Use of Thermodynamically Consistent Phase Equilibrium Data to Obtain a Generalized Padé-Type Model for the Henry’s Constants of Gases in Ionic Liquids

Abstract

A generalized Padé-type expression is proposed for Henry’s constant of gases in ionic liquids. The constants are determined using an equation of state, and generalized expressions for the Henry’s constants of the gases in the ionic liquids are proposed. The solute gases included in the study were oxygen, hydrogen, and carbon monoxide in three solvent ionic liquids ([MDEA][Cl], [Bmim][PF6], and [Hmim][TF₂N]). The Valderrama–Patel–Teja equation of state with the mixing rules of Kwak and Mansoori are employed to correlate the solubility data, to examine the thermodynamic consistency of the experimental data, and to determine the fugacity (f_i) for each concentration (x_i) of the solute gas in the liquid phase. From these data, the fugacity coefficients (f_i^L/x_i) are determined to obtain Henry´s constant as H_i = lim(f_i^L/x_i) when x_i→0. The calculated Henry’s constants are correlated in terms of the temperature and acentric factor of the gases to finally obtain a generalized expression for Henry´s constant, H_i.

1. Introduction

Previous studies have demonstrated that ionic liquids are good candidates for capturing different types of gases and possess several advantages over traditional industrial solvents [1,2]. The number of papers that have reported solubility data for gases in ionic liquids has increased in recent years, and several correlating and predicting methods have been proposed [3,4,5,6,7,8,9]. The values of the solubility of gases in ionic liquids, experimental or calculated, are needed for designing solvents, developing new gas–liquid processes, developing computer simulation methods [2,10], and obtaining better knowledge on the mechanisms involved in the absorption of gases in ionic liquids [4,11].

During the last ten years, research on the different aspects and applications of ionic liquids has notably grown; studies have appeared every day in the scientific literature, including studies on the synthesis of new ionic liquids, their environmental impact and toxicity, their technical and economic analysis, and experimental and theoretical research to determine their physical and physicochemical properties [12,13]. The phase behavior of mixtures involving ionic liquids has also received increasing attention, and several studies have been conducted on aspects such as liquid–liquid equilibrium, liquid–gas equilibrium, the thermodynamic consistency of data, solubility parameters, experimental difficulties, miscibility, and the effects of impurities [11,14,15,16,17,18]. General methods to correlate and predict the phase behavior of mixtures containing ILs are scarce, although the necessary thermodynamic tools are readily available. The main interest in recent years has been in carbon dioxide capture technology by ILs, and potential industrial applications have been analyzed. A good review of these studies and their applications was presented by Zhang et al. [19].

The present work considers the data of three gases (O₂, H₂, and CO) mixed in three ionic liquids ([Bmim][PF6], [Hmim][TF2N], and [DMEA][CI]), for which experimental data are available in the open literature. Some studies presented in the literature have discussed the difficulties in obtaining accurate data on the solubility of gases in ionic liquids, especially with some gases and in low concentration ranges. Lei et al. [11] discussed this subject and mentioned the following problems: (i) discrepancies in the variation in the solubility of CO as the temperature changes, which are caused by possible large errors in the experiments since the solubility of CO is low; and (ii) for oxygen, it is unclear how the variation in solubility occurs as the temperature increases. Most researchers agree that O₂ solubility decreases with increasing temperature, although some researchers argue that there is no influence of temperature; (iii) for hydrogen solubility, there is no agreement, and some authors stated that the solubility of H₂ in some ionic liquids first increases with increasing temperature and then decreases as the temperature increases further [20]. These factors impose an additional challenge to modeling solubility data, so a flexible model that can absorb this peculiar behavior is needed. Additionally, a consistency test could provide insight into these discrepancies.

Several different theoretical models, correlations, and equations of state have been used to correlate and predict the solubility of gases in ionic liquids. The concept of Henry´s law is applied in several approaches, including approaches that use the classical equations of state (EoSs) or incorporate activity coefficients. There are also applications of group contribution methods, quantum chemistry calculations, and statistical mechanics-based molecular approaches [21,22,23].

Anthony et al. [24] reported the solubility of nine gases (CO₂, C₂H₄, C₂H₆, CH₄, Ar, 0₂, C0, H_2, and N₂) in the ionic liquid [Bmim][PF₆] at T = 10 °C, 25 °C, and 50 °C, and they reported the associated Henry´s constants. Kumelan et al. [25] determined the solubilities of CO in the ionic liquid [Bmim][PF₆] for temperatures ranging from 293 to 373 K and at pressures up to 9.1 MPa. The solubility–pressure data were correlated through the extended Henry´s law. Kumelan et al. [5,26] reported experimental data on the solubility of H₂, CO, and O₂ in the ionic liquid [Hmim][TF2N] up to pressures of approximately 10 MPa and at temperatures ranging from 293 to 413 K, and the researchers correlated the data using the extended Henry´s law. Kumelan et al. [27] reported experimental data on the solubility of H₂ in the ionic liquid [Bmim][PF₆] for temperatures ranging from 313 K to 373 K and at pressures up to approximately 9 MPa.

