Ideal Adsorbed Solution Theory (IAST) of Carbon Dioxide and Methane Adsorption Using Magnesium Gallate Metal-Organic Framework (Mg-gallate)

Ideal Adsorbed Solution Theory (IAST) is a predictive model that does not require any mixture data. In gas purification and separation processes, IAST is used to predict multicomponent adsorption equilibrium and selectivity based solely on experimental single-component adsorption isotherms. In this work, the mixed gas adsorption isotherms were predicted using IAST calculations with the Python package (pyIAST). The experimental CO2 and CH4 single-component adsorption isotherms of Mg-gallate were first fitted to isotherm models in which the experimental data best fit the Langmuir model. The presence of CH4 in the gas mixture contributed to a lower predicted amount of adsorbed CO2 due to the competitive adsorption among the different components. Nevertheless, CO2 adsorption was more favorable and resulted in a higher predicted adsorbed amount than CH4. Mg-gallate showed a stronger affinity for CO2 molecules and hence contributed to a higher CO2 adsorption capacity even with the coexistence of a CO2/CH4 mixture. Very high IAST selectivity values for CO2/CH4 were obtained which increased as the gas phase mole fraction of CO2 approached unity. Therefore, IAST calculations suggest that Mg-gallate can act as a potential adsorbent for the separation of CO2/CH4 mixed gas.


Introduction
Single-component and mixed gas adsorption on porous solid materials plays an important role, particularly in the chemical, petrochemical and biochemical industries. The single-component adsorption isotherms (pure gas adsorption isotherms) are typically measured by using commercial instruments with high performance and accuracy. On the other hand, the measurement of multicomponent adsorption equilibrium (mixed gas adsorption isotherms) is one of the most complicated experimental techniques in the adsorption area since it is challenging, time-consuming and requires self-built instruments. Generally, the design and development of adsorptive separation require information of both single-component and multicomponent adsorption equilibria. Therefore, the ability to accurately predict the multicomponent adsorption equilibrium offers a great advantage.
Due to the aforementioned limitations in measuring multicomponent adsorption equilibrium, single-component adsorption isotherms are measured experimentally and used to calculate the mixture behaviors using Ideal Adsorbed Solution Theory (IAST). IAST, developed by Myers and Prausnitz, is a well-studied method to predict multicomponent adsorption equilibrium based only on the experimental data of single-component adsorption isotherms at the same operating temperature [1,2]. In this approach, the adsorbed phase is considered ideal without any interaction in the multicomponent system. Generally, IAST provides reliable predictions of the adsorption and selectivity of a gas mixture and has

Ideal Adsorbed Solution Theory (IAST)
IAST is a thermodynamic approach in which an ideal solution is considered to be formed by the adsorbed phase, corresponding to Raoult's law for vapor-liquid equilibrium [3]. To meet the ideal condition, there must be no interaction between the adsorbate molecules in the adsorbed phase, and the spreading pressures of the components must be equal at constant temperature [13]. The highlighted equations for deriving mixed gas isotherms are provided here for convenience, since the detailed explanations of IAST can be found in various sources [3,[13][14][15][16]. The spreading pressure can be calculated using the equation below: where π is the spreading pressure, A is the specific surface area of adsorbent (m 2 /g), R is the gas constant (8.314 J K −1 mol −1 ), T is the temperature (K) and n i is the adsorption of component i (mmol/g). The partial pressure (P i ) is defined using an analogue to Raoult's law: P i = y i P = x i P 0 i (π)(at constant T and ß) where P 0 i (π) is the partial pressure of pure component i calculated at the spreading pressure and temperature of the mixture. P i is the partial pressure of component i (bar), P is the total pressure (bar), y i and x i represent mole fractions of component i in the gas and adsorbed phases (dimensionless). The total adsorbed amount (n T ) can be determined as below: where N is the number of components in the mixture and n 0 i (mmol/g) is the standard state loading. Therefore, n i , which is the adsorption of component i (mmol/g), is calculated using this equation: The established adsorption selectivity based on IAST is defined by the following equation:

Result and Discussions
Mg-gallate with the chemical formula of Mg(C 7 O 5 H 4 )·2H 2 O is a type of crystal structure that contains magnesium ions as the secondary building unit (SBU) connected with oxygen atoms of gallic acid (organic linker) to form a three-dimensional framework [17]. The pore structure of Mg-gallate creates spaces or channels depending on its pore size within the framework that can accommodate the other molecules, in this case CO 2 or CH 4 molecules. The pore size of Mg-gallate can vary depending on the synthesis method and conditions used. Generally, the pore size ranges from a few amstrongs to nanometers. Its unique structure and pore size made it a promising CO 2 and CH 4 adsorbent. Figure 1 shows the structures of the building units and the resulting framework drawn using Material studio. Green, gray and red represent magnesium, carbon and oxygen respectively, while hydrogen atoms are omitted for clarity.
where ( ) is the partial pressure of pure component i calculated at the spreading pressure and temperature of the mixture.
is the partial pressure of component i (bar), is the total pressure (bar), and represent mole fractions of component i in the gas and adsorbed phases (dimensionless).
The total adsorbed amount ( ) can be determined as below: where is the number of components in the mixture and (mmol/g) is the standard state loading. Therefore, , which is the adsorption of component (mmol/g), is calculated using this equation: The established adsorption selectivity based on IAST is defined by the following equation:

Result and Discussions
Mg-gallate with the chemical formula of Mg(C7O5H4)·2H2O is a type of crystal structure that contains magnesium ions as the secondary building unit (SBU) connected with oxygen atoms of gallic acid (organic linker) to form a three-dimensional framework [17]. The pore structure of Mg-gallate creates spaces or channels depending on its pore size within the framework that can accommodate the other molecules, in this case CO2 or CH4 molecules. The pore size of Mg-gallate can vary depending on the synthesis method and conditions used. Generally, the pore size ranges from a few amstrongs to nanometers. Its unique structure and pore size made it a promising CO2 and CH4 adsorbent. Figure 1 shows the structures of the building units and the resulting framework drawn using Material studio. Green, gray and red represent magnesium, carbon and oxygen respectively, while hydrogen atoms are omitted for clarity.

