# A Brief Review of Formaldehyde Removal through Activated Carbon Adsorption
## 2 Thermodynamics and Kinetics for Adsorption of Formaldehyde on AC
### Thermodynamics and Adsorption Isotherms  
Thermodynamic analysis of formaldehyde adsorption on AC is important for determining whether adsorption or sorption is spontaneous or not. The value changes in standard enthalpy (\(\Delta H^{\circ}\)), standard entropy (\(\Delta S^{\circ}\)), and standard free energy (\(\Delta G^{\circ}\)) can provide evidence that the adsorption reaction is viable. The representative equations are shown below:  
\[\ln K_{d}=\frac{\Delta S^{\circ}}{R}-\frac{\Delta H^{\circ}}{RT} \tag{1}\]  
\[K_{d}=\frac{C_{AB}}{C_{B}} \tag{2}\]  
\[\Delta G^{\circ}=-RT\ln K_{d} \tag{3}\]  
where \(T\) is temperature; \(K_{d}\) is adsorption adsorption/sorption equilibrium constant. It has been reported that the adsorption of formaldehyde on Ag-AC, CaO-AC, and Fe\({}_{3}\)O\({}_{4}\)-AC was exothermic (\(\Delta H^{\circ}\) values was negative), and the negative values of \(\Delta G^{\circ}\) indicate the feasible and spontaneous nature of their adsorption process [21; 22]. The equations above also indicate that the adsorption capacity decreases with increasing adsorption temperature. Meanwhile, adsorption isotherms provide important information about the adsorption capacity of the adsorbent as a function of adsorbate partial pressure in the gas phase. According to the literature [21; 22; 23; 24; 25; 26; 27; 28; 29; 30], there are two major adsorption isotherm models describing formaldehyde behaviors on ACs (ACF or surface-modified ACs): the Langmuir model and the Freundlich model. These models depend on the surface properties, the pore structure of adsorbents, and adsorption conditions. In general, the Langmuir isotherm model is based on the ideally single-layer adsorption on the surface of the adsorbent. On the other hand, the Freundlich isotherm model is an empirical equation applicable to a heterogeneous surface and multi-layer adsorptions. Both cases also depend on the concentrations of adsorbate. For nonlinear isotherms, the Qi-LeVan model has been proposed [23]. Representative isotherm equations are shown below, in the order of Langmuir, Freundlich, and Qi-LeVan models. \[q_{e}=\frac{q_{m}\ K_{c}\ C_{e}}{1+K_{L}\ C_{e}} \tag{4}\]  
\[q_{e}=k_{f}C_{e}{}^{1/n} \tag{5}\]  
\[p=\frac{n}{\varepsilon_{0}+\varepsilon_{1}n+\varepsilon_{2}n^{2}+\varepsilon_{3 }n^{3}} \tag{6}\]  
where \(C_{e}\) is the formaldehyde concentration at the equilibrium, \(q_{e}\) is the formaldehyde amount adsorbed on adsorbents, \(q_{m}\) is the maximum adsorption amount of formaldehyde on adsorbents, which is recognized as the Langmuir's parameter, and \(K_{L}\) is the Langmuir's adsorption equilibrium constant. In Equation (6), \(p\) is the partial pressure, \(n\) is the surface loading, and \(\varepsilon_{i}\) is defined as the adsorbent-adsorbate pair parameter [23]. In Table 2, the adsorption isotherm models for formaldehyde on surface-modified AC are listed. It has been reported that the Langmuir model is more suitable than the Freundlich model when formaldehyde adsorbs on the surface of modified ACs (MnO\({}_{2}\)[24], potassium [25], EDA [26], and Ag [27]) and AC adsorptive filter media [28]. In these cases, a linear regression coefficient (R\({}^{2}\)) is often used to determine the best fit for isotherm models [24; 25; 26; 27; 28]. In addition, [PERSON] et al. [29] and [PERSON] et al. [30] also found that the Langmuir model is suitable for describing the adsorption isotherm of formaldehyde on the Ag-modified ACs. The reason why the Langmuir model showed better fit can be attributed to the hypothesis that the surface-modified ACs interact with formaldehyde by chemisorption through their functional groups on the surface and result in single-layer molecular adsorption [21; 24; 25; 26; 27; 28; 29].