Experimental values of molecular volume,
Abstract
The determination of the physicochemical properties of ionic liquids (ILs), such as density and refractive index, is essential for the design of processes that involve ILs. Density has been widely studied in ILs because of its importance whereas refractive index has received less attention even though its determination is rapid, highly accurate and needs a small amount of sample in most techniques. Due to the large number of possible cation and anion combinations, it is not practical to use trial and error methods to find a suitable ionic liquid for a given function. It would be preferable to predict physical properties of ILs from their structure. We compile in this work different methods to predict density and refractive index of ILs from literature. Especially, we describe the method developed by the authors in a previous work for predicting density of ILs through their molecular volume. We also correlate our experimental measurements of density and refractive index of ILs in order to predict one of the parameters knowing the other one as a function of temperature. As the measurement of refractive index is very fast and needs only a drop of the ionic liquid, this is also a very useful approach.
Keywords
- ionic liquids
- density
- refractive index
- ionic volume
- molecular volume
- prediction
- correlation
1. Introduction
Ionic liquids (ILs) are organic salts with low melting point so most of them are liquid at room temperature. During the last years, scientists and engineers have shown a huge interest for ILs in research and industrial fields because of their capacity for being chemical and biochemical reaction media. ILs are also of interest because they can be considered a new group of polar and nonaqueous solvents. Their most important advantage as solvents is that they possess negligible vapor pressure [1]. For that, they are well known as “green solvents” compared to conventional volatile organic compounds (VOCs). Other relevant properties of the ILs are that they are highly stable from the chemical and thermal points of view. However, ILs are mainly valued because of the possibility of modulating their physical and chemical properties, such as melting point, viscosity, density, hydrophobicity and polarity by selecting the appropriate anion and cation and, in this way, they can be used for a specific application. For this reason, some authors have called them “designer solvents.” Hence the number of different combinations of anions and cations that can be chosen to form potential ILs is enormous [2]. Some years ago, Álvarez-Guerra and Irabien [3] claimed that “more than 106 different ILs may be synthesized, with 1012 binary combinations and 1018 ternary systems possible,” while the number of traditional solvents widely used in industry is around a few hundred. The excellent properties of the ILs permit their application in many different fields such as synthesis, catalysis, electrochemistry, separation technology, analytical chemistry and nanotechnology [4].
In order to design processes involving ILs, it is essential to determine and understand some of their physicochemical properties, such as density and refractive index [5]. On the one hand, density of ILs is related to the mechanics and engineering components of a process and is usually used to determine parameters like rates of liquid-liquid phase separation, mass transfer, power requirements of mixing and pumping and equipment sizing [6]. In fact, density of ILs is necessary to solve material or energy balance equations of chemical processes in industry [7]. On the other hand, refractive index of ILs is more linked to specific chemical properties like the structuredness, polarity and relative hydrogen bonding donating and accepting ability which help to determine solubilities, partition constants and reaction rates [8]. In general, refractive index of a compound is especially used to verify a material and check its purity, or to determine the concentration of a mixture. It is also related to the forces between molecules or their behavior in solution [9] and can be easily correlated with certain properties of the material such as the dielectric constant, density and surface tension by means of thermodynamic equations [10].
We compile in this chapter different methods to predict density and refractive index of ILs from the literature. Especially, we describe the method developed by the authors in a previous work for predicting density of ionic liquids through their molecular volume. We also correlate our experimental measurements of density and refractive index for a set of ionic liquids in order to predict one of the parameters knowing the other one as a function of temperature [6].
2. Prediction methods of density of ionic liquids
Density (
As it has been mentioned before, nowadays the possible number of combinations of cation/anion to form ILs is huge. For this reason, it is almost impossible to measure experimentally the density of all the feasible ILs. In addition, it is not worthwhile to use trial and error methods to find the suitable IL for a specific application. Therefore, developing reliable predictive methods and correlations to estimate the density of ILs in a wide range of temperature is essential. Furthermore, this kind of method allows a better understanding of the influence of the structure of the ILs on the density and on other physicochemical properties [7].
During the last years, authors have developed different methods to estimate density of ILs. We have compiled in this chapter published studies concerning methods to predict or estimate density of ILs according to the classification established by Paduszyński and Domańska [7]. Briefly, some authors have used methods based on quantitative structure-property relationships (QSPRs) [11–15] and on artificial neural networks (ANNs) [16–18]. Other authors have developed estimation methods for density of ILs by adopting equation of state (EoS) [19–29]. There are also some studies in which the estimation of density of ILs is carried out by group contribution methods (GCMs) [6, 7, 10, 30–37] and correlations between density and other properties such as refractive index, molar refraction or surface tension [6, 38–42]. In order to improve the predicting capabilities, it is usual to find in the literature a combination of methods included in different categories of this classification.
