# Towards the equation of state for neutral ([C.sub.2][H.sub.4]), polar ([H.sub.2]O), and ionic ([bmim][[BF.sub.4]], [bmim][[PF.sub.6]], [pmmim][[Tf.sub.2]N]) liquids.

Despite considerable effort of experimentalists no reliable vapor-liquid coexistence at very small pressures and liquid-solid coexistence at high pressures have been until now observed in the working range of temperature 290 < T/K < 350 for ionic liquids. The measurements of high-pressure properties in low-temperature stable liquid are relatively scarce while the strong influence of their consistency on the phase equilibrium prediction is obvious. In this work we discuss the applicability of fluctuational-thermodynamic methodology and respective equation of state to correlate the properties of any (neutral, polar, ionic) liquids since our ultimate goal is the simple reference predictive model to describe vapor-liquid, liquid-liquid, and liquid-solid equilibria of mixtures containing above components. It is shown that the inconsistencies among existing volumetric measurements and the strong dependence of the mechanical and, especially, caloric derived properties on the shape of the functions chosen to fit the experimental data can be resolved in the framework of fluctuational-thermodynamic equation of state. To illustrate its results the comparison with the known experimental data for [bmim] [[BF.sub.4]] and [bmim][[PF.sub.6]] as well as with the lattice-fluid equation of state and the methodology of thermodynamic integration is represented. It corroborates the thermodynamic consistency of predictions and excellent correlation of derived properties over the wide range of pressures 0 < P/MPa < 200.1. Introduction

Behavior of low-melting organic salts or ionic liquids (ILs) [1-6] in the region of phase transitions is qualitatively similar to that either for high-temperature nonorganic molten salts or long-hydrocarbon-chain organic solvents and, even, for polymer systems. Such characteristic features as negligible vapor pressure [P.sub.[sigma]](T), undefined critical parameters [P.sub.c], [[rho].sub.c], [T.sub.c] for vapor-liquid (v, l)-transition, split of liquid-solid (l,s)-boundary onto melting [P.sub.m](T) and freezing [P.sub.f](T) branches, existence of glassy states make the problem of metastability to be quite complex but vital for many potential uses of ILs. In particular, thermodynamic modeling and computer simulation of the phase behavior in mixtures formed by ILs with water and low-molecular organic solvents such as ethylene can be of great importance for the further tuning of their operational parameters. If one proceeds from a pure to a mixed fluid, it is especially advantageous to develop the same format of reference equation of state (EOS) and the common format of reference pair potential (RPP) for each component and mixture.

As a first step toward consistent modeling of the phase behavior of IL and its solution we demonstrate in this work how the fluctuational-thermodynamic (FT) EOS [7-12] and the relevant finite-range Len-nard-Jones (LJ) RPP can be applied to model the underlying structure and properties of low-molecular ([C.sub.2][H.sub.4], [H.sub.2]O) and imidazolium-based (1-butyl-3-methylimidazolium tetrafluorob orate ([bmim][[BF.sub.4]]), 1-butyl-3-methylimida-zolium hexafluorophosphate ([bmim][[PF.sub.6]]), 2,3-dimethyl-1-propylimidazolium bis(trifluoromethylsulfonyl)imide ([pmmim][[Tf.sub.2]N])) solvents. For any pure component FT-model is based either on the measurable coexistence-curve input data [P.sub.[sigma]](T), [[rho].sub.v](T), [[rho].sub.l](T) (if they are achievable as for [C.sub.2][H.sub.4] and [H.sub.2]O) or on the also measurable one-phase density of liquid at atmospheric pressure [rho] ([P.sub.0] [approximately equal to] 0,1 MPa, T) for ILs. This methodology becomes purely predictive for density [rho](P,T) in any one-phase v,l,s-regions including their metastable extensions. Only the measurable isobaric heat capacity data [C.sub.P]([P.sub.0], T) have to be added to the set of input data for prediction of other caloric properties (isochoric heat capacity [C.sub.V](P,T), speed of sound W(P,T), and Gruneisen parameter Gr(P, T)) at higher pressures P > [P.sub.0] and lower T < [T.sub.m] or higher T > [T.sub.b] temperatures where [T.sub.b] is the hypothesized normal boiling temperature [T.sub.b]([P.sub.0]). Its existence itself is a debatable question because the thermal decomposition [T.sub.d] maybe former [T.sub.d] < [T.sub.b].

Such approach was proposed recently [7,8] to reconstruct the hypothetical (V, l)-diagram of any ILs in its stable and metastable regions on the base of only standard reference data on density [rho](T) at [P.sub.0] [1-4] and one free parameter, an a priori unknown value of the excluded volume [b.sub.0]. To our knowledge this is first attempt to predict simultaneously the whole set of one-phase and two-phase properties for ILs without the fit at any other pressures including the negative ones. It was argued that the particular low-temperature variant of the most general FT-EOS [9-12] should be used to obtain the consistent prediction of volumetric properties and the standard response functions [a.sub.P], [[beta].sub.T], [[gamma].sub.[rho]] = [a.sub.P]/[[beta].sub.T] by the following equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [b.sub.0] is the excluded molecular volume and a(T) is the T-dependent effective cohesive energy. The derivative da/dT affects the thermal expansion [[alpha].sub.P] and the thermal-pressure coefficient [[gamma].sub.[rho]] while the isothermal compressibility [[beta].sub.T] depends only on [b.sub.0]-value at the given pressure. The changeable sign of two thermal derivatives [[alpha].sub.P], [[gamma].sub.[rho]] offers a possibility to predict the properties of anomalous low-temperature substances (such as water, for example) too [7,8].

Fortunately we have obtained now [13-19] a possibility to test our predictions not only by the direct experimental one-phase data [14, 16, 18, 19] on [rho](P,T)- and W(P,T)-surfaces. Another possibility is offered by comparison of the predictions obtained by FT-EOS for the critical parameters of ILs ([bmim][[BF.sub.4]]: [T.sub.c] = 962,3 K, [P.sub.c] = 3503,9 kPa, [[rho].sub.c] = 438,565 [kg.m.sup.-3] with those predicted here by the Sanchez-Lacombe EOS for lattice fluid (LF) [15]: [T.sub.c] = 885,01 K, [P.sub.c] = 2829 kPa, [[rho].sub.c] = 248,565 [kg.m.sup.-3] as well as with those simulated by GEMC-methodology [6]: [T.sub.c] = 1252 K, [P.sub.c] = 390 kPa, [[rho].sub.c] = 181 [kg.m.sup.-3]. It seems that the relatively close location of ([T.sub.c],[P.sub.c])-parameters predicted by both EOSs is some guarantee of their reliability while [T.sub.c] and [P.sub.c] from [6] are significantly overestimated and underestimated, respectively. Interestingly, the known descriptive factor of compressibility [r.sub.t] = [P.sub.c]/([[rho].sub.t][RT.sub.c]) estimated by Guggenheim [20] in the vicinity of triple point [T.sub.t] for argon as [r.sub.t] = 0,108 is equal to close values [r.sub.t] = 0,082 for FT-EOS and [r.sub.t] = 0,072 for LF-EOS but only to very small value [r.sub.t] = 0,007 for result of GEMC-simulations if the common realistic estimate (see below) [[rho].sub.t] [approximately equal to] [[rho].sub.t] = 5,350646 [mol*dm.sup.-3] at T = 290 K is used. Moreover, it will be shown that the characteristic dimensional parameters [P.sup.*.sub.c], [T.sup.*.sub.c], [[rho].sup.*.sub.c] and another compressibility factor [r.sup.*.sub.c] = [P.sup.*.sub.c]/([[rho].aup.*.sub.c ][RT.aup.*.sub.c]) obtained by Machida et al. [14] by the fit to (P, [rho], T)-experimental data for [bmim][[BF.sub.4]] and [bmim][[PF.sub.6]] provide the structural estimates of hard-core volume, number of lattice sites in a cluster, and energy of near-neighbor pair interactions which are surprisingly close to ones independently predicted by the FT-model of a continuum substance.