The solubilities of gases (CO, CO₂, N₂, H₂, O₂) in [MDEA][Cl] have been reported by Zhao et al. [7], and the solubility data were correlated using the Krichevsky–Kasarnovsky equation. Raeissi et al. [2] experimentally investigated the solubility of CO in the ionic liquid [Bmim][TF2N] at several temperatures and pressures. Liu et al. [9] studied the solubility of eight common gases in [Emim][Tf₂N] using molecular dynamics simulation, and Ramdin et al. [28] used Monte Carlo simulations. Akbari et al. [29] applied a perturbed hard trimer chain (PHTC) equation of state (EoS) to model the solubilities of CO₂, H₂S, and R143a gases in several ionic liquids for temperatures ranging from 282 K to 353.15 K and at pressures up to 5 MPa. The proposed EoS was applied to estimate Henry’s constant and Gibbs energy of solvation of CO₂ in [C₈mim][PF₆]. The results for Henry’s constant on the mole fraction scale were compared with the experimental data. Eichenlaub et al. [30] studied the solubility of CO₂ in 62 ILs using three quantitative structure–property relationship models based on six descriptors. More recently, Jalili et al. [31] reported experimental data on the solubility of CO₂ and H₂S in the ionic liquid [Bzmim][TF2N] at temperatures ranging from 303 to 343 K and pressures up to approximately 3.1 and 1.6 MPa, respectively. The experimental data were correlated using the extended Henry´s law combined with a simplified Pitzer´s model.

In other developments, equations of state have received some attention. For example, Carvalho et al. [32,33] used the PR EoS with the Wong–Sandler mixing rules, including the UNIQUAC model, to calculate Henry´s constants of CO₂ in two ionic liquids. Shiffet and Yokozeki [34,35,36] published several works in which they measured the solubilities of different gases in different types of ILs and treated the data using the simple Redlich–Kwong EoS. Faúndez et al. [18,37,38] used the PR EoS with Kwak–Mansoori mixing rules to predict the solubility of NH₃, refrigerants, H₂S, and SO₂ in ILs. The SAFT equation of state and liquid phase models, such as Wilson’s equation, NRTL, UNIQUAC, and UNIFAC, have been used to describe the liquid phase in gas–ionic liquid mixtures [2,11,17,39]. The regular solution theory has also been considered for modeling gas solubilities in five ILs at low pressures and low mole fractions [14] and for correlating CO₂ solubility in imidazolium-type ionic liquids [40].

Some situations in industrial processes involve mixture phase equilibrium in which one or more substances in the mixture (but not all of them) are present at low concentrations in the liquid phase (e.g., below 20% on a molar basis). These situations are considered “solubility” problems, and in general, they are treated as special cases of phase equilibrium. In problems related to gases dissolved in liquids, the fundamental phase equilibrium relation, which is the equality of fugacity of the solute (component at low concentration) in both phases, is certainly valid:

$${\overline{f}}_i^V={\overline{f}}_i^L$$

(1)

The fugacity of a component in the vapor phase is usually expressed through the fugacity coefficient φ, while the fugacity of a component in the liquid phase is expressed through the activity coefficient (γ) or through the fugacity coefficient (φ). The fugacity is related to T, P, and the concentration through standard thermodynamic relations [21]. If φ is used for both phases, the method used to solve the phase equilibrium problem is known as the “equation of state method”. If γ is used for the liquid phase and φ is used for the gas phase, the procedure is known as the “gamma-phi method”.

As mentioned above, a simplified procedure is commonly used in solubility cases, although this procedure is also based on the fundamental Equation (1). At low concentrations, the fugacity of a component in the liquid phase follows a linear relation with the solute concentration [22], which can be written as follows:

$${\overline{f}}_i^L=H_i{x}_i$$

(2)

In this equation, $i$ represents the solute, and H_i is a parameter known as Henry´s constant, defined as follows:

$${\mathit{H}}_{\mathit{i}}=\underset{{\mathit{x}}_{\mathit{i}}\to 0}{\mathit{l}\mathit{i}\mathit{m}}\left(\frac{{\overline{\mathit{f}}}_{\mathit{i}}^{\mathit{L}}}{{\mathit{x}}_{\mathit{i}}}\right)\mathrm{ at\; constant\; T}$$

(3)

As observed, Equation (3) yields Henry’s constant at the vapor pressure of the solvent, which is corrected for pressure effects, as explained in what follows.