2.1. Estimation by quantitative structure-property relationships (QSPRs)
A QSPR model is a mathematical model that links the structure-derived features of a chemical compound to a physicochemical property. They are based on quantum chemistry calculations. This is their great advantage and, at the same time, their main drawback. While virtually any imagined compound can be studied with no previous experimental knowledge, usually the calculations are not easy and only can be developed by very specialized research groups.
In the literature, we can find several QSPR models to predict density of ILs. Trohalaki et al. [13] developed a QSPR model by the use of CODESSA software. They use three types of descriptors (electrostatic, quantum mechanical and thermodynamic) in order to predict the density of triazolium-based ILs. Palomar et al. [15] determined the density of 40 imidazolium-based ILs using COSMO-RS. In this model, thermodynamic data are obtained from the molecular surface polarity of the individual compounds of the mixture. A year later, they combined COSMO-RS with ANN to get a computational approach with a new descriptor which was useful to simulate the density of 45 imidazolium-based ILs [14]. Interestingly, this approach allows them to propose a design strategy which introduces the desired IL properties as input into inverse neural networks to obtain a selection of counterions. Lazzús [18] used a QSPR model with 11 descriptors based on semiempirical calculations to estimate the density of ILs as a function of temperature and pressure. Specifically, the range of temperature and pressure was 258–393 K and 0.09–207 MPa, respectively. Finally, El-Harbawi et al. [11] proposed a new QSPR model using MATLABTM software for the development of the algorithm and the same molecular descriptors used by Shen et al. [24]. The code was written based on a combination of multiple linear regression and polynomial equation.
2.2. Estimation by artificial neural networks (ANNs)
ANN is an especially efficient computer algorithm to approximate any function with a finite number of discontinuities by learning the relationships between input and output vectors [43]. ANNs are usually suitable to model chemical properties whose behavior is highly nonlinear because nonlinear relationships are well described with ANN. The predicting capabilities of this method depend on the quality of the algorithm for learning and very importantly on the quality, quantity and nature of experimental (or calculated) data used for the learning process. Some authors have developed approaches to estimate density of ILs based on ANN combined with a GCM.
Properties of chemical compounds like boiling point, critical temperature, critical pressure, vapor pressure, heat capacity, enthalpy of sublimation, heat of vaporization, density, refractive index, surface tension, viscosity, thermal conductivity, and acentric factor have been analyzed by ANN in the literature [44]. For instance, Valderrama et al. [17] proposed a combined GCM + ANN method which divides the molecule in defined groups but instead of determining the value of the group contributions, an ANN is used to get the relationship between the dependent and independent variables. In this way, the relationship between density and molecular structure was determined. Lazzús [16] also proposed in a first paper another combined (GCM + ANN) method, which uses a feedforward backpropagation neural network. It is shown that this network is very efficient in representing nonlinear relationships among properties. Specifically, the network, which was programmed with the software MATLAB, consists of a multilayer network, in which the flow of information spreads forward through the layers while the propagation of the error is back. In this way, he predicted the density of 72 ILs as a function of temperature and pressure. Later, Lazzús [18] published the estimation results obtained with another similar model replacing standard backpropagation with particle swarm optimization (PSO) because some authors had shown that PSO-based ANN led to a better training performance and predicting capacity and a faster convergence rate than the standard backpropagation algorithm. Briefly, PSO is a population-based optimization tool, where the system is initialized with a population of random particles and the algorithm searches for optima by updating generations [45]. In a PSO system, each particle is “flown” through the multidimensional search space, adjusting its position in search space according to its own experience and that of neighboring particles. The particle therefore makes use of the best position encountered by itself and that of its neighbors to position itself toward an optimal solution. The performance of each particle is evaluated using a predefined fitness function, which encapsulates the characteristics of the optimization problem [46].
2.3. Estimation by equation of state (EoS)
Equations of state are well known for determining the relationship between pressure, temperature, volume and composition of components providing a theoretical way to calculate some physical properties such as density. Therefore, some equations of state have been used to estimate density of ILs in the literature. Usually the proposed EoS contains some constants whose values have been obtained by previous fitting of the available experimental results. The main advantage is the simplicity in the use of these methods but a possible drawback is that to obtain the density of the desired compound usually some previous experimental (or calculated) data of certain critical properties (temperature, pressure, volume, …) are needed.
Goharshadi and Moosavi [19] proposed a simple EoS (called GMA EoS) to predict the density of phosphonium and imidazolium-based ILs at different temperatures and pressures. Valderrama and Zarricueta [21] used ten expressions based on the corresponding state principle to estimate the density of ILs and analyzed one of them in depth to obtain a new generalized, accurate and simple model using only the critical properties and the molecular weight of the ionic liquid. As several authors have claimed that the dependence between density and temperature is linear, the proposal of Valderrama and Zarricueta was to linearize the Valderrama and Abu-Sharkh model [47] to end up with a simple and totally predictive model, the linear generalized model, to estimate the density of ILs with acceptable accuracy.