Taking into account the compatibility of above results it is important to consider the presumable similarity between the square-well fluid (which may be thought of as a continuum analogue of the lattice-gas (LG) or lattice-fluid (LF) systems) on the one hand and the LJ-fluid of finite-range interactions (RPP) on the other. This conceptual analogy has been pointed out long ago for the critical region by Widom [21] who suggested that it is the propagation of attractive correlations in the LG which determines the peculiarities of criticality However, such unphysical LG-predictions at low temperatures of the ([rho], T)-plane as the nonexistence of a (l, s)-transition suggest that repulsive forces are not being treated properly by this RPP-model. In contrast with the discrete LG-model, it seems that both attractive and repulsive forces are being dealt with properly in the square-well continuum fluid because it exhibits both (v, l)- and (l, s)-transitions. The serious restriction of latter is however evident since any singularities of RPP imply an artificial jump of pair-distribution isotropic function g(r) at the point of cutoff radius [r.sub.c] for attractive interactions.

In this context only the shifted and smoothed at [r.sub.c]-point LJ-potential [5, 6] seems to be appropriate as RPP for a continuum system. Of course, the algebraic form of the respective reference EOS is essential too. In accordance with the statistical-mechanical arguments presented by Widom [21] there are the set of alternative forms including the original vdW-EOS and the LG-EOS in the well-known Bragg-Williams approximation which share the common restrictive feature. One may suppose that the probability of finding some prescribed value of the potential energy [TEXT NOT REPRODUCIBLE IN ASCII] at an arbitrary point in the fluid is independent of T at fixed [TEXT NOT REPRODUCIBLE IN ASCII]. Another simplifying assumption is that such EOS supposes only two types of fluid structure, one of the excluded (or hardcore) volume [Nv.sub.0] where the singular hard-sphere branch of potential is infinite and one of free volume (V - [Nv.sub.0]) where the potential is uniform, weak, and unrestricted (an infinite-range rectilinear well). It should be directly proportional to density e = U/N = -ap where U is the total configurational energy and a is the constant vdW-coefficient. These historical notes are important to explain how one can go beyond the above restriction of T-independency by adoption of linear [rho]-dependence for a generalized specific or molar energy (see also (8) below). Consider

a (T) = - [([partial derivative]e/[partial derivative][rho]).sub.T] (5)

Another aim of the developed FT-EOS follows from the possibility [7] to estimate the effective LJ-parameters without any fit. Indeed, their general T-dependent values,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6a)

[epsilon](T)/k = T (1- [Z.sub.l]) (6b)

are determined simply in the low-temperature range of all ILs where [b.sub.0] is constant in ((1)-(4)) while the compressibility factor of saturated liquid [Z.sub.t] = [P.sub.[sigma]]/([[rho].sub.l]RT) becomes negligible as well as the vapor pressure [P.sub.[sigma]](T) trends to zero. Taking into account this asymptotic behavior it is especially important to study the possible correlations of these parameters in the RPP-model of an effective LJ-potential for ILs as the functions of total molecular weight M. This concept is unusual for the conventional consideration of a separate influence of the anions [M.sub.a] and cations [M.sub.c] components. It may provide, in principle, the useful insight the nature of (v,l)-transition in ILs by effective capturing underlying pair interactions.

The distinction of both FT-EOS and LF-EOS [14] from the conventional hard-sphere reference EOS is crucial to provide the quantitative description of one-phase liquid. The formers include the quadratic in density contribution, which is dominating at high pressures along the isotherms. The latter considers this term as a small vdW-perturbation for the hard-sphere EOS. Such perturbation approach is not directly applicable to associating fluids such as water and alcohols for which presence of hydrogen bonding, anisotropic dipolar 1/[r.sup.3], or coulombic 1/r interactions in addition to isotropic dispersive 1/[r.sup.6] attractions is inconsistent with the main assumption of the perturbation methodology that the structure of a liquid is dominated by repulsive forces [15].

The FT-model promotes the more flexible approach in which the above factors of attraction and clustering can be effectively accounted by the a(T)-dependence. It was firstly confirmed by Longuet-Higgins and Widom and, then, by many authors that a combination of Carnahan-Starling EOS, for example, with the vdW-perturbation a[[rho].sup.2] is a reasonable approximation for the l- and s-phases but not the v-phase. Guggenheim [20] has concluded its applicability only to a liquid when large clusters are more important than small clusters (i.e., at low temperatures [T.sub.m] < T < [T.sub.b]). In contrast with this observation, the general FT-EOS provides the adequate representation of entire subcritical range [T.sub.m] < T < [T.sub.c] including the critical region and (v, l)-phase transition [9-12]. It will be shown below by FT-model without undue complexity of calculations.

2. Universal FT-EOS for Any Low-Themperature Fluids

2.1. General Form of FT-EOS for Subcritical Themperatures. It is often claimed that the original van der Waals (vdW)-EOS with two constant coefficients a, b determined by the actual critical-point properties [[rho].sub.c], [T.sub.c] is only an approximation at best and cannot provide more than qualitative agreement with experiment even for spherical molecules. However, it was proved recently [9-12] that the general FT-EOS with three T-dependent coefficients,

p = [[rho]RT [1 - c(T)]/1 - b(T) [rho]] - a(T) [[rho].sup.2], (7)

is applicable to any types of fluids including ILs. The measurable volumetric data of coexistence curve (CXC) have been used for evaluation of T-dependences without any fit. Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

where the reduced slope [A.sub.[sigma]](T) of [P.sub.[sigma]](T)-function is defined by the thermodynamic Clapeyron's equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

This fundamental ratio of the (v,l)-latent heat to the thermodynamic work of (v, l)-expansion is the main parameter of FT-coefficients determined by ((8)-(9)). It should be calculated separately in each of high-temperature ([T.sub.b] [less than or equal to] T [less than or equal to] [T.sub.c]) [9-12] v- and l-phases to obtain the reasonable quantitative prediction of one-phase thermophysical properties. The general FT-EOS is applicable to the entire subcritical range ([T.sub.t][less than or equal to] T [less than or equal to] [T.sub.c]) but it can be essentially simplified to the form of (1) if ([T.sub.t ][less than or equal to] T [less than or equal to] [T.sub.b]).

2.2. Particular Form of FT-EOS for Low Themperatures. An absence of input CXC-data [P.sub.[sigma]](T), [[rho].sub.g](T), [[rho].sub.l](T) for ILs is the serious reason to develop the alternative method for the evaluation of T-dependent FT-coefficients. The thermodynamically-consistent approach has been proposed in [7, 8] for the particular form of FT-EOS (1) applicable in the low-temperature range from the triple [T.sub.t] (or melting [T.sub.m]) point up to the [T.sub.b]-point. Former one is usually known for ILs while the latter one is, as a rule, more than temperature of thermal decomposition [T.sub.b] > [T.sub.d] ~ 650 K. The methodology was tested on two low-molecular-weight substances ([C.sub.2][H.sub.4], [H.sub.2]O) and two imidazolium-based ILs ([bmim][[PF.sub.6]], [pmmim][[Tf.sub.2]N]) with the promising accuracy of predictions even for the isothermal compressibility [[beta].sub.T] up to the pressure P = 200 MPa.

