Journal of Thermodynamics

Volume 2014, Article ID 496835, 15 pages

http://dx.doi.org/10.1155/2014/496835

## Towards the Equation of State for Neutral (C_{2}H_{4}), Polar (H_{2}O), and Ionic ([bmim][BF_{4}], [bmim][PF_{6}], [pmmim][Tf_{2}N]) Liquids

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

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

Academic Editor: Pedro Jorge Martins Coelho

Copyright © 2014 Vitaly B. Rogankov and Valeriy 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.

#### Abstract

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 / 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_{4}] and [bmim][PF_{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 /.

#### 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 , undefined critical parameters , , for vapor-liquid ()-transition, split of liquid-solid (*l,s*)-boundary onto melting and freezing 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 Lennard-Jones (LJ) RPP can be applied to model the underlying structure and properties of low-molecular (C_{2}H_{4}, H_{2}O) and imidazolium-based (1-butyl-3-methylimidazolium tetrafluoroborate ([bmim][BF_{4}]), 1-butyl-3-methylimidazolium hexafluorophosphate ([bmim][PF_{6}]), 2,3-dimethyl-1-propylimidazolium bis(trifluoromethylsulfonyl)imide([pmmim][Tf_{2}N])) solvents. For any pure component FT-model is based either on the measurable coexistence-curve input data , , (if they are achievable as for C_{2}H_{4} and H_{2}O) or on the also measurable one-phase density of liquid at atmospheric pressure ( MPa,* T*) for ILs. This methodology becomes purely predictive for density in any one-phase -regions including their metastable extensions. Only the measurable isobaric heat capacity data have to be added to the set of input data for prediction of other caloric properties (isochoric heat capacity , speed of sound , and Gruneisen parameter ) at higher pressures and lower or higher temperatures where is the hypothesized normal boiling temperature . Its existence itself is a debatable question because the thermal decomposition may be former .

Such approach was proposed recently [7, 8] to reconstruct the hypothetical -diagram of any ILs in its stable and metastable regions on the base of only standard reference data on density at [1–4] and one free parameter, an a priori unknown value of the excluded volume . 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 , , by the following equations:
where is the excluded molecular volume and is the -dependent effective cohesive energy. The derivative affects the thermal expansion and the thermal-pressure coefficient while the isothermal compressibility depends only on -value at the given pressure. The changeable sign of two thermal derivatives , 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 - and -surfaces. Another possibility is offered by comparison of the predictions obtained by FT-EOS for the critical parameters of ILs ([bmim][BF_{4}]: = 962,3 K, = 3503,9 kPa, = 438,565 kg·m^{−3} with those predicted here by the Sanchez-Lacombe EOS for lattice fluid (LF) [15]: = 885,01 K, = 2829 kPa, = 248,565 kg·m^{−3} as well as with those simulated by GEMC-methodology [6]: = 1252 K, = 390 kPa, = 181 kg·m^{−3}. It seems that the relatively close location of -parameters predicted by both EOSs is some guarantee of their reliability while and from [6] are significantly overestimated and underestimated, respectively. Interestingly, the known descriptive factor of compressibility estimated by Guggenheim [20] in the vicinity of triple point for argon as is equal to close values for FT-EOS and for LF-EOS but only to very small value for result of GEMC-simulations if the common realistic estimate (see below) = 5,350646 mol·dm^{−3} at* T* = 290 K is used. Moreover, it will be shown that the characteristic dimensional parameters , , and another compressibility factor obtained by Machida et al. [14] by the fit to ()-experimental data for [bmim][BF_{4}] and [bmim][PF_{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 ()-plane as the nonexistence of a ()-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 ()- and ()-transitions. The serious restriction of latter is however evident since any singularities of RPP imply an artificial jump of pair-distribution isotropic function at the point of cutoff radius for attractive interactions.