Ji and Adidharma [26] modeled the density of three families of ILs ([Cnmim][Tf2N], [Cnmim][BF4] and [Cnmim][PF6]) in a wide temperature and pressure range by the usage of a model published in previous works [48–50]. Specifically, this model uses a heterosegmented statistical associating fluid (hetero-SAFT) equation of state which can predict the properties of an ionic liquid based on the information of its alkyl substituent, cation head and anion. However, this model does not taken into account the electrostatic interactions. On the contrary, Wang et al. [23, 27] included an electrostatic term to residual Helmholtz energy expressed by mean spherical approximation (MSA) and they improved the results.
Abildskow et al. [22] used the statistical mechanical fluctuation solution theory to develop two models which provide a direct connection between integrals of the molecular direct correlation function and isothermal derivatives of pressure and density. In this way, they can predict densities of ILs at elevated pressures.
Shen et al. [24] extended the Valderrama and Robles [51] group contribution model for the critical properties to the estimation of densities of ILs at different temperatures and pressures representing the critical properties by the modified the Lydersen-Joback-Reid group contribution method and predicting the density by the Patel-Teja (PT) equation of state.
Patel and Joshipura [25] used a simple correlation presented by Nasrifar and Moshfeghian [52] in conjunction with the predictive-Soave-Redlich-Kwong (PSRK) equation of state to estimate density of ILs.
More recently, Mahboub and Farrokhpour [28] developed a molecular modeling of ILs incorporating the perturbed thermodynamic linear Yukawa isotherm regularity (LYIR) equation of state, which is derived based on an effective nearest neighboring pair attractive interaction of the Yukawa potential. They used this model to predict the densities of ILs up to high pressures (35 MPa) and in the temperature range 293.15–393.15 K. To use the LYIR for ILs, a simple molecular model was proposed to describe their molecular structure, in which they were considered as a liquid consisting of the ion pairs moving together in the fluid, and each ion pair was assumed to be a one-center spherical united atom. Farzi and Esmaeilzadeh [29] used the Esmaeilzadeh-Roshanfekr EoS obtained in a previous paper [53] to predict density of ILs. This EoS is based on the Patel-Teja EoS and Peng-Robinson EoS and offer better results than these two equations especially near the critical areas.
2.4. Estimation by group contribution methods (GCMs)
Other authors have preferred to develop GCMs to estimate the density of ILs. First of all, we will revise briefly the published studies in the literature and after that we will detail our own predictive method based on a GCM. In these methods, the desired property is obtained as individual contribution of each of the components of the final compound. Then, predictions are obtained using available simple equations. The main drawback of these methods is that their predictive capability depends on previous experimental (or calculated) data. With some exceptions, it is impossible to predict density if no data are available for both of the components of an ionic liquid. On the other hand, usually good predictions can be obtained very easily for any combination of known counterions in a few minutes.
Slattery et al. [32] supported that a strong relationship exists between molecular volumes and density. Thus they described this relationship and predict the density of ILs only from their molecular volumes and an anion-dependent correlation. They considered the molecular volume as the sum of the individual contributions of the anion and the cation. Ye and Shreeve [36] widened a predictive method proposed by Jenkins et al. [54] for estimation of thermochemical radius and closed packed volume of single ions and used it as GCM for ILs. The method provided good density prediction results but, unfortunately, is limited to room temperature and ambient pressure. To solve this disadvantage, Gardas and Coutinho [31] extended this method estimating three coefficients independent of the ionic liquid which take into account the influence of temperature and pressure on the molecular volume. In this way, they can predict the density of ILs in a wide range of temperature (273.15–393.15 K) and pressure (0.10–100 MPa). Jacquemin et al. [34, 35] studied the suitability of GCMs to predict density of ILs. The study was based on an assumption proposed by Rebelo et al. [55, 56] that the molar volume of the ILs can be obtained from the effective molar volumes of the ions considering the ionic liquid as an “ideal” mixture. Qiao et al. [30] introduced the interaction between several substitutes on the same center in the partition of groups. For that, the same group structure attached to different substitutes may have different group values. Paduszyński and Domańnska [7] proposed a simple and generalized correlation to estimate the density of ILs as a function of temperature and pressure. They divided the ILs in 177 functional groups which are classified in three subgroups: cation cores, anion cores and substituents. In this way, they obtained the group contributions to molar volume for each of the functional groups and the universal coefficients describing the relationship pressure-density-temperature. Lately, Keshavarz et al. [37] provided a simple correlation to predict density of ILs based on their size, structure and types of cations and anions. They introduced two correcting terms which take into account the effect that some specific cations and anion may have in the ionic packing leading to its increase or decrease and hence affect to the density values. Finally, Kermanioryani et al. [33] found new group contribution parameters using the Gardas and Coutinho model [31] to predict the density of ILs more accurately.