To illuminate the distinction between the particular (reference) and general form of FT-EOS let us discuss in brief the main steps of the proposed procedure. Its detailed analysis can be found elsewhere [7, 8]. The algorithm is as follows. Step 1. At the chosen free parameter [T.sub.b] one determines the orthobaric molar densities [[rho].sub.l]([T.sub.b]) = [rho]([T.sub.b],[P.sub.0]); [[rho].sub.g]([T.sub.b]) = [P.sub.0]/[RT.sub.b] to solve the transcendent equation:

[[rho].sub.g]/[[rho].sub.l] = 1 + y (x) [e.sup.-x]/ 1 + y (x) [e.sup.x], (12)

for the reduced entropy (disorder) parameter x([T.sub.b]) and the respective molar heat [r.sub.[sigma]]([T.sub.b]) of vaporization. Consider

x = [([s.sub.g] - [s.sub.l])/2R], (13a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13b)

Step 2. The universal CXC-function y(x) in (12) is determined by equalities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

and it provides the possibility to estimate a preliminary value of [b.sub.0],

[b.sub.0] = [x([T.sub.b]) - 1]/[[rho].sub.l]([T.sub.b]) [x([T.sub.b]) - 1/2], (15)

as well as to evaluate the orthobaric densities at any T if the function x(T) is known. Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16b)

Step 3 (A-variant of x(T)-prediction [7, 8]). To calculate its values one must obtain two densities [[rho].sup.[+ or -]] (at the assumption [P.sub.[sigma]] [approximately equal to] 0) from equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)

where [rho] = [rho]([T,P.sub.0]), [Z.sub.0] = [P.sub.0]/[rho]RT and the [[rho].sup.+](T)-function provides the preliminary estimate of x(T) for the low-temperature range (at the consistent assumption: [[rho].sup.+] >> [[rho].sub.g] [approximately equal to] 0). Consider

x(T) = 1 - [b.sub.0][[rho].sup.+]/2/1- [b.sub.0][[rho].sup.+]. (18)

Step 4. One substitutes x(T) from (18) in ((13a), (13b), (16a), (16b)) to calculate [r.sub.[sigma]](T), [[rho].sub.g](T), [[rho].sub.l](T), respectively. Step 5. A preliminary value of [a.sub.0](T) may be estimated then by the more restrictive assumption [P.sub.0] [approximately equal to] 0 (used also in the famous Flory-Orwoll-Vrij EOS developed for heavy n-alkanes). Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Step 6 (B-variant of x(T)-prediction). To control the consistency of methodology one may use instead of Step 3 (A-variant) the same equation (17) with the approximate equality [[rho].sub.l](T) [approximately equal to] [[rho].sup.+](T) to solve (16b) at the a priori chosen [b.sub.0]-value for determination of alternative x(T), and so forth, (Steps 4 and 5). Just this approach (B-variant) has been used below in the low-temperature range of [bmim][[BF.sub.4]].

Step 7. The self-consistent prediction of a hypothetical (v, l)-diagram requires the equilibration of CXC-pressures [P.sub.[sigma]](T, [[rho].sub.g]) = [P.sub.[sigma]](T, [[rho].sub.l]) by FT-EOS (1) with the necessary final change in [a.sub.0](T)-value from (19) to satisfy the equality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

Only in the low-temperature range T [less than or equal to] [T.sub.b] the distinction between the preliminary definition (18) and its final form (19) for a(T)-values is not essential at the prediction of vapor pressure [P.sub.[sigma]](T,[[rho].sub.g]).

3. Reference Equation of State, Effective Pair Potential, and Hypothetical Phase Diagram

To demonstrate universality of approach and for convenience of reader we have collected the coefficients of FT-EOS (1) for neutral ([C.sub.2][H.sub.4]) and polar ([H.sub.2]O) fluids [7, 8] in Table 1 and added in Table 2 to other ILs ([bmim][[PF.sub.6]], [pmmim][[Tf.sub.2]N] [7, 8]) the data for [bmim][[BF.sub.4]] obtained in this work (Table 3). When temperature is low [T.sub.m] < T < [T.sub.b] FT-model follows a two-parameter ([eplison](T), [[sigma].sub.0]) correlation of principle of corresponding states (PCS) on molecular level as well as a two-parameter (a(T), [b.sub.0]) correlation of PCS on macroscopic level.

One the most impressed results of FT-methodology is shown in Figure 1 where the comparison between such different high- and low-molecular substances as ILs and [C.sub.2][H.sub.4], [H.sub.2]O is represented. The results based on the coefficients of Tables 1 and 2 demonstrate that the proposed low-temperature model provides the symmetric two-value representation of vapor pressure [+ or -][P.sub.s](T) similar to that observed for the ferromagnetic transition in weak external fields.

To estimate the appropriate excluded molar volume [b.sub.0] (M = 225,82 g/mol) of FT-model we consider that it belongs to the range [v.sub.0] = M/[[rho].sub.0] [approximately equal to] 162, [v.sub.l] = M/[[rho].sub.l] [approximately equal to] 187[cm.sup.3]/mol]. The extrapolated to zero temperature T = 0 K "cold" volume [v.sub.0] = 162 cm /mol follows from (27). The fixed value: [b.sub.0] = 178 [cm.sup.3]/mol ([b.sub.0] [approximately equal to] 1,[1v.sub.0]) has been used in this work to demonstrate the main results of the proposed methodology. Such choice for [bmim][[BF.sub.4]] on the ad hoc basis is in a good correspondence with the respective values: [b.sub.0] = 195,3 [cm.sup.3]/mol for [bmim][[PF.sub.6]] and [b.sub.0] = 271,1 [cm.sup.3]/mol for [pmmim][[Tf.sub.2]N] where the empirical relationship [b.sub.0] [approximately equal to] 1,[1v.sub.0] was also observed [7, 8]. Our estimates of the effective LJ-diameters by (6a) for ILs: [sigma]([bmim][[BF.sub.4]]) = 5,208 A, [sigma]([bmim][[PF.sub.6]]) = 5,371 A, and [sigma]([pmmim][[Tf.sub.2]N]) = 5,992 A can be tested by comparison with the independently determined values [13] for anions [[sigma].sub.a]([[BF.sub.4]]) = 4,51 [Angstrom]; [[sigma].sub.a]([[PF.sub.6]]) = 5,06 [Angstrom]. We have verified Berthelots combining rule for spherical molecular ions (21a) and van der Waals' combining rule for chain molecules (21b) usually considered by van der Waals'-type of EOS for mixtures [22]. Consider

[sigma] = [[sigma].sub.c] + [[sigma].sub.a]/2, (21a)

[b.sub.0] = [b.sub.c] + [b.sub.a]/2. (21b)

The predicted by former rule of LJ-diameter for the same [bmim]-cation were close but still different, 5,906 [Angstrom] and 5,682 [Angstrom]. For the latter rule their values and distinction become even smaller, 5,757 [Angstrom] and 5,651 [Angstrom]. As a result, the chain rule (21b) seems preferable for ILs and its average value for [[sigma].sub.c][bmim] = 5,704 [Angstrom] can be used to estimate the LJ-diameter of [[Tf.sub.2]N]-anion: [[sigma].sub.a][[Tf.sub.2]N] = 6,254 [Angstrom] taking into account the equality: [M.sub.c] [bmim] = [M.sub.c] [pmmim] = 139 g/mol. The collected in Table 4 effective LJ-diameters are linear functions of [M.sub.a] in the set of ILs with different anions and cations if the molecular weight of latters [M.sub.c] is the same one.

Since the low-temperature compressibility factor [Z.sub.l](T) is very small for all discussed liquids their dispersive energies [eplison](T) (molecular attractions parameters) are comparable in accordance with (6b). However, the differences in cohesive energies a(T) (collective attraction's parameters) between the low-molecular substances ([C.sub.2][H.sub.4], [H.sub.2]O) and ILs are striking as it follows from Tables 1 and 2. The physical nature of such distinction can be, at the first glance, attributed to omitted in the reference LJ-potential influence of intramolecular force-field parameters and anisotropic (dipole-dipole and coulombic) interactions. At the same time, one must account the collective macroscopic nature of a(T)-parameter. It corresponds to the scales which are compatible or larger than the thermodynamic correlation length [xi]([rho], T). FT-model [9-12] provides an elegant and simple estimation of this effective parameter based on the concept of comparability between energetic and geometric characteristic of force field determined by the given RPP. Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

Taking into account the above results and the coefficients from Tables 1-3 we have used (22) at T = 300 K (T* = [k.sub.B]T/[eplison] [approximately equal to] 1) to compare the thermodynamic correlation length predicted for [bmim][[BF.sub.4]] (a = 8900,9 J*d[m.sup.3] /[mol.sup.2]; [b.sub.0] = 178 [cm.sup.3] /mol; [rho] = 5,322294 mol/d[m.sup.3]) and at T = 298,15 K for water (a = 548,27 J*d[m.sup.3]/[mol.sup.2]; [b.sub.0] = 16,58 cm /mol; [rho] = 55,444 mol/d[m.sup.3]) [24]. The dimensional and reduced ([xi]* = [xi]/[sigma]) values for former are, respectively, [xi] = 177,7 [Angstrom], [xi]* = 34,12 while for latter [xi] = 69,86 [Angstrom], [xi]* = 29,45. No more need be said to confirm the universality of FT-model.

