Vermeule $\frac{\overline{q}}{{q}_{e}}=\sqrt{1-\mathrm{exp}\left(-\frac{D{\text{π}}^{2}t}{{R}^{2}}\right)}$ $\mathrm{ln}\left(1-{\left(\frac{\overline{q}}{{q}_{e}}\right)}^{2}\right)=-\frac{D{\text{π}}^{2}t}{{R}^{2}}$ * 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 kid increases with increasing Co. To say the kinetics controlled by IPD model the line should pass through origin. [68] Bangham $\mathrm{log}\left(\mathrm{log}\frac{{C}_{e}}{{C}_{o}-q\cdot m}\right)=\mathrm{log}\left(\frac{{K}_{o}m}{2.303V}\right)+\alpha \mathrm{log}t$ o Assumes IPD to be the only rate-controlling step and used to check whether pore diffusion is the sole rate-controlling mechanism, where ko and α are constants. It also checks whether pore diffusion is the sole rate-controlling mechanism [69] Linear film diffusion $\frac{\text{d}c}{\text{d}t}=-kf\left(C-{C}_{s}\right)$ $\frac{C}{{C}_{o}}=\mathrm{exp}\left(-kft\right)$ o Cs = adsorbate concentration at the liquid?solid interface (mg/L). At short times, Cs is negligibly small (Cs ≈ 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, $\gamma =\frac{{C}_{o}-C}{{q}_{e}},k=\frac{4\text{π}{r}_{o}D}{\gamma }&\tau =\frac{{r}_{o}^{2}}{D\text{π}}$ $q={q}_{e}\frac{\mathrm{exp}\left(at+b{t}^{1/2}\right)-1}{{u}_{eq}\mathrm{exp}\left(at+b{t}^{1/2}\right)-1}$ 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, ro = particle radius (cm); D = film diffusivity (cm2/min) [71] Multi- exponential $q={q}_{e}\left[1-\frac{{\sum }_{i=1}^{N}{a}_{i}\mathrm{exp}\left(-{k}_{i}t\right)}{{\sum }_{i=1}^{N}{a}_{i}}\right]$ o Has multiple parallel routes that contribute to the total adsorbate uptake by different small and large sites, where ki is the rate coefficient for route i, ai 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]