CO₂ Capture on Prussian Blue Analogues (PBAs): Modeling Equilibrium, Kinetics, and Isosteric Heats of Adsorption

Abstract

The demand to alleviate climate change produced by greenhouse gases, such as CO₂, is a fact at present. The use of porous adsorbents such as Prussian blue analogues (PBAs) possessing fast removal velocities and improved uptake capacity properties has shown promise as an alternative in the solution to this problem. In this work, the structural characterization and CO₂ adsorption data of a potassium cobalt hexacyanoferrate (CoHCFM11) are presented. FTIR demonstrates the presence of a mixture of two chains, Fe³⁺–CN–Co²⁺ and Fe²⁺–CN–Co³⁺, produced during the synthesis procedure. XRD reveals a face-centered cubic crystal, spatial group Fm3̅m, and corroborates the presence of cavities ideal to capture CO₂ molecules. XPS shows the existence of two oxidation states of Co and Fe in PBA. TEM measurements confirm the cubic shape nature of CoHCFM11 with an average size of 80–130 nm. A deep understanding of the CO₂ adsorption mechanism by CoHCFM11 is obtained by mathematical models. The validation of models in predicting experimental isotherm and kinetic adsorption data sets is presented. The goodness of fit is collated through the residual and normal probability plots to discriminate the appropriate model. Normal probability plots confirm the best stability of the 2-S Langmuir model in agreement between the measurements and model. Kinetic data are fitted using five models. Goodness-of-fit statistics confirm that Avrami’s model better explains the data. Isosteric heats of adsorption are obtained through the Clausius–Clapeyron equation using the isosteres approach to which the heats obtained via the different isotherm model predictions are compared. The 2-S Langmuir model predicts closely the results obtained via the isosteres approach. The results of this study demonstrate the need to use goodness-of-fit statistics when selecting a model to recommend as the most appropriate in CO₂ adsorption data.

1. Introduction

The emission of greenhouse gases to the atmosphere, such as CO₂, has posed a concern in the last few decades. Different methodologies have been used to tackle this problem. These include the use of porous solids that are an attractive option to this end. These solids possess rapid uptake properties, increased adsorption capacities, enhanced attraction for CO₂ when other gases are present (i.e., improved selectivity), and stable regeneration characteristics.¹ A few reports indicate the application of Prussian Blue Analogs (PBAs) to remove CO₂. For instance, Roque et al.² employed a series of transition metals hexacyanocobaltates to capture CO_2, reporting a maximum uptake of 4.10 mol mol^–1; they used the Dubinin-Ashtakov isotherm to model the adsorption equilibrium. Their results indicated a dependence of the material properties on the metal used to form the 3D network. Ojwang et al.³ studied the CO₂ adsorption kinetics and equilibrium by TGA on two PBAs, K_2x/3Cu^II [Fe^II_x Fe^III_1–x (CN)₆]_2/3, with nominally K-free x = 0.0 and K-rich x = 1.0. The isosteric heats of adsorption were determined using the Clausius–Clapeyron equation. They found that three kinetic CO₂ adsorption curves could be described by a double exponential function reporting two collateral processes and differing rate constants. The highest CO₂ uptake for the two compositions was ∼ 4.5 mmol g^–1 at 1 bar and 273 K. The PBAs showed fast adsorption kinetics and stable cyclic performance at room temperature. Svensson et al.⁴ conducted studies on the CO₂ removal by several Cu-PBAs for nominal compositions, namely A_2/3Cu[Fe(CN)₆]_2/3 with A = vacant, Li, Na, K, Rb, Cs. Their results indicated that Na PBA exhibited a maximum adsorption uptake of ∼ 3.8 mmol g^–1 at 20 °C and 1 bar. Nonetheless, the just mentioned reports did not examine any specific adsorption isotherm nor kinetic modeling, which is essential to gain a comprehensive understanding of the adsorption process and can help in the design of industrial-scale equipment.

Any model applied in predicting experimental data sets requires to be validated. Regrettably, this condition fails to be observed, or little attention has been paid in many studies reported in the literature.⁵ In this regard, a significant quantity of studies used as a validation method the determination coefficient (R²), whereas some others based their validation methods on this coefficient combined with other parameters (e.g., Sum of Square Error, SSE; Chi-Square, χ²; or the Root Mean Square Error, RMSE, just to mention a few) as an extra basis to discriminate the suitable model for their data set.⁶⁻⁸ However, it has been reported that R² is insufficient to validate model fittings, and that is the most used validation point of reference in modeling adsorption equilibrium and kinetic data.^5,9 As a result of this, the sole use of this parameter does not ensure a good data fitting by the model; even high R² values do not guarantee model sufficiency. To overcome this situation, in the present work the goodness of fit statistics using graphical residual analysis provides a quantitative comparison of the different models.

Diverse types of residual plots can be used to check the validity of the underlying assumptions and provide information on how to improve the model.¹⁰ In this sense, it is also important to evaluate the distribution of errors before a recommended fit. Under this approach, the underlying statistical assumptions about residuals, such as constant variance, independence of variables, and normality of the distribution, can be examined. For these assumptions, the residuals must be randomly distributed around zero for a particular regression model to stay valid. These residual plots can be used to assess the quality of the regression. Graphical residual analysis is centered on the conduct of the residual of each data point to corroborate that certain modeling surmises related to the model error are authentic. These surmises consider that independence exists in the errors (i.e., there is randomness in their conduct), and that the error goes along with a normal distribution.¹¹ The first conjecture was proven by Simonin,¹² comparing adsorption rate kinetic data from the literature for the pseudo-first (PFO) and pseudo-second order (PSO) models. He constructed plots of residuals vs time and used them along with the values of R² and the average relative error (ARE) to analyze the acceptability of pseudo-first- and pseudo-second-order (PFO and PSO) models. Alternatively, Pavan et al.¹³ reported the use of residual probability plots to demonstrate the randomness of errors when their data presented a normal distribution. Along with the ARE and standard deviation parameters, they concluded that the Sips isotherm model best fitted their experimental data of methylene blue removal from aqueous solutions by fruit waste. The lack of validity of the aforementioned studies demonstrates the need to implement graphical acceptance methods to confirm the quality of data fit to a given model. In the present work, the values of R,² RMSE, GMRSE, GSEP, and PICPglobal parameters parameters in combination with the graphical residual analysis (residual and normal probability plots) were used to help recommend the most acceptable models to explain the isotherm and kinetic data of CO₂ adsorption on the Prussian Blue Analog CoHCF M11.

The construction of adsorptive gas separation units requires the knowledge of critical design parameters such as the isosteric heat of adsorption, also known as isosteric enthalpy of adsorption (q_st).¹⁴ The term isosteric heat of adsorption refers to the heat liberated when gas or liquid molecules are deposited on a solid surface at a given surface coverage.¹⁵⁻¹⁷ The solid material accumulates a portion of this heat, and then it becomes critical to estimate the heat of adsorption when performing energy budgets in the design of adsorption process units.¹⁸ There are two fundamental procedures to determine the isosteric heat of adsorption: (i) by molecular simulations and (ii) by experiments. Molecular simulations employ grand canonical Monte Carlo calculations to simulate adsorption isotherms and calculate the isosteric heat of adsorption by the ensemble fluctuation procedure.¹⁹ The experimental procedures can be divided into direct and indirect approaches. The first one employs calorimetric-volumetric devices to measure the released isosteric heat of adsorption directly. However, this method is scarcely used, as instruments are complex and costly. The indirect approach is the most widely used in the literature and employs data from experimentally measured isotherms at several temperatures. The experimental isotherm data are commonly obtained at 10–20 K intervals. The method is based upon the use of the Clausius–Clapeyron eq 1.