Powder X-ray Diffraction (PXRD) Pattern
The PXRD patterns of Mg-gallate were performed and are shown in Figure 2 to confirm its crystalline nature.  The PXRD patterns of Mg-gallate were performed and are shown in Figure 2 to confirm its crystalline nature.
The fresh Mg-gallate (as-synthesized) exhibited a PXRD pattern with three significant diffraction peaks around 11.35 • , 14.05 • and 24.53 • that corresponded to Miller indices (hkl) values of 010, 011 and 221, respectively, which showed that this porous material was in a good agreement with the previous reported work [18]. The signature of crystallinity of Mg-gallate could be detected by the sharp peaks. The average crystallite size for Mg-gallate was calculated to be 34.9 nm, which could be determined through X-ray diffraction line broadening by using the Debye-Scherrer formula [19]. There is a limited source of gallate-  The fresh Mg-gallate (as-synthesized) exhibited a PXRD pattern with th cant diffraction peaks around 11.35°, 14.05° and 24.53° that corresponded to M (hkl) values of 010, 011 and 221, respectively, which showed that this porous m in a good agreement with the previous reported work [18]. The signature of c of Mg-gallate could be detected by the sharp peaks. The average crystallite s gallate was calculated to be 34.9 nm, which could be determined through X-ray line broadening by using the Debye-Scherrer formula [19]. There is a limite gallate-based MOFs in the Joint Committee on Powder Diffraction Standards ( tabase and the highest peak search score obtained by using X'Pert HighScor ware was 55 with a reference number of 96-433-5642. On the other hand, the diffraction peaks of fresh Mg-gallate almost overl those of spent Mg-gallate. Generally, XRD pattern originates from the bulk rath surface of a material. CO2 and CH4 adsorption was classified as physisorptio adsorption), in which the molecules are attached to the surface of Mg-galla weak force known as Van der Waals force [20]. In addition, physisorption w lowed by incorporation into the crystal structure, which was confirmed by the XRD patterns between fresh Mg-gallate and after CO2 and CH4 adsorption.
The XRD pattern of the as-synthesized Mg-gallate exhibited peaks at aro 14.05°, 20.03°, 21.58°, 23.08°, 24.53°, 25.97°, 27.31°, 28.50°, 31.92°, 35.05°, 36. 39.88°, 42.41° and 43.30°, which well-agreed with the simulated pattern. The pattern was calculated based on the reference structure in the Cambridge graphic Data Centre (CCDC) with the database identifier of GELVEZ and number of 286498. However, the intensity of the peaks between the as-synth the simulated was different due to the elements of the as-synthesized Mg-ga not be uniformly distributed throughout the crystal structure. Typically, the the diffraction peaks is directly proportional to the amount of elements presen terial [21]. In addition, the as-synthesized peaks were broader compared to the On the other hand, the diffraction peaks of fresh Mg-gallate almost overlapped with those of spent Mg-gallate. Generally, XRD pattern originates from the bulk rather than the surface of a material. CO 2 and CH 4 adsorption was classified as physisorption (physical adsorption), in which the molecules are attached to the surface of Mg-gallate due to a weak force known as Van der Waals force [20]. In addition, physisorption was not followed by incorporation into the crystal structure, which was confirmed by the unaffected XRD patterns between fresh Mg-gallate and after CO 2 and CH 4 adsorption.
The XRD pattern of the as-synthesized Mg-gallate exhibited peaks at around 11 , which well-agreed with the simulated pattern. The simulated pattern was calculated based on the reference structure in the Cambridge Crystallographic Data Centre (CCDC) with the database identifier of GELVEZ and deposition number of 286498. However, the intensity of the peaks between the as-synthesized and the simulated was different due to the elements of the as-synthesized Mg-gallate might not be uniformly distributed throughout the crystal structure. Typically, the intensity of the diffraction peaks is directly proportional to the amount of elements present in the material [21]. In addition, the as-synthesized peaks were broader compared to the simulated peaks since the simulated crystallite size was calculated to be 46.4 nm. Broader peaks indicate a smaller size of crystallite [21].

