A predictive and a nonpredictive approach were applied and compared to represent ILs’ density and vapor pressure. Later, the non- predictive approach applied to binary phase equilibrium data with the species involved in the RWGS.

Thus, itwas applied to estimate the influence of the three ILs under study on the RWGS chemical equilibrium. This section presents the metrics used in the selection of three ionic liquids for this study and describes all models applied in the study, andadditional information can be found in the Supplementary Material.

## Ionic liquid selection

Three different ILs were selected for this study. The metrics considered in the ILs’ selection were the availability of experimental data, water solubility, and thermal stability.

The application of the nonpredictive thermodynamic methods was performed by regressing the model’s parameters to experimental data. Therefore, the selected ILs must have available experimental data in the open literature. The main required properties were the ILs’ density, vapor pressure, and thermal stability. Furthermore, experimental data for vapor-liquid or liquid-liquid equilibrium needed to be available for ILs binaries with CO2, H2, CO, and H2O. Furthermore, the availability of experimental studies about the IL’s application in CO2 conversion to CO was also considered.

Since water is one of the reaction’s products, the absorption of this species from the reacting system should shift the reaction toward the formation of more products. Therefore, ILs with different water sorption were selected for this study.

Although the previous studies reported that the addition of IL allowed the production of CO from CO2 hydrogenation reactions, moderate temperatures are still required (150°C) (Qadir et al. 2018, 2019; Yasuda et al. 2018). Therefore, the selected ILs should present suitable thermal resistance to this application.

## Thermodynamic models

A predictive and a nonpredictive model were considered in this study. These models were applied and compared to experimental data of pure ILs density and vapor pressure. The parameters of the nonpredictive model were simultaneously regressed to fit experimental data of density and vapor pressure. While the parameters of the non-predictive approach were estimated by Valderrama and Rojas group contribution (GC) method (2009).

The basic models applied on both approaches are quite similar to the Predictive Soave-Redlich-Kwong (PSRK) Equation of State (EoS).

The PSRK EoS was selected for being reported to predict phase equilibria of several systems containing organic species with low errors at high-pressure conditions (Holderbaum and Gmehling 1991). This model was developed from Soave-Redlich-Kwong (SRK) EoS and a modification of the Huron-Vidal mixing rule (Holderbaum and Gmehling 1991). This mixing rule requires the knowledge of an activity coefficient function. In the original study, the UNIFAC (Universal Functional-group Activity Coefficient) model was applied, while in this study the UNIQUAC (Universal Quasi-Chemical) model was used for the nonpredictive approach.

## Density

For the calculation of pure species’ density and vapor pressure, the PSRK can be simplified to the SRK EoS (Eq. 5).

$$\text{P}= \frac{\text{R}\text{T}}{{\text{v}}_{\text{m}}-\text{b}}-\frac{\text{a}}{{\text{v}}_{\text{m}}({\text{v}}_{\text{m}}+\text{b})}$$

5

Where \(P\) is pressure, \(T\) is temperature, \(R\) is the universal gas constant, \({\text{v}}_{\text{m}}\) is the molar volume, \(a\) is the attractive parameter, \(b\) is the co-volume parameter, and \({\alpha }\left(\text{T}\right)\) is the alpha function. The attractive and co-volume parameters were calculated for each component \(i\) by their correlations with the species' critical properties (\({P}_{c}\) and \({T}_{c}\)), according to Equations 6 and 7, where the subscript i denotes the pure species.

$${\text{a}}_{\text{i}}= 0.42747\frac{{\text{R}}^{2}{\text{T}}_{{\text{c}}_{\text{i}}}^{2}}{{{\text{P}}_{\text{c}}}_{\text{i}}}{\alpha }\left(\text{T}\right)$$

6

$${\text{b}}_{\text{i}}= 0.08664\frac{\text{R}{{\text{T}}_{\text{c}}}_{\text{i}}}{{{\text{P}}_{\text{c}}}_{\text{i}}}$$

7

Most of the ILs degrade at temperatures below the critical point. Therefore, experimental critical properties of the ionic liquids are nearly impossible to be experimentally measured.

For the nonpredictive approach, the ILs’ critical properties were regressed to fit the model to the experimental data of density. For the predictive approach, the group contribution method proposed by Valderrama and Rojas (2009) was applied. Their predictive group contribution method estimated several ILs’ critical properties by regressing these critical properties to fit another equation for density to experimental data. The density model applied by Valderrama and Rojas (2009) is not commonly used at high pressures and temperatures. Cubic equations of state, such as the PSRK EoS, are significantly more applied to engineering purposes since they accurately represent several systems at various pressures and temperatures.