Based on the definition given by Equation (3), Henry’s constant depends on temperature and pressure, considering that the fugacity of the solute in the liquid phase depends on these variables. The dependency of H_i on T and P is described in the literature, and details on the form in which Henry’s law can be applied to higher pressures have been reported [22]. Furthermore, Henry’s law is applicable at low concentrations of solute in liquid solvent (a vague value that depends on the type of solvent and the type of solute). Additionally, if Henry’s law is applied to higher concentrations, a correction term must be added to account for the higher solute concentration. To account for both factors, the Krichevsky–Kasarnovsky–Ilinskaya equation is used, as commonly described in the literature [21].

$$ln{H}_1\left(\frac{{\overline{f}}_1^L}{x_1}\right)=ln{H}_1\left(P_2^s\right)+\frac{v_1^{\infty }}{RT}\left(P-P_2^s\right)+\frac{A}{RT}\left(x_{\begin{array}{c}1\\ \end{array}}^2-1\right)$$

(4)

The second term on the right-hand side of Equation (4) was proposed by Krichevsky and Kasarnovsky, and the third term was proposed by Krichevsky and Ilinskaya [21]. In Equation (4), v₁^∞ is the partial molar volume at infinite dilution of the gas solute in the liquid phase, and P₂^S is the vapor pressure of the solvent (ionic liquid) at the system temperature. The term A is a parameter of the Margules equation used by Krichevsky and Ilinskaya to include the activity coefficient of the solute in the liquid solvent.

Of the many EoSs available in the literature, the so-called cubic equations of state derived from the proposal of van der Waals interactions are widely used to treat these types of systems [41]. Among the cubic equations, the Patel–Teja–Valderrama equation (VPT) proposed by one of the authors [42] is employed, and the parametrization developed by Kwak and Mansoori [43] is used. This modified expression of the VPT equation fulfills the basic requirement of the van der Waals model. Therefore, the parameters of the EoS are constant and not functions of the temperature.

The interesting and novel points of this paper include the combination of thermodynamic concepts such as thermodynamic consistency of solubility data, the use of an accurate cubic equation of state, the incorporation of flexible mixing rules, the determination of Henry´s constants of gases in ILs, and their generalization in terms of the temperature and of the acentric factor of the gases.

2. The Equation of State

The cubic EoS derived from the van der Waals proposal can be expressed in the following general form:

$$P=\frac{RT}{\left(V-b\right)}-\frac{a_c\mathsf{\alpha }\left(T\right)}{V\left(V+d\right)+c\left(V-d\right)}$$

(5)

Most of the cubic equations used in research and academia are obtained by introducing restrictions to the parameters of the general expression Equation (5). For the van der Waals equation, the parameters c and d are zero; for the Soave–Redlich–Kwong equation, c is zero and d = b; for the Peng–Robinson equation, d = c = b; and for the Patel–Teja–Valderrama equation, d equals b. Therefore, the VPT equation is:

$$\begin{gathered} P=\frac{RT}{\left(V-b\right)}-\frac{a_c\mathsf{\alpha }\left(T\right)}{V\left(V+b\right)+c\left(v-b\right)} \\ \mathrm{with} \\ {a}_c={\mathrm{\Omega }}_a\left(\frac{R^2{T}_C^2}{P_C}\right);{\mathrm{\Omega }}_b{=}_b\left(\frac{R{T}_c}{P_c}\right);{\mathrm{\Omega }}_c{=}_c\left(\frac{R{T}_c}{P_c}\right) \\ \mathsf{\alpha }\left(T_R\right)=\left[{\left[1+F\left(1-T_R^{0.5}\right)\right]}^2\right] \\ {\mathrm{\Omega }}_a=0.6612-0.7616Z_c \\ {\mathrm{\Omega }}_c=0.5777-1.8718Z_c \\ {\mathrm{\Omega }}_b=0.0221-0.2087Z_c \\ F=0.4628+3.5823\left(\mathsf{\omega }{Z}_c\right)8.1942{\left(\mathsf{\omega }{Z}_c\right)}^2 \end{gathered}$$

(6)

with

In the modifications presented by Kwak and Mansoori [43], the VPT equation is written as follows:

$$P=\frac{\mathrm{RT}}{V-b}-\frac{a+RTd-2\sqrt{adRT}}{V(\mathrm{V}+b)+c(V-b)}$$

(7)

and for mixtures:

$$P=\frac{\mathrm{RT}}{V-b_m}-\frac{a_m+RT{d}_m-2\sqrt{a_m{d}_{m}RT}}{V(V+b_m)+c_m(V-b_m)}$$

(8)

The combination rules for the mixture parameters (a_m, b_m, c_m, and d_m) are as follows:

$$a_m={\sum }_i{\sum }_j{y}_i{y}_j{a}_{ij}$$

(9)

$$a_{ij}=\sqrt{a_i{a}_j}\hspace{0.33em}\left(1-k_{ij}\right)$$

(10)

$$a_i=a\left(T_{Ci}\right)(1+F_i{)}^2$$

(11)

$$b_m={\sum }_i{\sum }_j{y}_i{y}_j{b}_{ij}$$

(12)

$$b_{ij}={\left(\frac{{b_i}^{1/3}+{b_j}^{1/3}}{2}\right)}^3\left(1-l_{ij}\right)$$