1

where q_st is the isosteric heat of adsorption, p is the absolute pressure, T is the absolute temperature, R is the ideal gas constant, and n is the amount of gas adsorbed. At least two isotherms measured at two different but close temperatures (∼10 K difference) are required to yield a basic calculation by selecting a common uptake amount. This method will provide a single value for the heat of adsorption, usually at a very low surface coverage. Substantial improvement in the precision of the calculations can be obtained if a set of isotherms at more than two temperatures and a range of adsorption quantities common to all isotherms are employed.²⁰ The latter means that the more isotherms used, the better the statistical significance obtained. On the other hand, numerous isotherm models for pure components have been used to assess the isosteric enthalpy of adsorption and are accessible in the literature; the option of the isotherm model has an impact on the appraisal of the isosteric enthalpy of adsorption. The most used isotherm models include Langmuir,^21,22 Toth,²² and multisite Langmuir.²³

To the best of our knowledge, no studies have been reported, including a statistical model validation of CO₂ adsorption data. The novelty of this work included the use of the residuals and normalized plots approach to discriminate model fitting for CO₂ adsorption. Most of the reports in the literature dealing with CO₂ adsorption, use either R² or Root Mean Squared Error (RMSE) or both parameters to discriminate model fitting. The residuals and normalized graph approach outperforms the approach using the R² and RMSE parameters when applied on CO₂ adsorption for several reasons, namely: (i) Better visualization of goodness of fit. This is especially important where the relationship between the amount of gas absorbed and the pressure can be complex and nonlinear. (ii) Residuals and normalized plots help in detecting patterns and trends not evident by the R² and RMSE parameters, such as randomness of data or systematic pattern of the errors. (iii) The residuals and normalized plots approach helps to identify normal distributions of the errors, outliers, and anomalous data. The aims of this work are 2-fold: (i) to employ the Goodness of Fit Statistics using graphical residual analysis to help elucidate more accurately which model explains better the isotherm and kinetic data, and (ii) to demonstrate that the estimated values of the isosteric enthalpy of adsorption using five different isotherm models strongly depend on the model selected when applied to the CO₂ adsorption on PBAs.

2. Materials and Methods

2.1. Materials

Co (NO₃)₂ was purchased from Fermont (Monterrey, MX) and K₃Fe(CN)₆ (99.0% purity) from J.T. Baker (Phillipsburg, NJ). The CO₂ (99.995% purity) and He (99.999% purity) were acquired from INFRA (Mexico City, MX) and Praxair Mexico (Guadalajara, MX), respectively.

2.1.1. Synthesis and Characterization of CoHCF M11

Our research group recently synthesized and characterized K–Co hexacyanoferrates (CoHCF) using three systems with varying reactant concentrations via the drop method.²⁴ In that work, it was found that sample CoHCF M11 presented the best CO₂ capture efficiencies and, consequently, is used in this study to perform all the model simulations. Briefly, sample CoHCF M11 was synthesized employing a molar ratio of K₃Fe(CN)₆:Co(NO₃)₂ of 1.71:2.5. 50 mL of a solution 0.175 M Co(NO₃)₂ was added dropwise to 300 mL of a solution 0.02 M K₃Fe(CN)₆ at time intervals of 2–3 s between each drop addition and at room temperature (25 °C). After addition of the nitrate solution, the suspension formed was stirred for about 5 min, followed by a rest period of 24 h to allow for system stability using amber bottles to avoid cobalt reduction by sunlight. The suspensions were separated by centrifugation for 20 min; the supernatant was discarded, and the remaining precipitate was washed with doubly distilled water. The resulting powder sample was dried at 60 °C for 72 h. The powder was ground by an Agatha mortar. The stoichiometric formula of CoHCF M11 K_0.12Co_1.52[Fe(CN)₆]*6.6H₂O was obtained from elemental analysis using a Leco TruSpec Micro CHNS elemental analyzer (St. Joseph, MI, USA) and by ICP-MS measurements (Agilent Technologies, 7500a model) after acid digestion with boiling H₂SO₄. The oxidation states of Fe and Co were determined from XPS measurements. Fourier transform infrared attenuated total reflection (FTIR-ATR) analysis was used to assess the nature of the surface functional groups of the CoHCF M11 sample using a Bruker Alpha FTIR-ATR system (Bruker Optics, Billerica, MA) with a 300 Golden Gate diamond ATR Model; a scanning range of 4000–450 cm^–1 at a resolution of 2 cm^–1 was employed. The crystalline characteristics of the CoHCMF M11 sample were obtained by Powder X-ray Diffraction (XRD) using a Rigaku diffractometer model Ultima IV with CuKα radiation with the following operation parameters: λ = 1.54051, 2θ range of 12°-72°, step size of 0.02, time step of 25 s, intensity of 30 mA and power of 40 kV. To generate the simulated XRD pattern and obtain the unit cell parameters corresponding to the phase of the CoHCF M11 sample, crystallographic data were taken from Ratuszna and Malecki.²⁵ Simulations were conducted using the freely accessible software Powder Cell for Windows V. 2.4 (PCW). The simulated XRD pattern was compared to the corresponding experimental pattern by a Rietveld refinement using the same software. XPS was used to validate the existence of Fe and Co and verify their oxidation states in the CoHCF M11 sample. Measurements were conducted with a Phoibos 150 spectrometer equipped with an XR50 M monochromatic Al Kα (hν = 1486.7 eV) X-ray source with a one-dimensional detector 1DDLD supplied by SPECS (Berlin, Germany) operated at 10–14 kV. A base pressure in the analyzer was <5.1 × 10^–9 Torr. An aluminum anode used as an X-ray source operated at 250 kW was used to collect the XPS spectra. The core level spectra were collected within the range of 50–70 eV (Co and Fe 2p) with the pass energy of 15 eV, and steps were 0.1 eV wide. The surface shape of the CoHCF M11 sample was analyzed by Scanning Electron Microscopy (SEM) with a Tescan Mira3 LMU high-resolution microscope operated at 15 kV; Transmission Electron Microscopy (TEM) measurements used a JEOL JEM 1010 100 kV microscope. For a detailed description of these techniques, the reader is referred to ref (24).

2.2. Equilibrium Adsorption Isotherms of CO₂

The adsorption isotherms for CO₂ on the adsorbent CoHCF M11 were measured in the range of 273–353 K at intervals of 10 K and equilibrium pressures ranging from 0.35 to 114 kPa. The experiments were conducted using the static volumetric method, as reported by Frias-Ureña et al.,²⁴ using an ASAP 2020 Micromeritics sorptometer (Nocross, GA) and a temperature-controlled bath ISO Controller with glycol-water coolant. The uncertainties generated by the instrument during the measurements were operating temperature ± 0.1 K and pressure ± 0.5 Pa; the adsorbed amounts were simultaneously measured based on pressure changes inside the adsorption system with an estimated error in the calculation of the uptake amount of ± 0.1 mg. Before each isotherm experiment, about 80–100 mg of previously dried sample was treated by degassing it under a vacuum of 10 μm Hg and at 150 °C for 6 h to remove all adsorbed impurities from the surface of adsorbents. The breakdown of CoHCF M11 does not occur at this temperature⁶⁶ and the disappearance of water is noticed. The equilibrium acquisition time was set at 5 s, since higher times yield excessive times in the collection of isotherm data.

