Modeling of CO₂ Adsorption on Surface-Functionalized Rubber-Seed Shell Activated Carbon: Isotherm and Kinetic Analysis

Abstract

Currently, adsorption is considered a promising technology for CO₂ separation with a wide range of adsorbents. A detailed study of equilibrium and kinetics plays a crucial role in the design and operation of industrial adsorption units. In this study, isotherm and kinetics of CO₂ adsorption on two RSS-derived AC samples previously prepared in our laboratory were evaluated using equilibrium experiments for pure CO₂ at 25 °C and 40 °C and 1 bar. Blank and IL-functionalized AC showed CO₂ adsorption capacity of 2.16 mmol/g, 1.96 mmol/g, 1.12 mmol/g and 1.71 mmol/g at 25 °C and 40 °C, respectively. Langmuir, Freundlich, and Temkin equations were used to model adsorption isotherm in low-pressure regions. The obtained results revealed that the Freundlich model provides an accurate fitting to the experimental findings, which indicate that the adsorption process occurs in a heterogeneous phase. Additionally, kinetic analysis was performed by using four empirical models, namely pseudo-first order, pseudo-second order, Elovich, and Avrami’s fractional models. Among the considered kinetic models, the pseudo-second order model fits best for both blank and IL-functionalized AC. Intra-particle and Boyd’s film diffusion models were evaluated for the adsorption mechanism.

1. Introduction

The rapid growth of modern industry has steadily increased the consumption of fossil fuels. The overreliance on fossil fuels has resulted in resource depletion and accelerated destruction of the environment. The huge release of CO₂ caused by the burning of fossil fuels has led to serious environmental issues such as melting ice caps, acid rain, rising sea levels, ozone layer depletion, and urban smog. In 2013, the use of fossil fuels caused an enormous global CO₂ emission of 36 billion tonnes [1]. The need to control CO₂ emissions has urged the development of new processes/materials, some of which are still under research for optimization and enhancement. Absorption is a popular separation technique that is widely used on a commercial scale for post-combustion CO₂ capture. The use of amine solvents, including monoethanolamine (MEA), diethanolamine (DEA), and methyldiethanolamine (MDEA), is very common. Despite its popularity, this process has shortcomings (such as equipment corrosion, harmful by-products due to amine leaching, and high regeneration cost) that require an imperative need for a viable alternative [2,3,4,5]. One suitable alternative technique for CO₂ separation, especially from point sources, is adsorption [6,7]. Many porous materials have been developed, including zeolites [8,9,10,11], molecular sieves [12], metal-organic frameworks (MOFs) [13,14], carbon nanotubes [15,16,17], and activated carbons [18,19,20,21]. Zeolites and MOFs are widely used for CO₂ separation because of their high selectivity for CO₂ [22]. Activated carbon (AC) has attracted much research attention due to its advantageous features, including good CO₂ adsorption capacity at low concentrations, high selectivity, tunable morphology, and surface functionalization [23,24]. Moreover, a wide range of biomass and agricultural waste are available as precursors. Rubber seed shell is a suitable precursor because of its high carbon content, low cost, and easy availability. Malaysia is the fifth largest producer of natural rubber. As per the Department of Statistics, Malaysia (DOSM), stocks of rubber production were recorded more than 40,000 tons each month between 2018 and 2021 [25]. A statistical overview of rubber production during this period is presented in Figure 1.

Recently, a new branch of hybrid adsorbents called supported ionic liquid phases (SILPs) became popular as they modify the surface and improve the inherent adsorption capacity of carbonaceous adsorbents by surface functionalization with ionic liquids (ILs). Researchers have employed many combinations of ILs and AC for surface functionalization studies [26,27,28]. In our previous study, we successfully transformed rubber seed shells (RSS) into AC using two different methods of chemical activation using K₂CO₃ [29]. Furthermore, a surface functionalization study was performed on prepared AC using IL via the wet impregnation technique. The prepared adsorbents were characterized and tested for CO₂ adsorption [30].

