Investigation of the CO₂ adsorption isotherms on activated carbon by statistical physics treatment

Abstract

The aim of this paper was to study CO₂ adsorption on the activated carbon (AC) prepared from olive waste. The AC was characterized via X-ray diffraction, Scanning Electron Microscopy (SEM) and N₂ adsorption-desorption. The adsorption isotherms were measured at various temperatures, namely, 298 K, 308 K, and 318 K. Based on the experimental result, a new model was generated and evaluated by means of a multilayer model with saturation. The physicochemical properties that characterize the CO₂ adsorption mechanism have been established using a physical statistical approach. The parameters which characterize the CO₂ adsorption isotherm, such as the number of carbon dioxide molecules per site (n), the receptor site densities $\:\left({D}_{m}\right)\:$, and the energetic parameters (ΔE₁, ΔE₂) were studied. In addition, thermodynamic functions such as internal energy (E_int) and Gibbs free energy (G), were established using a physics model. Thus, the obtained results indicated that the CO₂ adsorption on the AC was exothermic and physisorption in nature. Furthermore, the results achieved with this model can be used to design and optimize an adsorption unit for CO₂ capture.

Introduction

The IPCC (Intergovernmental Panel on Climate Change) mentioned that CO₂ emissions were a major role in global warming¹. Carbon capture and storage (CCS) technology is widely used to address these issues. This process involves the CO₂ capture and transport to storage sites². These technologies have been used to large point sources of CO₂ such as pre and post-combustion power plants and separating natural gas³. In this context, state-of-the-art available CCS technologies are absorption, adsorption, membrane separation, and cryogenic methods⁴. Adsorption technique is considered as the most promising to reduce the CO₂ due to easy operation and low energy requirements⁵. Many solid adsorbents such as activated carbon, zeolites, mesoporous silicates and metal-organic frameworks have been used for gas separation⁶. In this context, activated carbon was considered one of the most effective adsorbents for the CO₂ capture due to their higher adsorption capacity, tunable porosity and large surface area⁷. Many studies have investigated the operating parameters on the mechanism of CO₂ adsorption without being thorough on the thermodynamic parameters. Indeed, the objective of a theoretical model is to understand the adsorption mechanisms by determining the parameters of the model to which a physical meaning is associated. For these reasons, the modeling of the CO₂ adsorption process is the subject of various scientific works. Almoneef et al.⁸, have used the Langmuir and Freundlich models to study the adsorption of CO₂ on activated carbon (AC). K. Kumaraguru⁹ conducted a comparative study of the Langmuir and Freundlich adsorption models to fit CO₂ adsorption isotherms on activated carbon derived from sawdust and shrimp shells. Kishor Palle¹⁰ studied CO₂ adsorption on AC made from rice husk, analyzing the isotherms using the Langmuir, Freundlich, Sips, and Radke-Prausnitz models. Jarosław Serafin¹¹ developed a series of activated carbons from surgical mask waste for CO₂ capture. To determine the best fit for the experimental adsorption data, they applied various equilibrium isotherm models, including Langmuir, Freundlich, Toth, Sips, Unilan, Fritz-Schlunder, and Radke–Prausnitz. Saha et al.¹² reported the adsorption isotherms of CO₂ on activated carbons in fiber and powder form through the modified Tóth and Dubinin-Astakhov models. Karolina Kiełbasa et al.¹³ investigated CO₂ adsorption on AC derived from molasses. To model the experimental adsorption data, they evaluated several equilibrium isotherm models, including Dubinin–Radushkevich, Toth, Sips, Unilan, Fritz–Schlunder, and Temkin–Pyzhev.

Most of the cited works are missing quantitative interpretations of the physico-chemical parameters, which are indispensable. The classical models provide information on the saturation adsorption capacity and adsorption energy. Therefore, the objective of this paper is to present a new theoretical approach to the modeling of CO₂ adsorption by advanced statistical physics models.

The CO₂ adsorption on the AC were measured at different temperatures. The experimental data were fitted with statistical physics model. The fitting results were used to established the physicochemical parameters. In addition, the parameters characterizing the adsorption isotherm such as the number of molecules per site, the receptor site densities and the energetic parameter were estimated.