2.3. Adsorption Kinetic Measurements

The kinetic data of CO₂ adsorption onto CoHCF M11 were obtained at temperatures of 273, 298, and 313 K using the Rate of Adsorption (ROA) built-in option in the software of the ASAP 2020 sorptometer concurrently when the adsorption isotherms were measured. The method operates by administering a fixed quantity of CO₂ onto the sample and recording the amount of gas adsorbed and the corresponding pressures over time. After that, the data are transformed into transient uptakes to produce the kinetic data. The dosing quantity is taken from the BET monolayer capacity (v_m), and the adsorption equilibrium amount is taken as the last adsorption amount at the final temperature and pressure. The pressure readings were 200 per dose.^26,27

2.4. Modeling Methods (Isotherms, Isosteric Heats of Adsorption, and Kinetics)

The measured adsorption equilibrium data of CO₂ on CoHCF M11 were modeled and fitted to the fundamental and most popular isotherm equation of Langmuir and the empirical and semiempirical isotherms of Freundlich, Freundlich-Langmuir (Sips), and Toth and to the unusual two-site Langmuir model (2-S Langmuir). Every isotherm equation was tested for fitting in situations of constant temperature and temperature dependence. The saturation capacity parameter in all the isotherms was assumed to be independent of temperature.²⁸ The goodness of fits was determined using statistical concepts (e.g., residual and normal probability plots) to check for error variances and help discriminate the most appropriate model that could explain the data and recommend which model should be used. The isosteric heats of adsorption were calculated using two approaches, all based upon the Clausius–Clapeyron equation, namely, (i) the isosteres approach and (ii) the isotherm equations approach. The isosteres approach was taken here as a benchmark to which the isotherm was compared for the predictions. In the case of the kinetic experiments, five models were tested to explain the experimental data, pseudo-first-order (PSO), pseudo-second-order (PSO), intraparticle diffusion (IPD), Fickian diffusion (FDM), and Avrami’s models. Description of the kinetic models is presented in Section S4 of the Supporting Information.

2.4.1. Adsorption Isotherm Models

To make the most comprehensive inclusion of the temperature effects on the models, several issues should be considered, just to mention some: 1. Excessive simplification of the adsorption models. 2. Adsorption constants. In many adsorption models, these constants (such as K in the Langmuir model) are considered constants, which is not true. In this case, temperature can affect the adsorption constant, and if this is not considered in the simulations, the model can be inaccurate. 3. The adsorption energy. This parameter can vary with the temperature. If this variation is not properly considered in the simulations, the model may not be capable of predicting the adsorption at different temperatures. 4. Adsorption mechanisms. In this case, the adsorption mechanisms can change with temperature, and if not considered in the model, the predictions can lead to erroneous results of adsorption at different temperatures. However, in this work were considered the simplified temperature relationships reported in the literature, since considering all the effects involved in the temperature parameters will make the isotherm equations mathematically intractable and perhaps would not allow a simple comparison of the experimental data with the models. The following model selection criteria were considered in this study: (i) thermodynamic consistency. In this case, the models were selected based on the fact that adsorption enthalpy has a negative value for an exothermal process. (ii) The models reflect the adsorption on the internal surface of the adsorbent. (iii) The models reflect the presence of different adsorption sites to incorporate the heterogeneity of adsorbent.

2.4.1.1. Langmuir Model

This model is based upon the idea that the adsorption occurs on a homogeneous surface and that every adsorption site is equivalent. This is physically justified because it is assumed that the surface of CoHCF M11 is flat and uniform, which suggests that the adsorption sites are homogeneously distributed. Also, the model is justified because the adsorption of CO₂ on CoHCF M11 is a reversible process occurring at moderate temperatures and pressures. On the other hand, the chemical rationale is that CO₂ adsorption happens through the formation of a monolayer, and that weak chemical bonds are formed between the CO₂ and the surface sites of CoCHF M11since CO₂ can interact with the surface of CoCHF M11 via van der Waals and dipole–dipole forces. The Langmuir isotherm is expressed by the following equation:^29,30

2

where q is the amount of adsorbate at equilibrium with the solid phase (mmol g^–1), q_max is the maximum saturation adsorption capacity (mmol g^–1), p is the gas pressure at equilibrium (kPa), and K_L is called the affinity constant or Langmuir constant (kPa^–1), it is a measure of how strong an adsorbate molecule is attracted onto a surface. The affinity parameter as a function of temperature is given by

3

where K_Lo is a preexponential factor of the affinity constant (kPa^–1), E is the energy of adsorption of the single adsorbate (J mol^–1), R is the ideal gas constant, and T is the absolute temperature.

2.4.1.2. Freundlich Model

This isotherm is one of the most used models and can be applied to processes of multilayer adsorption occurring on heterogeneous surfaces. The model is physically rationalized because the surface of CoHCF M11 can contain defects and irregularities that affect the CO₂ adsorption. From the chemical point of view, this model is based upon the ideas that CO₂ adsorption occurs through both weak and strong chemical bonds between CO₂ and the surface of CoHCF M11, and that the surface of CoHCF M11 can have different adsorption sites with varying adsorption energies. This model provides an expression that defines surface heterogeneity and the exponential distribution of the active sites and their energies.^30,31 The equation for this model is given by

4

where q is the amount of adsorbate at equilibrium with the solid phase (mmol g^–1), K_F is the Freundlich adsorption constant (mmol g^–1) that represents the uptake capacity of gas by the solid, p is the gas pressure at equilibrium (kPa), and n represents the intensity of adsorption. Freundlich adsorption constant K_F as a function of temperature is given by

5

where K_Fo is a preexponential factor (mmol g^–1), A₀ is the characteristic adsorption potential, (J mol^–1), and α is an adjusting parameter.

2.4.1.3. Toth Model

An empirical variant of the Sips isotherm is the Toth equation, which can reproduce the data at very small pressures while showing asymptotic behavior for pressure values close to saturation. This isotherm model is based upon the idea that adsorption occurs on partially homogeneous and partially heterogeneous surface. The physical rationale behind this assumption is that the surface of CoHCF M11 can contain relatively flat and uniform areas, and areas which are rougher and more heterogeneous, whereas the chemical rationale is based upon the idea that CO₂ adsorption occurs through both weak and strong chemical bonds between CO₂ and the surface of CoHCF M11. Many systems showing submonolayer coverage can be well explained by this model;³² the following expression is given for this model:

6

where the parameters K_T (kPa^–1) and t are specific for adsorbate–adsorbent pairs. The Toth specific constant K_T and parameter t, are given as a function of temperature by

7

8

where K_T0 (kPa^–1) is the affinity at some reference temperature T₀, α and t₀ are the parameters of the temperature-dependent form of the Toth model, E is a measure of the heat of adsorption (J mol^–1).

2.4.1.4. Freundlich-Langmuir (Sips) Model

The problem of an increase in the amount adsorbed with increasing pressure in the Freundlich equation, was recognized by Sips (1948) and this led him to suggest an equation alike in form to the Freundlich equation, except that now it contains a finite limit when the pressure is high enough.³² The Sips model combines the Langmuir behavior at high pressures and the Freundlich behavior at low pressures, resulting in a three-parameter equation.³² The model is based upon the assumption that adsorption occurs on a surface with adsorption sites with different adsorption energies and that multiple layers are formed. The physical justification behind these assumptions rely on the fact that the CoHCF M11 sample can contain different adsorption sites with varying adsorption energies, while the chemical rationale is that both weak and strong chemical bonds can be formed between CO2 and the surface of the CoHCF M11 sample. This model has the following form:

9

where q_max (mmol g^–1) and K_FL (kPa^–1) have the same meanings as in the Langmuir model, and n is considered a parameter that includes the heterogeneity of the system.