For large-scale applications, a suitable adsorbent should demonstrate high CO₂ adsorption capacity and selectivity along with a rapid rate of adsorption. This requires an acceptable knowledge of the adsorption kinetics and mechanism. However, extensive research studies have been conducted to study the CO₂ adsorption performance of porous carbon. There is still fragmented literature available on isotherm, and kinetic studies focused on the mechanism of CO₂ adsorption, especially for IL-functionalized AC. Previous studies suggest many kinetic models for gas–solid adsorption. The accurate estimation of the kinetic parameters Of CO₂ adsorption is difficult due to the many mechanisms involved. Therefore, the simplest approach entails the fitting of experimental adsorption data by several proven kinetic models and choosing the best-fitted model. Generally, a typical kinetic model lumps all the mass transfer resistances into a single overall mass transfer coefficient. This makes the calculations of kinetic parameters simpler but does not provide information regarding the actual rate-controlling step [31].

This study aims to test the suitability of blank and surface-functionalized RSS-derived ACs by performing isotherm and kinetic analyses. The details of the synthesis and characterization of the prepared adsorbents are discussed in our previous study [30]. Batch adsorption experiments were conducted at 25 °C and 40 °C using pure CO₂ (99.98%). Experimental equilibrium data were fitted using three isotherm models, namely Freundlich, Langmuir, and Temkin isotherm models. For kinetic analysis, four kinetic models were used to estimate the rate parameters of CO₂ adsorption. The details of the isotherm and kinetic equations employed are presented in later sections. Finally, rate-limiting kinetic models, intra-particle diffusion, and Boyd’s film diffusion models were used for estimating the mass transfer mechanism controlling the CO₂ adsorption process on these adsorbents.

2. Materials and Methods

2.1. Preparation of Adsorbent

The RSS AC used in this study was synthesized from RSS via the chemical activation method using K₂CO₃. An impregnation ratio of 1:1 was used, and the activated RSS was carbonized at 800 °C for 120 min under a uniform flow of N₂ in a tube furnace. Further details on the synthesis of RSS AC are mentioned elsewhere [30]. The prepared AC sample had a BET surface area and total pore volume of 683.45 m²/g and 0.37 cm³/g with an average pore diameter of 2.16 nm. RSS AC sample is referred to as AC-blank in later sections.

Surface functionalization of AC was performed by [bmpy][Tf₂N] IL using the wet impregnation technique. The impregnation solution was made by mixing the desired amount (30 wt% AC) of IL with methanol. AC was mixed with the prepared impregnation solution and stirred at 350 rpm for 30 min. The mixture was left overnight and later dried in an oven at 60 °C for 8 h to remove the solvent. The prepared AC was stored in an airtight bottle and labeled as AC-functionalized.

2.2. CO₂ Adsorption Study

The prepared AC samples were tested for CO₂ (99.98% pure, provided by Linde Malaysia Sdn. Bhd.) adsorption using high-pressure volumetric analyzer (HPVA II) batch experiments at 25 °C and 40 °C. Each sample was weighed (0.2 g) and filled into the sample holder. Before the adsorption test, the sample was degassed at 150 °C for 4 h to remove impurities. Upon completion of degassing, the sample was cooled to room temperature and transferred to the analysis port. Helium was dosed for free space volume measurement at both ambient and analysis temperatures. Afterward, the sample cell was evacuated to remove helium gas completely, and CO₂ was dosed for the adsorption test. After the adsorption test was complete, desorption was performed by reducing the pressure in the sample cell. The temperature of the sample cell was maintained using a Julabo recirculating water bath. Pressure and temperature data were recorded throughout the experiment in a comprehensive data analysis package using Microsoft Excel macro (v.22.0.6) software. A schematic diagram of the HPVA system is shown in Figure 2 [30].

2.3. Adsorption Isotherm

The isotherm study of gas–solid adsorption is significant as it helps to understand the interactions between adsorbate molecules and solid adsorbent. Adsorption isotherm studies are also a useful tool to understand the adsorption performance and to estimate the equilibrium adsorption capacity in some cases. There are numerous adsorption isotherm models available in the literature. A suitable isotherm model is selected, which provides a precise fit of experimental data over the entire pressure range [13]. In this study, Langmuir, Freundlich, and Temkin adsorption isotherms were used. These empirical models describe the process of adsorption of gases on microporous solids, and their applicability for CO₂ adsorption has often been reported in the literature [19,32,33].