(13)

$$b_i=\frac{0.07780RT{c}_i}{P{c}_i}$$

(14)

$$c_m={\sum }_i{\sum }_j{y}_i{y}_j{d}_{ij}$$

(15)

$$c_{ij}={\left(\frac{{c_i}^{1/3}+{c_j}^{1/3}}{2}\right)}^3\left(1-m_{ij}\right)$$

(16)

$$d_m={\sum }_i{\sum }_j{y}_i{y}_j{d}_{ij}$$

(17)

$$d_{ij}={\left(\frac{{d_i}^{1/3}+{d_j}^{1/3}}{2}\right)}^3\left(1-n_{ij}\right)$$

(18)

$$d_i=\frac{a(T_{ci}){F_i}^2}{R{T}_{ci}}$$

(19)

The parameters included in Equation (8), that is, k_ij, l_ij, m_ij, and n_ij, are determined using experimental data obtained for the phase equilibrium (PTx data). The initial values of the parameters are estimated, and the phase equilibrium equation is applied to determine the bubble pressure. The deviations between the experimental and calculated pressures are determined. If deviations are above established limits, the parameters are adjusted, and calculations are performed again until deviations are within acceptable defined values. The Levenberg–Mardquart method was applied [44] to optimize the searching of the parameters. The method uses the average absolute deviation between the calculated and the experimental bubble pressure as an objective function. The optimum parameters are those values of k_ij, l_ij, m_ij, and n_ij that generate the lowest average deviation in bubble pressure and do not generate deviations higher than 10%.

The average absolute deviation between the calculated pressure P^cal and the experimental pressure P^exp is defined as follows:

$$\left|\%\mathsf{\Delta }P\right|=\frac{100}{N}{\sum }_{i=1}^N\frac{{\left|P^{cal}-P^{exp}\right|}_i}{P_i^{exp}}$$

(20)

Once the optimum parameters are determined, the values are automatically exported to an Excel sheet where a linear correlation between f_i^L/x_i and x_i is performed, and Henry´s constant is determined as the position coefficient of the linear model, following Equation (3).

3. Data and Equations Used

In this study, three gases (O₂, H₂, and CO) in ionic liquids ([MDEA][Cl], [Bmim][PF6], and [Hmim][TF2N]) were analyzed. The ranges of concentration and pressure of the data for each isotherm are presented in

Table 1. Ranges of temperature, solubility, and pressure for the gas + ionic liquid systems considered in this study.

Systems	T (K)	Δx (Gas in the IL)	ΔP (Bar)	References
O2 + [MDEA][Cl]	313	0.099–0.334	20.01–73.41	[7]
	318	0.099–0.317	21.20–74.11
	323	0.094–0.316	19.12–74.10
	328	0.092–0.291	20.23–69.28
	333	0.089–0.291	20.52–76.23
H2 + [MDEA][Cl]	313	0.032–0.131	12.93–54.44
	318	0.030–0.125	12.82–58.33
	323	0.030–0.110	15.82–62.02
	328	0.030–0.105	16.32–58.99
	333	0.029–0.100	16.37–58.70
CO + [MDEA][Cl]	313	0.078–0.233	26.13–82.68
	318	0.075–0.215	26.99–83.62
	323	0.066–0.205	25.03–81.90
	328	0.060–0.190	25.83–85.71
	333	0.057–0.176	25.95–83.88
O2 + [Bmim][PF6]	293	0.008–0.043	15.44–90.11	[4]
	313	0.006–0.043	11.60–90.58
	333	0.007–0.043	12.27–90.95
	353	0.007–0.044	13.49–89.48
	373	0.008–0.045	14.44–89.11
H2 + [Bmim][PF6]	313	0.004–0.022	16.72–90.99	[27]
	333	0.003–0.024	12.35–90.24
	353	0.004–0.026	14.68–91.19
	373	0.003–0.029	10.85–90.10
CO + [Bmim][PF6]	293	0.006–0.041	12.55–91.07	[25]
	313	0.008–0.039	16.75–89.63
	334	0.004–0.039	8.93–86.20
	354	0.008–0.040	14.76–89.38
	373	0.004–0.042	7.69–90.17
O2 + [Hmim][TF2N]	293	0.028–0.115	16.08–75.92	[26]
	333	0.024–0.121	14.40–85.93
	373	0.022–0.119	14.27–87.42
H2 + [Hmim][TF2N]	293	0.008–0.049	14.65–97.84
	333	0.014–0.056	22.61–93.07	[5]
	373	0.012–0.062	15.94–90.78
	413	0.014–0.074	16.27–94.99
CO + [Hmim][TF2N]	293	0.020–0.109	14.29–89.90
	333	0.020–0.104	15.65–91.98	[26]
	373	0.018–0.105	14.97–91.91
	413	0.025–0.110	19.57–98.11

. The critical properties needed to use the VPT equation of state for gases and to calculate Henry’s constants are obtained from the literature [45]. For the ionic liquids, the critical properties were determined by the method of Valderrama and Rojas [46]. A total of 286 data points with approximately 6 to 9 points for each of the forty isotherms were used in the analysis.

