Table 2 Kinetics and mechanism modeling of adsorption40,41,42,43,44,45,46,47,48,49,50,51.
Kinetic and mechanism models | Linear expression | Parameters nomenclature and description |
|---|---|---|
Pseudo first order (PFO) | \(log\left({q}_{e}-{q}_{t}\right)={logq}_{e}-\frac{{K}_{1}t}{2.303}\) (15) \({h}_{1}={k}_{1}{q}_{e}\) (16) | qe is the quantity of adsorbate at equilibrium per unit weight of the adsorbent (mg/g), qt is the amount of adsorbed at any time (mg/g) and k1 is the pseudo first-order rate constant (min−1) and h1 initial pseudo first-order rate constant (mg/g/min). qe and k1 were determined respectively from intercept and slope of the linear plot of loqe-qt vs t |
Pseudo second-order (PSO) | \(\frac{t}{{q}_{t}}=\frac{1}{{k}_{2}{q}_{e}^{2}}+\frac{t}{{q}_{e}}\)(17) \({h}_{2}={k}_{2}{q}_{e}^{2}\) (18) | k2 is the pseudo second-order rate constant (min−1) h2 is initial pseudo second-order adsorption rate constant (mg/g/min). qe and k2 were determined respectively from slope and intercept of the linear plot of t/qt vs t |
Elovich | \({q}_{t}=\frac{1}{\beta }\ln\left(\alpha \beta \right)+\frac{1}{\beta }\ln\left(t\right)\)(19) | qt is the amount of adsorbate per unit mass of adsorbent at time (t), and α and β are the constants slope and intercept of the determined from the linear plot of qt versus ln(t). ∝ is the initial adsorption rate (mg/g-min); β is the desorption constant (g/mg) during any one experiment. The slope is \(1/\beta\) while the intercept is \(1/\beta\ln(\alpha \beta )\) |
Fractional power (power function) | \(\log\hspace{0.33em}({q}_{t})=\mathit{log}(k)+v\mathit{log}(t)\)(20) | qt is the amount of adsorbate per unit mass of adsorbent, k is a constant, t is time, and v is a positive constant (< 1). The parameters v and k are obtainable from slope and intercept of a linear plot of log (qt) versus log (t) |
Intraparticle diffusion (IPD) | \({q}_{t}={k}_{id}{t}^{0.5}+C\)(21) | kid is the intraparticle diffusion rate constant (mg.g−1min0.5) and C is the thickness of the adsorbent determined from slope and intercept of linear plot of qt vs t0.5 |
Liquid film diffusion (LFD) | \(\mathit{ln}\left(1-F\right)=-{K}_{LFD}t\)+C (22) \(F=\frac{\left[{q}_{t}^{n}\right]}{\left[{q}_{e}^{n}\right]}\) (23) | F is fractional attainment to equilibrium and KLFD is the rate coefficient for particle-diffusion controlled process corresponding to the particle size of the adsorbent. -KLFD was determined from the linear plot of ln(1 − F) vs t |
