Estimation of Ultrasonic Velocity, Density, Internal Pressure, and Thermophysical Parameters of Ionic Liquid Mixtures: Application of Flory’s Statistical Theory

Flory’s statistical theory (FST) has been employed to estimate the ultrasonic velocity, density, internal pressure, and several important thermophysical parameters such as the energy of vaporization, the heat of vaporization, cohesive energy density, polarity index, and solubility for eight binary mixtures of ionic liquids and water within the temperature range of 288.15 to 308.15 K. The ionic liquids chosen for this investigation are [BMim][dca], [BMim][TfO], [BMpy][TfO], [BMpyr][dca], [BMpyr][TfO], [EEPy][ESO4], [HMim][dca], and [MPy][MSO4]. The predicted values of ultrasonic velocity and density show good agreement with the data reported in the literature. It endorses the applicability of FST to these binary mixtures. A comparative analysis of the internal pressure values (Pi) determined by using FST and the standard thermodynamic approach is also presented. The results obtained for Pi using both approaches show good agreement. Besides, for the mixtures under study, the correlation between ultrasonic velocity, density, and surface tension has also been examined. The variation of thermophysical parameters with concentration and temperature changes has been utilized to explore the nature and strength of the solute–solvent interactions prevalent in these mixtures. It is pointed out that A–A-type interactions dominate over A-B-type interactions in water-rich regions of the mixtures.


INTRODUCTION
Ionic liquids (ILs) have received the attention of many researchers and industrialists due to their exceptional properties, such as low vapor pressure, a wide range of viscosity, adjustable miscibility, and good thermal conductivity. 1,2Another promising feature of ILs is their tunable physical and chemical properties.ILs are often termed designer solvents, which makes them extremely useful for special applications in industry.Ultrasonic velocity is one of the prime properties of ILs.It is used in the formulation of equations of state and to derive many thermophysical properties.Recently, some researchers 3,4 calculated heat capacity, isothermal and adiabatic compressibility, molecular radius, and apparent isentropic compressibility for ILs and their mixtures using the speed of sound in conjunction with other thermophysical parameters.It is a matter of fact that for the industrial design processes of ILs, it is customary to determine their density and refractive index.Nowadays, density calculation is being used to solve the material or energy balance equations of chemical processes in industry. 5he study of molecular interactions is important in almost all fields of the physical and chemical sciences.These interactions provide valuable information about the molecular packing, orientation, and conformation of the molecules.Many researchers 6−9 have investigated the molecular interactions in ILs and some binary liquid mixtures.Fumino et al. 10 investigated the hydrogen bonding, Coulomb interactions, and dispersion forces in some ILs.Dhumal et al. 11 reported the molecular interactions of a Cu-based metal−organic framework with a confined imidazolium-based IL using experimental and computational techniques.Recently, Wei et al. 12 reported changes in molecular interactions in ILs with charged SiO 2 surfaces.It is reported 13−15 that the computational and experimental techniques are complementary for determining the structure, design, and thermophysical properties of liquids and their mixtures.Some researchers 16−18 employed density functional theory (DFT), molecular dynamics (MD), COSMO-R, and Flory's statistical theory (FST) to estimate these properties.Though all these theoretical formulations are found suitable to compute the thermophysical properties, FST is a valuable and powerful tool as a result of the limited input parameters and ease of calculations using a simple analytic expression.FST is a good candidate in the theoretical framework of industrial design to predict the thermodynamic properties. 19everal researchers 20−22 successfully employed FST to predict the ultrasonic velocity and density with reasonable accuracy for pure liquids.Pandey et al. 23 modified FST for ternary liquid mixtures to discuss their thermodynamic behavior.Oswal et al. 24 tested the validity of the FST, ERAS, and Rao theories for the alkyl amines.Gepert et al. 25 have compared FST and the Prigogine−Flory−Patterson (PFP) model for binary mixtures of hydrocarbons.Recently, Shrivastava et al. 26 estimated various thermodynamic parameters using FST for pure ILs at elevated pressure.However, despite the high demand for ILs and their mixtures in various fields, their physicochemical properties have not yet been systematically studied, particularly for the mixtures of ILs with molecular liquids. 27Thus, there is a need to compute thermophysical properties like the speed of sound, density, adiabatic compressibility, thermal expansion, internal pressure, etc., for these liquids.−30 In their review article, Isosaari et al., 31 have summarized the use of ILs for wastewater treatment.These findings motivated us to conduct in-depth investigations of the physical properties of water-based IL mixtures.
In this investigation, FST is employed to estimate ultrasonic velocity, density, and internal pressure for eight binary mixtures of ILs and water at various temperatures.The obtained results are compared to the literature values, and a reasonable agreement is found between them.The correlation of density, ultrasonic velocity, and surface tension was also investigated.Several researchers have reported the importance of such correlations. 8,32Besides, some important thermophysical parameters were determined to understand the molecular interactions prevalent in these binary mixtures.The concentration and temperature dependence of solubility parameters and several other thermophysical parameters were also been investigated.The data required for the present study has been taken from the literature. 33