Table 2. Properties of the solutes and solvents considered in this study. The data for gases are from Reid et al. [45], and the data for the ILs are from Valderrama and Rojas [46].

Components	M	Tc/K	Pc/MPa	Vc (m3/kmol)	ω
O2	31.99	154.60	5.04	0.07340	0.0250
H2	2.02	33.19	1.31	0.06415	0.2160
CO	28.01	132.90	3.50	0.09320	0.0660
[Bmim][PF6]	284.2	719.4	1.73	0.76250	0.7917
[Hmim][TF2N]	447.4	1298.8	2.39	1.10440	0.3893
[MDEA][Cl]	154.60	720.70	3.61	0.45160	1.1247

lists the values of these properties.

Within the pressure ranges of the mixtures studied (from 0.8 to 10 MPa), the second term on the right-hand side of Equation (4) can be important. Additionally, for the solubility ranges examined in this study (from 0.008 to 0.233 in mole fraction), the third term on the right-hand side of Equation (4) could reach values that affect the Henry’s constant, depending on the value of Margules parameter A, that was included in the third term of Equation (4). Preliminary calculations generated small, negligible values for (A/RT). These calculations were performed by fitting Henry’s constant at a fixed temperature in terms of pressure and mole fraction in the form $ln{H}_1=A+B(P-P_2^s)+C(x_1^2-1)$. Data from the literature [7] for similar systems indicate that A = ln ($H^o$) ≈ 5 to 6, ${B=v}_1^{\infty }/RT$ < 0.01, and C = (A/RT) < 0.001. Therefore, to achieve simplicity without losing accuracy, the third term is not further considered in this study. Equation (4) reduces to the following:

$$ln\left(\frac{{\overline{f}}_1^L}{x_1}\right)=ln{H}_1\left(P_2^s\right)+\frac{v_1^{\infty }}{RT}\left(P-P_2^s\right)$$

(21)

If Henry’s constant is calculated from isothermal data by extrapolating $({\overline{f}}_i^L/x_i)$ as

x_i → 0, the value obtained corresponds to Henry’s constant at the system temperature and at the pressure of the system when x_i = 0, that is, the saturation pressure of the ionic liquid (P₂^S). In the present study, the temperature ranged from 293 to 413 K, as the vapor pressure of the ionic liquid was very low, approximately between 10⁻⁴ and 10⁻² bars. Thus, the effect of pressure estimated using Equation (4) to correct Henry’s constant from P₂^S to P = 1 is negligible. Finally, ln H (P₂^S) ≈ ln H (P → 0) and Henry’s law are as follows:

$$ln\left(\frac{{\overline{f}}_1^L}{x_1}\right)=ln{H}_1\left(P\to 0\right)+\frac{v_1^{\infty }}{RT}\left(P-P_2^s\right)$$

(22)

An equation of state is used to determine the fugacity of the solute in the liquid phase that fulfills the fundamental equation of phase equilibrium. To apply an equation of state to mixtures, mixing rules must be used to describe the dependency of the EoS parameters on concentration and of combining rules to describe the interactions between molecules in the mixture.

4. Results

The data used in this work were first tested for thermodynamic consistency, following a method used in the past by the authors. Details of the consistency test can be found elsewhere [18,38,47,48,49] but are briefly summarized in

Table 3. Main expressions of the thermodynamic consistency test used in this work.

A	Gibbs–Duhem equation in terms of fugacity coefficients
	$\sum _{i=1}^I{x}_{i}d\left[\frac{G^R}{RT}\right]=-\frac{H^R}{R{T}^2}dT+\frac{v^R}{RT}dP$	(24)
B	At constant temperature:
	$\left[\frac{(Z-1)}{P}\right]dP=x_{1}d{(Ln\varphi }_1)+x_{2}d(Ln{\varphi }_2)$	(25)
	And arranging terms:
	$\frac{1}{P}dP=\frac{x_1}{{(Z-1)\varphi }_1}d{\varphi }_1+\frac{(1-x_1)}{{(Z-1)\varphi }_2}d{\varphi }_2$	(26)
	$\int \frac{1}{{Px}_1}dP=\int \frac{1}{{(Z-1)}_1}{d}_1+\int \frac{(1-x_1)}{{x_1(Z-1)}_2}{d}_2$	(27)
C	Experimental and estimated areas:
	$A_P=\int \frac{1}{{Px}_1}dP$	(28)
	$A_{\varphi }=\int \frac{1}{{(Z-1)\varphi }_1}d{\varphi }_1+\int \frac{(1-x_1)}{{x_1(Z-1)\varphi }_2}d{\varphi }_2$	(29)
D	Individual percent area deviation in the range [−20% to +20%]
	$A_i=100{\left[\frac{{A_{\varphi }-A}_P}{A_P}\right]}_i$	(30)
	Individual deviation in the system pressure in the range [−10% to +10%]
	$P_i=100{\left[\frac{P_i^{cal}-P_i^{exp}}{P_i^{exp}}\right]}_i$	(31)

to clarify the work presented in this paper.