Materials and methods

AC was prepared by treating olive waste with phosphoric acid (H₃PO₄) following a modified method⁸. The olive waste was treated with 85% phosphoric acid (H₃PO₄) at a 1:1 weight ratio. The mixture was then heated in an electric oven at 100 °C for 24 h. Following this treatment, the material was washed with distilled water and sodium bicarbonate (NaHCO₃) to neutralize any remaining acid. Finally, the sample was rinsed again with distilled water and dried at 600 °C to produce the activated carbon.

The adsorption isotherms of CO₂ on AC were measured using Micromeritics ASAP 2020 analysis in the pressure range of 0–20 bar and at various temperature (298, 308 and 318 K). Prior to the measurement, 0.09 g of the AC was degassed under vacuum at 250 °C for more than 13 h. The CO₂ with high purity (99.99%), was obtained from Sigma-Aldrich.

The X-ray diffraction (XRD) patterns of the AC were obtained using a Panalytical X’Pert Pro diffractometer with Cu Kα radiation, scanning over a 2θ range. The activated carbon, used as received, was characterized by scanning electron microscopy (SEM) using a Thermo Fisher FEI Q250 microscope operating at an acceleration voltage of 10–15 kV. The thermogravimetric analysis (TGA) was realized by using a SDTQ 600 thermal analyzer. The textural properties of the material were evaluated by nitrogen adsorption–desorption measurements using a Micromeritics ASAP 2010 analyzer. The specific surface area was calculated using the Brunauer–Emmett–Teller (BET) method.

Advanced modeling of CO₂ adsorption on AC

To better understand the CO₂ adsorption process on the AC, advanced adsorption models were investigated to confirm the results found using classical models, as well as to give more information on the behavior of the adsorbent-adsorbate system¹⁴. Three advanced models were used in this assessment and are listed in Table 1.

Table 1 The advanced statistical physics models M1, M2 and M3.

Results and discussion

Characterization of adsorbent

Different characterization techniques have been used to identify this adsorbent. The X-ray diffraction (XRD) pattern of the activated carbon prepared from Tunisian olive waste (Fig. 1) reveals two broad diffraction peaks centered at approximately 2θ ≈ 25° and 42°, which correspond to the (002) and (100) crystallographic planes of graphitic carbon, respectively. The broad nature of these peaks indicates a largely amorphous structure with low crystalline order²⁴. The (002) reflection at 25° is associated with the stacking of graphene-like layers, whereas the (100) peak near 42° corresponds to the in-plane structural ordering of carbon atoms²⁴.

Fig. 1

XRD of activated carbon.

Figure 2 shows scanning electron micrographs of the activated carbon surface at two magnifications. The images reveal a highly porous structure with irregularly distributed pores of varying sizes. At 50 μm magnification, the surface displays heterogeneous pores and roughness, while the 20 μm image provides more detail, highlighting porous structures. These features suggest a high specific surface area, beneficial for adsorption processes.

Fig. 2

SEM images of activated carbon.

Thermogravimetric Analysis of the AC was conducted to assess its thermal stability, as shown in Fig. 3. The AC exhibited an initial mass loss around 100 °C, which is attributed to the evaporation of physically adsorbed moisture²⁶. A significant weight loss was observed starting at approximately 500 °C, indicating the thermal decomposition of volatile organic compounds. The weight continued to decrease until about 670 °C, after which it remained relatively constant, suggesting the completion of major decomposition processes²⁶.

Fig. 3

TGA of the activated carbon.

Figure 4 illustrates the N₂ isotherms of the AC. The isotherm exhibits a rapid increase in N₂ adsorption at lower relative pressures, i.e. in the range 0 < $\:P/{P}_{0}$<0.3. This N₂ desorption is due attributed the presence of micropore. At $\:P/{P}_{0}$> 0.4, the isotherm reaches a plateau. Adsorption-desorption isotherms are reversible and hysteresis is detected. According to the IUPAC classification, the AC shows a type I isotherm, which characterizes the microporous structure¹⁴. The BET surface area and the volume total of the AC studied using N₂ adsorption-desorption isotherms are listed in Table 2.

