  
\[J_{f}{}^{0}=\left(\frac{D^{*}}{l}\right)\frac{\left([A]_{o}K[C]_{fo}\right)}{1+ K[C]_{fo}} \tag{19}\]  
The postulated mechanism indicates that \(J_{f}{}^{0}\) is related to the effective diffusion coefficient and formation constant of Cr(VI)-Aliquat 336 complex. In order to test the proposed relationship, Equation (19) was rearranged to a linear form, as in Equation (20). \[\frac{1}{J_{f}{}^{0}}=\left(\frac{l}{D^{*}}\right)\left(\frac{1}{[A]_{0}K} \right)\left(\frac{1}{[C]_{fo}}\right)+\left(\frac{l}{D^{*}}\right)\left( \frac{1}{[A]_{0}}\right) \tag{20}\]  
Transport experiments were carried out with \([C]_{fo}\) which varied in the range 0.2 \(\times\) 10\({}^{-3}\)-4 \(\times\) 10\({}^{-3}\) M. By plotting the values of \(1/J_{f}{}^{0}\) vs. \(1/[C]_{fo}\), the effective diffusion coefficient and formation constant of the complexes can be obtained from the slope and intercept, respectively. \[D^{*}=\frac{l}{\left[\overline{A}\right]_{o}\times\textit{intercept}} \tag{21}\]  
\[K=\frac{\textit{intercept}}{\textit{slope}} \tag{22}\]  
Figure 7 shows that the plot of \(1/J_{f}{}^{0}\) vs. \(1/[C]_{fo}\), is linear with high correlation coefficient (\(R^{2}\) = 0.99). Hence, the experimental results are in agreement with a mechanism in which the diffusion of the complex is the rate-determining step. Moreover, it confirms that the complex involved in the transport process has 1:1 stoichiometry as the HCrO\({}_{4}{}^{-}\)-Aliquat 336 complex characterized as Equation (10), since under the solution condition, over 90% of Cr(VI) species is HCrO\({}_{4}{}^{-}\). From the slope and intercept of the above result (Figure 7), the effective diffusion coefficient \(D^{*}\) is about 1.75 \(\times\) 10\({}^{-11}\) m\({}^{2}\) s\({}^{-1}\), and the formation constant K is about 5.68 mol\({}^{-1}\) L. By substituting the  
Figure 7: Plots of \(1/J_{f}{}^{0}\) vs. \(1/[C]_{fo}\), for the transport of Cr(VI) across the Aliquat 336/PolyHIPE membrane. Feeding phase: pH 4; stripping phase: pH 12. Stirring rate: 450 rpm. porosity (0.77), constrictive factor (3.3 \(\times\) 10\({}^{-3}\)), and tortuosity (1.18) into the equation \(D^{\star}\) = \(D\beta\Phi/\tau\), we can obtain the free bulk diffusion coefficient of Cr(VI)-complex (\(D\)) equal to 8.14 \(\times\) 10\({}^{-9}\) m\({}^{2}\) s\({}^{-1}\). For comparison, the free bulk diffusion coefficient for the Cr(VI)-Aliquat 336 complex was also estimated using Wilke-Chang equation (Equation (23)) [45].