## Vapor pressure

The IL’s negligible vapor pressure is one of the most important characteristics of ionic liquids for application in industrial processes. Therefore, it is expected that approximately no IL would be lost by evaporation. A reliable representation of this behavior can be performed by a suitable selection of the model’s alpha function [\({{\alpha }}_{\text{i}}\left(\text{T}\right)\)].

The alpha function selected for this study was the Mathias-Copeman correlation (Eq. 8) (Mathias and Copeman 1983). This correlation performed a modification of the original Soave alpha function by the addition of the quadratic and cubic terms, resulting in a function of three parameters (\({C}_{{1}_{i}}, {C}_{{2}_{i}}, and {C}_{{3}_{i}}\)). The Mathias-Copeman modification was reported to promote a better representation of systems containing species with high reduced temperatures (\(T/{T}_{{c}_{i}}\)), such as hydrogen at the conditions studied.

$${{\alpha }}_{\text{i}}\left(\text{T}\right)= {\left[1+ {{\text{C}}_{1}}_{\text{i}}\left(1-\sqrt{\frac{\text{T}}{{{\text{T}}_{\text{c}}}_{\text{i}}}}\right)+{{\text{C}}_{2}}_{\text{i}}{\left(1-\sqrt{\frac{\text{T}}{{{\text{T}}_{\text{c}}}_{\text{i}}}}\right)}^{2}+{{\text{C}}_{3}}_{\text{i}}{\left(1-\sqrt{\frac{\text{T}}{{{\text{T}}_{\text{c}}}_{\text{i}}}}\right)}^{3}\right]}^{2}$$

8

The predictive approach was applied by considering the \({C}_{{2}_{i}}, and {C}_{{3}_{i}}\) constants equal to zero. Then, the Mathias-Copeman is reduced back to the Soave original alpha function (Soave 1972) (Eq. 9), in which the \({C}_{{1}_{i}}\) parameter can be predicted by the Eq. 10 and the ILs’ acentric factor (\({\omega }\)). This parameter can also be estimated by the group contribution method (Valderrama and Rojas 2009).

$${{\alpha }}_{\text{i}}\left(\text{T}\right)= {\left[1+ {\text{m}}_{\text{i}}\left(1-\sqrt{\frac{\text{T}}{{{\text{T}}_{\text{c}}}_{\text{i}}}}\right)\right]}^{2}$$

9

$${{\text{C}}_{1}}_{\text{i}}={\text{m}}_{\text{i}}=0.48508+1.55171{{\omega }}_{\text{i}}-0.15613{{\omega }}_{\text{i}}^{2}$$

10

The nonpredictive approach was applied by regressing the \({C}_{{1}_{i}}, {C}_{{2}_{i}}, and {C}_{{3}_{i}}\) parameters to fit the model to the IL’s vapor pressure.

## Binary vapor-liquid and liquid-liquid equilibrium

Only the nonpredictive approach was applied to the binary phase equilibria representation. As aforementioned, the nonpredictive approach is based on the PSRK EoS. In this model, the \(a\) and \(b\) parameters of the mixtures are represented by a modification of the Huron-Vidal mixing rule (Eq. 11) (Holderbaum and Gmehling 1991), and an average weighting of each species co-volume parameter (Eq. 12)

$$\text{a}=\text{b}\left[\frac{{\text{G}}^{\text{E}}}{{\text{A}}_{1}}+\sum _{\text{i}}{\text{x}}_{\text{i}}\frac{{\text{a}}_{\text{i}}}{{\text{b}}_{\text{i}}}+ \frac{\text{R}\text{T}}{{\text{A}}_{1}}\sum _{\text{i}}{\text{x}}_{\text{i}}\text{l}\text{n}\frac{\text{b}}{{\text{b}}_{\text{i}}}\right]$$

11

$$\text{b}= \sum _{\text{i}}{\text{x}}_{\text{i}}{\text{b}}_{\text{i}}$$

12

$$\frac{{\text{G}}^{\text{E}}}{\text{R}\text{T}}=\sum _{\text{i}}{\text{x}}_{\text{i}}\text{ln}{{\gamma }}_{\text{i}}$$

13

Where \({\text{G}}^{\text{E}}\) is the excess Gibbs energy, \({\text{A}}_{1}\) is PSRK constant equal to \(-0.64663\), in the mixing rule (Eq. 11), \({\text{x}}_{\text{i}}\) is the mol fraction of the species \(\text{i}\) in the liquid or gas phase, and \(\text{n}\) is the number of mols.

