Table 1 Theoretical model equations.
From: Adsorption of Fe-modified peanut shell biochar for Pb(II) in mixed Pb(II), Cu(II), Ni(II) solutions
Models | Equations | Refs. |
|---|---|---|
Quasi-first (QFDM) | \(q_{t} = q_{e} \left( {1 - e^{{ - k_{1} t}} } \right)\) | |
Quasi-second (QSDM) | \(q_{t} = q_{e}^{2} k_{2} t/\left( {1 + q_{e} k_{2} t} \right)\) | |
Chi-square (χ2) | \(\chi^{2} = \frac{{\sum\nolimits_{i = 1}^{n} {\left( {q_{e,cal} - q_{e,\exp } } \right)_{i}^{2} } }}{{q_{e,\exp } }}\) | |
Intra-particle (IPDM) | \(q_{t} = k_{p} t^{1/2} + x_{i}\) | |
Elovich (EDM) | \(q_{t} = \frac{1}{\beta }\ln (\alpha \beta ) + \frac{1}{\beta }\ln t\) | |
Bangham (BDM) | \(q_{t} = q_{e} - q_{e} /e^{{k_{BH} t^{z} }}\) | |
Diffusion-chemisorption (DCDM) | \(q_{t} = \left( {t^{1/2} q_{e} k_{DC} } \right)/\left( {t^{1/2} k_{DC} + q_{e} } \right)\) | |
Langmuir (LIM) | \(Q_{e} = Q_{m} k_{L} C_{e} /\left( {1 + k_{L} C_{e} } \right)\) | |
Freundlich (FIM) | \(Q_{e} = k_{F} C_{e}^{{1/{\text{n}}}}\) | |
Langmuir–Freundlich (LFIM) | \(Q_{e} = Q_{LF} \left( {k_{LF} C_{e} } \right)^{{n_{LF} }} /\left( {1 + \left( {k_{LF} C_{e} } \right)^{{n_{LF} }} } \right)\) | |
Henry (HeIM) | \(Q_{e} = k_{H} C_{e} + C_{H}\) | |
Redlich-Peterson (RPIM) | \(Q_{e} = C_{e} k_{RP} /\left( {1 + a_{RP} C_{e}^{{n_{RP} }} } \right)\) | |
Koble-Corrigan (KCIM) | \(Q_{e} = A_{KC} C_{e}^{{n_{KC} }} /\left( {1 + k_{KC} C_{e}^{{n_{KC} }} } \right)\) | |
Sips (SIM) | \(Q_{e} = Q_{sips} \left( {k_{sips} C_{e} } \right)^{{n_{sips} }} /\left( {1 + k_{sips} C_{e}^{{n_{sips} }} } \right)\) | |
Hill (HiIM) | \(Q_{e} = Q_{Hill} C_{e}^{{n_{Hill} }} /(k_{Hill} + C_{e}^{{n_{Hill} }} )\) | |
Thermodynamic calculation | \(\Delta G = - RT\ln \left( {K_{C} } \right)\) | |
\(\ln \left( {K_{C} } \right) = \Delta S/R - \Delta H/RT\) | ||
\(K_{C} = C_{s} - C_{e}\) |