2.3.1. Langmuir Isotherm Model

The Langmuir isotherm is the simplest theoretical model for explaining the monolayer adsorption on a homogeneous surface. The basic assumptions of the model are as follows [34]:

• The number of localized active sites is fixed.
• Each adsorption site can accommodate only one adsorbate molecule, and once adsorbed, adsorbate molecules lose the freedom to migrate to other active sites.
• All adsorption sites are energetically equal, indicating a homogeneous surface.
• Adsorbate molecules adsorbed on neighboring active sites do not interact with each other.
• The adsorption process is in dynamic equilibrium, and the rate of adsorption and desorption are equal at equilibrium.

The mathematical expression of the model is expressed in Equation (1):

$$q_e=\frac{q_m{k}_L{P}_e}{1+k_L{P}_e},$$

(1)

where, q_e and q_m are the equilibrium and maximum monolayer adsorption capacity expressed as (mmol/g). P_e (bar) indicates the equilibrium pressure, and k_L is the Langmuir isotherm constant, and it represents the affinity between adsorbent and adsorbate molecules. A larger k_L value suggests maximum surface coverage and stronger affinity among the adsorbate molecules and the adsorbent. The value of k_L is obtained from the linearized form of the model presented in Equation (2).

$$\frac{P_e}{q_e}=\frac{1}{q_m}{P}_e+\frac{1}{k_L{q}_m},$$

(2)

2.3.2. Freundlich Isotherm Model

The Freundlich model is another empirical model which explains a non-ideal and reversible adsorption process. Unlike the Langmuir model, adsorption occurs by multilayer formation. Moreover, the adsorption energy exponentially increased with increasing surface coverage because of reduced active sites [35]. Therefore, as more adsorbate molecules are adsorbed, a higher energy will be required for the other adsorbate molecules to attach to the surface of the adsorbent. The mathematical expression of the model is expressed in Equation (3)

$$q_e=k_F{\left(P_e\right)}^{\frac{1}{n}},$$

(3)

where, k_F (mmol·g⁻¹·bar^−1/n) is the Freundlich isotherm constant. The value of k_F increases when the adsorption capacity is increased. n is the heterogeneity factor, also called the Freundlich coefficient, and represents the deviation in adsorption behavior from linearity. It also indicates the type of adsorption, whether it is physical adsorption (n > 1) or chemical adsorption (n < 1) [35]. 1/n is the Freundlich intensity parameter with a value between 0 and 10 and indicates the magnitude of the driving force for adsorption or surface heterogeneity [36]. Additionally, it also informs about the favorability of the adsorption process. When 1/n < 1, it indicates a spontaneous adsorption process [37]. Generally, the Freundlich isotherm model provides a reasonable fit at a low to mid-pressure range but fails at high pressure [38]. The reason for this behavior is that this model, unlike the Langmuir model, does not approach a limiting (fixed) adsorption capacity as the pressure approaches ∞. This means that, theoretically, an infinite surface coverage is estimated due to the absence of saturation limitation and shows multilayer adsorption on heterogeneous surfaces [38].

Freundlich isotherm constants are estimated using the linearized form expressed in Equation (4):

$$\mathit{log}\left(q_e\right)=\mathit{log}\left(k_F\right)+\frac{1}{n}\mathit{log}\left(P_e\right),$$

(4)

2.3.3. Temkin Isotherm Model

The Temkin model explains how an adsorption process is affected by adsorbate/adsorbate interactions. Its major assumption is that the heat of adsorption of adsorbate molecules in a single layer decreases linearly with the surface coverage. Temkin model is expressed by Equation (5):

$$q_e=\frac{RT}{b}ln{P}_e+\frac{RT}{b}ln{k}_T,$$

(5)

where, k_T is the Temkin isotherm constant expressed as (cm³/g·bar), and b is the Temkin constant related to the heat of adsorption expressed as (J/mol).

2.4. Adsorption Kinetics

2.4.1. Rate of Adsorption

The kinetic study is important given that the residence time for completion of the adsorption process, adsorption bed size, and the installation cost are all inherently linked with the rate of adsorption [39]. The ability of an adsorbent to withstand large feed flow rates and its efficiency in dynamic processes depend on the rate of adsorption. A slow adsorption rate makes an adsorbent unfavorable for industrial applications despite a high adsorption capacity and good selectivity. Among the available empirical models, two of the most extensively applied models, Lagergen’s pseudo-first order and pseudo-second order models, are used to study the kinetic behavior of CO₂ adsorption on prepared RSS AC. Additionally, the Elovich equation is also fitted to study the chemisorption behavior, and a fractional order kinetic model based on Avrami’s equation of particle nucleation is also used. Avrami’s model has been evaluated for amine-functionalized solid adsorbents [40]; therefore, it is relevant to our research focus.