THEORETICAL FORMULATION
−37 They have successfully employed it to compute the density (d), ultrasonic velocity (U), surface tension, and molar volume for organic liquids, polymers, and ILs.We extended the same formulation to IL mixtures.Herein, only those relations are reported that have been directly utilized in the present study.
Using a reduced equation of state, 38 reduced volume (V ̃), reduced temperature (T ̃), characteristic pressure (P*), characteristic temperature (T*), and characteristic volume (V*) for pure ILs can be deduced as per the equations given below.Herein, the characteristic volume (V*) and characteristic temperature (T*) are the molar volume and temperature at the zero pressure limit (P = 0) 39 where ( ) ( ) = * are the reduced volume and reduced temperature, respectively.The reduced volume V ̃in terms of the coefficient of thermal expansion (α) can be computed using the relation given below: where α can be calculated in terms of ultrasonic velocity (U) and density (d) at various temperatures and concentrations using a well-known relation available in the literature: P* can be evaluated from the knowledge of α and K T : where γ P is the thermal pressure coefficient at the zero pressure limit and K T is the isothermal compressibility.K T was determined using a formula taken from the literature. 40,41The above relations have been employed to compute P*, V*, and T* for mixture components.Thereafter, these parameters were used to determine the segment fractions (ψ), site fraction (θ), and interaction parameter (χ 12 ) for the binary mixtures.The segment fractions are calculated as follows: and where ψ 1 and ψ 2 are the segment fractions, V i * (i = 1, 2) is the characteristic volume, and X i (i = 1, 2) is the mole fraction of solute (water) and solvent (ILs), respectively.The site fraction is given by ( ) The interaction parameter is calculated as The segment fractions (ψ 1 and ψ 2 ), site fraction (θ 2 ), and interaction parameter (χ 12 ) are used to determine the values of characteristic pressure (P*) and characteristic temperature (T*) of mixtures based on the following relations: T P The characteristic surface tension (σ*) for binary mixtures can be calculated as where P* and T* are calculated using eqs 9 and 10 and k is the Boltzmann constant.The reduced surface tension [σ(V ̃)] is deduced as a function of the reduced volume (V ̃m) for binary mixtures as where M is the fractional change in the neighborhood cell count in the surface phase.Its value generally exists between 0.26 and 0.29.The surface tension of an IL mixture is computed using the following equation:

Estimation of Ultrasonic Velocity and Their
Correlation with Density and Surface Tension.Ultrasonic velocity plays a vital role in chemical ultrasonics.Simply, in conjunction with density, it can provide valuable information about liquid systems.In the present study, the density (d) and a diagnostic parameter, the surface tension (σ), are utilized to calculate the ultrasonic velocity of these systems using the standard correlation available in the literature: 44,45 Auerbach relation: Alternberg relation: Singh−Pandey−Sanguri relation: modified Auerbach relation:

FST and Estimation of Density.
The ideal reduced volume V ̃0 of the binary mixture can be obtained as V ̃0 is further used to calculate the ideal reduced temperature, T 0: The excess reduced volume is given by The reduced volume is then obtained as Finally, the molar volume (V m ) in terms of the characteristic volume of the mixture (V*) and the reduced volume (calculated using eq 21) can be obtained as V* can be calculated using the additive properties of the mixture: The density of the mixture (d) is then given by where M eff is the effective mass of the mixture and V m is calculated using eq 22.