Fourier Transform Infrared (FTIR) Spectrum
The FTIR spectra of Mg-gallate with the patterns that provide structural insights are illustrated in Figure 3.
For fresh Mg-gallate, a strong and broad band in the IR spectrum in the region of 3500-2800 cm −1 was found to be the stretching vibration of the O-H of a carboxyl group. The strong and narrow peak at 1622 cm −1 represented by a carbonyl (C=O) indicated the presence of a carboxyl group in the gallic acid. Three peaks located at 1554, 1462 and 1376 cm −1 were typical stretching vibrations of C-C bonds in an aromatic ring of the gallic acid. There were a few peaks in the region 1300-1000 cm −1 which indicated the stretching vibrations of C-O bonds and the bending vibration of O-H bonds in the aromatic ring of the gallic acid. C-H bonds of the aromatic ring were located at 748 cm −1 . Therefore, the IR spectrum identified the functional groups present in the structure of Mg-gallate. There is a very good agreement between the band positions of this work and the literature [19]. On the other hand, the IR spectra also displayed peaks identical to those of the spent Mggallate, which confirmed that CO 2 and CH 4 adsorption did not compromise the structure of Mg-gallate.
28, x FOR PEER REVIEW 5 of Figure 3. FTIR spectra of Mg-gallate.
For fresh Mg-gallate, a strong and broad band in the IR spectrum in the region 3500-2800 cm −1 was found to be the stretching vibration of the O-H of a carboxyl grou The strong and narrow peak at 1622 cm −1 represented by a carbonyl (C=O) indicated t presence of a carboxyl group in the gallic acid. Three peaks located at 1554, 1462 and 13 cm −1 were typical stretching vibrations of C-C bonds in an aromatic ring of the gallic ac There were a few peaks in the region 1300-1000 cm −1 which indicated the stretching brations of C-O bonds and the bending vibration of O-H bonds in the aromatic ring of t gallic acid. C-H bonds of the aromatic ring were located at 748 cm −1 . Therefore, the spectrum identified the functional groups present in the structure of Mg-gallate. There a very good agreement between the band positions of this work and the literature [1 On the other hand, the IR spectra also displayed peaks identical to those of the spent M gallate, which confirmed that CO2 and CH4 adsorption did not compromise the structu of Mg-gallate.   For fresh Mg-gallate, a strong and broad band in the IR spectrum in th 3500-2800 cm −1 was found to be the stretching vibration of the O-H of a carb The strong and narrow peak at 1622 cm −1 represented by a carbonyl (C=O) in presence of a carboxyl group in the gallic acid. Three peaks located at 1554, 14 cm −1 were typical stretching vibrations of C-C bonds in an aromatic ring of th There were a few peaks in the region 1300-1000 cm −1 which indicated the st brations of C-O bonds and the bending vibration of O-H bonds in the aromati gallic acid. C-H bonds of the aromatic ring were located at 748 cm −1 . There spectrum identified the functional groups present in the structure of Mg-gall a very good agreement between the band positions of this work and the lit On the other hand, the IR spectra also displayed peaks identical to those of th gallate, which confirmed that CO2 and CH4 adsorption did not compromise t of Mg-gallate. Figure 4 shows the TGA curve to describe the thermal stability of Mg-g The thermal stability of Mg-gallate is reported in the form of onset temp and decomposition temperature (Tmax). T0 is obtained from the intersection of and the tangent of the sample weight against the temperature, while Tmax is t ture at the point where the maximum weight loss of the sample occurs [22 Figure 4, the Mg-gallate started to decompose at T0 = 417.7 K, which indicate ning of the first stage of weight loss. Tmax for the first stage occurred at 462.9 K The thermal stability of Mg-gallate is reported in the form of onset temperature (T 0 ) and decomposition temperature (T max ). T 0 is obtained from the intersection of the baseline and the tangent of the sample weight against the temperature, while T max is the temperature at the point where the maximum weight loss of the sample occurs [22]. Based on Figure 4, the Mg-gallate started to decompose at T 0 = 417.7 K, which indicated the beginning of the first stage of weight loss. T max for the first stage occurred at 462.9 K when 10.5% was lost due to the release of the remaining water and ethanol. Meanwhile, the decomposition of the gallic acid (organic linker) started at the second stage of the weight loss, at T 0 = 535.7 K. Approximately 24.9% of Mg-gallate was reported as the maximum weight loss at T max = 561.5 K. It also showed that the structure of Mg-gallate was stable up to 561.5 K. Therefore, Mg-gallate can stand and operate at high-temperatures. Figure 5 shows the adsorption-desorption isotherms of nitrogen (N 2 ) and pore size distribution. At the initial stage, a rapid increment in N 2 uptake was observed, followed by a plateau phase, which represented the monolayer adsorption. Due to the further adsorption, a multilayer was produced. In addition, the existence of a small hysteresis loop in the N 2 adsorption-desorption curves could be seen at the region P/P 0 0.15-0.7. This is due to capillary condensation in which gas adsorbed in pores at low density, spontaneously condenses into a liquid-like state inside the pores of a framework. In other words, hysteresis occurs when desorption does not occur in the same way as adsorption.

Porous Properties
= 535.7 K. Approximately 24.9% of Mg-gallate was reported as the maximum weight loss at Tmax = 561.5 K. It also showed that the structure of Mg-gallate was stable up to 561.5 K. Therefore, Mg-gallate can stand and operate at high-temperatures. Figure 5 shows the adsorption-desorption isotherms of nitrogen (N2) and pore size distribution. At the initial stage, a rapid increment in N2 uptake was observed, followed by a plateau phase, which represented the monolayer adsorption. Due to the further adsorption, a multilayer was produced. In addition, the existence of a small hysteresis loop in the N2 adsorption-desorption curves could be seen at the region P/P0 0.15-0.7. This is due to capillary condensation in which gas adsorbed in pores at low density, spontaneously condenses into a liquid-like state inside the pores of a framework. In other words, hysteresis occurs when desorption does not occur in the same way as adsorption. The Brunauer-Emmett-Teller (BET) equation was used to calculate the specific surface area in the relative pressure range of 0.00-0.09. The BET surface area of Mg-gallate was found to be 512.38 m 2 /g. Furthermore, the pore volume was estimated using the t-plot method and the value was 0.187 cm 3 /g.