One may note that our estimates of correlation length are significantly larger than those usually adopted for the dimensional or reduced cutof radius ([r.sub.c] or [r*.sub.c] = [r.sub.c]/[sigma]) of direct interactions at computer simulations. As a result, the standard assumption [[xi]*.sub.c] [approximately equal to] [r*.sub.c] may become questionable in the comparatively small (mesoscopic) volumes of simulation [L.sup.3] < [xi][([rho],T).sup.3]. At this condition the simulated properties are mesoscopic although their lifetime may be essentially larger than its simulated counterpart. The key point here is the same as one near a critical point where the problem of consistency between the correlation length for statics and the correlation time for dynamics becomes crucial. In any case, the computer study of possible nongaussian nature of local fluctuations within the thermodynamic correlation volume [[xi].sup.3] may be quite useful. The relevant inhomogeneities in the steady spacial distributions of density and enthalpy can affect, first of all, the simulated values of volumetric ([[alpha].sub.P], [[beta].sub.T]) and caloric ([C.sub.p],[C.sub.v]) derived quantities. Simultaneously, an account of internal degrees of freedom and anisotropy by the perturbed RPP may change the correlation length itself.

The above described by ((12)-(20)) FT-methodology has been used to reconstruct the hypothetical phase diagram (HPD) for [bmim][[BF.sub.4]] shown in Figures 2, 3, and 4 and represented in Table 3. Both (T, [rho]) (Figure 2) and (P, T) (Figure 4) projections contain also the branches of classical spinodal calculated by the LF (Sanchez-Lacombe)-EOS obtained in [14]. Its top is the location of a respective critical point. It seems that the relatively close ([P.sub.c],[T.sub.c])-parameters predicted independently by FT-EOS and by LF-EOS (see Section 1) are reasonable.

The FT-model provides a possibility to estimate, separately, the coordination numbers of LJ-particles in the orthobaric liquid [[rho].sub.l](T)- and vapor [[rho].sub.g](T)-phases. An ability to form the respective "friable" ([N.sub.l,g] + 1)-clusters is defined by the ratio of effective cohesive and dispersive molar energies at any subcritical temperature. Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

The term "friable" is used here to distinguish the clusters formed by the unbounded LJ-particles at the characteristic distance l* = l/[sigma] [approximately equal to] [cube root of 2] > 1 from the conventional "compact" ones with the bonding distance l* < 1 studied, in particular, by the GEMC-methodology [25] to model of molecular association. It is straightforwardly to obtain the low-temperature estimates based on the assumptions.

[Z.sub.l] << [Z.sub.g] [approximately equal to] 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24b)

and to find the critical asymptotics based on the difference of classical ([a.sup.0] ,[b.sup.0] ,[c.sup.0]) and nonclassical (a, b, c) T-dependent FT-EOS' coefficients [9-12]. Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25b)

The crucial influence of excluded-volume in (24a) and its relative irrelevance in (24b) for [N.sub.l,g,] -predictions are illustrated by Figure 5 where [N.sub.l](T) function is shown also for the entire l-branch based on the evaluated in the present work HPD. For comparison, the low-temperature ability to form the ([N.sub.l] + 1)-clusters in liquid water [7, 24] is represented in Figure 5 too.

In according with ((25a), (25b)) the "friable" clusters can exist only as dimers in the classical critical liquid phase ([N.sub.l.sup.c] [approximately equal to] 1). It is not universal property in the meaning of scaling theory but it corresponds to the PCS-concept of similarity between two substances ([H.sub.2]O and [C.sub.4]mim [[BF.sub.4]], e.g.) if their Z.sub.c]-values are close. On the other side, the scaling hypothesis of universality is confirmed by the FT-EOS' estimates in the nonclassical critical vapor phase. For the set of low-molecular-weight substances studied in [9] (Ar,[C.sub.2][H.sub.4], [CO.sub.2], [H.sub.2]O); for example, one obtains by (25b) the common estimate ([N.sup.c].sub.g] [approximately equal to] 2,5) which shows a significant associative near-mean-field behavior.

It is worthwhile to note here the correspondence of some FT-EOS'-estimates with the set of GEMC-simulated results. One may use the approximate estimate of critical slope [A.sub.c] [approximately equal to] 7,86 [9] for [bmim][[BF.sub.4]] based on the similarity of its [Z.sub.c]-value with that for [H.sub.2]O [24]. In such case, the respective critical excluded volume [b.sub.c] [approximately equal to] 220 [cm.sup.3]/mol becomes much more than vdW-value 1/3[[rho].sub.c] = [b.sub.c.sup.0] [approximately equal to] [b.sub.0] [approximately equal to] 178[cm.sup.3]/mol. Another observation seems also interesting. Authors [25] have calculated (see Figure 3 in [25]) for the "compact" clusters at [l.sup.*] = 0,7; 0,5; 0,45; the ([T.sup.*], [[rho].sup.*])-diagram of simple fluids. One may note that only the value [l.sup.*] = 0,7 corresponds to the shape of strongly-curved diameter shown in Figure 2 for the HPD while the smaller values: l* = 0,5; 0,45 give the shape of HPD and the nearly rectilinear diameter strongly resembling those obtained by the GEMC-simulations [6] for the complex ILs force-field. If this correspondence between the "friable" and "compact" clustering is not accidental one obtains the unique possibility to connect the measurable thermophysical properties with the both characteristics of molecular structure in the framework of FT-EOS.

4. Comparison with the Empirical Tait EOS and Semiempirical Sanchez-Lacombe EOS

The empirical Tait EOS is based on the observation that the reciprocal of isothermal compressibility [[beta].sup.-.sub.T] for many liquids is nearly linear in pressure at very high pressures. Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (26)

where some authors [14, 19] omit the T -dependence in coefficient C and ignore the value [P.sub.0] [approximately equal to] 0 [14]. Such restrictions transform the Tait EOS into the empirical form of two-parameter (B(T),C) PCS because the sets of C-values for different ILs become close one to another. For example, Machida et al. [14] have found the sets C = 0,09710 for [bmim][[PF.sub.6]], C = 0,09358 for [bmim][[BF.sub.4]], and C = 0,08961 for [bmim][OcSO.sub.4] which is rather close to the set obtained by Matkowska and Hofman [19] C = 0,088136 for [bmim][[BF.sub.4]] and C = 0,0841547 for [bmim][MeS[O.sub.4]]. At the same time, Gu and Brennecke [3] have reported the much larger T-dependent values C(298,2 K) = 0,1829 and C(323,2 K) = 0,1630 for the same [bmim][[PF.sub.6]].

Two other reasons of discrepancies in the Tait methodology is the different approximations chosen by authors for the reference input data [rho]([P.sub.0], T) and for the compound-dependent function B(T). Some authors [4, 14, 18] prefer to fit the atmospheric isobars [rho]([P.sub.0],T) and[C.sub.p]([P.sub.0],T) with a second-order or even third-order polynomial equation while the others [1, 2, 16, 19] use a

linear function for this aim.. As a result, the extrapolation ability to lower and higher temperatures of different approximations becomes restricted.

In this work we have used for [bmim] [[BF.sub.4]] the simplest linear approximation of both density and heat capacity,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (28)

taken from [1]. The extrapolated to zero of temperature value [[rho].sub.0](0 K) = 1394,65 [kg*m.sup.-3] [1] is in a good correspondence with that from [14] [[rho].sub.0](0 K) = 1393,92 [kg*m.sup.-3], in reasonable correspondence with that from [19] [[rho].sub.0](0 K) = 1416,03 [kg.m.sup.-3] and that from [16] [[rho].sub.0](0 K) = 1429[kg*m.sup.-3] but its distinction from value [[rho].sub.0](0 K) = 1476,277 [kg*m.sup.-3] reported by authors [18] is rather large. The similar large discrepancy is observable between [C.sup.0.sub.p] (0 K) = 273,65 J/mol*K from [1] and [C.sup.0.sub.p] (0 K) = 464,466 J/mol*K from [18].