Recently, we have proposed our own method to predict density of ILs [6]. It can be considered as a GCM and it is based on the prediction of the molecular volume of ILs from the ionic volume. It will be described in detail in the following paragraphs.
293.15 | 0.2561 | 5.93 | 0.3119 | 5.96 | 0.3676 | 6.06 |
303.15 | 0.2577 | 0.3134 | 0.3698 | |||
313.15 | 0.2592 | 0.3152 | 0.3721 | |||
323.15 | 0.2607 | 0.3171 | 0.3744 | |||
333.15 | 0.2623 | 0.3190 | 0.3766 | |||
343.15 | 0.2638 | 0.3209 | 0.3789 | |||
293.15 | 0.4234 | 6.14 | 0.3112 | 6.08 | 0.3691 | 6.19 |
303.15 | 0.4260 | 0.3131 | 0.3714 | |||
313.15 | 0.4286 | 0.3150 | 0.3737 | |||
323.15 | 0.4313 | 0.3169 | 0.3760 | |||
333.15 | 0.4339 | 0.3189 | 0.3784 | |||
343.15 | 0.4366 | 0.3208 | 0.3808 | |||
293.15 | 0.5401 | 6.09 | 0.3442 | 6.13 | 0.3996 | 6.14 |
303.15 | 0.5434 | 0.3463 | 0.4021 | |||
313.15 | 0.5467 | 0.3485 | 0.4045 | |||
323.15 | 0.5500 | 0.3506 | 0.4070 | |||
333.15 | 0.5534 | 0.3528 | 0.4096 | |||
343.15 | 0.5568 | 0.3549 | 0.4121 | |||
293.15 | 0.4556 | 6.18 | 0.4264 | 6.65 | 0.4834 | 6.72 |
303.15 | 0.4584 | 0.4292 | 0.4866 | |||
313.15 | 0.4612 | 0.4321 | 0.4903 | |||
323.15 | 0.4641 | 0.4349 | 0.4933 | |||
333.15 | 0.4670 | 0.4378 | 0.4966 | |||
343.15 | 0.4699 | 0.4408 | 0.5003 | |||
293.15 | 0.5397 | 6.71 | 0.5959 | 6.78 | 0.3160 | 5.47 |
303.15 | 0.5434 | 0.5999 | 0.3177 | |||
313.15 | 0.5470 | 0.6040 | 0.3195 | |||
323.15 | 0.5507 | 0.6081 | 0.3212 | |||
333.15 | 0.5544 | 0.6122 | 0.3230 | |||
343.15 | 0.5581 | 0.6164 | 0.3248 |
It is known that volume is more informative about structure and packing efficiency than density, and the conclusions and predictions obtained for volume can be immediately translated to density values. The molecular volume
If we define the molecular volume (
In Eq. (2), where
Regarding the cations, our data (see Figure 1 and Table 1) reveal a linear increase of
IL | ||||
---|---|---|---|---|
[emim+][NTf2−] | 391.31 | 280.133 | 0.270 | 1035.995 |
[bmim+][NTf2−] | 419.37 | 0.287 | 976.073 | |
[hmim+][NTf2−] | 447.43 | 0.298 | 940.991 | |
[omim+][NTf2−] | 475.47 | 0.308 | 909.818 | |
[bmim+][PF6−] | 284.18 | 144.961 | 0.148 | 978.144 |
[hmim+][PF6−] | 312.24 | 0.158 | 919.803 | |
[omim+][PF6−] | 340.29 | 0.168 | 864.922 | |
[emim+][TfO−] | 260.23 | 149.062 | 0.155 | 960.451 |
[bmim+][TfO−] | 288.29 | 0.173 | 861.132 | |
[emim+][BF4−] | 197.97 | 86.802 | 0.100 | 867.153 |
[bmim+][BF4−] | 226.02 | 0.116 | 748.939 | |
[hmim+][BF4−] | 254.08 | 0.126 | 691.099 | |
[omim+][BF4−] | 282.13 | 0.135 | 641.078 | |
[emim+][EtSO4−] | 236.29 | 125.126 | 0.160 | 782.037 |
[bmim+][OcSO4−] | 348.50 | 209.280 | 0.344 | 608.372 |
We can study the effect of the volume of the ion in the global
We now propose an alternative way to assign the ionic volume, and so to explain the experimental results of density. Volume of a given chemical structure can be theoretically calculated. We used the web page chemicalize.org [60] to obtain the van der Waals volume of the ions studied. This volume is not comparable with the ionic volume, because it only takes into account the isolated molecular structure and it is independent of the temperature. We propose below a way to assign the ionic volumes from the theoretically calculated volumes of the components and the experimental measurements of density, which takes into account the effect of temperature.