Porous Properties
The classification of porous materials is usually based on the diameter of their pores, microporous (<2 nm), mesoporous (2-50 nm) and macroporous (>50 nm) according to the International Union of Pure and Applied Chemistry (IUPAC) [23]. Due to the nature and mode of MOFs' preparation, MOFs generally have a non-uniform range of pores, hence an average pore size is used. The average pore size of Mg-gallate is 6.66 nm and it is classified as mesoporous (2-50 nm). The pore size distribution was interpreted based on the relationship between the pore width (w) and dV/dlog(w) using the Barrett-Joyner-Halenda (BJH) model [24]. The porous properties of Mg-gallate are tabulated in Table 1. Given its porous properties, Mg-gallate was expected to exhibit a remarkable performance in CO2/CH4 adsorptive separation.  The Brunauer-Emmett-Teller (BET) equation was used to calculate the specific surface area in the relative pressure range of 0.00-0.09. The BET surface area of Mg-gallate was found to be 512.38 m 2 /g. Furthermore, the pore volume was estimated using the t-plot method and the value was 0.187 cm 3 /g.
The classification of porous materials is usually based on the diameter of their pores, microporous (<2 nm), mesoporous (2-50 nm) and macroporous (>50 nm) according to the International Union of Pure and Applied Chemistry (IUPAC) [23]. Due to the nature and mode of MOFs' preparation, MOFs generally have a non-uniform range of pores, hence an average pore size is used. The average pore size of Mg-gallate is 6.66 nm and it is classified as mesoporous (2-50 nm). The pore size distribution was interpreted based on the relationship between the pore width (w) and dV/dlog(w) using the Barrett-Joyner-Halenda (BJH) model [24]. The porous properties of Mg-gallate are tabulated in Table 1. Given its porous properties, Mg-gallate was expected to exhibit a remarkable performance in CO 2 /CH 4 adsorptive separation.

Single-Component Gas Adsorption
An equilibrium distribution is reached if the adsorbent and guest molecules come into contact and can be explained quantitatively. The equilibrium behavior is expressed in terms of the amount of adsorbate as a function of partial pressure at a fixed temperature. This equilibrium can be illustrated in what is called an isotherm. Adsorption isotherms are important for the description of how the adsorbate will interact with the adsorbent based on the pore surface properties and affinity, which results in the adsorption capacity. Single-component gas adsorption capacity is determined from the pure gas sorption isotherm at a certain temperature and pressure. The adsorption capacity is an important parameter in the evaluation of MOFs for CO 2 capture and is the main evaluation tool for gas capture applications.
In order to investigate the potential impact of the porous nature of Mg-gallate on CO 2 and CH 4 behaviors, single-component gas adsorption was conducted at three different temperatures. Figure 6 illustrates the experimental CO 2 single-component adsorption isotherms at three different temperatures with S-shaped isotherms (sigmoidal).
An equilibrium distribution is reached if the adsorbent and guest mole into contact and can be explained quantitatively. The equilibrium behavior i in terms of the amount of adsorbate as a function of partial pressure at a fixe ture. This equilibrium can be illustrated in what is called an isotherm. Adso therms are important for the description of how the adsorbate will interact w sorbent based on the pore surface properties and affinity, which results in the capacity. Single-component gas adsorption capacity is determined from the pu tion isotherm at a certain temperature and pressure. The adsorption capacit portant parameter in the evaluation of MOFs for CO2 capture and is the main tool for gas capture applications.
In order to investigate the potential impact of the porous nature of Mg-gal and CH4 behaviors, single-component gas adsorption was conducted at thr temperatures. Figure 6 illustrates the experimental CO2 single-component ads therms at three different temperatures with S-shaped isotherms (sigmoidal). The amount of CO2 adsorbed on Mg-gallate increased as the pressure in all temperatures. However, the isotherms decreased as the temperatures incr 273 to 313 K. The reason behind is that the surface adsorption energy and the diffusion rate increase with the temperature, causing the gas molecules to move and making it difficult for them to attach to the surface of the adsorben sequently, a higher temperature condition is not advantageous for the adsorpt and may cause a reduction in the adsorbed amount. This phenomenon in wh sorbed amount increases with pressure and decreases with temperature is due telier's principle [26]. According to Le Chatelier's principle, the exothermic higher temperature prefers conditions that produce less heat. In addition, sin tion is known as an exothermic process, it is expected to have a higher adsorb at a lower temperature due to a higher affinity between the adsorbent and the which results in the adsorbent's surface being covered with more adsorbates.
The influence of the porous nature of Mg-gallate on the CO2 adsorptio could be observed at 1 bar, showing that Mg-gallate offered a CO2 adsorption 5.06, 4.92 and 4.60 mmol/g at 273, 298 and 313 K, respectively. MOFs are consi petitive adsorbents once they can offer a CO2 adsorption capacity of 3.0 mmol [27]. Figure 7 illustrates the experimental CH4 single-component adsorption i The amount of CO 2 adsorbed on Mg-gallate increased as the pressure increased for all temperatures. However, the isotherms decreased as the temperatures increased from 273 to 313 K. The reason behind is that the surface adsorption energy and the molecular diffusion rate increase with the temperature, causing the gas molecules to unsteadily move and making it difficult for them to attach to the surface of the adsorbent [25]. Subsequently, a higher temperature condition is not advantageous for the adsorption process and may cause a reduction in the adsorbed amount. This phenomenon in which the adsorbed amount increases with pressure and decreases with temperature is due to Le Chatelier's principle [26]. According to Le Chatelier's principle, the exothermic process at higher temperature prefers conditions that produce less heat. In addition, since adsorption is known as an exothermic process, it is expected to have a higher adsorbed amount at a lower temperature due to a higher affinity between the adsorbent and the adsorbate, which results in the adsorbent's surface being covered with more adsorbates.
The influence of the porous nature of Mg-gallate on the CO 2 adsorption capacity could be observed at 1 bar, showing that Mg-gallate offered a CO 2 adsorption capacity of 5.06, 4.92 and 4.60 mmol/g at 273, 298 and 313 K, respectively. MOFs are considered competitive adsorbents once they can offer a CO 2 adsorption capacity of 3.0 mmol/g or higher [27]. Figure 7 illustrates the experimental CH 4 single-component adsorption isotherms at three different temperatures. The experimental CH4 single-component adsorption isotherms also di same pattern in which the amount of CH4 adsorbed on Mg-gallate increased w ing pressure for all temperatures with a linear shape isotherm. However, th decreased as the temperature increased from 273 to 313 K. The linear shape also known as Henry adsorption isotherm. Henry isotherm is considered the sorption isotherm since the partial pressure of adsorptive gas corresponds to of adsorbate [28]. Based on the isotherms, the equilibrium adsorbed amoun Mg-gallate was substantially proportional to the gas partial pressure. Mg-galla a CH4 adsorption capacity of 0.438, 0.226 and 0.158 mmol/g at 273, 298 and 31 tively and at 1 bar. It is worth noting that Mg-gallate is able to prevent CH4 adsorbed because of low gas uptake due to poor affinity between the framew molecules.
The CO2 adsorption capacity of Mg-gallate is greater than that of CH4 und condition due to the thermodynamic equilibrium effect. The thermodynamic effect arises due to the difference in the affinity/interaction of various gas mo adsorbed on the adsorbent surface. The affinity between the adsorbent and th depends on the different physical properties of the gas molecules, such as p and quadrupole moment. This large gap between the adsorption capacity va and CH4 verified that CO2 is a strong adsorbate and adsorbs more favorably on due to a stronger interaction between CO2 and Mg-gallate. It is known tha larger polarizability (29.1 × 10 −25 cm 3 for CO2, 25.9 × 10 −25 cm 3 for CH4) and moment (4.30 × 10 −26 esu cm 2 for CO2, 0 for CH4) compared to CH4 [29]. Th polarizability of the adsorbate, the higher the interaction with the adsorbent s interaction is favorable since the quadrupole moment of CO2 is complementa larization of Mg-gallate. The CO2 adsorption capacity was greatly affected by tions of Mg-gallate and CO2 molecules, which occurred at two binding sites. T ecules firstly occupied the open metal sites of the secondary building unit (SB interacted with the organic linker. Based on this phenomenon, the separ CO2/CH4 mixture is feasible. The experimental CH 4 single-component adsorption isotherms also displayed the same pattern in which the amount of CH 4 adsorbed on Mg-gallate increased with increasing pressure for all temperatures with a linear shape isotherm. However, the isotherms decreased as the temperature increased from 273 to 313 K. The linear shape isotherm is also known as Henry adsorption isotherm. Henry isotherm is considered the simplest adsorption isotherm since the partial pressure of adsorptive gas corresponds to the amount of adsorbate [28]. Based on the isotherms, the equilibrium adsorbed amount of CH 4 on Mg-gallate was substantially proportional to the gas partial pressure. Mg-gallate provided a CH 4 adsorption capacity of 0.438, 0.226 and 0.158 mmol/g at 273, 298 and 313 K, respectively and at 1 bar. It is worth noting that Mg-gallate is able to prevent CH 4 from being adsorbed because of low gas uptake due to poor affinity between the framework and CH 4 molecules. The CO 2 adsorption capacity of Mg-gallate is greater than that of CH 4 under the same condition due to the thermodynamic equilibrium effect. The thermodynamic equilibrium effect arises due to the difference in the affinity/interaction of various gas molecules to be adsorbed on the adsorbent surface. The affinity between the adsorbent and the adsorbate depends on the different physical properties of the gas molecules, such as polarizability and quadrupole moment. This large gap between the adsorption capacity values of CO 2 and CH 4 verified that CO 2 is a strong adsorbate and adsorbs more favorably on Mg-gallate due to a stronger interaction between CO 2 and Mg-gallate. It is known that CO 2 has a larger polarizability (29.1 × 10 −25 cm 3 for CO 2 , 25.9 × 10 −25 cm 3 for CH 4 ) and quadrupole moment (4.30 × 10 −26 esu cm 2 for CO 2 , 0 for CH 4 ) compared to CH 4 [29]. The higher the polarizability of the adsorbate, the higher the interaction with the adsorbent surface. This interaction is favorable since the quadrupole moment of CO 2 is complementary to the polarization of Mg-gallate. The CO 2 adsorption capacity was greatly affected by the interactions of Mg-gallate and CO 2 molecules, which occurred at two binding sites. The CO 2 molecules firstly occupied the open metal sites of the secondary building unit (SBU) and then interacted with the organic linker. Based on this phenomenon, the separation of the CO 2 /CH 4 mixture is feasible.