The Sips specific constant K_FL is given as a function of temperature by

10

where E (J mol^–1) is different from the heat of adsorption presented in the Langmuir equation, where it resulted equal to the isosteric heat invariant with the surface loading; in the case of the Sips equation, this parameter represents solely a measure of the adsorption heat.³²

2.4.1.5. 2-S Langmuir Model

To describe adsorption differences caused by heterogeneous adsorbents, two different sites are available in the 2-S Langmuir model, where K_L1(T) and K_L2(T) are modeled as separate equilibrium constants for each site. The adsorption on surfaces with two types of sites with different adsorptions is assumed, and that a monolayer is formed on every site. This is physically explained because the two different types of surface sites can have different adsorption energies that can affect the CO2 adsorption. The chemical rationale in this model is like the two previous models in that both weak and strong chemical bonds can be formed between CO₂ and the surface of the CoHCF M11 sample. In this case, a 2-S Langmuir adsorption isotherm better describes the adsorption data of adsorbents interacting through more than one mechanism or distinct adsorption energies. The equation for this model is given by

11

where q_max1 (mmol g^–1) and K_L1 (kPa^–1) are the saturation capacity and the affinity parameter on the first set of sites, respectively, and q_max2 (mmol g^–1) and K_L2 (kPa^–1) are the analogous parameters on the second set of sites. The affinity parameters as a function of temperature are given as

12

13

where E₁ (J mol^–1) and E₂ (J mol^–1) are the adsorption energies of sites 1 and 2, respectively, and b₀₁ (kPa^–1) and b₀₂ (kPa^–1) are the corresponding preexponential factors.

2.4.2. Isosteric Heat of Adsorption

In the Clausius–Clapeyron equation calculations, two main approaches are used: (i) the isosteres method and (ii) the isotherm derivatives approach. In the present work, the results of adsorption heat with an isostere approach and different isotherm models were obtained and compared to gain insights related to the rational determination of the adsorption heat in CoHCF M11. These two approaches are described below.

2.4.2.1. q_st by the Isosteres Method

In this approach, eq 1 was used in the form:

14

Plots of ln p vs 1/T produce straight lines for constant n, called isosteres, with slope m = q_st/R from which q_st can be readily obtained. In the cases where more than two isotherms are used in the calculation of the isosteric heat of adsorption, this necessitates choosing the adsorption isotherms where the range of adsorbed quantities should be common for each of the isotherms being used, encompassing all analysis temperatures. This process allowed us to find the lower and upper limits of adsorption quantities. In the present work, seven isotherms were used from which the upper limit was found by taking the isotherm of lowest ultimate volume, in this case, the corresponding to the highest temperature of 353 K; a value of 0.627 mmol g^–1 was obtained (dotted line in Figure 1). On the contrary, the lower limit for the adsorption quantity was found by taking the isotherm of the highest initial quantity; in this case, at the lowest adsorption temperature of 273 K, a value of 0.043 mmol g^–1 was obtained, as shown by the dotted line of the inset of Figure 1.²⁰ This range was input into the MicroActive software of ASAP 2020 to calculate the isosteric heat of adsorption by the isosteres method.

Figure 1.

Isotherms of CO₂ adsorption on the CoHCF M11 sample used to obtain the adsorption limits to calculate the isosteric heat of adsorption by the isosteres approach.

2.4.2.2. q_st by the Isotherm Equations Approach (Analytical)

In the analytical approach, the measured isotherm data must be fitted first with the same model. Then, the isotherm equation models must be rearranged to have pressure p as a function of the amount adsorbed q. Afterward, analytical expressions for the isosteric heat of adsorption are derived. Lastly, by introducing the fitted parameters of the isotherm models into the derived analytical equations, the isosteric heat of adsorption can easily be acquired using eq 1. This work employed five isotherm models: Langmuir, Freundlich, Freundlich-Langmuir (Sips), Toth, and 2-S Langmuir. The isosteric heats of adsorption for each model are given below; details of the derivation of each equation are shown in the Supporting Information section S1–S3.

a) q_st by the Langmuir Model

In this model, eq 2 was used along with eq 3 to obtain an explicit expression of pressure p as a function of the amount adsorbed q and T. Then, applying the Clausius–Clapeyron eq 1, it was obtained:

15

b) q_st by Freundlich Model

In this case, the isosteric heat of adsorption was obtained by substitution of the Freundlich isotherm eqs 4 and 5 in the Clausius–Clapeyron eq 1 to have^33,34

16

c) q_st by Toth Model

In this model, eqs 6, 7, and 8 were used in eq 1 to obtain the isosteric heat of adsorption as a function of the fractional loading defined as , as follows:³²

17

In the case of zero fractional loading, q_st becomes infinity, whereas in the case of high loadings, q_st becomes minus infinity. These two extreme situations restrict the application of the Toth equation in the calculations of the isosteric heat of adsorption.³² When the fractional load is equal to zero, E = (q_st)_θ=0.

d) q_st by Freundlich-Langmuir (Sips) Model

eqs 9 and 10 were used in eq 1 to calculate q_st as a function of fractional loading θ ; the result is shown in eq 18:^32,35

18

In this situation, when the fractional load θ equals 1/2, the isosteric heat of adsorption q_st becomes equal to E. Parameter E defined in eq 9 is the isosteric heat at the fractional loading of 0.5, E = (q_st)_θ=1/2.

e) q_st by 2-S Langmuir Model

Using more complex isotherm equations to calculate the isosteric heat of adsorption is not easy. In the case of the 2-S Langmuir isotherm model, an explicit analytic expression for the pressure p was required as a function of the amount adsorbed q and obtained from eq 11. Then, the expression for q_st was obtained using eqs 11, 12 and 13. The analytic expression for q_st in this model, as reported elsewhere,²³ is as follows (see Supporting Information):

19

where:

2.5. Methods of Parameter Assessment and Validation of Results

The isotherm equation models were fitted with and without the influence of the temperature parameter; both isotherm and kinetic models were fitted using the nonlinear forms given in Sections 2.4.1 and S4 of the Supporting Information, respectively. The Levenberg–Marquardt algorithm implemented in the software Origin Pro ver. 2021 was employed in the present work to find the parameters by nonlinear regression of the experimental data. The linearization of the equations of isotherm and kinetic models was avoided as this procedure has demonstrated that it produces statistical bias of the regressed data.^36,37 The goodness of fit of the isotherm models was assessed by using the nonlinear coefficient of determination, R,² the root-mean-squared-error, RMSE, the Geometric Mean Relative Squared Error, GMRSE, and the Global Standard Error of Prediction, GSEP, and the uncertainty was calculated in a global manner for each model by the Prediction Interval Confidence Percentage, PICP global, with 95% confidence level, given by

20

21

22

23

24

where y_i meas is the measured value, y_i, pred is the predicted value by the model, y_avg is the average value of the data, N is the total number of data points, and q^calc and q^meas indicate the calculated and the experimental amounts adsorbed, respectively, and Z is the value of the standard distribution for a confidence level chosen (Z= 1.96 for 95% confidence level was used in this work). Here, GMRSE is a measurement of the model precision based on the geometric mean of the relative squared errors. A small value of GMRSE indicates that the model is accurate and that the predictions are close to the observed values; a high value of GMRSE indicates a lack of accuracy of the model. GSEP is a measure of the prediction errors of the model. A small value of GSEP indicates that the model is accurate and that the predictions are close to the observed values. PICP global is a measure of uncertainty of a predictive model. A high value of this parameter results in a lower accuracy, whereas a small value indicates that the model is more accurate.