Estimation of Internal Pressure and FST.
−50 Due to the complex procedure of its experimental measurements, many empirical/semiempirical relations have been proposed to compute the internal pressure based on ultrasonic velocity, molecular radius, and thermal expansion at different temperatures for pure as well as mixtures.The thermodynamic method is the most prominent approach to computing the internal pressure (P i ), which is rewritten below in its usual form: where α and K T are the thermal expansivity and isothermal compressibility of the given mixtures, which can be deduced using the following standard relations: 51 Pandey and co-workers computed the internal pressure (P FST ) for liquid mixtures at 303.15 K using FST.Later, the same approach was opted by many workers to compute the internal pressure for different liquid mixtures.The mathematical formulation to compute the P FST is reproduced below: In the above relation, α FST and (K T ) FST are the thermal expansivity and isothermal compressibility calculated using Flory's parameters as In the above relations, V ̃m and P* are the reduced volume and characteristic pressure of mixtures; their values have been computed using eqs 2 and 9. Herein, we have also employed eq 28 for the first time to compute the P i values for the binary mixtures of water and ILs.The obtained results are then compared with the P i values obtained by using the thermodynamic method.
The cohesive energy density basically depends upon ΔE V , n, and the molar volume (V m ) of the mixture.The general form of the relation is given below: where here the polarity index is n = 1.The solubility parameter is given by ced = (34)   Hildebrand originally developed the solubility parameter relations for organic liquids.Here, we are investigating its application to binary mixtures of water and ILs.Theoretically obtained values of the ultrasonic velocity are reported in Table S1 (Supporting Information) as well as plotted against the mole fraction (X 1 ) of water at different temperatures and depicted in Figure 1a−h.The close look of Figure 1a indicates that for water + [BMim][dca] binary mixture, the U values obtained using Auerbach relation (U A ), Singh−Pandey−Sanguri relation (U SP ), and modified Auerbach relation (U MA ) are in reasonable agreement with ultrasonic velocity (U*) data reported in the literature. 33However, the Alternberg relation (U AR ) reports a strong deviation with the increase in the mole fraction of water.This deviation is much more pronounced for the higher mole fractions of water.Figure 1b−h also indicates similar results, which confirm that except for U AR , the other three relations; U A , U MA , and U SP , can be employed to compute the ultrasonic velocity for water + 1e], the deviation of U A , U MA , and U SP with respect to U* increased for X 1 > 0.7.It is also noticed for the same system that the Alternberg relation (U AR ) shows reasonable agreement until X 1 = 0.7.

[BMim][TfO], water + [BMpy][TfO], water + [BMpyr][dca], water + [EEpy][ESO 4 ], water + [HMim][dca], and water + [MMpy][MSO 4 ] mixtures. In the case of water+[BMpyr][TfO] [Figure
The reasonable agreement of U A , U MA , and U SP with U* also affirms the U−d−σ correlation and the ability of FST to compute the ultrasonic velocity for the given water + IL mixtures.The higher deviation of U AR values from the literature data is due to the noncompliance of the assumptions made for the derivation of this relation.The general agreement of U A , U MA , and U SP with U* is further confirmed from the absolute percentage deviation (Table 1).It is pertinent to mention here that, however, the general agreement is reached between theoretical models (U A , U MA , and U SP ) with U*, but the deviation is still toward the higher side.It is attributed to the fact that these relations were originally derived for organic mixtures having component size is almost similar, but in the present systems, the size and structure of ILs and water molecules are different, which leads to a higher deviation.It further suggests that these relations also need modification, considering the   complex structure of IL to estimate the ultrasonic velocity for given mixtures.