Isotherm Models
The experimental single-component adsorption isotherms of CO 2 and CH 4 should be first fitted using a proper model in order to perform the integration required by IAST. It is to discrete data so that the uncertainty in the multicomponent predictions always comes from this fit of data [3]. There are no restrictions on the choice of the isotherm models as long as they can fit precisely. Phyton can offer several isotherm models such as the Langmuir, quadratic, BET, Henry, approximated Temkin, and dual-site Langmuir to fit the experimental single-component adsorption isotherms [15]. The model parameters are tabulated in Table 2.  The differences between the studied isotherm models appeared as a function of how accurately the models fit the experimental single-component adsorption isotherms. It is important to emphasize that if IAST is used using a poorly fitted model, then the multicomponent prediction is expected to be inaccurate even though IAST itself is accurate for the system of interest. According to Table 2, the Langmuir model gave the best fit for all single-component adsorption isotherms of CO 2 and CH 4 at 273-313 K in terms of RMSE value with relevant parameter values. Figure 8 shows the experimental single-component adsorption isotherms and Langmuir fitted-isotherms of Mg-gallate for CO 2 and CH 4 at three different temperatures in terms of loading (mmol/g) versus pressure (bar).
The Langmuir model is a model to describe the adsorption behavior of gases and is defined as below [30,31]: where, q e (mmol/g) represents the amount of the adsorbed gas per unit mass of adsorbent at equilibrium, M (mmol/g) is the maximum adsorption capacity, K L (1/bar) is the Langmuir constant related to the free energy of adsorption and P e (bar) is the equilibrium pressure. It is a widely used isotherm model due to its simplicity, effectiveness and reasonable explanation of its parameters [32]. The Langmuir model was developed based on the assumption of monolayer adsorption, no interactions between adsorbed molecules, equal energy of adsorption and molecules adsorbed at fixed sites that do not migrate over the surface [33]. It is a widely used isotherm model due to its simplicity, effectiveness and reas explanation of its parameters [32]. The Langmuir model was developed based on sumption of monolayer adsorption, no interactions between adsorbed molecules energy of adsorption and molecules adsorbed at fixed sites that do not migrate o Based on Table 2, the maximum adsorption capacity (M) values of CO 2 molecules on Mg-gallate increased as the Langmuir constants (K L ) decreased from 273 to 313 K. K L explains the adsorption energy and the affinity between the adsorbates and the adsorption sites of the adsorbent in whose reduction of this factor causes the increase of M [34]. Normally, K L decreases with increasing temperature since adsorption is an exothermic process. All the values of K L for CO 2 are much higher than those for CH 4 , resulting in higher M values for CO 2 . Meanwhile, the values of M and K L for CH 4 fluctuated.