It is important to evaluate the distribution of errors before selecting a recommended fit by using Goodness of Fit Statistics. The underlying statistical assumptions about residuals, such as constant variance, independence of variables, and normality of the distribution, can be examined. In this study, two graphs were included in the analysis: (i) plots of residuals vs independent variable (i.e., residuals vs pressure p, kPa). In this case, the residuals were obtained by residuals = y_i– y = observed value of y – predicted value of y. Here, the pattern structures of residual plots not only help to check the validity of a regression model but also can provide hints about how to improve it. For these assumptions to hold true for a particular regression model, the residuals must be randomly distributed around zero. In other words, no trend should be observed in the plots. (ii) Normal probability plot of the residuals (expressed as percentiles). The percentiles are obtained by first searching for the class where , k = 1, 2, 3···99 is in a table of cumulative frequencies; then, the following expression is applied to obtain the positions:

25

where L_i is the lower limit of the class where the position is located, N is the summation of absolute frequencies in the class where the position is located, F_i-1 is the cumulative frequency before the position, f_i is the absolute frequency, a_i is the amplitude of class. Here, a normal probability plot of the residuals can be used to check whether the variance is normally distributed as well; if the resulting plot exhibits approximate linearity, then the assumption is made that the error terms follow a normal distribution. The plot is constructed based on percentiles vs ordered residuals.

3. Results and Discussion

3.1. Structural characterization of CoHCF M11

3.1.1. FTIR Study

The characterization of CoHCFs is regularly carried out by FTIR.^38,39Figure 2a shows the FTIR spectrum of the CoHCF M11 sample. Some important aspects can be obtained from this figure. For instance, the frequency range between 2200 and 2000 cm^–1 is usually attributed to the CN group and used to partially ascertain the oxidation states of the metallic ions comprising the CoHCF structures.^37,40 In this sense, the two bands at ca. 2160 and 2110 cm^–1, indicate the presence of a mixture of two chains Fe³⁺–CN–Co²⁺ and Fe²⁺–CN–Co³⁺, produced during the synthesis procedure.²⁴ When aqueous solutions of K₃[Fe^III(CN)₆] and Co²⁺(NO₃)₂ are mixed during the coordination polymer formation, a redox reaction occurs, to form a mixed-valence solid, K_xCo^III_yCo²⁺_z[(Fe^III)_2-x(Fe^II)_x]. This inner redox reaction progresses during the activation reaction (heating) when the sample is conditioned for the adsorption experiment. The XPS studies confirm the presence of Fe²⁺/Fe³⁺ and Co²⁺/Co³⁺ ions. It has been reported that the generation of the Fe²⁺–CN–Co³⁺ chain is due to a spontaneously occurring electron transfer amid Fe³⁺ and Co²⁺ ions caused by the reaction Fe³⁺–CN–Co²⁺ → Fe²⁺–CN–Co³⁺.^37,40 The signals at 544 and 425 cm^–1 and a weak signal at 590 cm^–1 are also representative of the presence of the mixture of these chains.^41,42

Figure 2.

Structural characterization of the CoHCF M11 sample: a) FTIR spectrum and b) XRD diffraction pattern; solid blue lines at the bottom show the difference between observed and calculated patterns.

3.1.2. XRD Results

The crystalline structure of the CoHCF M11 sample was studied by X-ray diffraction. The XRD patterns (Figure 2b) reveal sharp, intense diffraction peaks featuring cubic crystal structures representative of PBAs. It can be observed that the Fe oxidation state influences the formation of the type of structure of this sample and, ultimately, the CO₂ uptake efficiencies. In this sense, the CoHCF M11 sample containing K⁺ ions and a mixture of Fe²⁺ and Fe³⁺ ions, forms face-centered cubic (FCC) cells with cell parameter a = 10.2613(1) Å and depicts a spatial group Fm3̅m.^43,44 Some impurities are detected in this sample, whose source could not be assigned; however, the recognized peaks correspond to the FCC state. Crystallite size calculated by the Scherrer equation is 30.72 nm. The main structural variables of the CoHCF M11 sample obtained from the Rietveld refinement with the PCW software include atomic coordinates, Wyckoff positions, and site occupancies (Table 1). The results of the difference between the observed and calculated patterns indicate that our phase presents preferential growth in the crystallographic planes (200), (400), (420), (220), and (600) in the CoHCF M11 sample (Figure 2b, bottom blue pattern). Metal ions Fe II are located at the 4a (0, 0, 0) crystallographic position; the Co II ions fill up the 4b (1/2, 1/2, 1/2) crystallographic position, while the 8c (1/4, 1/4, 14) positions are taken up by the K ions. The 24e (x’, 0, 0) sites are filled up by both N and C. Partial occupancy of these sites is due to the presence of [Fe (CN)₆] vacancies. It is estimated that ∼ 50% of [Fe(CN)₆] vacancies result in two types of water bound molecules, namely coordinated and noncoordinated. The number of oxygen atoms obtained from the refinement process indicates the total number of water molecules contained in the unit cell. The refinement results indicate two types of oxygen atoms generated. The first one fills up the 24e (x’, 0, 0) position and is ascribed to the coordinated water, whereas the second occupies the 32f (x’, 0, 0) position and is due to the zeolitic water.⁴⁵ The schematic crystal structure of the unit cell is shown in Figure 3. On the other hand, it has been reported⁴⁶ that in the cases of face-centered cubic cells of PBAs, the 3D lattice is comprised of open channels of ∼ 3.2 Å of diameter, cubelike cavities (interstitial sites) of 4.6 Å and larger hollow spaces of ∼ 9 Å diameter, created by vacancies of the [Fe(CN)₆] components. The dimensions of these pores and cavities indicate that they can easily accommodate CO₂ molecules (kinetic diameter = 3.3 Å) and are ideal to facilitate their diffusion through the PBA structure. Consequently, these open pore structures are attractive in applications of greenhouse gas removal (CO₂).

Table 1. Rietveld Refinement Results of the CoHCF M11 Sample: K_0.12Co^II₁Co^III_0.52[Fe^II_0.67Fe^III_0.33(CN)₆] · 6.6H₂O^a (x’, y’, and z’ Are Fractional Atomic Coordinates).

			atomic coordinates
cellparameter, Å	element	Wyckoff position	x’	y’	z’	occupancy	discrepancy factors (%)
a = 10.2613 (1)	Co	4b	0.5	0.5	0.5	0.5	Rp = 3.28,bRexp = 1.64c
	Fe	4a	0	0	0	0.5
	K	8c	0.25	0.25	0.25	0.25
	C	24e	0.814	0	0	0.5
	N	24e	0.701	0	0	0.5
	O1	24e	0.691	0	0	0.25
	O2	32f	0.774	0.7740	0.774	0.125

^aStoichiometric coefficients determined from ICP-MS after acid digestion; oxidation states were obtained from XPS.

^bRp = profiling factor.

^cR exp = expectation factor.

Figure 3.

Schematic depiction of the unit cell generated after the Rietveld refinement of the XRD pattern data of the CoHCF M11 sample.

