Name Isotherm Equation Uses and notes Ref. Adsorption reaction kinetics PFO $\frac{\text{d}q}{\text{d}t}={K}_{1}\left({q}_{e}-q\right)$ $\mathrm{log}\left({q}_{e}-{q}_{t}\right)=\mathrm{log}{q}_{e}-\frac{{K}_{1}}{2.303}t$ Modified $\frac{\text{d}q}{\text{d}t}{K}_{1}=\frac{{q}_{e}}{q}\left({q}_{e}-q\right)$ $\frac{q}{{q}_{e}}+\mathrm{ln}\left({q}_{e}-q\right)=\mathrm{ln}\left({q}_{e}\right)-{K}_{1}t;\begin{array}{c}{K}_{1}\end{array}=\frac{D{\text{π}}^{2}}{{r}^{2}}$ o Valid only at the initial stage of adsorption, where, k1 decrease with increasing Co, time & particle size Its qe is often much farther from the experimental value Affected by reaction conditions (pH, Concentration)  PSO $\frac{\text{d}q}{\text{d}t}={K}_{1}{\left({q}_{e}-q\right)}^{2}$ $\frac{t}{{q}_{t}}=\frac{1}{{k}_{2}{q}_{e}^{2}}+\frac{1}{{q}_{e}}t$ o qe is often less than, but close to, the experimental value and K2 decrease with increasing initial concentrations, time and particle size  Elovich $\frac{\text{d}q}{\text{d}t}=\alpha \mathrm{exp}\left(-\beta q\right)$ $q=\frac{1}{\beta }\mathrm{ln}\left(\alpha \beta \right)+\frac{1}{\beta }\mathrm{ln}t$ o Suitable for kinetics far from equilibrium where desorption does not occur, where α is the initial sorption rate (mg/g・min), β is a desorption constant related to the extent of θ & Ea for chemisorption, mostly both increases with increasing Co   First-order reversible $\frac{\text{d}{C}_{B}}{\text{d}t}=-\frac{\text{d}{C}_{A}}{\text{d}t}={k}_{a}{C}_{A}-{k}_{d}{C}_{B}$ & $\frac{{K}_{a}}{{K}_{d}}-\frac{{C}_{Be}}{{C}_{Ae}}$ $\mathrm{ln}\left(1-\frac{{C}_{AO}-{C}_{A}}{{C}_{AO}-{C}_{e}}\right)=-\left({k}_{a}+{k}_{d}\right)t$ o Limiting form for Langmuir kinetics model when adsorption is in the Henry regime, where ka is adsorption rate constant & kd is desorption rate constant and CA0, CA and Ce are initial bulk, at time t and equilibrium concentrations (mg/L)  Avrami $\frac{\text{d}q}{\text{d}t}=k\cdot n\cdot {t}^{n-1}\left({q}_{e}-q\right)$ * $q={q}_{e}-{q}_{e}\mathrm{exp}\left(-k{t}^{n}\right)$ $\mathrm{ln}\left(\mathrm{ln}\frac{{q}_{e}}{{q}_{e}-q}\right)=n\cdot \mathrm{ln}t+n\cdot \mathrm{ln}k$ o Kinetic system that describes a time-dependent rate coefficient (fractal-like kinetics), where n is a model constant related to the adsorption mechanism and its value can be integer or fraction.  General $\frac{\text{d}q}{\text{d}t}={k}_{n}{\left({q}_{e}-q\right)}^{n}$ ** $q={q}_{e}-\frac{{q}_{e}}{{\left({k}_{n}{q}_{e}^{n-1}t\left(n-1\right)+1\right)}^{\frac{1}{n-1}}};n\ne 1$ o Developed to compensate for the deficiencies of PFO and PSO, n can be an integer or non-integer rational number, and must be determined by an experiment.  Combined (Avrami* and General**) $\frac{\text{d}q}{\text{d}t}={k}_{n}{t}^{m-1}{\left({q}_{e}-q\right)}^{n}$ $q={q}_{e}-\frac{1}{{\left(\frac{{k}_{n}\left(n-1\right){t}^{m}}{m}+\frac{1}{{q}_{e}^{n-1}}\right)}^{\frac{1}{n-1}}};n\ne 1$ Adsorption?diffusion model Crank $\frac{q}{{q}_{e}}=1+\frac{2R}{\text{π}r}\sum _{n-1}^{\infty }\frac{{\left(-1\right)}^{n}}{n}\mathrm{sin}\frac{n\text{π}r}{R}\mathrm{exp}\left(\frac{-D{n}^{2}{\text{π}}^{2}t}{{R}^{2}}\right)$ * $\overline{q}=\frac{3}{{R}^{2}}{\int }_{0}^{R}q{r}^{2}\text{d}r$ ** inserting* into ** becomes $\frac{\overline{q}}{{q}_{e}}=1-\frac{6}{{\text{π}}^{2}}\sum _{n-1}^{\infty }\frac{1}{{n}^{2}}\mathrm{exp}\left(\frac{-D{n}^{2}{\text{π}}^{2}t}{{R}^{2}}\right)$ o D and r, respectively denote the intraparticle diffusivity (cm2/min) and the radial distance (cm) from the center of the spherical particles. $\overline{q}$ is average value of q in the spherical particle of radius R at a time, t. External diffusion and surface reaction are assumed to be more rapid than IPD