The different choices of an approximation function for B(T) (so authors [14] have used the exponential form while authors [19] have preferred the linear form) may distort the derivatives [[beta].sub.T] and [[alpha].sub.P] calculated by the Tait EOS (26). The problem of their uncertainties becomes even more complex if one takes into account the often existence of systematic distinctions of as much as 0,5% between the densities measured by different investigators even for the simplest argon [23]. Machida et al. [14], for example, pointed out the systematic deviations measured densities from those reported by the de Azevedo et al. [18] and Fredlake et al. [1] for both [bmim][[BF.sub.4]] and [bmim][[PF.sub.6]]. Matkowska and Hofman [19] concluded that the discrepancies between the different sets of calculated [[beta].sub.T-] and [[alpha].sub.P-]derivatives increase with increasing of T and decreasing of P due not only to experimental differences in density values but also result from the fitting equation used. The resultant situation is that the expansivity [[alpha].sub.P] of ILs reported in literature was either nearly independent of T [18] or noticeably dependent of T [3,19].

We can add to these observations that the linear in molar (or specific) volume Tait Eos (26) is inadequate in representing the curvature of the isotherm P([rho]) at low pressures. It fails completely in description of (v, l)-transition where the more flexible function of volume is desirable. However, this has been clearly stated and explained by Streett for liquid argon [23] that the adjustable T-dependence of empirical EOS becomes the crucial factor in representing the expansivity [[alpha].sub.P] and, especially, heat capacities [C.sub.P],[C.sub.v] at high pressures even if the reliable input data of sound velocity W(P, T) were used.

From such a viewpoint, one may suppose that the linear in temperature LF-EOS proposed by Sanchez and Lacombe,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (29)

is restricted to achieve the above goal but can be used as any unified classical EOS common for both phases to predict the region of their coexistence. Such conjecture is confirmed by the comparison of FT-EOS with LF-EOS presented in Figures 2 and 4 and discussed below. The obvious advantage of former is the more flexible T-dependence expressed via the cohesive-energy coefficient a(T). On the other hand, the LF-EOS is typical form of EOS (see Section 1) in which the constraint of T-independent potential energy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is inherent [21].

One may consider it as the generalized variant of the well-known Bragg-Williams approximation for the ordinary LG presented here in the dimensional form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (30)

Such generalization provides the accurate map of phenomenological characteristic parameters T*, P*, [rho]* which determine the constant effective number of lattice sites [N.sub.l] ]occupied by a complex molecule,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (31)

into the following set of molecular characteristic parameters for a simple molecule ([N.sub.l] = 1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (32)

where [v.sub.0] is the volume of cell and z is the coordination number of lattice in which the negative [eplison] is the energy of attraction for a near-neighbor pair of sites. In the polymer terminology [eplison]* from (31) is the segment interaction energy and v* is the segment volume which determines the characteristic hard core per mole M/[rho]* (excluded volume b in the vdW-terminology).

Another variant of described approach is the known perturbed hard-sphere-chain (PHSC) EOS proposed by Song et al. [15] for normal fluids and polymers

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (33)

where g(d) = (1 - [eta]/2)/[(1 - [eta]).sup.3] is the pair radial distribution function of nonbonded hard spheres at contact and the term with ([N.sub.l] - 1) reflects chain connectivity while the last term is the small perturbation contribution. Though the PHSC-EOS has the same constraint of the potential energy field [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] authors [15] have introduced two universal adjustable [[PHI].sub.a](T)- and [[PHI].sub.b](T)-functions to improve the consistency with experiment. The vdW-type coefficients were rescaled as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (34a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (34b)

where s([N.sub.l]) is the additional scaling function for T* = [eplison]/[k.sub.B]. It provides the interconnection of molecular LJ-type parameters ([eplison], [sigma]) with the phenomenological vdW-ones (a, b). The resultant reduced form of PHSC is [15]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (35)

where the following characteristic and reduced variables are used:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (36)

It was compared with the simpler form of LF-EOS (29). Their predictions of the low-temperature density at saturation [[rho].sub.l](T) are comparable but, unfortunately, inaccurate (overestimated) even for neutral low-molecular liquids. The respective predictions of the vapor pressure [P.sub.[sigma]](T) are reasonable [15] excepting the region of critical point for both EOSs. Our estimates based on the LF-EOS [14] shown in Figures 2 and 4 are consistent with these conclusions.

The comparison of volumetric measurements and derived properties [14,18] with the purely predictive (by the FT-EOS) and empirical (by the Tait EOS and LF-EOS) methodologies used for [bmim][[BF.sub.4]] is shown in Figures 6-9. Evidently, that former methodology is quite promising. Machida et al. [14] have reported two correlations of the same (P, [rho], T)-data measured for [bmim][[BF.sub.4]] at temperatures from 313 to 473 and pressures up to 200 MPa. To examine the trends in properties of ILs with the common cation [bmim] the Tait empirical EOS was preliminarily fitted as the more appropriate model. The estimate of its extrapolation capatibilities for (P,[rho], T)-surface in the working range (290 < T/K < 350) follows from the compatibility of experimental points (where those measured by de Azevedo et al. [18] in the range of temperature 298 < T/K < 333 and pressure (0,1 < P/MPa < 60) were also included) with the thick curves in Figure 6. It is noticeable, for example, that the extrapolated Tait's isotherm T = 290 K coincides practically with isotherm T = 298,34 K from [18] because the measured densities of latter source are systematically higher than those from [14]. Density data of Fredlake et al. [1] for [bmim][[BF.sub.4]] (not shown in Figure 6) are also systematically shifted from measurements [14].

The consequence of such discrepancies is also typical for any simple liquids (Ar, Kr, Xe) [23] at moderate and high pressures. It is impossible to reveal an actual T-dependence of volumetric (mechanical) derived functions [[alpha].sub.P], [[gamma].sub.[rho]] due to systematic deviations between the data of different investigators. In such situation an attempt "to take the bull by the horns" and to claim the preferable variant of EOS based exclusively on volumetric data maybe erroneous. Indeed, since the Tait EOS is explicit in density while the LF-EOS--in temperature the direct calculation of [[alpha].sub.P], [[beta].sub.T]-derivatives for former and [[alpha].sub.P], [[gamma].sub.[rho]]-derivatives for latter are motivated. To illustrate the results of these alternative calculations we have used in Figures 6-9 the coefficients of LF-EOS reported by Machida et al. [14] for the restricted range of moderate pressures 0,1 < P/MPa < 50. The thick dashed curves represent the boundaries of working range where the extrapolation to T = 290 K is again assumed. One may notice the qualitative similarity of FT-EOS (the thin curves) and LF-EOS which can be hypothesized as an existence of certain model substance at the extrapolation to higher pressures P/MPa > 50. It demonstrates the smaller compressibility [[beta].sub.T] (Figure 8) and expansivity [[alpha].sub.P] (Figure 7) than those predicted by the Tait EOS while the value of thermal-pressure coefficient [[gamma].sub.[rho]] for FT-EOS (Figure 9) becomes larger. It determines the distinctions in the calculated internal pressure. The choice of the FT-models substance as a reference system for the perturbation methodology provides the set of advantages in comparison with the LF-EOS.

It follows from Figure 6 that at moderate pressures P/MPa < 50 the predictive FT-EOS is more accurate than the fitted semiempirical LF-EOS [14] although the discrepancies of both with the empirical Tait EOS [14] become significant at the lowest (extrapolated) temperature T = 290 K. The Tait's liquid has no trend to (v, l)-transition (as well as polymers) in opposite to the clear trends demonstrated by FT-EOS and LF-EOS. One may suppose [26] a competition between vaporization of IL (primarily driven by the isotropic dispersive attraction 1/[r.sup.6] in RPP) and chain formation (driven mainly by the anisotropic dipolar interactions 1/[r.sup.3]) reflected by the Tait EOS fitted to the experimental data. Of course, such conjecture must be, at least, confirmed by the computer simulations and FT-model provides this possibility by the consistent estimate of RPP-parameters ([eplison], [sigma]) at each temperature.