2.4.2. Pseudo-First Order Model

In the nineteenth century, Lagergren presented an empirical model which states that the rate of adsorption is proportional to the number of available active sites. The mathematical expression of the model is presented in Equation (6):

$$\frac{d{q}_t}{dt}=k_1(q_e-q_t),$$

(6)

where, q_e and q_t (mmol g⁻¹) represent the quantity of adsorbate adsorbed at equilibrium and at any time (min), respectively. k₁ (min⁻¹) is the first-order rate constant. Integrating Equation (6) under the boundary conditions t = 0, q_t = 0, and t = ∞, q_t = q_e yields Equation (7) and is used to calculate k₁.

$$q_t=q_e[1-exp(-k_{1}t\left)\right]$$

(7)

The pseudo-first order model represents the reversible interactions among adsorbent and adsorbate molecules and better explains the CO₂ adsorption on physical adsorbents such as activated carbon and zeolites [41,42].

2.4.3. Pseudo-Second Order Model

The pseudo-second order equation was presented by Blanchard et al. to study the adsorption of heavy metals on zeolites [43]. This model assumes that the rate of adsorption is proportional to the square of unoccupied active sites and is mathematically expressed by Equation (8).

$$\frac{d{q}_t}{dt}=k_2{(q_e-q_t)}^2,$$

(8)

where, k₂ (mmol g min⁻¹) is the second-order rate constant. Integrating Equation (8) under the boundary conditions t = 0, q_t = 0 and t = ∞, q_t = q_e, yields Equation (9).

$$q_t=\frac{q_e^2{k}_{2}t}{1+q_e{k}_{2}t},$$

(9)

Under the assumption of this model, the adsorption process involves strong binding of gas molecules to the surface of the adsorbent. Therefore, it provides a better fit when the adsorption involves chemical interactions or a combination of both physisorption and chemisorption, such as functionalized adsorbents [44]. It also explains solid diffusion rate-controlled processes not accurately explained by the pseudo-first order model. Based on Equation (9), four linearized forms of the model are obtained and used for the calculation of kinetic parameters.

$$\frac{t}{q_t}=\frac{1}{q_e}t+\frac{1}{k_2{q}_e^2},$$

(10)

$$\frac{1}{q_t}=\frac{1}{k_2{q}_e^{2}t}+\frac{1}{q_e},$$

(11)

$$q_t=-\left(\frac{1}{k_2{q}_e}\right)\left(\frac{q_t}{t}\right)+q_e,$$

(12)

$$\frac{q_t}{t}=-k_2{q}_e{q}_t+k_2{q}_e^2,$$

(13)

2.4.4. Elovich Kinetic Model

For gas–solid adsorption processes involving chemical interactions, the rate of adsorption decreases over time because of increasing surface coverage. Once equilibrium is reached, desorption starts, and the net rate of adsorption is reduced to zero. One of the suitable kinetic models to describe such a process is the Elovich equation presented in Equation (14):

$$\frac{d{q}_t}{dt}=\alpha exp(-\beta q_t),$$

(14)

The model parameters are evaluated from the linear form expressed in Equation (15).

$$q_t=\frac{1}{\beta }\mathit{ln}\left(\alpha \beta \right)+\frac{1}{\beta }ln\left(t\right),$$

(15)

2.4.5. Avrami’s Kinetic Model

This kinetic model was initially proposed to simulate phase transitions and crystal growth [45]. The mathematical expression of the model is presented in Equation (16):

$$\frac{d{q}_t}{dt}=k_A^{n_A}{t}^{n_A-1}\left(q_e-q_t\right),$$

(16)

where, k_A is Avrami’s kinetic constant and n_A is Avrami’s exponent that reflects changes in the adsorption mechanism during the process and the dimensional growth of adsorption sites. n_A is equal to 1, 2, or 3 for one-dimensional, two-dimensional, and three-dimensional growth [46]. The integrated form of Equation (16) is used for the calculation of kinetic parameters and is expressed by Equation (17):

$$q_t=q_e(1-\mathit{exp}\left(-{\left(k_{A}t\right)}^{n_A}\right)),$$

(17)