3.1.3. XPS Results

XPS was used to confirm the existence of transition metals and verify their oxidation states in the CoHCF M11 sample. The C 1s transition window (Figure 4a) depicts two peaks at 283.55 and 284.03 eV ascribed, respectively, to C–Fe²⁺ and C–Fe³⁺ bonds.^45,47 These findings suggest that electron activity by charge transfer occurred within this sample during the synthesis procedure. The peaks at 283.13 and 285.04 eV, are due to the carbon of the sample holder and adventitious carbon, respectively. The O 1s core level spectra (Figure 4b) need three peaks to fit the signals: the peaks at 532.4, 530.91, and 531.88 eV are due to water molecules,⁴⁸ Fe_xO_y,^49,50 and Co(NO₃)_x,⁵¹ respectively. The core level spectra of N 1s (Figure 4c) reveal peaks at 397.21 eV caused by the N–Co²⁺ bond, at 397.7 eV attributed to the N–Co³⁺ bond,⁴⁵ and at 396.55 eV due to a compound of the type Co_xN_y.⁵² The core level spectrum of K 2p (Figure 4d) needs two peaks to fit the signal, at 292.41 and 292.62 eV, allotted to K in the complex K_nCo^2+,3+[Fe (CN)₆];⁴⁵ satellites appear in this sample. Fe 2p core level spectra are shown in Figure 4e resolved as individual peaks. Three main peaks can be observed, one at 712.72 eV and agrees with the C–Fe²⁺ bond,⁵³ a second peak at 715.27 eV designated to the C–Fe³⁺ bond,⁵³ and the third one at 711.7 eV attributed to Fe_xO_y.⁵⁴ Results of Fe 2p spectra show that if the Fe³⁺ ion in ferricyanide is used in the synthesis, this ion suffers electron activity reflected in a change of its oxidation state. In the case of Co 2p core level spectra (Figure 4f) two peaks at 785.54 and 786.36 eV are attributed to N–Co^2+49,55 and N–Co³⁺,³⁷ respectively. Once again, Co also suffers spontaneous electron transfer, resulting in a change of its oxidation state. The XPS results confirm the FTIR observations in that a change of the oxidation state caused by spontaneous electron transfer occurred in the two transition metals Fe and Co of CoHCF M11 during the synthesis procedure in addition to the different bonds characterized in its structure, and that in the end, this will have an influence on the CO₂ removal efficiencies.

Figure 4.

Results of XPS core-level spectra of the CoHCF M11 sample. a) C 1s, b) O 1s, c) N 1s, d) K 2p, e) Fe 2p, f) Co 2p.

3.1.4. Surface Morphology Results

The surface shape of the CoHCF M11 sample was analyzed by SEM (Figure 5a) and TEM (Figure 5b). The clustering of this sample can be distinguished with clarity by SEM. A more detailed picture of the typical cubic structure of this sample can be observed in the TEM micrograph. Particle sizes ranged between 80 and 130 nm.²⁴

Figure 5.

Micrographs of surface morphology of the CoHCF M11 sample studied by a) SEM, b) TEM.²⁴

3.2. Henry’s Constants

Henry’s constant provides the adsorbate–adsorbent adsorption characteristics, including the adsorption strength and the isosteric heat of adsorption. This constant represents the slope of a tangent line to the isotherm as p → 0. This constant can be obtained by either numerically deriving the experimental data or using an isotherm equation. However, according to Schell et al.,⁵⁶ using the derivative method of the experimental data, complications are frequently presented because a great amount of data points are needed in the low p range, which is experimentally demanding, and the data are especially susceptible to error.⁵⁶ Consequently, in this work, Henry’s constants were calculated using the virial type isotherm equation⁵⁷⁻⁶⁰ given by

26

In the limit when q → 0, eq 26 can be truncated after de virial coefficient A₁ to give:

27

where A₁ is the second virial coefficient.

eq 27 was used to perform nonlinear regressions of the isotherm data using the software Origin Pro 2021 to obtain K_H values for all temperatures studied, and the results are shown in Table S1 (the results for all temperatures of the nonlinear regressions to extract K_H0, are shown in Figure S1 of the Supporting Information). In general, Henry’s constant values are obviously decreased with increasing temperature since the adsorption surface sites of CoHCF M11 get higher intensity interactions with the CO₂ molecules at low temperatures, which assist them in binding stronger onto the anchoring sites. The temperature dependence of K_H can be expressed using a Van′t Hoff form as follows:

28

where ΔH^o_ads is the change of adsorption enthalpy obtained at very low adsorptions, and K_H0 corresponds to the limiting value when T → ∞. The obtained value of K_H was fitted using the nonlinear eq 28 as a function of temperature to extract K_H0, and the obtained value was 4.36 × 10^–6 mmol g^–1 kPa^–1, and that of -ΔH^o_ads = 22.4 kJ mol^–1 (Table S1).

3.3. Fitted Results and Comparison to Isotherm Models

The experimentally measured isotherm data and the predictions, including constant-temperature (solid lines) and temperature-dependent (dashed lines) models of the models evaluated, are shown in Figure 6. All data in the left plots are plotted on a linear scale, whereas graphs to the right are plotted on a log scale. The Langmuir and Freundlich models poorly fit the data for constant and temperature-dependent situations, especially at low pressures. Including the effect of temperature in the Langmuir model does not produce significant improvements to the fittings, whereas, in the case of the Freundlich model, a slight improvement is observed. When the two models are compared, the temperature dependent Freundlich model causes more significant deviations to the data, especially at very low pressures, where it is known that this isotherm model fails in conforming with Henry’s law.³² Consequently, the Langmuir and Freundlich models insufficiently capture the CO₂ adsorption onto CoHCF M11, despite incorporating the temperature effects in the isotherm equations. The constant-temperature Toth model fittings satisfactorily match the data at all pressures and temperatures. However, when the effect of temperature is incorporated in the model, more significant deviations from the data are produced and are more notorious at low temperatures and pressures (Figure 6 and Table S4). The Toth isotherm has correct limits when P approaches either zero or infinity; i.e., Henry’s law is obeyed at lower pressures by this model, as can be legibly noticed in Figure 6. This linear relationship is visually apparent at low pressures (∼0.35–7.56 kPa), as shown in Figure 6.

Figure 6.

Experimentally measured CO₂ isotherms on CoHCF M11 (symbols) were compared to predictions by different isotherm models under two different situations: constant temperature (solid lines) and temperature dependence (dashed lines). Experimentally measured CO₂ isotherms on CoHCF M11 (symbols) compared to predictions by different isotherm models under two different situations: constant temperature (solid lines) and temperature dependence (dashed lines).

Despite the Toth model catches the right thermodynamic behavior, it deviates significantly from the measured data when the effect of temperature is included in the equation (right panel in Figure 6). In general, the Toth model outperforms both the Langmuir and Freundlich models. In the case of the Freundlich-Langmuir (Sips) model, as expected, it fails at low pressures. This equation model possesses a finite saturation limit when the pressure is sufficiently high. However, it still shares the same disadvantage with the Freundlich isotherm at low pressure, i.e., they do not give Henry’s law correct limit. Again, the inclusion of the temperature effect worsens the fittings of data. A similar situation is presented in the 2-S Langmuir model than in the Toth equation: this model can satisfactorily predict the experimental data when constant-temperature equations are tested. However, it fails when the temperature-dependent model is tested. The data at low pressures are matched by the slope and magnitude of this model, as observed in the log-scale plot in Figure 6. The constant-temperature model continues to match the data with increasing pressure, which is not the case when the temperature-dependent model is tested, where a significant deviation of the data is observed in Figure 6.