As detailed in these several papers, the consistency test for isothermal data is reduced to check the fulfillment of Equation (29) in Table 3:

$$\int \frac{1}{P{x}_1}dP=\int \frac{1}{(Z-1){\phi }_1}d{\phi }_1+\int \frac{(1-x_1)}{x_1(Z-1){\phi }_2}d{\phi }_2$$

(23)

The left-hand side is calculated from purely experimental data (pressure P and solubility x₁), while the right-hand side is determined by calculating the compressibility factor Z and the fugacity coefficients (φ₁ and φ₂) from an equation of state. If a set of data is consistent, the left-hand side should be equal to the right-hand side of Equation (23), within acceptable defined deviations [47].

Table 4. Optimum values of binary interaction parameters k₁₂, l₁₂, m₁₂, n₁₂, and deviations in the correlated bubble pressure and in the area test. In the last column are the results of the area test.

Systems	T (K)	k12	l12	m12	n12	∣P%∣	Pmax%	∣A%∣	Results
O2 + [MDEA][Cl]	313	0.0935	−0.2831	0.9772	−0.2284	1.3	−3.3	9.7	TC
	318	−1.9379	−0.1677	−0.7994	−1.0891	2.8	4.6	15.2	NFC
	323	−1.4627	−0.1710	−0.3144	−1.3602	0.8	−1.9	8.1	NFC
	328	−1.3057	−0.3855	−0.1885	0.8291	1.3	−1.6	8.0	TC
	333	−0.9675	−0.2131	0.1288	−1.0606	0.5	0.9	5.5	TC
H2 + [MDEA][Cl]	313	−0.8914	−0.4110	1.0074	0.6583	0.7	−2.1	8.0	TC
	318	−1.0617	−0.3060	0.9274	−0.8499	1.4	−4.3	12.7	NFC
	323	−1.3801	−0.4279	0.7854	0.9863	3.2	6.9	15.1	NFC
	328	−1.2594	−0.2808	0.8802	−1.5749	1.2	2.7	22.5	TI
	333	−0.6554	−0.3194	1.0469	−0.7202	1.0	3.0	13.5	TI
CO + [MDEA][Cl]	313	−0.1329	−0.2489	0.8007	−0.0584	0.4	−0.9	5.8	TC
	318	−0.3208	−0.2794	0.6323	0.3438	0.2	−0.4	2.6	TC
	323	0.6315	−0.0958	1.3610	−1.5004	0.4	−0.7	7.5	TC
	328	0.7043	−0.2612	1.3864	0.5223	0.4	0.7	5.4	TC
	333	0.0512	−0.2831	0.9072	0.4285	2.0	−3.5	7.9	TC
O2 + [Bmim][PF6]	293	0.9370	0.8163	−1.4508	−1.3471	0.6	−1.5	5.1	TC
	313	0.1187	0.3718	−1.3672	−0.7725	0.2	0.4	2.0	TC
	333	−0.3721	0.0417	−1.0753	−0.7178	0.3	0.5	2.8	TC
	353	0.4983	0.5920	−1.4202	−0.3341	0.4	0.8	2.1	TC
	373	−0.3822	0.0141	−0.9040	−0.7361	0.3	0.5	2.2	TC
H2 + [Bmim][PF6]	313	0.4047	0.0482	−1.4958	0.9822	0.4	−1.5	4.2	TC
	333	0.2782	0.0433	−1.3495	0.7720	0.2	−0.4	2.4	TC
	353	−1.1105	−0.1328	−0.8160	−0.9612	0.4	0.9	2.9	TC
	373	−0.9477	0.0350	−1.0469	−0.9507	0.2	0.4	2.3	TC
CO + [Bmim][PF6]	293	0.4407	−0.6569	1.2148	−0.5628	0.8	−1.6	4.6	TC
	313	−0.2501	−1.2953	1.3716	0.8582	0.7	1.4	7.3	TC
	334	0.6404	−0.3389	0.8296	−1.0046	0.3	0.7	2.9	TC
	354	0.8075	0.4370	−0.6824	−0.3793	1.0	2.4	4.3	TC
	373	−0.3740	0.0885	−0.9212	−0.6212	0.4	−1.5	3.6	TC
O2 + [Hmim][TF2N]	293	−0.0655	−0.6700	0.4426	−0.1842	2.0	−3.4	28.7	NFC
	333	−0.5649	−0.0699	−0.5849	−3.4782	3.8	−4.3	8.7	TC
	373	−0.0822	−0.7106	0.4930	0.0886	2.8	−3.4	6.1	TC
H2 + [Hmim][TF2N]	293	−0.7862	−0.5961	0.5086	−1.8160	0.5	−1.2	36.4	TI
	333	−0.0850	−1.0845	1.4951	1.5627	0.2	0.6	34.3	TI
	373	−0.8501	−1.0356	1.0029	1.3714	0.2	−0.5	31.8	NFC
	413	−1.4295	−0.7772	0.4772	−0.3354	0.2	0.5	8.5	NFC
CO + [Hmim][TF2N]	293	−0.6299	−0.2578	−0.5352	−2.1154	2.0	−3.0	79.6	TI
	333	−1.2862	−0.5464	−1.6584	1.4298	0.7	−1.1	38.6	NFC
	373	−0.9619	−0.5485	−0.5671	−0.0595	1.1	−2.5	14.7	NFC
	413	−1.3618	−0.6958	−1.2675	2.1638	0.5	−1.9	12.0	NFC