Cation | Structural volume/nm3 (theoretical) | Ionic volume/nm3 (this work) | Ionic volume/nm3 (Slattery et al. [32]) |
---|---|---|---|
[emim+] | 0.11553 | 0.19142 | 0.156 |
[bmim+] | 0.14945 | 0.24762 | 0.196 |
[hmim+] | 0.18341 | 0.30388 | 0.242 |
[omim+] | 0.21735 | 0.36012 | 0.288 |
From the values of the theoretical van der Waals volume obtained for the set of alkyl methylimidazolium cations used in this work (see Table 3), it is straightforward to assign a molecular volume to one methylene group (–CH2–),
For comparison purposes, Table 3 includes ionic volume for the cations used in this work proposed by Slattery et al. [32]. Their values are systematically lower than ours. In addition they propose a
Using Eq. (1), we can obtain the ionic volume occupied by the anions at 293.15 K (see Table 4).
Anion | Structural volume/nm3 (theoretical) | Ionic volume/nm3 (this work) | % increment |
---|---|---|---|
[BF4−] | 0.05420 | 0.0634 ± 0.0008 | 17 |
[PF6−] | 0.07135 | 0.0952 ± 0.0007 | 33 |
[TfO−] | 0.08520 | 0.1201 ± 0.0010 | 41 |
[EtSO4−] | 0.09639 | 0.1242 | 29 |
[NTf2−] | 0.15671 | 0.2349 ± 0.0002 | 50 |
[OcSO4−] | 0.19820 | 0.2919 | 47 |
We find various advantages in this procedure. First, we are able to obtain a plausible ionic volume for the anions participating in the ionic liquid studied. Our results show that the ionic volume of the anion is not affected significantly by the length of the alkyl chain of the cation.
The second consequence of our approximation is that there is a very good correlation between the molecular volume with both, volume of the cation and also with the volume of the anion.
Indeed, in Figure 2, we observe that volume of the anion is the fundamental factor to determine the molecular volume (the higher the volume of the anion, the higher
Using our treatment, we can obtain the increment of anionic volume with respect to the theoretical volume of the molecular structure (Table 4). This increment takes into account interionic distances and effect of temperature and could be taken as a measure of the packing quality. If we assume this idea, we conclude that packing quality is higher for small molecules, but when the size of the anion becomes similar to the cation, the packing quality decreases, reaches a certain level and probably other factors than volume become significant.
Finally, this treatment allows us to predict with high accuracy the densities of all the combinations at all temperatures of these anions and cations by obtaining molecular volumes of ILs after assigning proper ionic volumes to its constituents. Indeed, following the method we have used for
Ionic volume/nm3 | S.D. | % increase | |
---|---|---|---|
293.15 | 0.0281 | 0.0004 | 65.7 |
303.15 | 0.0283 | 0.0004 | 66.8 |
313.15 | 0.0285 | 0.0005 | 67.8 |
323.15 | 0.0287 | 0.0004 | 69.1 |
333.15 | 0.0289 | 0.0005 | 70.1 |
343.15 | 0.0291 | 0.0005 | 71.3 |
Knowing this quantitative dependence the ionic volume of alkyl imidazolium cations with different alkyl chain lengths can be properly assigned for any temperature. We propose the following equation:
Or in terms of number of carbons
In Eq. (3),
Regarding the dependence of the ionic volumes of the anions studied in this work on temperature, in all cases we find a very good linear dependence, with a tendency to higher slopes for higher ionic volumes (see Figure 4 and Table 6 for numerical values). From our results, the ionic volume of these ions can be predicted for any temperature:
In Eqs. (5–10),
To check our procedure we have compared the calculated densities from Eqs. (1–10) with our experimental values and some others found in the literature [34, 39, 40, 61–68]. As an illustration, some results are plotted in Figure 5 (see also Tables 7 and 8 for numerical values).