The results of the accuracy of goodness of fit of each model are shown in Table 2 (the values of the raw isotherm experimental data, the nonlinear regressed parameters, and the uncertainties of the measurements generated by the instrument are shown in Tables S2 and S3, respectively, of the Supporting Information). The results and discussions of the kinetic experiments are shown in Section S5 of the Supporting Information. It can be observed that when no temperature effect was included in the fitting, the values of R² were all above 0.99, except in the case of the Freundlich model, which indicates a poor fitting to this model. For the Langmuir model, fitting was acceptable according to the R² values (with 95% confidence and p < 0.01), yet it was poor according to the RMSE value (0.0449). When the number of fitting parameters increased, fitting improved for all models. However, it comes at the expense of greater model complexity and the probability of false fitting artifacts made by a larger number of parameters (i.e., overfitting). Furthermore, it was not always the case that a high R² value is good for the regression model. The quality of the coefficient depends on several factors, including the units of measure of the variables, the nature of the variables employed in the model, and the applied data transformation. Thus, a high coefficient can sometimes indicate issues with the regression model and overfitting. Indeed, for instance, in the case of the Toth model at constant temperature, a value of R² = 0.9996 was obtained; however, this model could not predict the saturation capacities (q_max) of the adsorbent satisfactorily, especially at low temperatures where values of this parameter were far from the measured values (q_max calc = 8.27 ± 0.15 mmol g^–1 vs q_max exp = 2.45 mmol g^–1, see Table S3 of the Supporting Information). This situation was also present in the case of the Freundlich-Langmuir (Sips) model, which depicts R² values close to unity (0.9999) (with 95% confidence and p < 0.01), but less notorious because the saturation capacity deviated slightly from the experimental values. These results are consistent with reports from the literature.⁶¹ In the 2-S Langmuir model, the increase in fitting parameters significantly improved the results, as observed from the R² (with 95% confidence and p < 0.01) and RMSE values. This model could satisfactorily predict the saturation capacities (Table S3). On the other hand, when the effect of temperature was included in the models, the fitting results generally did not improve the values of R.² In this case, the fittings were poorer, as can be seen from the values of the RMSE, GMRSE, GSEP, and PICPglobal parameters, which show an increasing trend instead of decreasing. As can be seen, these results suggest that the temperature did not significantly affect the CO₂ adsorption process. Overall, error analysis shows that the highest values of the GSEP and PICP parameters correspond to the Langmuir, Freundlich, and Toth models, both at constant and temperature dependent, and smallest values of 0.005, ± 0.010 and 0.004, ± 0.007 correspond to models Freundlich-Langmuir and 2-S Langmuir with constant temperature, respectively. These values are very close to each other. Therefore, it is not possible to discriminate solely on the basis of the values of the R,² RMSE, GSEP, and PICP parameters between the Freundlich-Langmuir and 2-S- Langmuir models to recommend the most appropriate model to explain the data. As a result, these results leave the Langmuir and 2-S Langmuir models as the most promising models to explain the experimental data. In this sense, the goodness of fit statistics was applied to validate further all models assessed, as discussed in Section 3.3.

Table 2. Results of the Goodness of Fit for Classic Isotherm Models and Their Comparison with Temperature-Dependent Isotherm Models.

model	no. of fitting parameters	R2	RMSE	GMRSE	GSEP	PICPglobal
Langmuir	2	0.9944	0.0449	0.045	0.045	±0.088
Langmuir-Ta	3	0.9943	0.0386	0.039	0.039	±0.076
Freundlich	2	0.9780	0.0891	0.089	0.090	±0.177
Freundlich-Ta	3	0.9928	0.0498	0.050	0.050	±0.098
Toth	3	0.9996	0.0117	0.012	0.012	±0.023
Toth-Ta	5	0.9914	0.0544	0.054	0.055	±0.107
Freundlich-Langmuir (Sips)	3	0.9999	0.0053	0.005	0.005	±0.010
Freundlich-Langmuir (Sips)-Taa	5	0.9957	0.0380	0.038	0.038	±0.075
2-S Langmuir	4	0.9999	0.0035	0.004	0.004	±0.007
2-S Langmuir-Ta	6	0.9957	0.0386	0.039	0.039	±0.076

^aModel includes temperature effects.

In summary, the models that best explained the data for all temperature and pressure situations were Freundlich-Langmuir and 2-S Langmuir, both at a constant temperature. In a first attempt to discriminate which of these two models better explains the experimental data, a comparison of the fitting parameters is shown in Table 2, namely, the discrimination coefficient R,² RMSE, GMRSE, GSEP, and PICPglobal parameters. It can be observed that it is not possible to discriminate between the models since these parameters are not consistently smaller for both models. It has been reported that R² is insufficient to validate model fittings despite R² is the most used validation point of reference in modeling adsorption equilibrium and kinetic data.^5,9 The goodness of fit statistics quantitatively compare the different models to overcome this situation, as described in the next section.

3.4. Results of Goodness Fit Statistics for CO₂ Adsorption Isotherms on CoHCF M11

Figures 7-8 show the residuals vs independent variable plots and the normal probability plots of the residuals for the constant-temperature models, respectively, at the lowest temperature of 273 K. The results for the remaining temperatures were all similar and are not shown here. In the case of constant temperature models, the residuals vs independent variable plots (Figure 7a-e), only models Toth, Freundlich-Langmuir and 2-S Langmuir depict randomness of data since no trends are observed (Figure 7c-e), whereas in the case of the normal probability plots (Figure 8a-e), only the 2-S Langmuir model (Figure 8e) shows linearity of the data, with one outlier detected (at 0.007 mmol g^–1). The opposite effect is observed when the temperature is included (Figures S2–S3 Supporting Information), where all models show neither randomness nor linearity of the data. Additionally, the normal probability plots for the Toth and Freundlich-Langmuir models (Figure 7c-d) show no linearity of the data, which disqualifies these as good-fitting models. Variations in the parameters, including temperature, produced worse fit results, especially at the lowest pressures and temperatures (Table S4 Supporting Information). Consequently, on the basis of the results of the residual plot of the constant temperature 2-S Langmuir model, this model did not show either trend or patterns in the error distribution, and the normalization plots indicated that the residuals followed a normal distribution. This model complies with the constraints of random distribution of errors and is a straightforward model that describes the measured equilibrium isotherm data more accurately, covering the whole range of temperatures and pressures studied. These results agree with the reports of Andonova et al.⁶² who explained that distinct types of sites are generated by the K⁺ cations effectively embedded in a similar porous PBA (i.e., K-rich-NiFe-PBA) and act synergistically to generate highly specific binding sites for capturing CO₂ which can increase the selective capture of CO₂.

Figure 7.

Plots of residuals vs independent variable for constant-temperature models. a) Langmuir, b) Freundlich, c) Toth, d) Freundlich-Langmuir, and e) 2-S Langmuir.

Figure 8.

Normal probability plots of the residuals for the constant-temperature models. a) Langmuir, b) Freundlich, c) Toth, d) Freundlich-Langmuir, e) 2-S Langmuir.

3.5. Comparison of Heat of Adsorption Data to Predictions

Due to the lack of calorimetric experimental data of CO₂ adsorption on CoHCF M11, the isosteric heats of adsorption obtained using the isosteres approach are taken here as a reference point. The computations were obtained in the region where no extrapolation of the data was needed, as opposed to the case of the derivatives approach (analytic), where extrapolations were applied. A range of q_st of 24.07–25.80 kJ mol^–1 and an average value of 24.34 kJ mol^–1 is obtained, characteristic of the range corresponding to physical adsorption (Figure 9a).⁶³ The high level of heterogeneity and changes in the adsorbate–adsorbent and adsorbate–adsorbate interactions of CoHCF M11 produces variations in the trend of q_st with the loadings in that first a decrease to a minimum is attained, followed by a modest increase and eventually a decrease again is obtained.⁶⁴Figure 9b-f shows the results of q_st evaluated through isotherm models compared to the isosteres method. It is observed that the model that best approaches the isosteres method corresponds to the 2-S Langmuir model (Figure 9f). These results are consistent with those obtained in the isotherm fittings, where it was found that the 2-S Langmuir model best predicted the experimental data. To observe the two different adsorption sites in the q_st calculations, it was necessary to include higher CO₂ loadings in the adsorbent.

