Vermeule | $\frac{\overline{q}}{{q}_{e}}=\sqrt{1-\mathrm{exp}\left(-\frac{D{\text{\pi}}^{2}t}{{R}^{2}}\right)}$ $\mathrm{ln}\left(1-{\left(\frac{\overline{q}}{{q}_{e}}\right)}^{2}\right)=-\frac{D{\text{\pi}}^{2}t}{{R}^{2}}$ * | 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 | [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 k | [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 C | [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, $\gamma =\frac{{C}_{o}-C}{{q}_{e}},k=\frac{4\text{\pi}{r}_{o}D}{\gamma}\&\tau =\frac{{r}_{o}^{2}}{D\text{\pi}}$ $q={q}_{e}\frac{\mathrm{exp}\left(at+b{t}^{1/2}\right)-1}{{u}_{eq}\mathrm{exp}\left(at+b{t}^{1/2}\right)-1}$ 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, r | [71] |

Multi- exponential | $q={q}_{e}\left[1-\frac{{\displaystyle {\sum}_{i=1}^{N}{a}_{i}\mathrm{exp}\left(-{k}_{i}t\right)}}{{\displaystyle {\sum}_{i=1}^{N}{a}_{i}}}\right]$ | o Has multiple parallel routes that contribute to the total adsorbate uptake by different small and large sites, where k | [35] |