Table 2 Sorption kinetics models and their linear and non-linear forms with corresponding plots.

Pseudo first order

\({q}_{t}={q}_{e}\left(1-{e}^{-{k}_{1}t}\right)\)

\(log\left[{q}_{e}- {q}_{t}\right]=log{q}_{e}- \left[\frac{{k}_{1}}{2.303}\right]t\)

k₁ (min⁻¹): rate constant for pseudo first order

q_e (mg g⁻¹): sorption capacity at equilibrium

q_t (mg g⁻¹): Sorption capacity at time “t”

\(log\left[{q}_{e}- {q}_{t}\right]\) vs \(t\)

Pseudo second order

\({q}_{t}=\frac{\left({k}_{2}{q}_{e}^{2}.t\right)}{\left(1+{k}_{2}{q}_{e}^{2}.t\right)}\)

\(\frac{t}{{q}_{t}}= \frac{1}{{k}_{2}{\left({q}_{e}\right)}^{2}}+ \frac{1}{{q}_{e}}\left(t\right)\)

k₂ (g mg⁻¹ min⁻¹): rate constant for pseudo first order

\(\frac{t}{{q}_{t}}\) vs \(t\)

Elovich

\({q}_{t}=\beta ln\left(\alpha \beta t\right)\)

\({q}_{t}=\frac{1}{\beta }ln\left[\alpha \beta \right]+ \frac{1}{\beta }ln\left[t\right]\)

α (mgg⁻¹ min⁻¹): rate of sorption process

β (gmg⁻¹): activation energy for chemical sorption process

\({q}_{t}\) vs \(ln\left[t\right]\)

Intraparticle diffusion

\({q}_{t}= {K}_{ip}{t}^{0.5}\)

\({q}_{t}= {K}_{ip}{t}^{0.5}+ I\)

K_ip (mg g⁻¹ min^0.5): constant for intraparticle diffusion

I (mg g⁻¹): constant related with thickness of boundary layer

\({q}_{t}\) vs \({t}^{0.5}\)

Liquid film diffusion

–

\(ln\left[1-F\right]= -{k}_{fd}\left(t\right)\)

F: (q_t/q_e): fractional attainment of equilibrium

k_fd (min⁻¹): rate constant of liquid film diffusion

\(-ln\left[ 1-F\right]\) vs \(t\)