Prediction of CO 2 and CH 4 by IAST Calculations
Based on the experimental single-component adsorption isotherms, the multicomponent adsorption isotherms predicted using IAST with the Phyton package for the CO 2 /CH 4 gas mixture for different compositions at 273, 298 and 313 K and up to 1 bar are shown in Figure 9.
Based on the experimental single-component adsorption isotherms, the multicomponent adsorption isotherms predicted using IAST with the Phyton package for the CO2/CH4 gas mixture for different compositions at 273, 298 and 313 K and up to 1 bar are shown in Figure 9.   Figure 9 shows that the predicted amount of adsorbed CO2 and CH4 on Mg-gallate is lower than the experimental amount. This is due to competitive adsorption between the different components of the gas mixture in which the CO2 and CH4 molecules competed with each other for the adsorption sites. Consequently, the presence of CH4 in the gas mixture slightly affected the adsorption of CO2 on Mg-gallate. Nevertheless, CO2 was still dominant and favorably adsorbed compared to CH4 due to a stronger interaction between  Figure 9 shows that the predicted amount of adsorbed CO 2 and CH 4 on Mg-gallate is lower than the experimental amount. This is due to competitive adsorption between the different components of the gas mixture in which the CO 2 and CH 4 molecules competed with each other for the adsorption sites. Consequently, the presence of CH 4 in the gas mixture slightly affected the adsorption of CO 2 on Mg-gallate. Nevertheless, CO 2 was still dominant and favorably adsorbed compared to CH 4 due to a stronger interaction between CO 2 and Mg-gallate, resulting in a lower predicted amount of CH 4 due to the previously mentioned thermodynamic equilibrium effect. The predicted CO 2 adsorption isotherms exhibited the same pattern as the experimental single-component adsorption isotherms in that they increased with pressure but decreased as temperatures increased from 273 to 313 K. In addition, it is worth noting that the predicted adsorbed amount of CO 2 increased as the composition of CO 2 in the CO 2 /CH 4 gas mixture increased. Moreover, the predicted amount of CH 4 is too low and almost showed a plateau pattern. Indeed, the multicomponent adsorption isotherms using IAST led to the evaluation of the capability of Mg-gallate.
At initial stage of adsorption (lower pressure range), the CO 2 isotherms demonstrated a rapid increase in gas uptake, especially at a higher gas-phase mole fraction of CO 2 in which the isotherms became steeper but less steep as temperature increased. It means that the fastest initial stage of adsorption was achieved by the 75:25 composition adsorption at 273 K since it formed the steepest curve in the lowest pressure range. In addition, the steep curve was followed by the plateau gradient, indicating the saturation of the monolayer, which proved the occurrence of the gas adsorption limit. It is worth noting that the predicted CO 2 adsorption isotherms fulfilled the characteristics and belonged to Type I, while the CH 4 adsorption isotherms almost corresponded to the linear isotherms.
The isotherm shape can be used to determine whether an adsorption system is favorable or unfavorable. Based on the isotherm shapes, the predicted CO 2 adsorption systems were more favorable as the gas-phase mole fraction of CO 2 approached unity and at lower temperature. In addition, the predicted CH 4 adsorption systems could be considered approached unfavorable. The isotherm shapes in this work followed the pattern mentioned by the literature as shown in Figure 10 [35]. exhibited the same pattern as the experimental single-component adsorption isotherm that they increased with pressure but decreased as temperatures increased from 27 313 K. In addition, it is worth noting that the predicted adsorbed amount of CO2 increa as the composition of CO2 in the CO2/CH4 gas mixture increased. Moreover, the predi amount of CH4 is too low and almost showed a plateau pattern. Indeed, the multicom nent adsorption isotherms using IAST led to the evaluation of the capability of Mg-gal At initial stage of adsorption (lower pressure range), the CO2 isotherms dem strated a rapid increase in gas uptake, especially at a higher gas-phase mole fractio CO2 in which the isotherms became steeper but less steep as temperature increase means that the fastest initial stage of adsorption was achieved by the 75:25 composi adsorption at 273 K since it formed the steepest curve in the lowest pressure range addition, the steep curve was followed by the plateau gradient, indicating the satura of the monolayer, which proved the occurrence of the gas adsorption limit. It is w noting that the predicted CO2 adsorption isotherms fulfilled the characteristics and longed to Type I, while the CH4 adsorption isotherms almost corresponded to the li isotherms.
The isotherm shape can be used to determine whether an adsorption system is fa able or unfavorable. Based on the isotherm shapes, the predicted CO2 adsorption syst were more favorable as the gas-phase mole fraction of CO2 approached unity and at lo temperature. In addition, the predicted CH4 adsorption systems could be considered proached unfavorable. The isotherm shapes in this work followed the pattern mentio by the literature as shown in Figure 10   The degree of favorability in this work was supported by the important feature o Langmuir isotherm that can be expressed in terms of a dimensionless constant separa factor or equilibrium parameter ( ).
represents the isotherm shape and the natur the adsorption. It can be defined as below [36]: The value of determines whether the nature of the isotherm is irreversible ( 0), favorable (0 < < 1), linear ( = 1) or unfavorable ( > 1). The values of for di ent CO2/CH4 compositions at different temperatures are illustrated in Figure 11. The ues of in the range of 0-1 verified that CO2 adsorption was favorable. In addition, adsorption was less reversible in all CO2 compositions, confirmed by the lower value . Furthermore, CH4 adsorption was less favorable based on higher values of c pared to CO2. Additionally, the highest values of for CH4 that approached unity Figure 10. Nature of the adsorption isotherms [35].
The degree of favorability in this work was supported by the important feature of the Langmuir isotherm that can be expressed in terms of a dimensionless constant separation factor or equilibrium parameter (R L ). R L represents the isotherm shape and the nature of the adsorption. It can be defined as below [36]: The value of R L determines whether the nature of the isotherm is irreversible (R L = 0), favorable (0 < R L < 1), linear (R L = 1) or unfavorable (R L > 1). The values of R L for different CO 2 /CH 4 compositions at different temperatures are illustrated in Figure 11. The values of R L in the range of 0-1 verified that CO 2 adsorption was favorable. In addition, CO 2 adsorption was less reversible in all CO 2 compositions, confirmed by the lower values of R L . Furthermore, CH 4 adsorption was less favorable based on higher values of R L compared to CO 2 . Additionally, the highest values of R L for CH 4 that approached unity (reversible) represented the ease of adsorbed CH 4 molecules being desorbed from Mg-gallate. This is the reason behind the lower CH 4 uptake compared to CO 2 . In short, the degree of favorability is moving towards zero, indicating the completely ideal irreversible process than unity, meaning a completely reversible process [36]. s 2023, 28, x FOR PEER REVIEW of favorability is moving towards zero, indicating the completely ideal irreversi cess than unity, meaning a completely reversible process [36].