The PSRK is considered a predictive model for estimating the required activity coefficient (\({{\gamma }}_{\text{i}}\)) function by the predictive UNIFAC model (Holderbaum and Gmehling 1991). In this study, the UNIFAC was replaced by the UNIQUAC activity coefficient model. The UNIQUAC \({r}_{i}\) and \({q}_{i}\) parameters for the cations and anions in the liquid ionic were obtained from the UNIFAC-Lei model (Lei et al. 2009). The UNIQUAC model binary (\({\tau }_{ij}\)) parameters (Eq. 14) were regressed to fit experimental vapor–liquid and liquid-liquid equilibrium data.

$${{\tau }}_{\text{i}\text{j}}={\text{e}\text{x}\text{p}}^{\left({\text{a}}_{\text{i}\text{j}} + \frac{{\text{b}}_{\text{i}\text{j}}}{\text{T}} + {\text{c}}_{\text{i}\text{j}}\text{l}\text{n}\text{T} + {\text{d}}_{\text{i}\text{j}}\text{T} + \frac{{\text{e}}_{\text{i}\text{j}}}{{\text{T}}^{2}} \right)}$$

14

Detailed calculations for vapor-liquid equilibrium are presented in the Supplementary Material.

Figure 1 summarizes the procedure for the application of the predictive and nonpredictive methods. The critical constants and the alpha function are first estimated by both approaches for calculations of the ILs density and vapor pressure. The results are then compared with vapor pressure and density experimental data, and the model with better performance is selected for further calculations. The UNIQUAC binary parameters are regressed to fit the PSRK EoS to the available binary phase equilibrium data. The final approach is used to estimate the chemical equilibrium of IL-containing RWGS systems to evaluate the CO2 conversion to CO at different IL content, pressure, and temperatures.

All predictive properties were estimated using an Excel sheet adapted from Valderrama and Rojas (2009). All parameter regressions were performed with Aspen Plus V8.8 software.

All data were evaluated by the average absolute relative deviations (AARD) from the experimental data, calculated according to Eq. 15.

$$\text{A}\text{A}\text{R}\text{D}=\frac{1}{\text{n}}\sum \frac{\left|{\text{x}}_{\text{e}\text{x}\text{p}}-{\text{x}}_{\text{c}\text{a}\text{l}}\right|}{{\text{x}}_{\text{e}\text{x}\text{p}}}$$

15

## Chemical equilibrium

All chemical equilibria calculations were performed by the minimization of Gibbs energy of the systems. Previous studies reported that, if allowed, CO2 is hydrogenated mostly to methane, from a thermodynamic perspective (Müller et al. 2014; André Pacheco 2017; Marques and Guirardello 2019). From a thermochemical perspective, the CO2 hydrogenation to methane (Eq. 2) releases a significant amount of energy (-164.7 kJ/mol), which should enhance its conversion at low temperatures (Sandler 1989). Although the methanation reaction is exergonic and exothermic, methane formation is negligible in some IL-containing CO2 hydrogenation systems (Qadir et al. 2018, 2019; Yasuda et al. 2018). The authors suggested that ILs can significantly influence the reaction’s kinetics; presumably, facilitating the formation of products without significant CH4 unwanted formation.

One of the major challenges of direct CO2 conversion into high-chain products is the activation of the stable molecule of CO2 (Aresta 2010; Müller et al. 2014). The CO2 conversion into CO is a possible pathway for this activation and has been used to enhance the conversion of FTS synthesis (Zang et al. 2021).

To study the influence of the addition of ILs in RWGS reaction systems, a constraint related to hydrocarbon formation was applied. Therefore, no other reaction aside from the RWGS was allowed to occur. A similar consideration was performed on methane formation in a previous study about thermodynamic analysis of CO conversion to hydrocarbons and produced interesting insights for the FTS synthesis (Marques and Guirardello 2019).

The chemical equilibrium calculations were performed using the Gibbs reactor block in Aspen Plus V8.8 software.