Table 1 Kinetics and isotherms models of adsorption process.

Pseudo-first-order kinetic

ln (q_e − q_t) = ln q_e − k₁t

(5)

ln(q_e − q_t) vs. t

k₁ is the pseudo-first-order adsorption rate constant; q_e is the amount of antibiotic adsorbed at saturation per gram of adsorbent (mg g⁻¹), q_t is the amount of antibiotic adsorbed at time t per gram of adsorbent (mg g⁻¹)

Pseudo second-order kinetic

\(\frac{\mathrm{t}}{{\mathrm{q}}_{\mathrm{t}}}=\left[\frac{1}{{\mathrm{k}}_{2}{\mathrm{qe}}^{2}}\right]+\frac{1}{{\mathrm{q}}_{\mathrm{e}}}\mathrm{ t}\)

(6)

t/q_t vs. t

k₂ is adsorption rate constant of the pseudo-second-order

Intraparticle diffusion kinetic

q_t = k_i t^1/2 + C

(7)

q_t vs. t^1/2

k_i (mg g^-1 min^−1/2) is the intraparticle diffusion rate constant, which is the slope of the straight line of q_t versus t^1/2; C is the value of intercept, which is a constant reflecting the significance of the boundary layer or mass transfer effect

Langmuir isotherm

\(\frac{{\mathrm{q}}_{\mathrm{e}}}{{\mathrm{C}}_{\mathrm{e}}}=\frac{1}{{\mathrm{K}}_{\mathrm{L}}{\mathrm{q}}_{\mathrm{m}}}+\frac{{\mathrm{C}}_{\mathrm{e}}}{{\mathrm{q}}_{\mathrm{m}}}\)

(8)

(C_e/q_e) vs. C_e

q_e is the solid-phase concentration in equilibrium with the liquid-phase; concentration C_e is expressed in mole L⁻¹; q_m is the maximum monolayer adsorption capacity (mg g⁻¹); and K_L is an equilibrium constant (L mol⁻¹)

Freundlich isotherm

\({\mathrm{lnq}}_{\mathrm{e}}=\mathrm{ ln}{\mathrm{K}}_{\mathrm{f}}+\frac{1}{\mathrm{n}}\mathrm{ ln}{\mathrm{C}}_{\mathrm{e}}\)

(9)

ln q_e vs. ln C_e

plotting ln q_e versus ln C_e gives a straight line with slope of 1/n, where n is a constant related to adsorption intensity and its magnitude shows an indication of the favorability of adsorption; the intercept is ln K_f where K_f is constant (function of energy of adsorption and temperature)