The differences of calculated expansivity [[alpha].sub.P] in Figure 7 are especially interesting. FT-EOS predicts even less variation of it with temperature than that for the Tait EOS. This result and crossing of [[alpha].sub.P](P)-isotherms are qualitatively similar to those obtained by de Azevedo et al. [18] although the pressure dependence of all mechanical ([[alpha].sub.P], [[beta].sub.T], [[gamma].sub.[rho]]) and caloric ([C.sub.P],[C.sub.v]) derivatives (see Figures 10,11, and 12) is always more significant for the FT-EOS predictions. It seems that the curvature of the [rho](T)-dependence following from the LF-EOS (29) is not sufficient to predict the [[alpha].sub.P](P) behavior (Figure 7) correctly. The strong influence of the chosen input [rho]([P.sub.0], T)-dependence is obvious from Figures 7-9.

The prediction of caloric derivatives ([C.sub.P],[C.sub.v], [C.sub.P]/[C.sub.v], Gr) is the most stringent test for any thermal (P, [rho], T) EOS. It should be usually controlled [23] by the experimental (W, [rho], T)-surface to use the thermodynamical identities,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (37)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (38)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (39)

in addition to the chosen input [C.sub.P]([P.sub.0], T)-dependence. de Azevedo et al. [18] applied this strategy to comprise the approximated by the Pade-technique measured speed of sound data for [bmim][[PF.sub.6]] and [bmim][[BF.sub.4]] (Figure 13) with the evaluated at high pressures heat capacities.

Our predictive strategy is based [17] on the differentiation of a(T)-dependence to evaluate directly the most subtle ([C.sub.V], P, T)-surface in a low-temperature liquid

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (40)

where the influence of the consistence for the chosen input [rho]([P.sub.0], T)- and[C.sub.p]([P.sub.0], T)-dependences (via (37) used for estimate of [C.sub.V]([P.sub.0], T) at the atmospheric pressure [P.sub.0]) becomes crucial. The use of first derivative da/dT (even by its rough approximation in terms of finite differences: [DELTA]a/[DELTA]T) to calculate simultaneously by ((3), (4), (37), (40)) all volumetric and caloric derivatives is the important advantage over the standard integration of thermodynamic identities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (41a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (41b)

To illustrate such statement it is worthwhile to remind the situation described by the Streett [23] for liquid argon. Since isotherms of [[alpha].sub.P](P) cross over for many simple liquids (Ar, Kr, Xe), this author concludes that the sign of [([[partial derivative].sup.2]V/[[partial derivative]T.sup.2]).sub.P] changes also from positive to negative at the respective pressure. This conclusion is not valid because to change the sign of derivative [([partial derivative][[alpha].sub.P]/[partial derivative]T).sub.P] it is enough to account for the exact equality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (42)

in which [([[partial derivative].sup.2]V/[[partial derivative]T.sup.2]).sub.P] can be always positive. In this case one would expect the monotonous decrease of [C.sub.p] with increasing P in accordance to ((41a), (41b)) while the presence of extremum (minimum or maximum of [C.sub.p](P)-dependence) seems to be artificial.

There is the variety of pressures reported by different investigators as a presumable cross-point for the same ILs. Machida et al. [14] have estimated it to be about 10 MPa on the base of Tait EOS for [bmim][[PF.sub.6]] but have not found it (Figure 7) for [bmim][[BF.sub.4]]. For latter our estimate by the FT-EOS is: P = 20,6 MPa. de Azevedo et al. [18] have reported the mild decrease of [[alpha].sub.P](T)-dependence and the sharp decrease [[alpha].sub.P](P)-dependence while a presumable cross-point is located between about 100 and 120 MPa for [bmim][[BF.sub.4]]. Taking into account the above distinction in the evaluated ([[alpha].sub.P], P, T)-surface it is interesting to consider their consequences for caloric ([C.sub.v],P,T)-([C.sub.p],P,T)- and[C.sub.v]/[C.sub.p]-surfaces shown in Figures 10-12.

The remarkable qualitative and even quantitative (<8%) correspondence between the predicted by FT-EOS[C.sub.v]-values and those reported by de Azevedo et al. [18] follows from Figure 10. At the same time, although the discrepancies between[C.sub.p]-values [18] and those predicted by the FT-EOS are again within acceptable limits (<10%) the formers demonstrate the weak maximum and very small pressure dependence for [bmim][[BF.sub.4]] (for [bmim][[PF.sub.6]] this[C.sub.p](P)-dependence is monotonous as well as that predicted by the FT-EOS). It seems that the resultant ratio of heat capacity [C.sub.p]/[C.sub.v] shown in Figure 12 which demonstrates the irregular crossing of isotherms [18] is questionable. It suggests that their pressure dependence either needs the more accurate approximation or reflects the realistic distinction of reference FT-EOS from the actual behavior of [bmim][[BF.sub.4]].

The lock of noticeable variations in pressure is the common feature of integration methodology [18] based on the given (W, P, T)- and ([rho], P, T)-surfaces. The unavoidable accumulation of uncertainties at each stage of calculations in the set W - [C.sub.p] - [C.sub.v] may cause the unplausible behavior of adiabatic exponent [C.sub.p]/[C.sub.v] in liquid. The same is true for the set [C.sub.v ] - [C.sub.p] - W used in the FT-methodology. It is the most appropriate explanation of significant discrepancies for W(P)-dependence shown in Figure 13. Let us remind also that the precise mechanical measurements of speed velocity [18] in the very viscous IL cannot be attributed exactly to the condition of constant entropy.

Thus, strictly speaking, the measured (W, P, T)-surface reflects the strong dispersive properties of media and must be less than its thermodynamic counterpart in the ideal (without a viscosity) liquid.

5. Conclusions

There are the structure-forming factors related to the above-discussed thermodynamic characteristic. Despite the certain discrepancies between the predicted and derived properties for FT-EOS and LF-EOS, both ones provide the close estimates of structure factors represented in Table 5.

Our aim here is to show that the thermodynamically-consistent predictions of thermodynamical properties by the FT-EOS yields also the molecular-based parameters which are, at least, realistic (see also Table 3). The estimate of average T-dependent well-depth [bar.[eplison] by (6b) as well as estimate of average value [[bar.N].sub.l] by (24a) is related to the middle of temperature range: T = 320 K. The distinction of [bar.[eplison] from the respective [eplison]*-parameters of LF-EOS [14] can be attributed to the difference between nonbonded interactions in the discrete (LF-EOS) and continuum (FT-EOS) models of fluid. Our estimate of cohesive-energy density [[eplison].sub.coh] by equality,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (43)

represented in Table 6 seems also physically plausible. Mag-inn et al. [5, 6] have determined it within the framework of GEMC-simulations by the knowledge of [rho]([P.sub.0], T) and the internal energy difference between an ideal-gas ion pair and the average internal energy of an ion pair in the liquid state.

Such definition indicates that cohesive energy densities of many ILs are on the order of 500-550 J/[cm.sup.3] (see, for comparison, Table 6) and demonstrate a slight decrease as temperature increases.

Another relevant characteristic is the internal pressure determined by the derivative of molar (or specific) internal energy,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (44)

which is compared to ones calculated by different authors [14, 16] for [bmim][[BF.sub.4]] in Table 6. As in the other cases, the FT-EOS predicts the much faster change of both cohesive energy density [[eplison].sub.coh] and internal pressure [([partial derivative]e/[partial derivative]v).sub.T] as temperature increases.

One should collect a large number of precise experimental measurements to reconstruct the thermodynamic surface of a substance. FT-methodology provides a possibility of preliminary reliable estimates of relevant macroscopic and molecular-based correlations. Its thermodynamic consistency provides the serious advantage in comparison with the purely empiric treatment of any volumetric measurements at the description of derived heat capacities.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] C. P. Fredlake, J. M. Crosthwaite, D. G. Hert, S. N. V. K. Aki, and J. F. Brennecke, "Thermophysical properties of imidazolium-based ionic liquids," Journal of Chemical and Engineering Data, vol. 49, no. 4, pp. 954-964, 2004.