[emim+] | [bmim+] | [hmim+] | [omim+] | [BF4−] | [NTf2−] | [TFO−] | [EtSO4−] | [PF6−] | [OctSO4−] | |
---|---|---|---|---|---|---|---|---|---|---|
293.15 | 0.1914 | 0.2476 | 0.3039 | 0.3601 | 0.0640 | 0.2356 | 0.1206 | 0.1246 | 0.0959 | 0.2925 |
303.15 | 0.1927 | 0.2492 | 0.3059 | 0.3625 | 0.0642 | 0.2372 | 0.1213 | 0.1250 | 0.0964 | 0.2942 |
313.15 | 0.1939 | 0.2508 | 0.3078 | 0.3648 | 0.0644 | 0.2390 | 0.1220 | 0.1256 | 0.0969 | 0.2959 |
323.15 | 0.1953 | 0.2527 | 0.3101 | 0.3674 | 0.0645 | 0.2404 | 0.1225 | 0.1259 | 0.0972 | 0.2973 |
333.15 | 0.1966 | 0.2543 | 0.3121 | 0.3698 | 0.0648 | 0.2421 | 0.1232 | 0.1264 | 0.0978 | 0.2991 |
343.15 | 0.1980 | 0.2506 | 0.3143 | 0.3724 | 0.0649 | 0.2437 | 0.1238 | 0.1268 | 0.0980 | 0.3007 |
|
0.1309 | 0.1693 | 0.2078 | 0.2462 | 0.0198 | 0.1617 | 0.0629 | 0.0451 | 0.0425 | 0.1644 |
|
0.1530 | 0.1979 | 0.2829 | 0.2878 | 0.0587 | 0.1882 | 0.1022 | 0.1114 | 0.0835 | 0.2443 |
0.9996 | 0.9996 | 0.9996 | 0.9996 | 0.9828 | 0.9994 | 0.9979 | 0.9962 | 0.9924 | 0.9997 |
Experimental (different sources) | Exp. Average ± σ | Error* | Error+ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
[NTf2−] | |||||||||||
[emim+] | 1.52381 | 1.51972 | 1.52293 | 1.52343 | 1.52604 | 1.5232 | 0.0023 | 1.5215 | −0.0016 | −0.11 | |
[bmim+] | 1.44031 | 1.44082 | 1.44273 | 1.44023 | 1.43895 | 1.44055 | 1.4406 | 0.0012 | 1.4408 | 0.0003 | 0.02 |
[hmim+] | 1.37631 | 1.37552 | 1.37513 | 1.37546 | 1.37506 | 1.3755 | 0.0005 | 1.3770 | 0.0015 | 0.11 | |
[omim+] | 1.32481 | 1.32342 | 1.32453 | 1.32814 | 1.3252 | 0.0020 | 1.3251 | −0.0001 | −0.01 | ||
[c10mim+] | 1.28282 | 1.28243 | 1.2826 | 0.0003 | 1.2821 | −0.0005 | −0.04 | ||||
[c12mim+] | 1.24902 | 1.2490 | 1.2459 | −0.0031 | −0.25 | ||||||
[PF6−] | |||||||||||
[bmim+] | 1.37071 | 1.36792 | 1.36983 | 1.36813 | 1.37045 | 1.37167,8 | 1.3698 | 0.0015 | 1.3734 | 0.0036 | 0.26 |
[hmim+] | 1.29731 | 1.29642 | 1.29797,8 | 1.2972 | 0.0008 | 1.2964 | −0.0008 | −0.06 | |||
[omim+] | 1.24021 | 1.23782 | 1.23967,8 | 1.2392 | 0.0012 | 1.2385 | −0.0007 | −0.06 | |||
[BF4−] | |||||||||||
[emim+] | 1.28331 | 1.2833 | 1.2869 | 0.0036 | 0.28 | ||||||
[bmim+] | 1.20311 | 1.20292 | 1.20493,9 | 1.20383 | 1.2037 | 0.0009 | 1.2041 | 0.0004 | 0.04 | ||
[hmim+] | 1.14751 | 1.1475 | 1.1467 | −0.0008 | −0.07 | ||||||
[omim+] | 1.10641 | 1.1064 | 1.1044 | −0.0020 | −0.18 | ||||||
[TfO−] | |||||||||||
[emim+] | 1.38841 | 1.3884 | 1.3848 | −0.0036 | −0.26 | ||||||
[bmim+] | 1.2966x | 1.30352 | 1.30133 | 1.290010 | 1.2979 | 0.0060 | 1.2998 | 0.0019 | 0.15 |
Experimental (different sources) | Exp. Average ± σ | Error* | Error+ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
[NTf2−] | ||||||||||||
[emim+] | 1.50371 | 1.49962 | 1.50293 | 1.50343 | 1.50384 | 1.5027 | 0.0018 | 1.5005 | −0.0022 | −0.15 | ||
[bmim+] | 1.42021 | 1.42162 | 1.42306 | 1.41985 | 1.42145 | 1.42353 | 1.42123 | 1.4215 | 0.0015 | 1.4210 | −0.0005 | −0.04 |
[hmim+] | 1.35801 | 1.35722 | 1.35693 | 1.35706 | 1.3573 | 0.0005 | 1.3581 | 0.0008 | 0.06 | |||
[omim+] | 1.30701 | 1.30552 | 1.30703 | 1.30564 | 1.3063 | 0.0008 | 1.3070 | 0.0007 | 0.05 | |||
[c10mim+] | 1.26532 | 1.26533 | 1.2653 | 0.0000 | 1.2646 | −0.0007 | −0.06 | |||||
[c12mim+] | 1.23182 | 1.2318 | 1.2289 | −0.0029 | −0.23 | |||||||
