Name	Isotherm Equation	Uses and notes	Ref.
Adsorption reaction kinetics
PFO	$\frac{d q}{d t} = K_{1} (q_{e} - q)$ $\log (q_{e} - q_{t}) = \log q_{e} - \frac{K_{1}}{2.303} t$ Modified $\frac{d q}{d t} K_{1} = \frac{q_{e}}{q} (q_{e} - q)$ $\frac{q}{q_{e}} + \ln (q_{e} - q) = \ln (q_{e}) - K_{1} t; \begin{matrix} K_{1} \end{matrix} = \frac{D π^{2}}{r^{2}}$	o Valid only at the initial stage of adsorption, where, k₁ decrease with increasing C_o, time & particle size Its q_e is often much farther from the experimental value Affected by reaction conditions (pH, Concentration)	[62]
PSO	$\frac{d q}{d t} = K_{1} {(q_{e} - q)}^{2}$ $\frac{t}{q_{t}} = \frac{1}{k_{2} q_{e}^{2}} + \frac{1}{q_{e}} t$	o q_e is often less than, but close to, the experimental value and K₂ decrease with increasing initial concentrations, time and particle size	[62]
Elovich	$\frac{d q}{d t} = α \exp (- β q)$ $q = \frac{1}{β} \ln (α β) + \frac{1}{β} \ln t$	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 & E_a for chemisorption, mostly both increases with increasing C_o	[5] [63]
First-order reversible	$\frac{d C_{B}}{d t} = - \frac{d C_{A}}{d t} = k_{a} C_{A} - k_{d} C_{B}$ & $\frac{K_{a}}{K_{d}} - \frac{C_{B e}}{C_{A e}}$ $\ln (1 - \frac{C_{A O} - C_{A}}{C_{A O} - C_{e}}) = - (k_{a} + k_{d}) t$	o Limiting form for Langmuir kinetics model when adsorption is in the Henry regime, where k_a is adsorption rate constant & k_d is desorption rate constant and C_A₀, C_A and C_e are initial bulk, at time t and equilibrium concentrations (mg/L)	[64]
Avrami	$\frac{d q}{d t} = k \cdot n \cdot t^{n - 1} (q_{e} - q)$ * $q = q_{e} - q_{e} \exp (- k t^{n})$ $\ln (\ln \frac{q_{e}}{q_{e} - q}) = n \cdot \ln t + n \cdot \ln k$	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.	[65]
General	$\frac{d q}{d t} = k_{n} {(q_{e} - q)}^{n}$ ** $q = q_{e} - \frac{q_{e}}{{(k_{n} q_{e}^{n - 1} t (n - 1) + 1)}^{\frac{1}{n - 1}}}; n \neq 1$	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.	[66]
	Combined (Avrami* and General**) $\frac{d q}{d t} = k_{n} t^{m - 1} {(q_{e} - q)}^{n}$ $q = q_{e} - \frac{1}{{(\frac{k_{n} (n - 1) t^{m}}{m} + \frac{1}{q_{e}^{n - 1}})}^{\frac{1}{n - 1}}}; n \neq 1$
Adsorption?diffusion model
Crank	$\frac{q}{q_{e}} = 1 + \frac{2 R}{π r} \sum_{n - 1}^{\infty} \frac{{(- 1)}^{n}}{n} \sin \frac{n π r}{R} \exp (\frac{- D n^{2} π^{2} t}{R^{2}})$ * $\bar{q} = \frac{3}{R^{2}} \int_{0}^{R} q r^{2} d r$ ** inserting* into ** becomes $\frac{\bar{q}}{q_{e}} = 1 - \frac{6}{π^{2}} \sum_{n - 1}^{\infty} \frac{1}{n^{2}} \exp (\frac{- D n^{2} π^{2} t}{R^{2}})$	o D and r, respectively denote the intraparticle diffusivity (cm²/min) and the radial distance (cm) from the center of the spherical particles. $\bar{q}$ 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	[66]