2.5. Validity of Kinetic Model

The reliability of kinetic models is evaluated by the non-linear coefficient of determination (R²) and a normalized standard deviation error function (Δq). The value of R² shows how well the experimental data points are fitted by the model and is calculated using Equation (18). The error function (Δq) determines the deviation between experimental data and those predicted by a kinetic model equation. The error for each kinetic model is calculated using Equation (19) [40].

$$R^2=1-\left[\frac{\sum _{i=1}^m{\left(q_{i,exp}-q_{i,cal}\right)}^2}{\sum _{i=1}^m{\left(q_{i,exp}-q_{exp}\right)}^2}\right]\times \left(\frac{m-1}{m-p}\right),$$

(18)

$$∆q=\sqrt{\frac{\sum {[(q_{i,exp}-q_{i,cal})/q_{i,exp}]}^2}{N-1}}\times 100,$$

(19)

where, q_i,exp, and q_i,_cal are the experimental and calculated values of adsorbate uptake per mass of the adsorbent expressed as (mmol/g), respectively. q_exp is the average of the experimental adsorption capacity. Furthermore, m represents the number of experimental data points until equilibrium is reached (i.e., q_t/q_e = 1), and p is the number of parameters of the kinetic model [40]. Fitting of all four kinetic models was performed using Excel Solver Tool (Microsoft Excel (Version 2308 Build 16.0.16731.20182)). The value of R² closer to 1 and the lower value of the error function suggest that the CO₂ adsorption process is successfully described by the given kinetic model.

2.6. Rate-Limiting Kinetic Models

The generic kinetic models described in the previous section are convenient for predicting the rate of adsorption by estimating the kinetic parameters. However, the identification of the rate-limiting step is difficult with these models because all the resistances to mass transfer are lumped together in a single step. For porous adsorbents such as AC and silica, the physical meaning of the evaluated rate constant helps to know the mass transfer mechanisms involved. A typical CO₂ adsorption process includes the following steps (i) transport of CO₂ molecules from bulk gas phase towards the gas–solid interface, (ii) film or boundary layer diffusion where CO₂ molecules diffuse through the gas film to reach the surface of the adsorbent, (iii) inter-particle diffusion involving the diffusion of CO₂ molecules among the s voids of the agglomerated adsorbent particles, (iv) intra-particle diffusion where CO₂ molecules are transported into inner pores of the adsorbent particle, and finally (v) surface adsorption where adsorbed CO₂ molecules interact with affinity groups in internal active sites. Generally, the rate of CO₂ adsorption is controlled by either film diffusion, intra-particle diffusion, or both in some cases [47,48,49,50,51]. In this study, intra-particle diffusion and Boyd’s film diffusion models are evaluated to investigate the CO₂ adsorption mechanism.

2.6.1. Intra-Particle Diffusion Model

This model was originally presented by Weber and Morris to describe the pore diffusion effect and the adsorption behavior for spherical adsorbent particles and is expressed by Equation (20):

$$q_t=k_{ipd}{t}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+C,$$

(20)

where, k_ipd is the rate constant expressed as (mmol g⁻¹ min^−0.5) and C shows the boundary layer thickness. Intra-particle diffusion is the sole rate-controlling mechanism if q_t vs. t^1/2, called the Weber–Morris plot, is a straight line passing through the origin. A non-linear plot with an intercept indicates that other mechanisms also contribute as a rate-limiting step of adsorption [41].