[c14mim+] | 1.20062 | 1.2006 | 1.1985 | −0.0021 | −0.17 | |||||||
[PF6−] | ||||||||||||
[bmim+] | 1.35401 | 1.35132 | 1.35333 | 1.35183 | 1.35345 | 1.35287 | 1.3528 | 0.0010 | 1.3562 | 0.0034 | 0.25 | |
[hmim+] | 1.28151 | 1.28042 | 1.2810 | 0.0008 | 1.2801 | −0.0009 | −0.07 | |||||
[omim+] | 1.22501 | 1.22242 | 1.2237 | 0.0018 | 1.2227 | −0.0010 | −0.08 | |||||
[BF4−] | ||||||||||||
[emim+] | 1.26811 | 1.2681 | 1.2721 | 0.0040 | 0.31 | |||||||
[bmim+] | 1.19051 | 1.18872 | 1.19013 | 1.18893 | 1.19086 | 1.19078 | 1.1900 | 0.0009 | 1.1898 | −0.0002 | −0.01 | |
[hmim+] | 1.13371 | 1.1337 | 1.1328 | −0.0009 | −0.08 | |||||||
[omim+] | 1.09291 | 1.0929 | 1.0908 | −0.0021 | −0.19 | |||||||
[TfO−] | ||||||||||||
[emim+] | 1.37161 | 1.3716 | 1.3673 | −0.0043 | −0.31 | |||||||
[bmim+] | 1.28081 | 1.28772 | 1.28563 | 1.29247 | 1.2866 | 0.0048 | 1.2832 | −0.0034 | −0.26 |
Tariq et al. [39] showed the experimental density of some ILs composed with combinations of cations and anions not used for our research group. A further test for our method is to predict volumes of these ions. For
To have an idea of the quality of the predictions we can use the average deviation from experimental values. Then, we can compare our values (see Tables 7 and 8) with those obtained by Tariq et al. [39]. Those average deviations are between σave = 0.21 and σave = 1.29. Our method gives σave = 0.13.
As a summary, with our results we should be able to predict, prior to synthesis, the density and molecular volumes for IL with alkyl imidazolium cations and with different anions (and very likely combinations of ions of similar chemical structure) in a wide range of temperatures from easily accessible theoretical calculations of the structure of the proposed ions.
2.5. Estimation by correlations between density and other properties
The last approach to estimate density is through correlations between density and other properties. Obviously, no prediction can be obtained if these correlations have not been previously established. But these methods can be especially interesting if the measurement of the property used is easier or cheaper than the measurement of the density. A good example is the refractive index.
There are some authors who have studied the relationship between density and refractive index in order to predict one of the properties from the other one due to the experimental measurement of refractive index only needs a drop of the ionic liquid and it is very fast. However, in the case of density the determination requires higher volumes of ionic liquid and it takes more time. Tariq et al. [39] and Soriano et al. [38, 40] used different empirical models (Lorentz-Lorenz, Dale-Gladstone, Eykman, Oster, Arago-Biot and Newton equations) to predict density of ILs from their refractive index. Other authors such as Deetlefs et al. [41] and Gardas et al. [42] used the parachor to estimate density. Parachor is a surface-tension-weighted molar volume, which constitutes a link between the structure, density and surface tension of the ILs. Deetlefs et al. [41] also used the refraction molar to estimate density.
We detail in the following paragraphs our prediction of the density from the refractive index.
