# A Brief Review of Formaldehyde Removal through Activated Carbon Adsorption
## 2 Thermodynamics and Kinetics for Adsorption of Formaldehyde on AC
### Kinetic Models  
Many researchers have reported diverse adsorption kinetic models for formaldehyde, such as the pseudo-first-order model (Lagergren equation), the pseudo-second-order model (HO and McKay equation) and the Bangham model (Bangham equation) [22; 25; 26; 27; 28; 31].  
Kinetic models for the adsorption of formaldehyde on surface-modified AC are presented in Table 3. Representative equations are shown below in the order of pseudo-first-order, pseudo-second-order, and Bangham model.  
\[\frac{dq_{t}}{dt}=k_{2}(q_{e}-q_{t})^{2} \tag{7}\]  
\[\log(q_{e}-q_{t})=\log q_{e}-\frac{k_{1}}{2.303}t \tag{8}\]  
\[q_{t}=q_{e}-\frac{q_{e}}{e^{k_{B}/^{2}}} \tag{9}\]  
where \(q_{e}\) ad \(q_{t}\) are the amount of adsorbate uptake per mass unit of adsorbent at equilibrium and at a certain time \(t\), respectively; \(k_{i}\) is the rate constant; \(k_{B}\) and \(z\) are Bangham parameters. According to kinetic equations (Equations (7)-(9)), sorption capacity (\(q_{e}\) and \(q_{t}\)), adsorption rate constant (\(k_{1}\) and \(k_{2}\)), and correlation coefficient (often referred to as R\({}^{2}\) value) can be obtained by fitting experimental data. In specific, when the surfaces of ACs were modified by PEI [31], potassium [25], Ag [27], CaO, and Fe\({}_{3}\)O\({}_{4}\) [30], pseudo-second-order kinetic models showed the best fit. On the other hand, the Bangham model was the best fit for ethylenediamine-modified ACs [26].  
Therefore, various factors should be evaluated in kinetic models of formaldehyde adsorption. Various factors include operational temperature, partial pressure, the amount of adsorbents, and so on. In many cases, chemisorption and physisorption occur at the same time. The surface functional group will contribute more chemisor  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Modification AC** & **Adapted Model** & **Concentration (ppm)** & **Ref.** \\ \hline CaO-AC & & & \\ Fe\({}_{3}\)O\({}_{4}\)-AC & Pseudo-second-order & 5–50 & [22] \\ Potassium-AC & Pseudo-second-order & 0.25, 0.5, 0.75, 0.9 & [25] \\  & Pseudo-first-order & 2.45, 8.15 & [26] \\  & Bangham & & \\ Ag-AC & Pseudo-second-order & 0–14 & [27] \\ AC adsorptive filter media & Pseudo-second-order & 0.25, 0.56, 0.79 & [28] \\  * PEI-AC & Pseudo-second-order & 50 & [31] \\ \hline \hline \end{tabular}  
* PEI: Polyethyleneimine.  
\end{table}
Table 3: Kinetic models for adsorption of formaldehyde on surface-modified AC reported in the literature.  
is still possible as long as adsorptive surfaces of activated carbon participate. Adsorption is affected by pore size and pore volume. As the adsorption proceeds, the inlet of pores may be blocked and may inhibit the adsorbate to diffuse within the inner pores. In this case, physisorption kinetics may be unfavorably affected, and the delay or the prevention of adsorption will also negatively affect overall adsorption kinetics. Meanwhile, pseudo-first-order adsorption may occur when very low concentration of adsorbate are applied. It can be expected that the increase of adsorbate will result in the linear increase of overall adsorption in low concentration ranges.