[2] C. Cadena, J. L. Anthony, J. K. Shah, T. I. Morrow, J. F. Brennecke, and E. J. Maginn, "Why is CO2 so soluble in imidazolium-based ionic liquids?" Journal of the American Chemical Society, vol. 126, no. 16, pp. 5300-5308, 2004.

[3] Z. Gu and J. F. Brennecke, "Volume expansivities and isothermal compressibilities of imidazolium and pyridinium-based ionic liquids," Journal of Chemical and Engineering Data, vol. 47, no. 2, pp. 339-345, 2002.

[4] B. Gonzalez, E. Gomez, A. Dominguez, M. Vilas, and E. Tojo, "Physicochemical characterization of new sulfate ionic liquids," Journal of Chemical and Engineering Data, vol. 56, no. 1, pp. 14-20, 2011.

[5] C. Cadena, Q. Zhao, R. Q. Snurr, and E. J. Maginn, "Molecular modeling and experimental studies of the thermodynamic and transport properties of pyridinium-based ionic liquids," The Journal of Physical Chemistry B, vol. 110, no. 6, pp. 2821-2832, 2006.

[6] N. Rai and E. J. Maginn, "Vapor-liquid coexistence and critical behavior of ionic liquids via molecular simulations," Journal of Physical Chemistry Letters, vol. 2, no. 12, pp. 1439-1443, 2011.

[7] V. B. Rogankov, "Equation of state for ionic liquids," High Themperature, vol. 47, no. 5, pp. 656-663, 2009.

[8] V. B. Rogankov, V. I. Levchenko, and Y. K. Kornienko, "Fluctuational equation of state and hypothetical phase diagram of superheated water and two imidazolium-based ionic liquids," Journal of Molecular Liquids, vol. 148, no. 1, pp. 18-23, 2009.

[9] V. B. Rogankov and L. Z. Boshkov, "Gibbs solution of the van der Waals-Maxwell problem and universality of the liquid-gas coexistence curve," Physical Chemistry Chemical Physics, vol. 4, no. 6, pp. 873-878, 2002.

[10] V. A. Mazur and V. B. Rogankov, "A novel concept of symmetry in the model of fluctuational thermodynamics," Journal of Molecular Liquids, vol. 105, no. 2-3, pp. 165-177, 2003.

[11] V. B. Rogankov, O. G. Byutner, T. A. Bedrova, and T. V. Vasiltsova, "Local phase diagram of binary mixtures in the near-critical region of solvent," Journal of Molecular Liquids, vol. 127, no. 1-3, pp. 53-59, 2006.

[12] V. B. Rogankov, "Asymmetry of heterophase fluctuations in nucleation theory," in Nucleation Theory and Applications, J. W. P. Schmelzer, G. Ropke, and V. Priezshev, Eds., chapter 22, pp. 227-241, Joint Institute for Nuclear Research, Dubna, Russia, 2011.

[13] A. B. McEwen, H. L. Ngo, K. LeCompte, and J. L. Goldman, "Electrochemical properties of imidazolium salt electrolytes for electrochemical capacitor applications," Journal of the Electrochemical Society, vol. 146, no. 5, pp. 1687-1695, 1999.

[14] H. Machida, Y. Sato, and R. L. Smith Jr., "Pressure-volume-temperature (PVT) measurements of ionic liquids ([[bmim.sup.+]] [[PF.sub.-6]], [[bmim.sup.+]][[BF.sub.-4]], [[bmim.sup.+]][[OcSO.sub.-4]]) and analysis with the Sanchez-Lacombe equation of state," Fluid Phase Equilibria, vol. 264, no. 1-2, pp. 147-155, 2008.

[15] Y. Song, S. M. Lambert, and J. M. Prausnitz, "A Perturbed Hard-Sphere-Chain equation of state for normal fluids and polymers," Industrial & Engineering Chemistry Research, vol. 33, no. 4, pp. 1047-1057,1994.

[16] A. Kumar, "Estimates of internal pressure and molar refraction of imidazolium based ionic liquids as a function of temperature," Journal of Solution Chemistry, vol. 37, no. 2, pp. 203-214, 2008.

[17] L. P. N. Rebelo, J. N. C. Lopes, J. M. S. S. Esperanca, and E. Filipe, "On the critical temperature, normal boiling point, and vapor pressure of ionic liquids," The Journal of Physical Chemistry B, vol. 109, no. 13, pp. 6040-6043, 2005.

[18] R. G. de Azevedo, J. M. S. S. Esperanca, V. Najdanovic-Visak et al., "Thermophysical and thermodynamic properties of 1-Butyl-3-methylimidazolium tetrafluoroborate and 1-Butyl-3-methylimidazolium hexafluorophosphate over an extended pressure range," Journal of Chemical & Engineering Data, vol. 50, no. 3, pp. 997-1008, 2005.

[19] D. Matkowska and T. Hofman, "High-pressure volumetric properties of ionic liquids: 1-butyl-3-methylimidazolium tetrafluoroborate, [C.sub.4]mim][[BF.sub.4]], 1-butyl-3-methylimid-azolium methylsulfate [[C.sub.4]mim][[MeSO.sub.4]] and 1-ethyl-3-methylimidazolium ethylsulfate, [[C.sub.2]mim][[EtSO.sub.4]]," Journal of Molecular Liquids, vol. 165, pp. 161-167, 2012.

[20] E. A. Guggenheim, "The new equation of state of Longuet-Higgins and Widom," Molecular Physics, vol. 9, no. 1, pp. 43-47, 1965.

[21] B. Widom, "Some topics in the theory of fluids," The Journal of Chemical Physics, vol. 39, pp. 1808-1812, 1963.

[22] R. L. Scott and P. H. van Konynenburg, "Static properties of solutions: van der Waals and related models for hydrocarbon mixtures," Discussions of the Faraday Society, vol. 49, pp. 87-97, 1970.

[23] W. B. Streett, "Thermodynamic properties of liquid argon at high pressures, calculated from PVT and sound-velocity data," Physica, vol. 76, no. 1, pp. 59-72, 1974.

[24] A. Saul and W. Wagner, "International equations for the saturation properties of ordinary water substance," Journal of Physical and Chemical Reference Data, vol. 16, pp. 893-901, 1987.

[25] F. Bresme, E. Lomb, and J. L. F. Abascal, "Influence of association on the liquid-vapor phase coexistence of simple systems," The Journal of Chemical Physics, vol. 106, no. 4, pp. 1569-1575, 1997.

[26] R. van Roij, "Theory of chain association versus liquid condensation," Physical Review Letters, vol. 76, no. 18, pp. 3348-3351, 1996.

Department of Physics, Odessa State Academy of Refrigeration, Dvoryanskaya Street 1/3, Odessa 65082, Ukraine

