In order to develop an equation describing the relation between the adsorption yield and three adsorption variables shown in Table 2, a BBD was conducted. Quadratic model and reduced cubic model were developed to correlate the variables to the response. Predicted values for yield by using reduced cubic model (Table 4) were closer to observed values than those by using quadratic model (Table 3). In quadratic model (Table 5), F-value of the model 5.46 implied that the model was statistically significant. There was only a 3.81 percent chance that a model F-value this large could occur due to noise. The fit of the model was checked by the determination coefficient (R2). In this case, the value of the determination coefficient (R2 = 0.9076) indicated that 9.24 percent of the total variable was not explained by the model. The value of adjusted determination coefficient (adjusted R2 = 0.7413) was low and it was not in reasonable agreement with the adjusted R2. Negative predicted R2 implied that the overall mean was a better predictor of the response than the current model. Significant lacks of fit and high value of the coefficient of variation were found. Probability values (greater than 0.1000) indicated that some of the model terms are not significant. But, reduced cubic model (Table 6) produced the closest predicted values, predicted R2 and adjusted R2 equal to 1, no lack of fit and low coefficient of variance. The model F-value implied that the model was significant, and there was only a 0.01 percent chance occurs due to noise. Thus, as a result of the statistical analysis, reduced cubic model was found satisfactory for describing the process and useful for developing empirical relation.
Lead removal showed to be very sensitive to changes in the temperature both in dilute and in concentrated solutions. The removal capacity of mordenite was sharply increased when the adsorption temperature increased from 20 to 50°C in dilute solutions; as it was also reported by Wang  (from 20 to 40°C; in a solution of 40 mg lead/L). No comparison can be made with the research by Dai et al.  since the temperature set constant at 25°C. For moderately concentrated solutions (1005 mg/L) this increase in yield was only 5 percent. The increase in yield due to increase in adsorption temperature in diluted solutions was more dominated than one in concentrated solutions.
Initial lead ion concentration was another parameter that has high effect on the response. As can be seen on Figure 1, the concentration of the aqueous solution increases the removal of lead decreases. About 84 percent yield was obtained in diluted solutions (10 mg/L). When the initial lead concentration increased to 100 mg/L, the yield decreased to 81 percent. Dai et al.  also observed that increase in initial lead concentration from 3 to 200 mg/L decreased the lead removal down to 70 percent. Also (Figure 1), the results of the study showed that this yield could also be achieved up to 500 mg/L of initial lead concentrations.
Adsorption time has little effect on lead removal. It was found that nearly 80–90 minute was enough to obtain highest yield in both dilute and concentrated solutions. The results obtained were in agreement with the work done by Dai  which reported that optimum time required to reach the equilibrium was 100 min. Also Wang  reported that adsorption time has little effect on yield and that the adsorption required 90 min.
Optimum conditions for the adsorption process were searched by numerical optimization section of the software by choosing different targets for initial lead ion concentration. As shown in Table 8, for dilute solutions (initial lead ion concentration up to 10 mg/L), the best local maximum was found to be at adsorption time of 85–90 min, adsorption temperature of 50°C, lead removal of nearly 84 percent, and desirability of 1.000. High desirability shows that the estimated function may represent the experimental model and desired conditions. As the concentration increases, desirability and lead removal percentage were found to be decreasing (Table 8). Finally, 97 percent lead adsorption was achieved by using HDTMA-mordenite at the same optimum conditions of unmodified-mordenite; i.e. initial lead ion concentrations: 10–100 mg/L. It can be concluded that, modification of mordenite or adjustment of adsorption medium (lowering up to pH 3)  produce nearly the same adsorption yield. But lowering the pH of the medium was useful for solutions containing 3 mg lead /L , whereas modification of mordenite was applicable up to 100 mg/L of lead solutions.
Fitting of adsorption data showed that (Table 9), Langmuir isotherm, thus monolayer adsorption, solely, was not suitable for the process. Equilibrium adsorption data were best represented by the Freundlich isotherm. Heterogeneous adsorption of lead on mordenite was also stated in the literature before . The obtained value of (1/n) (0.1 < 1/n < 1) demonstrated that favourable nature of both lead and the heterogeneity of the mordenite sites. The 1/n value of the present study was higher than those obtained (0.537 at 30°C; 0.555 at 40°C)  showing that higher adsorption intensity of the mordenite used. Negative Gibbs free energy (Table 10) indicated the spontaneous nature of adsorption at those temperatures. These results were well-matched with literature . Also, negative ∆H˚ values showed the adsorption of lead ions was an exothermic process. A negative enthalpy values were also reported for the adsorption of lead ions onto wollastonite, bentonite and mordenite . A positive ∆S˚ value corresponded to an increase in both the randomness at the solid-solution interface and the degree of freedom of the adsorbed species. Adsorption reaction rate was found 4.4 at 0°C (Figure 2). In a different study, lead adsorption data (at 20-40°C) onto a local mordenite were fitted well to several equations; pseudo-second order, parabolic diffusion and Elovich equations . It seemed that the steps of adsorbate transport from the solution to the surface of mordenite; such as film diffusion, pore diffusion, surface diffusion and adsorption are strongly affected on temperature.