Estimation of Density Using FST.
Density is an important thermodynamic parameter.It is presently used to characterize liquid mixtures in industry. 19,38,52It is also used in conjunction with ultrasonic velocity and temperature to compute important physical parameters, such as the coefficient of thermal expansion.In the present study, the density (d FST ) of eight binary IL mixtures is computed using FST, and the obtained results are compared with the literature values (d*).Density data, computed by using FST, and their percentage deviation from the literature values at various concentrations and temperatures are reported in Table 2.A perusal of Table 2 reveals good agreement between the computed and literature values of density for all the systems under study.The mean percentage deviation (MPD) for each system at different temperatures is reported in Table 3.It is obvious from Table 3 that MPD is less than 1.5% for all the binary mixtures of ILs at all temperatures.It confirms the validity of FST to predict density in the given concentration and temperature range.
3.3.Estimation of Internal Pressure Using FST.Internal pressure has been computed using the thermodynamic method (P i ) and FST (P FST ) for eight binary mixtures of water and ILs at various temperatures.The results obtained are reported in Table 4. Additionally, thermal expansivity and isothermal compressibility using the thermodynamic method (α, K T ) and FST [α FST , (K T ) FST ] are computed and tabulated in Table S2 [Supporting file].Due to the nonavailability of the experimental data of Pi for these mixtures, the Pi values obtained by using the thermodynamic method are considered standard to check the validity of FST in predicting the internal pressure.The ratio of P FST /P i is plotted against the mole fraction (X 1 ) for eight IL mixtures at different temperatures in Figure 2. As is obvious from Figure 2, the P FST /P i ratio is closer to unity for all of the mixtures under study.It confirms the applicability of FST to predict the Pi values for these mixtures.To generalize the validity of FST, this approach can be applied to these and similar ILs that are mixed with other suitable organic solvents.
3.4.Thermophysical Parameters, Molecular Interactions, and FST.Some other important physical parameters, such as energy of vaporization (ΔE V ), heat of vaporization (ΔH), cohesive energy density (ced), surface tension (σ), and polarity index (n), have been evaluated using FST in the given range of concentrations and temperatures.The obtained results of ΔE V , ΔH, and σ are reported in Table S3.In the current work, σ has been used in the calculation of the ultrasonic velocity using   eqs 14−17.−55 The energy of vaporization (ΔE V ) is defined as the energy utilized in the evaporation of one mol of the liquid by breaking all the associated forces, whereas the enthalpy of vaporization (ΔH) is a sum of the pressure-volume work done and the internal energy of the system.In the current study, these parameters are evaluated for given mixtures at given temperatures and concentrations, as reported in Table S1.The portrayal of Table S1 indicates that both ΔE v and ΔH v represent a similar decrease with an increase in the concentration of water in the mixtures.The decreasing trend in both of these parameters hints at a decrease in the cohesive forces with the addition of the first component. 56The ced represents both specific and nonspecific intermolecular interactions, which overall contribute to the total intermolecular interaction energies, whereas internal pressure counts only the specific interactions present in the liquid state.In the present study, ced is computed for eight IL mixtures, as reported in Table 5.The perusal of Table 5 indicates that ced decreases with a decrease in concentration of the ILs in the given mixtures at all temperatures.The decreasing values of ced suggest a decrease in the cohesion present within the liquid mixture and, hence, an increase in the molecular interactions. 57he Hildebrand solubility parameter (δ) is the square root of the ced and is important to access the intermolecular interactions in the liquid system.Several workers 58,59 have calculated δ for organic liquid mixtures and polymer mixtures to analyze their solubility.Recently, Pandey and co-worker 32,50 computed δ for pure ILs.In the present study, δ is calculated for given mixtures of water at various concentrations and temperatures, as reported in Table 5.The portrayal of Table 5 indicates that δ values gradually decrease in the water-rich region in these binary mixtures.This indicates that the solute−solute (A−A) interactions dominate over the solute−solvent (A−B) interactions in the water-rich region of the mixtures.

CONCLUSIONS
In the current study, density, internal pressure, ultrasonic velocity, and some important thermophysical properties of eight binary mixtures of water and ILs have been evaluated at three different temperatures using FST.The very low mean percentage deviation of computed density values (Table 3) and the P FST /P i ratio being closer to unity (Figure 2) for all the mixtures under study confirm the applicability of FST for the evaluation of density and internal pressure of the liquid mixtures.

Figure 1 .
Figure 1.Comparison of ultrasonic velocity estimated using the Auerbach relation (U A ), Alternberg relation (U AR ), Singh−Pandey−Sanguri relation (U SP ), and modified Auerbach relation (U MA ) along with the literature values (U*) for binary mixtures.Solid line is a guide for eye.

Figure 2 .
Figure2.Ratio of internal pressure ratio (P FST /P i ) obtained the thermodynamic method and the FST method.Solid line is a guide for eye.
The reasonable agreement of the Auerbach relation (U A ), Singh−Pandey−Sanguri relation (U SP ), and modified Auerbach relation (U MA ) with the literature values (U*) validates the U− d−σ correlation for the given systems.The variations of the solubility parameter (δ) in the given concentration at different temperatures indicate the dominance of A−A interactions over A−B in the water-rich region of the water + IL mixtures.■ ASSOCIATED CONTENT * sı Supporting Information

Table 2 .
33mputed Density (d FST ) and Literature (d*)33Density along with Percentage Deviations for Eight Binary Mixtures of Water and Ionic Liquids at Different Temperatures a

Table 3 .
Mean Percentage Deviation of the Computed Value of Density for Eight Binary Mixtures of Water and ILs

Table 4 .
Internal Pressure Computed Using the Thermodynamic Approach (P i ) and Flory's Theory (P FST ) at Different Temperatures for Eight IL Mixtures