Figure 9.

Isosteric heats of adsorption predicted by the different isotherm models: a) Isosteres, b) Langmuir, c) Freundlich, d) Toth, e) Langmuuir- Freundlich, and f) 2-S Langmuir.

It can be observed that in the case of the Langmuir model (Figure 9b), the saturation capacity is independent of temperature, and as a result, the heat of adsorption is constant, independent of loading, and equal to q_st = E = 22.48 kJ mol^–1. The values of q_st predicted by all of the isotherm models present a weak temperature dependence. When the Freundlich model (Figure 9c) was used, it produced exceedingly high values of q_st (= 37.92 kJ mol^–1) at very low loadings. This behavior occurs because the isosteric heat is a linear function of the logarithm of the amount adsorbed. In this case, when q → 0, q_st → ∞. This thermodynamic point of view is not based on simplifying surmises; however, from this viewpoint, extrapolation of the Freundlich equation to the zone of very low CO₂ loadings is not acceptable because by eq 16 at q ∼ 0, q_st → ∞ which has no physical meaning.⁶⁵ Concerning the Freundlich-Langmuir (Sips) model (Figure 9e), this equation asserts that as the pressure (or loading) increases, the isosteric heat of adsorption decreases. In the case of zero adsorption, the isosteric heat of adsorption takes a value of infinity and, at saturation, equals minus infinity.³² In other words, this equation is useful within an intermediate pressure range, despite its correct prediction of finite capacity at large pressures. On the other hand, the equation of isosteric heat of adsorption eq 18 discloses the physical meaning of parameter E in eq 9. This is because the isosteric heat of adsorption and parameter E become equal in the case of halfway fractional loadings. Term E contained in the affinity constant K_FL of eq 9 equals the isosteric heat of adsorption at 1/2 of the fractional loading. In the case of the CO₂ adsorption on CoHCF M11, a value of E = (q_st)_θ=1/2 = 18.6 kJ/mol is obtained. A characteristic feature of the Freundlich-Langmuir isotherm is observed and consists of all seven isotherms converge at the same point when the fractional loading equals 1/2. For the Toth isotherm (Figure 9d), like the Sips equation, the dependence of the isosteric heat of adsorption on the pressure (or loading) is observed. At the two extreme limits, this isotherm approaches infinity at zero loadings and minus infinity at high loadings. These constraints restrict the use of the Toth isotherm when the isosteric heat of adsorption is computed. The Toth isotherm resembles the Sips equation in that the isosteric heat decreases as loading increases and the temperature dependence is also insignificant. In the case of the 2-S Langmuir model (Figure 9f), there is an inflection point in the isosteric heat of adsorption. This point occurred at a loading of 2.25 mmol g^–1. Considering the range of the extrapolation loading region, the isosteric heat of adsorption drops from 22.48 kJ mol^–1 (which corresponds to the heat of adsorption developed by the strong sites) to 4.65 J mol^–1 and is assigned to the heat of adsorption of the weak sites (Table 3).

Table 3. Calculated Isosteric Heat of Adsorption (q_st) of CoHCF M11 by Using Different Approaches.

method	qst, kJ mol–1	explanation
No extrapolation loading region
isosteres	24.07–25.80	computed using isosteres and no fitting equation
Extrapolation loading region
analytic	22.48	computed using Langmuir fitting isotherm
	18.22- 37.92	computed using Freundlich fitting isotherm
	15.61- 21.09	computed using Toth fitting isotherm
	5.61- 26.72	computed using Freundlich-Langmuir fitting isotherm
	4.65- 22.48	computed using 2-S Langmuir fitting isotherm

The uncertainty generated in the estimation of the heats of adsorption at zero coverage when the derivatives of isotherms at different temperatures are employed, as well as the extrapolation to high loadings, is clearly observed from Figure 9 and Table 2, where exceedingly high and low values of q_st can be observed, especially in the case of the Freundlich, Freundlich-Langmuir, and 2-S Langmuir models. Regardless of the small errors obtained during the fitting of the isotherms in all models, when extrapolations to zero loadings are performed, the values predicted of isosteric heats of adsorption differ significantly depending upon the model used (e.g., Freundlich and Freundlich-Langmuir models) (see Table 2). In summary, the values of q_st obtained are strongly dependent on the computation method employed. In this sense, experimenters must determine which method is more appropriate and compare it with experimental measurements. Before comparing the used methods, it is essential to differentiate between the q_st values acquired using isotherm derivatives (analytic) and isosteric experimental data (isosteres) since the q_st values attained via indirect (analytic) approaches are frequently built on experimental data.

4. Conclusions

The CO₂ adsorption data on a PBA (CoHCF M11) were obtained at 273–298 K and modeled with five different isotherm models to determine which model constitutes the best approximation of the experimental data. The Goodness of Fit Statistics using residuals and normalized plots along with the most used statistical parameters R,² RMSE, GMRSE, GSEP, and PICPglobal parameters was used to confirm that the 2-S Langmuir model approached the experimental data. This was based upon the results of residuals plot of this model which did not show either trend nor patterns in the error distribution, and the normalization plots indicated that the residuals followed a normal distribution. The presence of two different energetic sites contained in the 2-S Langmuir model helped explain the data in that these distinct types of sites are generated by the K⁺ cations effectively embedded in a porous PBA and act synergistically to generate highly specific binding sites to capture CO₂ which can increase the selective capture of this gas. The CO₂ adsorption kinetic data were obtained at three temperatures of 273, 298, and 313 K and matched with five different kinetic models. The residual and normalized percentile plots proved that Avrami’s model presented the best match with the experimental data. The isosteric heats of adsorption were determined using two approaches: the isosteres and the use of isotherms derivatives. The results of this study revealed that it is important to evaluate the distribution of errors when predicting kinetic and equilibrium adsorption data before selecting a recommended fit using Goodness of Fit Statistics, since the use of conventional statistical parameters such as the determination coefficient R² or the RMSE, do not provide a thorough assessment of the fitting. It was obtained that the 2-S Langmuir model better approached the q_st values when compared to the isosteres approach. Also, it was obtained that the values of q_st obtained are strongly dependent on the computation method employed. Consequently, the results of these calculations demonstrated that before comparing the methods used, it is essential to differentiate between the q_st values acquired using isotherm derivatives (analytic) and isosteric experimental data (isosteres) since the q_st values attained via indirect (analytic) approaches are frequently built on experimental data.

Data Availability Statement

This manuscript does not contain associated data.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.4c11489.

• In this work, CO₂ adsorption data on PBAs was used to model equilibrium, kinetic and isosteric heats of adsorption and are presented here. In this file, are reported the raw isotherm adsorption data, values of the regressed fitting parameters, goodness of fit plots for kinetic and equilibrium data, isotherm model equations at constant temperature and temperature dependent, and derivation equations of the isosteric heats of adsorption (PDF)

Author Contributions

P.M.F.-U.: sample synthesis, carried out the BET, CO2 capture, experiments, XPS data processing, conceived the original idea and writing. M.B.-S.: supervised the project. E.O.-G.: carried out heats of adsorption data processing. V.S.: performed and interpreted XRD data. K.C.: performed XRD data refinement. J. B–C.: XPS data processing and assignments. J. R. R.-O.: supervised the project, data analysis. I. F.C.: carried out heats of adsorption data processing. S.G.-S.: conceived the original idea, supervised project, and performed writing. All authors have read and agreed to the published version of the manuscript.

This work was supported by the University of Guadalajara through the program PRO SNI-2022.

The authors declare no competing financial interest.