Table 3 Evaluated kinetic models for biosorptive removal of BF dye by Gel@GO-F-ZrSiO₄@Gel.

Pseudo-first order

ln (qe − qt) = ln

qe − k₁t

qe and qt are the eliminated quantity of BF dye (mg g⁻¹) at equilibrium and time t (min, respectively, while k₁ is pseudo-first order rate constant (min⁻¹)

ln(qe − qt)

versus time (t)

Pseudo-second order

t/qt = 1/k₂qe² + t/qe

qe and qt are the eliminated quantity of BF (mg g⁻¹) at equilibrium and at time t (min), respectively, while k₂ is the pseudo second order rate constant (g/(mg min)

(t/qt) versus

time (t)

Intraparticle diffusion

qt = k_id t ^½ + C

kid is the intraparticle diffusion rate constant (mg g⁻¹ min ^−1/2). C is the boundary layer thickness (mg g⁻¹)

(q_t) against

(t^1/2)

Elovich

q_t =\(\frac{1}{\upbeta }\) ln(αβ) + \(\frac{1}{\upbeta }\) ln t

β is the activation energy, α is the initial elimination rate (mg g⁻¹ min) and the surface coverage of

chemisorption

(q_t) versus ln t