IAST Selectivity of CO2 and CH4
The IAST method is also applicable to calculate the mixed gas selectivity (IAS tivity) [37]. Adsorption selectivity occurs due to the difference in affinity of the components of the gas mixture to be adsorbed on the pore surface of adsorbents. S ity can be simply understood as the affinity of Mg-gallate to adsorb CO2 compared The adsorption selectivity established based on IAST calculations are shown in Fi The high IAST selectivity values for CO2/CH4 represented that CO2 was a str sorbate and adsorbed more favorably on Mg-gallate compared to CH4. A high se for CO2 over the other components of a gas mixture is necessary in CO2 capture tions. The IAST selectivity increased rapidly as the gas-phase mole fraction of proached unity. However, it could be said that IAST overpredicted the selectivit CO2/CH4 mixture, especially at 273 and 298 K. It usually happens because one com

IAST Selectivity of CO 2 and CH 4
The IAST method is also applicable to calculate the mixed gas selectivity (IAST selectivity) [37]. Adsorption selectivity occurs due to the difference in affinity of the various components of the gas mixture to be adsorbed on the pore surface of adsorbents. Selectivity can be simply understood as the affinity of Mg-gallate to adsorb CO 2 compared to CH 4 . The adsorption selectivity established based on IAST calculations are shown in Figure 12.
Molecules 2023, 28, x FOR PEER REVIEW 13 of 1 of favorability is moving towards zero, indicating the completely ideal irreversible pro cess than unity, meaning a completely reversible process [36].

IAST Selectivity of CO2 and CH4
The IAST method is also applicable to calculate the mixed gas selectivity (IAST selec tivity) [37]. Adsorption selectivity occurs due to the difference in affinity of the variou components of the gas mixture to be adsorbed on the pore surface of adsorbents. Selectiv ity can be simply understood as the affinity of Mg-gallate to adsorb CO2 compared to CH4 The adsorption selectivity established based on IAST calculations are shown in Figure 12 (a) (b) (c) The high IAST selectivity values for CO2/CH4 represented that CO2 was a strong ad sorbate and adsorbed more favorably on Mg-gallate compared to CH4. A high selectivity for CO2 over the other components of a gas mixture is necessary in CO2 capture applica tions. The IAST selectivity increased rapidly as the gas-phase mole fraction of CO2 ap proached unity. However, it could be said that IAST overpredicted the selectivity of th CO2/CH4 mixture, especially at 273 and 298 K. It usually happens because one componen (CO2) strongly adsorbed compared to the other component (CH4). Furthermore, uncer tainty arose as IAST was applied in regimes that required extrapolation beyond the ex perimental data. Despite these drawbacks, IAST is known as the established standard in multicomponent adsorption predictions [3].
As the temperature decreased, the IAST selectivity increased, confirming that Mg gallate performed better at low temperature for both single-component and mixed ga adsorption. Adsorption is an exothermic process in which low temperature conditions ar The high IAST selectivity values for CO 2 /CH 4 represented that CO 2 was a strong adsorbate and adsorbed more favorably on Mg-gallate compared to CH 4 . A high selectivity for CO 2 over the other components of a gas mixture is necessary in CO 2 capture applications. The IAST selectivity increased rapidly as the gas-phase mole fraction of CO 2 approached unity. However, it could be said that IAST overpredicted the selectivity of the CO 2 /CH 4 mixture, especially at 273 and 298 K. It usually happens because one component (CO 2 ) strongly adsorbed compared to the other component (CH 4 ). Furthermore, uncertainty arose as IAST was applied in regimes that required extrapolation beyond the experimental data. Despite these drawbacks, IAST is known as the established standard in multicomponent adsorption predictions [3].
As the temperature decreased, the IAST selectivity increased, confirming that Mggallate performed better at low temperature for both single-component and mixed gas adsorption. Adsorption is an exothermic process in which low temperature conditions are favored. Since adsorption is inversely proportional to temperature, the adsorbate molecules tend to desorb from the adsorbent under high temperature conditions, and this process is called desorption. This is the reason behind lower adsorption uptake under high temperature conditions. The IAST selectivity of Mg-gallate in this work is compared to the other gallate-based MOFs at 298 K and 1 bar for CO 2 /CH 4 composition of 50:50. This Mg-gallate could offer an IAST selectivity of 5236, higher than Ni-gallate (3171), Mggallate (2497) and Co-gallate (198) under the same operating conditions, done in previous work [38].
Even at high content of CH 4 (low content of CO 2 ), Mg-gallate could still offer the potential IAST selectivity. The composition of CH 4 in natural gas is normally in the range of 75-98 vol% [39]. Therefore, these outcomes are relevant to the practical challenges of natural gas purification and Mg-gallate can be considered as the promising adsorbent for CO 2 /CH 4 separation.

