## 3 Results and Discussion
### Transport Study of Cr(VI) through Aliquat 336/PolyHIPE Membrane
#### 3.4.3 Modeling and Parameters of Cr(VI) Diffusion  
Since the aforementioned \"apparent rate constant\" is a function of Cr(VI)-Aliquat 336 complex reaction and the solute diffusion rate, determination of the Cr(VI)-Aliquat 336 formation constant and diffusion coefficient is required in order to elucidate the transport behavior of Cr(VI) through PolyHIPE membrane. In this study, since the bulk phase is effectively stirred, i.e., when the aqueous diffusion layer is compressed to a minimum, the film diffusion of Cr(VI) between bulk phase and membrane interface is a fast process, aqueous analytical concentrations [\(C\)]; and [\(C\)]\({}_{\rm s}\) are uniform throughout the feeding and stripping phases, respectively, and the concentration of Cr(VI) complex changes linearly inside the membrane [45; 46; 47]. Assuming that the rate-determining step is the diffusion of Cr(VI) through the membrane, the formation and dissociation of the Cr(VI)-Aliquat 336 complex at the interfaces are fast processes [35]. The effective diffusivity (\(D^{\star}\)) is key, and the most convenient parameter to describe the process of diffusion through porous membrane, since the value of \(D^{\star}\) is directly related to the pore structure characteristics of PolyHIPE membrane due to the tortuosity and constrictivity effects. The determination of effective diffusion coefficient can generally be achieved from the relationship observed between the flux and the concentrations of diffusant in the feeding phase [46]. From [PERSON]'s first law, the diffusion flux inside the membrane (\(J_{m}\)) can be defined as Equation (15). \[J_{m}=D^{\star}\frac{d[AC]}{dl}=D^{\star}\frac{[AC]_{f}-[AC]_{s}}{l} \tag{15}\]  
where \(D^{\star}\) is the effective diffusion coefficient of Cr(VI)-Aliquat 336 complex inside the membrane, [\(AC\)] is the concentration of Cr(VI)-Aliquat 336 complex in the membrane phase, subscripts \(f\) and \(s\) refer to the feeding and stripping interfaces of the membrane, and \(l\) is the membrane thickness. Since the stripping phase is at very high pH (pH 12), the dissociation of Cr(VI)-Aliquat 336 is a fast process, and the concentration of complex [\(AC\)]\({}_{\rm S}\) is practically nil. On the other hand, the formation of complex is a fast heterogeneous equilibrium [37]. Therefore, the concentration of complex [\(AC\)]\({}_{\rm f}\) can be related to the concentration of Cr(VI) in the feeding phase according to the mass action law, Equation (16):  
\[[AC]_{f}=K[A]\ [C]_{f} \tag{16}\]  
where \(K\) is the formation constant of the Cr(VI)-Aliquat 336 complex, [\(A\)] and [\(C\)]\({}_{\rm f}\) are the concentrations of the Aliquat 336 in the membrane phase and the concentration of Cr(VI) in the feeding phase, respectively. The total carrier concentration [\(A\)]\({}_{o}\) immobilized in the membrane is constant, equal to the sum of the concentrations [\(A\)] and [\(AC\)]\({}_{\rm f}\) (Equation (17)). Therefore, the concentration of complex [\(AC\)]\({}_{\rm f}\) can be related to the concentration of Cr(VI) in the feeding phase by Equation (18). \[[A]_{o}=[A]+[AC]_{f} \tag{17}\]