shows the results of the consistency test applied to all the mixtures considered in this work (40 sets of data). As shown in the table, twenty-five systems were thermodynamically consistent (TC), ten sets of data were not fully consistent (NFC), and only five sets were thermodynamically inconsistent (TI). These sets of data were not considered further in this study for determining and generalizing Henry´s constants. This consideration is in agreement with the meaning of the test based on the Gibbs–Duhem equation; if a set of data does not pass the test, the data are thermodynamically inconsistent and probably incorrect.

After the thermodynamic consistency of the data was checked, Henry’s constants were determined as explained above. This was achieved by using the definition of Henry´s constant (Equation (3)).

To evaluate this limit, the two data points with the lowest solubility (x_i→0) for each isotherm were considered. In the (f_i^L/x_i)-vs.-x_i plot, a straight line passing through the two points is fitted, so the position coefficient generates Henry´s constant $H^o$ at the saturation pressure ($P_2^s$) of the ionic liquid for the given constant temperature.

After $H^o=H_1$($P_2^s$) is found, the effect of pressure at a constant given temperature is calculated using the Krichevsky–Kasarnovsky relation:

$$ln\left(\frac{{\overline{f}}_1^L}{x_1}\right)=lnH\left(P\right)=ln{H}^o+\frac{v_1^{\infty }}{RT}\left(P-P_2^s\right)$$

(32)

In this equation, $H^o$ is known, so v₁^∞ can be directly calculated from Equation (32). This calculation is simply performed by fixing the position coefficient equal to $ln{H}^o$ when P = ($P_2^s$) and calculating v₁^∞ from the slope of the line.

The results for all cases studied are listed in

Table 5. Values of the calculated Henry´s constants $H^o$(bar) obtained for all mixtures studied.

Systems	T (K)	${\mathit{H}}^{\mathit{o}}$(Bar)	v1∞ (m3/kmol)
O2 + [MDEA][Cl]	313	194.41	0.0313
	318	206.37	0.0325
	323	193.02	0.0597
	328	212.31	0.0343
	333	219.59	0.0550
H2 + [MDEA][Cl]	313	392.75	0.0446
	318	417.45	0.0664
	323	518.49	0.0530
	328	NC	NC
	333	NC	NC
CO + [MDEA][Cl]	313	327.39	0.0270
	318	342.96	0.0397
	323	362.71	0.0346
	328	413.78	0.0312
	333	444.85	0.0262
O2 + [Bmim][PF6]	293	1765.30	0.0312
	313	1786.10	0.0341
	333	1791.90	0.0356
	353	1791.90	0.0350
	373	1767.20	0.0381
H2 + [Bmim][PF6]	313	4133.20	0.0210
	333	3711.99	0.0264
	353	3352.03	0.0255
	373	3061.10	0.0183
CO + [Bmim][PF6]	293	1964.60	0.0288
	314	1903.10	0.0456
	334	1926.99	0.0490
	354	1883.67	0.0601
	373	1971.35	0.0384
O2 + [Hmim][TF2N]	293	558.43	0.0364
	333	588.48	0.0534
	373	636.34	0.0470
H2 + [Hmim][TF2N]	293	NC	NC
	333	NC	NC
	373	1351.44	0.0446
	413	1338.8	0.0456
CO + [Hmim][TF2N]	293	NC	NC
	333	739.58	0.0552
	373	797.93	0.0394
	413	757.31	0.0702

. In the table, NC indicates that values were not calculated because the available data were found to be thermodynamically inconsistent.

Henry´s constant for the mixtures considered in this study has been reported by other authors, although the ranges of temperature are not the same in all the cases. Therefore, for some of the mixtures, only a general non-quantitative comparison can be carried out.