2.6.2. Boyd’s Film Diffusion Model

This model assumes that gas film adjacent to the surface of the adsorbent offers the main resistance for diffusion of adsorbate molecules, commonly referred to as film or boundary layer resistance. The mathematical expression of the model is represented by Equation (21).

$$F=1-\frac{6}{{\pi }^2}{\sum }_{n=1}^{\mathrm{\infty }}\frac{1}{n^2}exp(-n^2{B}_t),$$

(21)

For F < 0.85:

$$B_t={\left(\sqrt{\pi }-\sqrt{\pi -\frac{{\pi }^{2}F}{3}}\right)}^2,$$

(22)

For F > 0.85:

$$B_t=-0.4977-\mathit{ln}\left(1-F\right),$$

(23)

where F is the fractional adsorption capacity expressed as:

$$F=\frac{q_t}{q_e}$$

(24)

Equations (21)–(24) are used to identify the mechanism steps controlling the adsorption process. Whether the adsorption process is controlled by pore diffusion or film diffusion is decided by the graph of B_t vs. time. Pore diffusion is the only rate-controlling step if the B_t vs. time graph is a straight line passing through the origin. In the case of a linear or non-linear plot that does not pass through the origin, film diffusion also significantly affects the rate of adsorption in addition to pore diffusion [50,52,53]. In Boyd’s film diffusion model, the overall resistance to mass transfer is divided into two main components: the film or boundary layer resistance and the internal pore diffusion resistance within the adsorbent material. B_t is called the Biot number and is defined as the ratio of the film resistance to the pore resistance. The magnitude of B_t indicates which resistance dominates in determining the rate-limiting step of the adsorption process.

3. Results and Discussion

3.1. Adsorption Isotherm Analysis

Experimental data of CO₂ adsorption obtained from HPVA were fitted to three empirical models to study the adsorption isotherm. The applicability of isotherm models is estimated by the magnitude of the R² value. The calculated isotherm model parameters are summarized in

Table 1. Isotherm model parameters for CO₂ adsorption on blank and IL-functionalized AC.

Sample	AC-Blank		AC-Functionalized
Temperature (°C)	25	40	25	40
Langmuir isotherm
qmax (mmol/g)	2.5907	2.7196	1.3535	2.0117
kL (1/bar)	3.2991	1.7239	2.9635	3.1927
R2	0.9871	0.9789	0.9815	0.9708
Freundlich isotherm
n	1.6821	1.6681	2.1079	2.2681
1/n	0.5945	0.4409	0.4744	0.4409
kF (mmol/g·bar)	2.1871	1.5619	1.0454	1.5619
R2	0.9889	0.9964	0.9968	0.9998
Temkin isotherm
Bt (J/mol)	130.5117	103.1559	222.0873	144.0238
kT (mol/g·bar)	59.8707	20.8912	36.1641	44.6218
R2	0.9608	0.9626	0.9738	0.9536

. A comparison between experimental isotherm and fitted models is also presented in Figure 3. Langmuir and Freundlich isotherm constants k_L and k_F represent the affinity among adsorbent and adsorbate molecules and show an increasing trend with higher CO₂ adsorption capacity. The values of both constants are reduced at high temperatures for AC-blank, indicating a physisorption behavior. On the other hand, AC-functionalized shows higher values of both k_L and k_F at high temperatures and does not follow the usual trend. This is due to the higher CO₂ adsorption capacity of AC-functionalized at 40 °C than at 25 °C. An increase in the value of isotherm constants indicates that AC-functionalized does not adsorb CO₂ entirely by physisorption and has some chemical interactions involved due to IL functionalities.

The Langmuir model parameters for AC-functionalized do not fit the experimental findings accurately as q_m representing the maximum monolayer adsorption capacity increases with temperature despite a reduced physisorption behavior at 40 °C. The value of the n parameter of the Freundlich model is greater than 1 for all samples representing physisorption as the main process of adsorption [54]. The values of the Freundlich heterogeneity parameter (1/n) are less than 1 and confirm that the adsorption process is spontaneous. Moreover, these values do not vary much after functionalization, which indicates that the surface of the adsorbent is energetically homogenous after IL impregnation.

The Temkin model represents the adsorbate–adsorbent interactions. The value of the Temkin constant (k_T) decreases with temperature for the AC-blank, indicating that the CO₂ molecules interact with AC physically (likely weak van der Waals interaction). On the other hand, k_T values for AC-functionalized indicate stronger interaction between CO₂ and IL functionalities and follow an increasing trend with temperature. Overall, based on the value of R², the Freundlich model fits better with experimental data of CO₂ adsorption for both blank and IL-functionalized AC. Moreover, the Freundlich model parameters agree with the experimental findings, implying that the adsorption process occurs in a heterogeneous phase and is not restricted to monolayers.

