Table 1.

Types of adsorption equations

Isotherm	Equation
Single component dsorption	$q=\frac{(C_o-C_f)V}{m}$
Langmuir isotherm	$q_e=\frac{q_m{K}_L{C}_e}{1+K_L{C}_e}$
Brunauer-Emmett eller (BET) isotherm	$\frac{1}{W\left(\frac{P_0}{P}-1\right)}=\frac{1}{W_{m}C}+\frac{\left(C-1\right)}{W_{m}C}\frac{P_0}{P}$
Freundlich isotherm	$q_e=K_F{C}_e^{\frac{1}{n}}$
Gibbs isotherm	$d\gamma =-\sum \left(\frac{n_i}{A}\right)d{\mu }_i$
Temkin Isotherm	$q_e=\left(\frac{\text{RT}}{B_T}\right)\text{ln}\left(A_T{C}_e\right)$
Dubinin-Radushkevich DRK) isotherm	$q_e=q_s\text{exp}(-K_{\text{ad}}{\epsilon }^2)$

Here q = the metal concentration retained in the sorbent phase (mg/g); C₀ = the initial concentrations of the metal ion in solution (mol/l); C_f = the initial and final concentrations of the metal ion in solution (mol/l); V = the solution volume (liters); m = the mass of sorbent (g); q_e = the quantity of metal adsorbed on the surface of the adsorbent (mg/g); C_e = the amount of metal present in the solution at equilibrium condition (mg/L); q_m = the maximum adsorption capacity of the adsorbent (mg/g); K_L = the Langmuir constant related to energy of adsorption (L/mg); K_F = the Freundlich constant; W = weight of gas adsorbed; P/P₀ = relative pressure; W_m = weight of adsorbate as monolayer; C = BET constant; γ = interfacial tension; (n_i/A) = the number of moles of component adsorbed per unit area; μ_i = the chemical potential of the surfactant solution; A_T = Temkin isotherm equilibrium binding constant (L/g); B_T = Temkin isotherm constant; R = universal gas constant (8.314 J/mol/K); T = Temperature (K); qs = theoretical isotherm saturation capacity (mg/g); K_ad = Dubinin–Radushkevich isotherm constant (mol²/kJ²) and ε = Dubinin–Radushkevich isotherm constant