### Where ki

Where ki (mg/g min1/2) is the intraparticle diffusion rate coefficient, the ?t value is a mathematical function of F = Qt/Qe. Di is the effective diffusion coefficient, and ? and k? are constants. Qe (mg/g) and Qt (mg/g) are the sorption capacity at equilibrium and after time t (min). Particle diffusion mechanism for sorption of Br.G, To.B and Tr.B onto [email protected] was studied with Morris–Weber model (Weber and Morris, 1963). Plotting Qt vs. t1/2 give straight line did not pass through the origin with Ki=0.74, 0.85 and 0.57 mg/g min1/2, (R2 = 0.53, 0.59, 0.73) as shown in fig.1b. The intercept values (film thickness) were 1.01, 0.71 and 1.03 for the sorption of Br.G, To.B and Tr.B. The intercepted values are due to streaming fluid for sorption of Br.G, To.B and Tr.B dyes onto [email protected] and the higher the value of it, the greater the boundary layer effect (Salem et el., 2016). The deviation of the linear plot consist of two steps (Moawed et el., 2014). The first step, the diffusion rate is high then, decreased with the time pass in the second step. The values of ki(1) , ki(2) of these steps are (2.74 , 0.089 mg g-1 min1/2) for Br.G , (2.65 , 0.032 mg/g min1/2) for To.B and (1.35 , 0.2 mg/g min1/2) for Tr.B. ki(2) decrease as the active centers were blocked by sorbed dye molecules. The first step is a transport step after transfer the adsorbed solution into adsorbent, it is too fast step. The second step is equilibrium step which represents diffusion of adsorbate molecules from the external of the adsorbent into the pores of adsorbent molecules (Tran et el., 2017). Rate of diffusion of To.B > Br.G >Tr.B which mean ki dependent on the molecular size of dyes. The lighter molecule’s weight as To.B will diffuse faster in contract Tr.B which has heavier molecule’s weight. Larger ki, better adsorption mechanism which related to improved bonding between adsorbate molecules with adsorbent particles.

The kinetic data used further to know the slowness of adsorption step according to Bangham model (Nowak and Bangham, 1996). The double logarithmic plots against time doesn’t give perfect line with a correlation coefficient (R2= 0.69, 0.74 and 0.77) for sorption of Br.G, To.B and Tr.B dyes onto [email protected], respectively (Fig.1CS). This means that the film diffusion is not the sole rate-controlling step. The values of ? (Table 3) are 0.47, 0.55 and 0.40 for Br.G, To.B and Tr.B dyes which means ? value independent of the dye size.

The plot of Bt versus t is straight line do not pass through the origin (Fig.1DS) proving that mass transfer is involved the values of the effective diffusion coefficient (Di) by plot F VS. t0.5 f=((6)?(R)((D_i t)?(?)))1/2.The results show that Di values are 8.74 ×10-7, 1.15 ×10-7 and 1.36 ×10-7 cm/min for Br.G, To.B and Tr.B dyes, respectively and it is independent of the size of the dyes