Synthesis of Mg-Gallate
Mg-gallate was prepared by hydrothermal synthesis according with previously reported work [17]. 50 mmol of MgCl 2 and 100 mmol of gallic acid were added to 250 mL of 0.5 M KOH aqueous solution in a round-bottomed flask. Then, the mixture was heated and refluxed at 80 • C and ambient pressure with continuous stirring for 24 h. After being naturally cooled, the product was collected and washed two times with deionized water. Then, the product was immersed in ethanol with double replenishment.

Characterization of Mg-Gallate
The Powder X-ray Diffraction (PXRD) pattern for Mg-gallate was recorded using X'Pert Powder Panalytical equipped with Cu Kα radiation (λ = 1.54 Å). The data was collected in the range of 5-50 • (2θ angle range) with a scan step size of 0.02 • .
The Fourier Transform Infrared Spectroscopy (FTIR) spectrum was performed with a potassium bromide flakelet (KBr) method using a Thermo Scientific Nicolet IS5 in the wavenumber range of 500-4000 cm −1 with 16 scans. Background scanning was performed to determine the presence of air impurities such as the carbon dioxide peak.
The Thermogravimetric Analysis (TGA) of Mg-gallate was measured using a Perkin Elmer STA 6000. Approximately 5.0 mg of sample was weighed in a crucible pan and placed on the sample holder. The measurement was conducted at a 283 K/min heating rate under a temperature range of 323-973 K under N 2 condition.
The porous properties of Mg-gallate were investigated via N 2 adsorption-desorption isotherms measured at 77 K using 3FLEX Micromeritics Surface Characterization.
For every characterization analysis, Mg-gallate was degassed at 393 K for 24 h under an ultrahigh vacuum.

Single-Component Gas Adsorption
Prior to the adsorption measurement, Mg-gallate was first degassed at 393 K for 24 h under an ultrahigh vacuum. The single-component isotherms for both CO 2 and CH 4 were measured at 273, 298 and 313 K using 3FLEX Micromeritics Surface Characterization.

IAST Calculation
The mixed gas adsorption isotherms were predicted from experimental CO 2 and CH 4 single-component adsorption isotherms using IAST calculations with the Python package (pyIAST), as studied in previous work [15]. The detailed methodology was explained there, here the highlighted points were simplified.
First, pyIAST characterized the experimental CO 2 and CH 4 single-component adsorption isotherms by fitting the suitable analytical models such as the Langmuir, quadratic, BET, Henry, approximated Temkin, and Dual-site Langmuir to the data. Second, pyIAST linearly interpolated the data. Finally, pyIAST performed IAST calculations to predict the mixed gas adsorption isotherms. The compositions of the CO 2 /CH 4 mixed gas were varied to 10:90, 25:75, 50:50 and 75:25. From the mixed gas adsorption isotherms, the IAST selectivity was calculated.

Conclusions
IAST is a method to describe the multicomponent adsorption equilibrium in which adsorption selectivity can be predicted solely based on experimental single-component adsorption isotherms. The IAST calculation came with model fittings of the experimental data where the Langmuir model gave the best fit based on the RMSE value with relevant parameter values. The predicted amount of adsorbed CO 2 and CH 4 on Mg-gallate is lower than experimental amount due to competitive adsorption among the different components of the gas mixture. The stronger interaction between CO 2 and Mg-gallate contributed to the higher predicted adsorbed amount of CO 2 than CH 4 , which increased as the compositions of CO 2 /CH 4 increased. It is also confirmed that the CO 2 adsorption was favorable based on the values of R L . IAST selectivity increased rapidly as the gas-phase mole fraction of CO 2 approached unity. Therefore, Mg-gallate is a promising adsorbent for separation of CO 2 /CH 4 based on IAST calculations. In summary, the main purpose of this work was achieved that is using IAST calculations to predict the potential selectivity of Mggallate based on simple measurements of single-component adsorption isotherms, since selectivity is the important factor when working with multicomponent gas adsorption. These predicted multicomponent adsorption behaviors can be applied in the design of practical gas adsorption and separation processes.  Data Availability Statement: The raw and processed data required to reproduce these findings cannot be shared as the data is part of the ongoing study.