Fig. 4

N₂ adsorption - Desorption.

Table 2 Textural parameters of the Activated Carbon.

Statistical physics models analysis

The experimental isotherm is validated by a statistical physics treatment. The best fitting adsorption isotherm was determined by the adjustment coefficient R² and the residual Root Mean Square Error (RMSE). Tables 3,and 4 highlights the R² and RMSE values for the three models and corresponding adjustment parameter values.

Table 3 R² and RMSE values for the three models.

Table 4 Adjustment parameter values corresponding to our fit model.

The RMSE, which estimates the standard error between the fitted model and experimental data, is expressed as¹⁵ :

$$\:RMSE=\:\sqrt{\frac{RSS}{{m}^{{\prime\:}}-{p}^{{\prime\:}}}}$$

(1)

where RSS indicates the sum of residual squares $\:{m}^{{\prime\:}}$ is the no. of points on the experimental isotherm and $\:{p}^{{\prime\:}}$ is a variable parameter.

Therefore, it can be assumed that model 3 shows a good correlation with the results observed (Fig. 4). This model assumes that adsorption of CO₂ molecules on AC involves a non-fixed number of layers with two types of energy¹⁵. The first energy variable (-ε1) describes the CO₂ molecules interactions on the AC surface and it’s to be independent of the degree of coverage. The second adsorption energy (-ε2) is linked to interactions between CO₂ molecules in the following layers¹⁵. As a result, the total number of adsorption layers is $\:{N}_{c}=1+{N}_{2}$. Figure 5 shows the experimental data for the CO₂ adsorption isotherms on AC.

Fig. 5

Experimental data for the CO₂ adsorption isotherms on activated carbon fitted with our model.

Steric parameter interpretation

$\:\mathbf{n}$ parameters

The n parameters represent the number of CO₂ molecules captured per binding site. The calculated values of this coefficient are reported in Table 5. As can be observed, the n values calculated at each temperature are greater than unity, which indicates a non-parallel anchoring position for the CO₂ adsorption on the AC. Figure 6 shown the relationship between n and the temperature. It was found that the number of CO₂ molecules captured per site increased with temperature, ranging from 1.41 to 1.98. According to n values, we can deduce that the adsorption system is thermally activated and that temperature has an effect on the aggregation process¹⁶.

Table 5 Comparative of CO₂ adsorption by various activated carbons.

Fig. 6

Variation in the number of CO₂ molecules captured versus the temperature.

$\:{\varvec{D}}_{\varvec{m}}$ parameters

The density of receptor sites occupied per unit area of adsorbent $\:\left({D}_{m}\right)$ as a function of temperature is given in Fig. 7. This coefficient changes inversely with temperature. The temperature decrease in this parameter is clarified by the increase in the number of CO₂ captured per AC adsorption site. In this way, as the number of CO₂ molecules captured rises, the number of empty adsorbent sites available for CO₂ adsorption diminishes, leading to an increased density of receptor sites¹⁷.

Fig. 7

Variation in the density of receptor sites versus the temperature.

$\:{\varvec{N}}_{\varvec{L}}$ parameters

The parameter $\:{N}_{L}$ refers to the total number of adsorbed CO₂ layers (1 + N) produced during the adsorption processes. The relationship between the temperature and $\:{N}_{L}$ parameter is shown in Fig. 8. On the basis of the results shown in Fig. 7, it can be seen that the overall number of layers formed for the system is between 1.62 and 1.89. This suggests that CO₂ adsorption is obtained by the formation of two adsorbate layers, which increase with the adsorption temperature. It may be noted that the increase in the number of layers formed may be due to the decrease in interaction energies. In this case, the reduction in the number of active sites available for CO₂ capture at the surface is partially counterbalanced by a multilayer adsorption mechanism.

Fig. 8

Variation in the on the total number of formed layers of CO₂ molecules versus the temperature.

