Kinetic adsorption models based on mass transfer^a.

Kinetic model	Equation	Eq. number
Mathews and Weber		(13)
Furusawa and Smith		(14)
Weber and Morris	q t = kWM(t)1/2 + c	(15)
Short time approach		(16)
Vermeulen		(17a)
Fractal-like Vermeulen		(17b)
Homogeneous internal diffusion model		(18)

^aWhere C is the concentration of the sorbate in the bulk of the solution; C₀ is the initial concentration; F = q/q_e the fraction of attainment of equilibrium at time t; R_p is the particle radius where the particle is assumed to be spherical, 0.015 cm; m_s is the adsorbent mass per unit of the volume of solution (g L⁻¹); ε_p is particle porosity; ρ_p is the particle density (g L⁻¹); S_A is the external particle area per unit of the volume of solution cm⁻¹; k_WM (mg g⁻¹ min^−0.5) is the internal diffusion constant and c (mg g⁻¹) is the constant related to the external mass transfer resistance in the boundary layer. k_MW and k_FS are the external mass transfer coefficient (m s⁻¹) of the Mathews and Weber, and Furusawa and Smith models, respectively. K is a constant (L g⁻¹). D_st, D_app, D_V, and D_Vf (m² s⁻¹) are the effective diffusion coefficient of the solute in the solid phase given by the short time approach, homogeneous intraparticular, Vermeulen and fractal-like Vermeulen models, respectively.