Correspondence should be addressed to Vitaly B. Rogankov; vrogankov@yandex.ua

Received 5 August 2014; Accepted 4 November 2014; Published 16 December 2014

Academic Editor: Pedro Jorge Martins Coelho

Copyright [c] 2014 V. B. Rogankov and V. I. Levchenko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Table 1: Coefficients of FT-EOS (1) for neutral ([C.sub.2][H.sub.4]) and polar ([H.sub.2]O) substances. [b.sub.0] [[dm.sup.3]/mol], a [[J.dm.sup.3]/[mol.sup.2]], T [K] [C.sub.2][H.sub.4] [H.sub.2]O [b.sub.0] = 0,04181 [b.sub.0] = 0,01658 T a T a 103,99 1557,12 273,16 517,162 105 1459,76 278,15 527,171 110 1127,30 283,15 534,941 115 935,490 288,15 540,955 120 812,151 293,15 545,767 125 725,634 298,15 548,270 130 661,817 303,15 549,648 135 613,602 308,15 549,424 140 575,528 313,15 548,755 145 545,191 318,15 547,172 150 520,785 323,15 544,760 155 500,788 328,15 541,609 160 484,652 333,15 537,810 165 471,466 338,15 533,455 169,35 462,006 343,15 529,044 348,15 524,605 353,15 519,789 358,15 514,676 363,15 509,674 368,15 504,480 373,15 499,159 Table 2: Coefficients of FT-EOS (1) for ILs: [bmim][[PF.sub.6]], [pmmim] [[Tf.sub.2]N], and [bmim][[BF.sub.4]]. [b.sub.0] [[dm.sup.3]/mol], a [[J.dm.sup.3]/[mol.sup.2]], T [K] [bmim][[PF.sub.6]] [pmmim][[Tf.sub.2]N] [bmim][[BF.sub.4]] [b.sub.0] = 0,1953 [b.sub.0] = 0,2711 [b.sub.0] = 0,178 T a T a T a 285 8433,52 288,15 11724,8 290 9470,0 290 8214,24 295,15 12224,5 300 8900,9 295 8015,04 304,25 12279,0 310 8435,0 300 7833,44 313,65 11728,9 320 8047,5 305 7667,34 324,35 11308,4 330 7720,8 310 7514,97 335,15 10672,3 340 7442,2 315 7374,83 344,65 10051,1 350 7202,6 320 7245,64 350,15 9709,50 325 7126,31 330 7015,89 335 6913,55 340 6818,59 345 6730,37 350 6648,36 Table 3: Predicted hypothetical (v, l)-transition in the low-temperature range for FT-model of [bmim][[BF.sub.4]] based on the experimental data [1, 2] treated by FT-EOS (B-variant of x(T)-prediction). T (K) [[rho].sub.l] (mol/[dm.sup.3]) [[rho].sub.g] (mol/[dm.sup.3]) 290 5,350646 3,09E - 08 300 5,322170 2,1E - 07 310 5,293693 1,01E - 06 320 5,265215 3,74E - 06 330 5,236735 1,13E - 05 340 5,208254 2,92E - 05 350 5,179771 6,61E - 05 360 5,151287 0,000135 370 5,122802 0,000253 380 5,094315 0,000441 390 5,065828 0,000725 400 5,037338 0,001131 410 5,008848 0,001689 420 4,980355 0,002428 430 4,951862 0,003379 440 4,923367 0,004569 450 4,894871 0,006027 460 4,866373 0,007778 470 4,837873 0,009843 480 4,809372 0,012244 490 4,780869 0,014997 500 4,752365 0,018119 510 4,723859 0,021622 511 4,721008 0,021993 512 4,718157 0,022369 513 4,715307 0,022748 514 4,712456 0,023132 515 4,709605 0,023519 516 4,706754 0,023910 517 4,703903 0,024306 T (K) [P.sub.[sigma]] (kPa) [r.sub.[sigma]] (J/mol) 290 7,45E - 05 53082 300 5,24E - 04 49866 310 0,002 47230 320 0,009 45032 330 0,031 43175 340 0,082 41587 350 0,192 40216 360 0,404 39022 370 0,778 37974 380 1,393 37048 390 2,347 36225 400 3,754 35488 410 5,740 34826 420 8,446 34228 430 12,017 33684 440 16,604 33188 450 22,359 32733 460 29,432 32314 470 37,968 31926 480 48,105 31566 490 59,977 31230 500 73,704 30914 510 89,400 30618 511 91,082 30589 512 92,785 30561 513 94,509 30532 514 96,254 30504 515 98,020 30476 516 99,806 30448 517 101,615 30420 T (K) a (J.[dm.sup.3]/[mol.sup.2]) [eplison]/k (K) 290 9470,0 290 300 8900,9 300 310 8435,0 309,99 320 8047,5 319,99 330 7720,8 329,99 340 7442,2 339,99 350 7202,6 349,99 360 6994,6 359,99 370 6813,0 369,98 380 6653,5 379,96 390 6512,7 389,94 400 6387,8 399,91 410 6276,6 409,86 420 6177,4 419,79 430 6088,7 429,70 440 6009,1 439,59 450 5937,6 449,45 460 5873,2 459,27 470 5815,4 469,05 480 5763,2 478,79 490 5716,3 488,49 500 5674,1 498,13 510 5636,1 507,72 511 5632,5 508,67 512 5629,0 509,63 513 5625,4 510,58 514 5622,0 511,54 515 5618,6 512,49 516 5615,2 513,44 517 5611,8 514,40 Table 4: Effective LJ-diameters of FT-model for ILs determined by (6a), (6b), and (21b) on the base of estimates [7,13] and the choice [b.sub.0] = 178 [cm.sup.3]/mol for [bmim][[BF.sub.4]] in this work. IL M (g/mol) [sigma]([Angstrom]) [bmim][[BF.sub.4]] 225,82 5,208 [bmim][[PF.sub.6]] 284 5,371 [7] [pmmim][[Tf.sub.2]N] 419,1 5,992 [7] IL [M.sub.c]/[M.sub.c] [bmim][[BF.sub.4]] 139/86,82 [bmim][[PF.sub.6]] 139/145 [pmmim][[Tf.sub.2]N] 139/280,1 IL [[sigma].sub.c]/[[sigma].sub.a] [bmim][[BF.sub.4]] 5,757/4,51 [13] [bmim][[PF.sub.6]] 5,651/5,06 [13] [pmmim][[Tf.sub.2]N] 5,704/6,254 Table 5: Comparison of excluded volumes (M/[rho]* [14] and [b.sub.0]), characteristic interaction energy ([eplison]* [14] and [bar.[eplison]]), and efffective number of bonded units (MP*/RT*[rho]* [14,15] and [[bar.N].sub.l]) (see text). Compound M/[rho]* [b.sub.0] [eplison]* ([cm.sup.3]/mol) ([cm.sup.3]/mol) (J/mol) [bmim][[BF.sub.4]] 175,3 178 5642 [bmim][[PF.sub.6]] 196,2 195,3 5658 Compound [bar.[eplison]] MP*/RT*[rho]* [[bar.N].sup.l] (J/mol) [bmim][[BF.sub.4]] 2661 17,6 17,0 [bmim][[PF.sub.6]] 2661 18,8 18,1 Table 6: Comparison of internal pressure ([partial derivative]e/[[partial derivative]v).sub.T] for [bmim][[BF.sub.4]] based on the LF-EOS [14] and FT-EOS (this work) with the values estimated [16] by experimental data on speed of sound W, density [rho], and isobaric heat capacity [C.sub.p]. I [14] ([partial derivative]e/[[partial derivative]v).sub.T] (K) (MPa) 313,1 482,78 332,6 471,50 352,6 460,35 372,7 450,11 392,8 440,00 412,9 429,23 432,6 419,35 452,3 408,84 472,2 399,19 I [14] T [16] ([partial derivative]e/[[partial derivative]v).sub.T] (K) (K) (MPa) 313,1 283,15 459,91 332,6 288,15 459,31 352,6 293,15 459,20 372,7 298,15 457,79 392,8 303,15 455,76 412,9 308,15 453,73 432,6 313,15 451,20 452,3 318,15 448,06 472,2 323,15 446,64 328,15 444,61 333,15 443,09 338,15 441,68 343,15 439,75 I [14] T [FT] ([partial derivative]e/[[partial derivative]v).sub.T] (K) (K) (MPa) 313,1 290 743,76 332,6 310 573,16 352,6 330 463,91 372,7 350 388,61 392,8 370 333,84 412,9 390 292,27 432,6 410 259,66 452,3 430 233,40 472,2 450 211,77 470 193,64 490 178,19 510 166,76 517 162,41 I [14] [[eplison].sub.coh] (K) (J/[cm.sup.3]) 313,1 555,15 332,6 486,41 352,6 437,85 372,7 401,57 392,8 373,35 412,9 350,67 432,6 331,94 452,3 316,13 472,2 302,52 290,60 280,01 270,46 267,31

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Title Annotation: | Research Article |
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Author: | B. Rogankov, Vitaly; I. Levchenko, Valeriy |

Publication: | Journal of Thermodynamics |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jan 1, 2014 |

Words: | 10860 |

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