$\:{\varvec{Q}}_{\varvec{S}}$ parameters

The CO₂ uptake capacity ($\:{Q}_{S}$) is expressed as the maximal quantity that can be adsorbed at any temperature. This parameter reflects the potential of the adsorbent surface to absorb CO₂ molecules. In addition, it is related to the number of molecules per site, the total number of layers formed and the density of receptor sites, as expressed as follows¹⁷ :

$$\:{Q}_{S}=n{.D}_{m}.NL$$

(2)

The influence of temperature on the amount of saturation adsorption is plotted in Fig. 9. We noted that the increase in temperature leads to a decline in $\:{Q}_{S}$, which is ascribed to the exothermic nature of the adsorption process and the impact of temperature on the aggregation mechanism. The reduction in the amount of saturated adsorption is the product of the reduced strength between the AC surface and the CO₂.

Fig. 9

Variation in carbon dioxide adsorption capacity at saturation versus the temperature.

Surface Adsorption Energy

The adsorption energy of CO₂ gas on the surface of the AC surface was obtained using the model investigated. The two half-saturation pressures are related to the two adsorption energies ($\:{\Delta\:}{E}_{1}$) and ($\:{\Delta\:}{E}_{2}$), which can be expressed as follows¹⁷:

$$\:-\varDelta\:{E}_{1}=-RTln\left(\frac{{P}_{vs}}{{P}_{1}}\right)$$

(3)

$$\:-\varDelta\:{E}_{2}=-RTln\left(\frac{{P}_{vs}}{{P}_{2}}\right)$$

(4)

Where $\:{P}_{vs}$ denotes the saturation vapour pressure of CO₂.

We note that the first adsorption energy (-$\:{\Delta\:}{E}_{1}$) is related to interactions between the AC surface and the CO₂ molecules, whereas the second adsorption energy (-$\:{\Delta\:}{E}_{2}$) describes the CO₂-CO₂ interactions of the following layers. We can see that the first energy is greater than the second, owing to the lower interactions between CO₂-CO₂ molecules. In addition, the adsorption energies obtained were below 30 kJ/mol, suggesting the existence of a physisorption process caused by van der Waals interactions. The relationship between temperature and adsorption energies is depicted in Fig. 10. We can see that adsorption energy rise with temperature, in the 298–318 K range. This pattern can be attributed to by the temperature effect, which moves the atoms in the site and allows the adsorbed atoms to lodge easily in the volume of the site. In fact, as temperature rises, the availability of receptor sites is higher, further increasing the average energy of the active site. The CO₂ molecules are adsorbed on sites with a higher adsorption energy. Once these sites are occupied, CO₂ molecules adsorb on sites with low adsorption energy values¹⁷. Figure 11 shows the evolution of the internal energy, Eint, versus the pressure at different temperatures.

Fig. 10

Variation in Surface Adsorption Energy versus the temperature.

Fig. 11

Evolution of the internal energy, Eint, versus the pressure at different temperatures.

Thermodynamic analysis

Internal Energy

The internal energy gives the interactions between the adsorbate and the adsorbent, which is defined by the following equation¹⁸:

$$\:{E}_{int}=-\frac{\partial\:Ln({Z}_{gc}}{\partial\:\beta\:}+\frac{\mu\:}{\beta\:}\left(\frac{\partial\:ln{Z}_{gc}}{\partial\:\mu\:}\right)$$

(5)

Figure 10 shows the variation in internal energy ($\:{E}_{int}$) at different temperatures. We can mark from this figure that the internal energy increases with increasing temperature, suggesting the existence of CO₂ interactions on the surface of AC. Moreover, the $\:{E}_{int}\:$values are less than zero for all temperatures. This indicates that the system releases energy during the adsorption, indicating the exothermic nature of the process. (Fig. 11)

Gibbs free energy

Gibbs free energy gives the spontaneity of the system is derived by¹⁸:

$$\:G=\:\mu\:*{Q}_{a}$$

(6)

Figure 12 depicts the change in Gibbs energy as a function of pressure at several temperatures. The values of G are less than zero which indicates the spontaneous process of the adsorption. Moreover, G declines inversely with the increase of the temperature, which implies a reduction in the feasibility of the adsorption.