Table 6. Comparison of the calculated Henry´s constants for all the systems considered in this study.

Case	Systems	Range of T (K)	${\mathit{H}}^{\mathit{o}}$(Bar)
1	O2 + [MDEA][Cl]	313–333	194–220	This work
		313–333	190–220	Zhao et al. [7]
2	H2 + [MDEA][Cl]	313–323	393–519	This work
		313–333	440–560	Zhao et al. [7]
3	CO + [MDEA][Cl]	313–333	327–445	This work
		313–333	320–440	Zhao et al. [7]
4	O2 + [Bmim][PF6]	293–373	1765–1792	This work
		293–343	1024–1610	Jacquemin et al. [50]
5	H2 + [Bmim][PF6]	373–413	3061–4133	This work
		293–373	3890–4350	Sharma et al. [51]
6	CO + [Bmim][PF6]	293–373	1883–1971	This work
		283–343	1201–1356	Jacquemin et al. [50]
7	O2 + [Hmim][TF2N]	293–373	558–636	This work
		313–373	623–790	Shi et al. [52]
8	H2 + [Hmim][TF2N]	373– 413	1339–1351	This work
		293–343	1258–1984	Costa-Gomez [53]
		293– 353	1501–1981	Raessi et al. [54]
9	CO + [Hmim][TF2N]	333–413	740–798	This work
		293–413	319–347	Kumelan et al. [26]

presents the values reported by these authors and the results obtained from the generalized correlations proposed in this work.

For the three gases in [MDEA][Cl], Henry’s constants were reported by Zhao et al. [7]. For the three gases in the ionic liquid [Bmim][PF6], the values are provided by Jacquemin et al. [50] and Sharma et al. [51]. For the three gases in [Hmim][TF2N], the values of Henry’s constant are reported by Shi et al. [52], Costa-Gomez [53], Raessi et al. [54], and Kumelan et al. [26].

Comparisons are difficult due to the differences in the temperature range of the data used by each author. For the gases in the ionic liquids [MDEA][Cl], the values of Henry’s constants are determined using data in the same temperature ranges for the literature values and those calculated in this work. The reported values in both cases have deviations lower than 5%. For the gases in [Bmim][PF6], the values show a slightly greater discrepancy but are still acceptable considering that the temperature ranges are different. Not much can be said of the results for the three gases in [Hmim][TF2N] because the temperature ranges, between literature and calculated values, are very different.

5. Generalized Correlations

Henry´s constants determined in this work, using only thermodynamically consistent data, were modeled as a function of the temperature for the three gases. The acentric factor is the property used to distinguish between the gases. The model has the form of a Padé expression with two variables, for each ionic liquid:

$$ln{H}^o=\frac{a+b\cdot \omega \cdot T}{c+d\cdot \omega }$$

(33)

Table 7. Expressions for the Henry’s constants of the gases in three ionic liquids.

Ionic liquid	$\mathit{ln}{H}^o$ for any of the gases
[Bmim][PF6]	$\mathit{ln}{H}^o$= (18.1 − 0.040⋅ω⋅T)/(2.44 − 2.68⋅ω)
[MDEA][Cl]	$\mathit{ln}{H}^o$= (−2.69 + 12.81⋅ω⋅T)/(2.08 + 652.6⋅ω)
[Hmim][TF2N]	$\mathit{ln}{H}^o$= (597.8 + 0.7934⋅ω⋅T)/(93.66 + 5.91⋅ω)

shows the expression for each of the three ionic liquids considered in this study. This equation allows estimating the Henry’s constants for oxygen, hydrogen, and carbon monoxide in the ionic liquids [Bmim][PF6], [MDEA][Cl], and [Hmim][TF2N].

The Henry’s constants for the twelve mixtures calculated from the equation of state ($H^o$_EOS) and those estimated from the Padé equations presented in Table 7 ($H^o$_Padé) are plotted in Figure 1. In this figure, dots are experimental data and lines are calculated values.

As observed in Figure 1, some slight dispersion is found, but these deviations may be acceptable for primary estimations. More importantly, the feasibility of generalizing properties such as Henry´s constants opens great possibilities for future research in the area of fluid property estimation for ionic liquids and mixtures containing ionic liquids.

6. Conclusions

From the results presented and discussed in this paper, the following main conclusions can be drawn: (i) the VPT equation of state accurately correlates with the solubility of gases in ionic liquids (as indicated by the low deviations in bubble pressure values presented in Table 4); (ii) the Kwak–Mansoori method has the required flexibility to converge in all cases studied (as observed by the wide range of values of the binary interaction parameters k₁₂, l₁₂, m_12, and n_12, presented in Table 4, which range from −3.48 to 2.16); and (iii) the general modeling of Henry´s constants (one equation for any of the gases) opens great possibilities for future research in this area of mixture properties, demonstrating that some generalization is possible.