Table 1 Mathematical representation of the four adsorption kinetic models investigated in the present study.

From: Adsorptive removal of phosphate from water with biochar from acacia tree modified with iron and magnesium oxides

Model

Equation

Linear form

XY plot

Pseudo 1st order

\(\frac{{dq_{t} }}{dt} = k_{1} (q_{e} - q_{t} )\)

\(ln(q_{e} - q_{t} ) = ln(q_{e} )\) − k1t

qt vs ln t

Pseudo 2nd order

\(\frac{{dq_{t} }}{dt} = k_{2} (q_{e} - q_{t} )^{2}\)

\(\frac{t}{{q_{t} }} = \frac{1}{{k_{2} q_{e}^{2} }} + \left( {\frac{1}{{q_{e} }}} \right)t\)

qt vs ln t

Elovich

\(\frac{{dq_{t} }}{dt} = \alpha \;\exp ( - \beta q_{t} )\)

\(q_{t} = \frac{1}{\beta }ln(\alpha \beta ) + \left( {\frac{1}{\beta }} \right)ln(t)\)

qt vs ln t

Intra-particle diffusion

\(q_{t} = K_{IPD} \sqrt t + c\)

\(q_{t} = K_{IPD} \sqrt t + c\)

qt vs \(\sqrt t\)

  1. Where adsorption capacity at any given time (t) as well as equilibrium adsorption capacity are denoted by qt and qe, respectively (in mg/g). Moreover, rate constants corresponding to the pseudo 1st and 2nd order models (in 1/min and g/mg/min) are denoted by k1 and k2, respectively. Moreover, Elovich and intra-particle diffusion model’s parameters (α: initial rate of adsorption (in g/mg) as well as desorption (in mg/g/min) were used. Similarly, intraparticle intercept (in mg/g) as well as intraparticle intercept constant (in mg/g·min) were denoted by KIPD as well as C, respectively37.
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