## 3 Analysis of the Final Equilibrium State
### Calculation of Final Equilibrium State
We should point out that our aim is to reduce \(\lambda\) as much as possible because this would mean that the toxins would be efficiently removed from the largest part of the duct. It is, therefore, evident that \(\lambda\) decreases with the increase in potential as well as with the increase in the charge of toxins. It is observed (Figure 6) that regarding the first and second toxins with z = 2.318 and z = 2.392 (red and green lines, respectively), for potential \(\varphi(0)=0.1\) V, \(\frac{\lambda}{\lambda_{D}}\) is of order \(10^{-2}\). Therefore, \(\lambda\) is two orders of magnitude smaller than \(\lambda_{D}\) (i.e., according to Table 2 \(\sim 10^{-7}\) m), for the third toxin with z = 1.176 (blue line) one order of magnitude smaller (i.e., \(\sim 10^{-6}\) m ), whereas for the fourth toxin with z = 7.644 (black line) four orders of magnitude smaller (i.e., \(\sim 10^{-10}\) m). This means that \(\lambda\) is almost negligible compared to the L (total width of the duct) and it is considered at least \(L\sim 10^{-4}\) m. Conclusively, the largest part of the duct remains with the target toxin concentration \(C_{M}\), after the application of the electric field. Moreover, using a smaller width duct inside the larger duct in which the electric field is applied, the decontaminated solution can be collected, as previously presented in Figure 2.  
Throughout the previous analysis, it is considered that the distribution of charged toxins along the duct follows the Boltzmann distribution. Obviously, this is true outside the compact layer. Moreover, there is symmetry at the two opposite electrodes. When the toxins are negatively charged (i.e., the distribution near the positively charged electrode), toxin concentration is expressed as:  
\[C_{-}=C_{M}e^{+\frac{\frac{\pi}{RT}}{\varphi}} \tag{15}\]  
where potential \(\varphi\) is given by Equations (8) and (9). As already mentioned, \(C_{M}\) is the concentration at the center of the duct after the application of the electric field and after the final equilibrium state has occurred. We consider it equal to the upper limit of the concentration we want to achieve, i.e.,  
\[C_{M}=56\times 10^{-9\text{ mol}/\text{m}^{3}}\]  
Let's symbolize with \(C_{Ref}\) the uniform concentration of the toxin before applying the field. From the principle of the mass conservation, it can easily be derived that it is associated with the \(C_{M}\) with the relationship:  
\[C_{Ref}=\frac{C_{M}\int_{\lambda_{s}}^{L-\lambda_{s}}e^{+\frac{\pi}{RT}\varphi} dy}{L}\]  
or  
\[\frac{C_{M}}{C_{Ref}}=\frac{L}{\int_{\lambda_{s}}^{L-\lambda_{s}}e^{+\frac{ \pi}{RT}\varphi}dy} \tag{16}\]  
In Figures 7-10 the fraction \(\frac{C_{M}}{C_{Ref}}\) is represented as a function of the width of the duct for various potentials for the four toxins with the desired final concentration \(C_{M}=56\times 10^{-9\text{ mol}}\). Figure 8: \(\frac{C_{M}}{C_{M}}\) as a function of \(L\) (m) for toxin 2, concentration \(C_{M}=56\times 10^{-9}\frac{\text{mol}}{\text{m}^{3}}\) for various potentials \(\varphi(0)\) (V). Considering the numerator constant in the fraction \(\frac{C_{M}}{C_{brf}}\), this becomes smaller the greater the concentration \(C_{brf}\) is, before the application of the field. Therefore, the smaller this ratio becomes, the better results are achieved in terms of reducing the amount of initial toxin. From the Figures 7-10, it is evident that the higher potential and the lower width of the duct decrease the ratio \(\frac{C_{M}}{C_{brf}}\). We observe in all graphs that, for the lowest value of potential \(\varphi(0)\) (V), we do not have a noticeable decrease in concentration (black lines). The lowest value of potentials in all diagrams represents the limit at which the linear approximation applies (the linear approximation is developed is Section 4). However, for potential ten times and twenty times larger than the lowest value, and for ducts less than 1 cm wide, the decrease in concentration of ion toxins in the main volume of the solution is enormous.