3.2. Adsorption Kinetic Analysis

The CO₂ adsorption kinetics study was performed by fitting four kinetic model equations. Theoretically, either one of the considered kinetic models could explain the CO₂ adsorption process on given adsorbents. Nevertheless, all models might not be a proper fit for both adsorbents depending on the nature of the adsorption process. All four kinetic models are fitted for AC-blank and AC-functionalized. The best-fitted model is used to calculate kinetic constants. The calculated values of kinetic parameters and the corresponding R² values and errors are recorded in

Table 2. Kinetic model parameters of CO₂ adsorption on blank and IL-functionalized AC at 25 °C and 40 °C and 1 bar for pure CO₂ feed.

		Units	AC-Blank		AC-Functionalized
Temperature		°C	25	40	25	40
qe,exp		cm3/g	1.0802	4.6064	2.5351	3.5421
Two parameter models
Pseudo-second order model	qe,cal	cm3/g	0.8450	4.6512	2.3115	3.5044
	k2	g·min/cm3	9.8422	21.9141	7.3616	5.2338
	R2	-	0.9981	0.9998	0.9997	1.000
	Δq	%	23.45	5.88	4.98	5.06

. It has been established that both pseudo-first order and pseudo-second order models have shortcomings in predicting CO₂ adsorption on porous adsorbents such as AC. Various findings in the literature confirm that the pseudo-first order model overestimates the CO₂ uptake in the early stage of adsorption, whereas the calculated CO₂ uptake is underestimated in the later stages close to equilibrium [31,55]. The fitted kinetic models for blank AC at 25 °C are presented in Figure 4. Both Elovich and the pseudo-second order model provide a good fit for experimental data. The pseudo-second order model provides a better fit to experimental data with an R² close to 1. This shows that adsorption capacity increases exponentially with the unoccupied surface active sites [42]. The obtained results agree with earlier findings of CO₂ adsorption on biomass-derived physical adsorbents [19,56]. Similarly, CO₂ adsorption is better fitted by a pseudo-second order model for AC-functionalized, as evident from the fitted kinetic model shown in Figure 5 at both temperatures. Additionally, the R² values for the pseudo-second order model are much closer to unity compared with other kinetic models. As stated earlier, the pseudo-second order model better suits the adsorption processes involving chemical interactions [57]. This model fits better for AC-functionalized due to chemical interactions among CO₂ molecules and IL functionalities.

Temperature has a positive effect on the rate of mass transfer. The values of k₁ and k₂ increase with temperature due to an increase in the rate of diffusion of CO₂ molecules and their mobility. Due to higher mobility, CO₂ molecules move faster inside the pores, which, in turn, increases the rate of adsorption. However, in some cases, k₁ and k₂ can decrease with temperature. It is important to mention that these observations are specific to a given system and may not always follow the same trend for different systems and under different conditions. The behavior of adsorption kinetics can be influenced by various factors, such as the nature of adsorbate–adsorbent interactions, surface properties, and the presence of chemical interactions. As shown in Table 2, calculated values of k₁ and k₂ increase with temperature for both blank and functionalized AC. High temperature improves the adsorption rate by minimizing the film layer resistance, but its effect on desorption is more pronounced [58]. At high temperatures, CO₂ molecules have a greater tendency to escape from the surface of the adsorbent by overcoming the weak van der Waals forces for physisorption. As desorption starts, the net rate of adsorption is reduced at high temperatures. Similar observations were made in another study for CO₂ adsorption, which concluded that the time to reach equilibrium at 75 °C is less than that required at low temperatures (25–40 °C) [41]. The shorter equilibrium time is due to faster movement of gas molecules, higher rate of collisions, and lower adsorbed volume. The values of k₂ observed for AC-functionalized do not follow this trend because of a higher CO₂ adsorption capacity at 40 °C, possibly due to the involvement of strong chemical interactions and high reaction rate. For functionalized AC, higher temperature results in better distribution of IL molecules due to lower viscosity and surface tension. This makes active sites more accessible, and CO₂ molecules are also more mobile at high temperatures, which results in an increase in the adsorption capacity. Similar findings were reported by Erto et al. [26] for commercial AC functionalized with two different ILs. Experimental findings showed a higher CO₂ adsorption capacity of functionalized AC at 80 °C compared with bare AC despite a poor performance at room temperature due to better IL dispersion at elevated temperatures. Overall, AC samples show promising results in terms of CO₂ adsorption when compared with previous studies. A detailed comparison of the adsorption performance of prepared adsorbents reported in the literature is provided in our previous publication [30].

