Vermeule

$\frac{\overline{q}}{q_e}=\sqrt{1-\exp \left(-\frac{D{\text{π}}^{2}t}{R^2}\right)}$

$\ln \left(1-{\left(\frac{\overline{q}}{q_e}\right)}^2\right)=-\frac{D{\text{π}}^{2}t}{R^2}$ *

o Used to check whether IPD is the sole rate-limiting step, a straight line on a plot of $1-{\left(\overline{q}/q_e\right)}^2$ vs. t passing through the origin indicates IPD is the sole rate-controlling step.

[67]

Weber- Morris

$q=K_{id}\sqrt{t}+B$

o The third most common candidate model after PFO and PSO models, used for modeling adsorption kinetics limited by IPD, where B is the initial adsorption and k_id increases with increasing C_o. To say the kinetics controlled by IPD model the line should pass through origin.

[68]

Bangham

$\log \left(\log \frac{C_e}{C_o-q\cdot m}\right)=\log \left(\frac{K_{o}m}{2.303V}\right)+\alpha \log t$

o Assumes IPD to be the only rate-controlling step and used to check whether pore diffusion is the sole rate-controlling mechanism, where k_o and α are constants. It also checks whether pore diffusion is the sole rate-controlling mechanism

[69]

Linear film diffusion

$\frac{\text{d}c}{\text{d}t}=-kf\left(C-C_s\right)$

$\frac{C}{C_o}=\exp \left(-kft\right)$

o C_s = adsorbate concentration at the liquid?solid interface (mg/L). At short times, Cs is negligibly small (C_s ≈ 0)

[70]

MSRDCK

$\frac{\text{d}q}{\text{d}t}=k\left(1+\frac{{\tau }^{1/2}}{t^{1/2}}\right)\left(C_o-\gamma q\right)\left(q_e-q\right)$

Where, $\gamma =\frac{C_o-C}{q_e},k=\frac{4\text{π}{r}_{o}D}{\gamma }\&\tau =\frac{r_o^2}{D\text{π}}$

$q=q_e\frac{\exp \left(at+b{t}^{1/2}\right)-1}{u_{eq}\exp \left(at+b{t}^{1/2}\right)-1}$

where $u_{eq}=1-\frac{C_e}{C_o},a=k{C}_o\left(u_{eq}-1\right)$ &

$b=2k{C}_o{\tau }^{1/2}\left(u_{eq}-1\right)$

o A kinetic model that includes the surface reaction and film diffusion, where control the constant a accounts surface reaction & b surface diffusion and film diffusion, r_o = particle radius (cm); D = film diffusivity (cm²/min)

[71]

Multi- exponential

$q=q_e\left[1-\frac{{\displaystyle {\sum }_{i=1}^N{a}_i\exp \left(-k_{i}t\right)}}{{\displaystyle {\sum }_{i=1}^N{a}_i}}\right]$

o Has multiple parallel routes that contribute to the total adsorbate uptake by different small and large sites, where k_i is the rate coefficient for route i, a_i is the weight coefficient that reflects the share of route i (N) and for N = 1, this model is reduced to the PFO model

[35]