In this context only the shifted and smoothed at -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 at an arbitrary point in the fluid is independent of at fixed : . Another simplifying assumption is that such EOS supposes only two types of fluid structure, one of the excluded (or hard-core) volume where the singular hard-sphere branch of potential is infinite and one of free volume where the potential is uniform, weak, and unrestricted (an infinite-range rectilinear well). It should be directly proportional to density where is the total configurational energy and is the constant vdW-coefficient. These historical notes are important to explain how one can go beyond the above restriction of -independency by adoption of linear -dependence for a generalized specific or molar energy (see also (8) below). Consider

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,are determined simply in the low-temperature range of all ILs where is constant in ((1)–(4)) while the compressibility factor of saturated liquid becomes negligible as well as the vapor pressure 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 . This concept is unusual for the conventional consideration of a separate influence of the anion’s and cation’s components. It may provide, in principle, the useful insight the nature of -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 , or coulombic interactions in addition to isotropic dispersive 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 -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 is a reasonable approximation for the - and -phases but not the -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 ). In contrast with this observation, the general FT-EOS provides the adequate representation of entire subcritical range including the critical region and ()-phase transition [9–12]. It will be shown below by FT-model without undue complexity of calculations.

#### 2. Universal FT-EOS for Any Low-Temperature Fluids

##### 2.1. General Form of FT-EOS for Subcritical Temperatures

It is often claimed that the original van der Waals (vdW)-EOS with two constant coefficients , determined by the actual critical-point properties , 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 -dependent coefficients,
is applicable to any types of fluids including ILs. The measurable volumetric data of coexistence curve (CXC) have been used for evaluation of -dependences without any fit. Consider
where the reduced slope of -function is defined by the thermodynamic Clapeyron’s equation:
This fundamental ratio of the -latent heat to the thermodynamic work of -expansion is the main parameter of FT-coefficients determined by ((8)-(9)).* It should be calculated separately in each of high-temperature * [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 but it can be essentially simplified to the form of (1) if .

##### 2.2. Particular Form of FT-EOS for Low Temperatures

An absence of input CXC-data , , 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 (or melting ) point up to the -point. Former one is usually known for ILs while the latter one is, as a rule, more than temperature of thermal decomposition K. The methodology was tested on two low-molecular-weight substances (C_{2}H_{4}, H_{2}O) and two imidazolium-based ILs ([bmim][PF_{6}], [pmmim][Tf_{2}N]) with the promising accuracy of predictions even for the isothermal compressibility 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 one determines the orthobaric molar densities ; to solve the transcendent equation:
for the reduced entropy (disorder) parameter and the respective molar heat of vaporization. Consider

*Step 2. *The universal CXC-function in (12) is determined by equalities
and it provides the possibility to estimate a preliminary value of ,
as well as to evaluate the orthobaric densities at any if the function is known. Consider

*Step 3 (-variant of -prediction [7, 8]). *To calculate its values one must obtain two densities (at the assumption ) from equation
where , and the -function provides the preliminary estimate of for the low-temperature range (at the consistent assumption: ). Consider

*Step 4. *One substitutes from (18) in ((13a), (13b), (16a), (16b)) to calculate , , , respectively.

*Step 5. *A preliminary value of may be estimated then by the more restrictive assumption (used also in the famous Flory-Orwoll-Vrij EOS developed for heavy* n*-alkanes). Consider

*Step 6 (-variant of -prediction). *To control the consistency of methodology one may use instead of Step 3 (-variant) the same equation (17) with the approximate equality to solve (16b) at the a priori chosen -value for determination of alternative , and so forth, (Steps 4 and 5).* Just this approach (**-variant) has been used below in the low-temperature range of [bmim][BF*_{4}*].*

*Step 7. *The self-consistent prediction of a hypothetical -diagram requires the equilibration of CXC-pressures by FT-EOS (1) with the necessary final change in -value from (19) to satisfy the equality

Only in the low-temperature range the distinction between the preliminary definition (18) and its final form (19) for -values is not essential at the prediction of vapor pressure .

#### 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_{2}H_{4}) and polar (H_{2}O) fluids [7, 8] in Table 1 and added in Table 2 to other ILs ([bmim][PF_{6}], [pmmim][Tf_{2}N] [7, 8]) the data for [bmim][BF_{4}] obtained in this work (Table 3). When temperature is low FT-model follows a two-parameter correlation of principle of corresponding states (PCS) on molecular level as well as a two-parameter correlation of PCS on macroscopic level.