Avrami’s model provides a reasonable fit for only the AC-blank. The unique quality of the fit of this model at both low and high surface coverage is linked with its ability to accommodate complex reaction pathways. Both samples are best represented by a pseudo-second order model with the lowest error function compared with others.

3.3. CO₂ Adsorption Mechanism

The study of adsorption kinetics is useful for estimating the rate of adsorption. However, conventional kinetic models do not provide much information regarding the actual mechanism that governs gas–solid adsorption. For this purpose, rate-limiting models have been utilized to investigate the rate-controlling step for CO₂ adsorption.

For the intra-particle diffusion model, a plot of q_e vs. t^0.5 is presented in Figure 6 for both samples at 25 °C and 40 °C. The plot is non-linear for both samples with an intercept. Ideally, if pore diffusion is the sole rate-controlling step, a straight line passing through the origin is obtained. In any other case, other mechanisms control the overall rate of adsorption. The absence of a straight line here indicates that the other mechanisms are affecting the CO₂ adsorption for both blank and IL-functionalized AC. Based on the curve-shaped plot, a three-step mechanism can be considered for CO₂ adsorption: (i) bulk transport of CO₂ molecules across the gas–solid interface to the surface of adsorption mainly controlled by film or boundary layer diffusion, (ii) gradual adsorption stage where intra-particle diffusion takes place and (iii) finally equilibrium stage where intra-particle diffusion is slowed down due to saturation of the active sites.

The contribution of film diffusion or external mass transfer is further confirmed via Boyd’s plot presented in Figure 7. Non-linear plots are obtained at both temperatures for blank and IL-functionalized AC. This confirms that film diffusion plays a significant role in controlling the rate of adsorption in addition to pore diffusion. The plot for AC-functionalized is identical at 25 °C and 40 °C except for a higher slope at 40 °C. This is expected because, at high temperatures, the film resistance is reduced due to faster mobility of CO₂ molecules and a less viscous IL layer impregnated on AC. This leads to a higher CO₂ adsorption rate and faster kinetics. From the above results, it is evident that, overall, two mechanisms control CO₂ adsorption. At a given time, only one mechanism governs the adsorption process. Initially, film diffusion is the main resistance to mass transfer and controls the adsorption rate. Afterward, the fine porosity of the adsorbent makes intra-particle diffusion the rate-limiting mechanism until equilibrium is reached.

4. Conclusions

This study confirms the promising usage of surface-functionalized AC derived from biomass for CO₂ adsorption. Overall, the Freundlich isotherm model provided the best fit for both samples under the tested conditions. The isotherm analysis confirmed that the CO₂ was adsorbed by multilayer formation in a heterogeneous phase by spontaneous adsorption. The evaluated isotherm constants agree with earlier findings of biomass-derived AC reported in the literature. The best fit of experimental data of CO₂ adsorption on blank AC over the tested conditions was obtained by applying a pseudo-second order model. The respective rate constant k₂ increased with temperature. Similarly, CO₂ adsorption using IL-functionalized AC was also best represented by a pseudo-second order model. The analysis of kinetic parameters indicated that CO₂ adsorption mainly occurred through physisorption for blank AC. Whereas, for IL-functionalized AC, CO₂ was adsorbed by a combination of both physisorption by the support and chemisorption by IL functionalities. Furthermore, the analysis of diffusion models revealed that mass transfer during CO₂ adsorption proceeds with a diffusion-controlled process involving intra-particle diffusion and film diffusion mechanisms. Film diffusion dominates initially during the early stage of adsorption, and pore diffusion controls the overall rate of adsorption in later stages until equilibrium is reached. Lastly, the faster kinetics of CO₂ adsorption on these adsorbents was confirmed, thus providing support to the conclusion drawn in our previous study regarding the suitability of surface-functionalized RSS-derived AC for CO₂ separation. For future studies, cyclic experiments to estimate selectivity for CO₂ adsorption and the lifetime of adsorbent should be performed.