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# In general up to date no study has considered thermodynamic

In general, up to date, no study has considered thermodynamic modeling to estimate the HyperScribe Poly (A) Tailing Kit of SO2 by different DESs. Since experimental measurements are time consuming and expensive, general and accurate theoretical models are vital to estimate SO2 solubilities in different DESs over wide ranges of temperatures. Since no theoretical models are available in open literature to estimate SO2 solubility in DESs, it was the goal of this study to propose a general thermodynamic model to reliably estimate the phase behavior of binary mixtures of SO2 and different types of DESs over wide ranges of temperatures.

Theory
The two major routes of thermodynamic phase equilibrium calculations are the approaches of ϕ−ϕ and γ−ϕ. Both of these approaches are widely used for vapor-liquid equilibrium calculations. Even though ϕ−ϕ is a good approach for calculations of vapor-liquid equilibria, especially at high pressures, this approach is highly dependent on the system under investigation and the EoS being used. For certain cases, the γ−ϕ approach is more recommended [31]. Because conventional cubic EoSs are not suitable for the estimation of liquid phase non-idealities, and are furthermore inaccurate in estimating liquid densities, it is sometimes recommended to calculate liquid phase equilibria using activity coefficient models, thus the γ−ϕ approach. However, it is important to mention that because of the pressure-independency of activity coefficient models, the γ−ϕ should not be used at high pressure conditions [31].
The fugacity coefficient is calculated by an equation of state (EoS) while the activity coefficient is estimated by an activity coefficient model. Eqs. (1), (2) show the basic equations used in γ−ϕ approach calculations, equating the fugacities in the vapor and liquid phases of each component [32].
In the case of DESs in particular, because of the negligible vapor pressure of a DES, the vapor phase can be assumed to be pure, for example, for a binary system of DES+SO2, one can assume yDES=0. Therefore, the phase equilibrium calculations will be simplified to Eq. (3).
The exponential expression is the Poynting correction factor, which is especially important at high pressures. ϕ is the saturated fugacity coefficient of SO2 which is calculated by an EoS at the investigated temperature, T, and the corresponding saturated vapor pressure of SO2, P. For SO2, the saturated vapor pressure is calculated by Eq. (4)[33].
Where, P and T are in bar and kelvin, respectively.
ϕ is the fugacity coefficient of SO2 at the investigated temperature and corresponding total pressure, which, the same as ϕ, is calculated by an EoS. The SO2 activity coefficient,γ, is calculated by activity coefficient models.

Investigated compounds
In this study, the SO2 solubility data in different DESs have been collected from open literature. This consists of data over wide ranges of temperatures, at atmospheric pressure, for 14 different DESs with different natures and various HBA to HBD molar compositions [3], [9], [29]. A total of 84 solubility data points made up the collected data bank. Table 1 presents the temperatures and SO2 solubility ranges of the systems considered, as well as the number of data and references for all of the investigated DESs in this study.

Results and discussion
As explained in detail in the Theory section, one of the highly recommended approaches in thermodynamics to model vapor-liquid equilibria at low pressure conditions is the γ−ϕ approach. Following this approach, in order to have greatly reliable results for the challenging systems of SO2+DESs, we have tried to incorporate powerful thermodynamic models for both the vapor and liquid phases. The CPA EoS for the vapor phase, and both the NRTL and UNIQUAC models for the liquid phase, were chosen with this idea in mind. To the best of our knowledge from open literature, this is the first study which considers the CPA EoS for SO2. Therefore, Constant regions was necessary to obtain the CPA parameters of SO2. In the particular case of vapor-liquid equilibria of mixtures of SO2 and DESs, because of the negligible vapor pressure of DESs, the vapor phase is practically purely SO2 which allows for simplifying assumptions in this respect. Regarding the association scheme, SO2 is an inert in the vapor phase.