Fig. 12

Evolution of the Gibbs free energy versus pressure at different temperatures.

CO₂ adsorption kinetics

Many models describe gas adsorption kinetics on adsorbents. The first model to relate adsorption rate to adsorption capacity was Lagergren’s pseudo-first-order equation. However, CO₂ adsorption kinetics can be effectively described by first-order kinetics, which is expressed as²⁶:

$$\:\frac{\partial\:q}{\partial\:t}={k}_{eff}({q}^{*}-q)$$

(4)

With $\:{k}_{eff}$ represent the mass transfer value. $\:{q}^{*}$ was the CO₂ equilibrium amount which correspond to the CO₂ concentration of gas phase at some temperature. $\:q$ was the quantity of CO₂ absorbed at time t.

Figure 13 presents the CO₂ adsorption kinetics on AC under varying pressures and temperatures. Initially, CO₂ uptake rises sharply, driven by the abundance of active adsorption sites and a strong concentration gradient facilitating mass transfer. As the process continues, the adsorption rate gradually decreases until equilibrium is reached, marking the saturation of available sites and a balance between adsorption and desorption²⁴.

Fig. 13

Adsorption kinetics curves of CO₂ on the AC at 1 (a), 3 (b), and 15 bar (c).

The higher pressures lead to a notable increase in adsorption capacity because the elevated CO₂ partial pressure enhances the driving force for adsorption, increased temperatures generally reduce the equilibrium adsorption capacity since CO₂ adsorption on AC is exothermic, and higher temperatures favor desorption. Temperature changes also influence CO₂ diffusion rates within AC pores, which can either promote or inhibit adsorption kinetics depending on the balance between molecular mobility and adsorption affinity²⁶.

In addition, the presence of surface functional groups on activated carbon significantly influences both CO₂ adsorption capacity and adsorption kinetics. Oxygen-containing groups, typically introduced during chemical activation, tend to reduce adsorption performance by increasing surface polarity and enhancing moisture affinity²⁷. In contrast, nitrogen-containing groups, introduced through amination or thermal treatment, substantially improve CO₂ uptake through acid–base interactions^27,28.

Comparison of adsorption capacity with literature

To assess the effectiveness of the AC as an adsorbent for CO₂ capture, it is crucial to compare its maximum adsorption capacity (qₘₐₓ) with those of other adsorbent materials reported in the literature. Table 5 provides a comparative summary of qₘₐₓ values obtained under comparable experimental conditions. Notably, adsorbents such as palm kernel shell-derived carbon²¹ and Norit SX2²³, exhibit relatively high CO₂ uptake, reflecting their promising adsorption capabilities. Nevertheless, the AC synthesized in this work demonstrates a significantly higher qₘₐₓ than several other materials, including pistachio shell¹⁹, activated carbon from waste plastics²², and algae-based adsorbents²⁵. These results underscore the superior adsorption potential of the prepared AC and reinforce its suitability as a high-efficiency material for CO₂ capture applications.

Conclusion

AC derived from olive waste was used as an adsorbent in this investigation for CO₂ capture. The AC were measured at 298, 308 and 318 K and characterize with SEM, XRD and N₂ adsorption desorption. A multilayer model was used to investigate the behavior of CO₂ adsorption isotherms and the corresponding adsorption process. The physicochemical parameters related to the adsorption process, namely the number of CO₂ molecules per site (n), the density ($\:{D}_{m}$), and the two energy parameters (ΔE1, ΔE2), were obtained from experimental isotherm curves. The adsorption capacity Qs ranges from 28.34 mmol/g at 298 K to 26.33 mmol/g at 313 K. The adsorption energy values obtained by the suggested model range from 9.52 kJ/mol to 14.05 kJ/mol, which are similar to those of the physisorption band. In addition, the proposed model has been applied to evaluate thermodynamic parameters such as internal energy and Gibbs free energy. Thus, the obtained results indicated that the CO₂ adsorption on the AC was exothermic and physisorption in nature. In addition, the obtained results with this model provides more in-depth information on the mechanism of CO₂ adsorption on activated carbon, and can be used to design and optimize an adsorption unit for CO₂ capture.