It is known that the density and refractive index (
where
System | L-L | D-G | Eykman | Oster | A-B | Newton |
---|---|---|---|---|---|---|
[emim+][BF4−] | 0.1948 | 0.3220 | 0.4289 | 1.9418 | 1.1130 | 0.7752 |
[bmim+][BF4−] | 0.2125 | 0.3524 | 0.4687 | 2.1276 | 1.1966 | 0.8518 |
[hmim+][BF4−] | 0.2251 | 0.3738 | 0.4969 | 2.2585 | 1.2586 | 0.9055 |
[omim+][BF4−] | 0.2359 | 0.3923 | 0.5212 | 2.3719 | 1.3102 | 0.9523 |
[emim+][TfO−] | 0.1881 | 0.3128 | 0.4155 | 1.8914 | 1.0442 | 0.7594 |
[bmim+][TfO−] | 0.2027 | 0.3375 | 0.4481 | 2.0415 | 1.1208 | 0.8204 |
[bmim+][OcSO4−] | 0.2620 | 0.4407 | 0.5825 | 2.6791 | 1.3885 | 1.0865 |
[bmim+][PF6−] | 0.1813 | 0.2996 | 0.3992 | 1.8059 | 1.0404 | 0.7203 |
[hmim+][PF6−] | 0.1947 | 0.3224 | 0.4291 | 1.9450 | 1.1052 | 0.7775 |
[omim+][PF6−] | 0.2061 | 0.3418 | 0.4546 | 2.0638 | 1.1607 | 0.8263 |
[emim+][NTf2−] | 0.1679 | 0.2784 | 0.3703 | 1.6809 | 0.9457 | 0.6729 |
[bmim+][NTf2−] | 0.1789 | 0.2969 | 0.3947 | 1.7933 | 1.0031 | 0.7186 |
[hmim+][NTf2−] | 0.1888 | 0.3137 | 0.4168 | 1.8959 | 1.0526 | 0.7605 |
[omim+][NTf2−] | 0.1969 | 0.3275 | 0.4350 | 1.9797 | 1.0952 | 0.7946 |
[emim+][EtSO4−] | 0.2286 | 0.3853 | 0.5088 | 2.3446 | 1.2020 | 0.9525 |
0.0032 | 0.0020 | 0.0025 | 0.0016 | 0.0099 | 0.0018 |
All these equations assume that the relation between density and refractive index is temperature-independent. But, from our results (see Tables 1 and 9) it can be deduced that
Nevertheless, as a first approximation, and assuming that the dependence is small, it is worth finding the equation that best fits our experimental results. In this approximation we shall assume that
As mentioned above, a small dependence of
ILs | S.D. | |||
---|---|---|---|---|
[emim+][BF4−] | 4.709 | 0.9632 | 0.9882 | 0.0009 |
[bmim+][BF4−] | 5.632 | 1.0174 | 0.9992 | 0.0003 |
[hmim+][BF4−] | 5.829 | 1.0731 | 0.9994 | 0.0002 |
[omim+][BF4−] | 6.139 | 1.1149 | 0.9999 | 0.0001 |
[emim+][TfO−] | 4.712 | 0.8942 | 0.9996 | 0.0002 |
[bmim+][TfO−] | 5.140 | 0.9572 | 0.9971 | 0.0005 |
[bmim+][OcSO4−] | 6.189 | 1.1916 | 0.9991 | 0.0003 |
[bmim+][PF6−] | 4.717 | 0.8904 | 0.9998 | 0.0001 |
[hmim+][PF6−] | 4.888 | 0.9496 | 0.9997 | 0.0001 |
[omim+][PF6−] | 5.221 | 0.9946 | 0.9997 | 0.0001 |
[emim+][NTf2−] | 4.622 | 0.7986 | 1.0000 | 0.00003 |
[bmim+][NTf2−] | 4.695 | 0.8538 | 0.9981 | 0.0003 |
[hmim+][NTf2−] | 5.123 | 0.8896 | 0.9999 | 0.0001 |
[omim+][NTf2−] | 5.351 | 0.9250 | 0.9997 | 0.0002 |
[emim+][EtSO4−] | 4.831 | 1.0482 | 0.9952 | 0.0006 |
0.0003 |
As a consequence, we think that a very good and simple description of the correlation between density and the refraction index can be obtained by including the dependence on
3. Prediction methods of refractive index of ionic liquids
The refractive index (
Some of the studies related to the estimation of refractive index are listed here. The first prediction model in the literature was developed by Deetlefs et al. [41]. They calculated the molar refraction to predict the refraction indices of ILs from their surface tension. However, all the parameters involved in this model should be experimentally measured or correlated with other experimental properties; thus, it is not suitable to predict refractive indices of new ILs. In addition, it was developed only from 9 ILs so it is far away of being a universal method. The next model was published by Gardas and Coutinho [10] who developed a GCM to predict refractive index of ILs from 245 experimental values of refractive index of ILs in the temperature range from 283.15 to 363.15 K. All the ILs used to model the refractive index were based on imidazolium salts (with different anions) thus this model is not appropriate to estimate the refractive index of ILs with another cation. The same method was used by Freire et al. [70] and Soriano et al. [38] who broadened the method by incorporating some new ILs. Xu et al. [71] synthesized a new ionic liquid ([C3mim][Val]) and measured its refractive index in the temperature range of 298.15–333.15 K. After that, they estimated the refractive index for the homologue ILs ([C
4. Conclusions
Many efforts have been done during the last years to develop a great variety of prediction methods of density of ILs. Thus they have reached a high degree of perfection. Prediction methods of density of ILs can be classified in five categories although the methods are sometimes a combination of more than one category: i) QSPR, ii) ANN, iii) EoS, iv) GCM, and v) correlations between density and other properties.
QSPR models are based on quantum chemistry calculations and this is their great advantage and, at the same time, their main drawback. While virtually any imagined compound can be studied with no previous experimental knowledge, usually the calculations are not easy and only can be developed by very specialized research groups.
ANN is an especially efficient computer algorithm whose main advantage is that it is usually suitable to model chemical properties whose behavior is highly nonlinear because nonlinear relationships are well described with ANN. However, the predicting capabilities of this method depend on the quality of the algorithm for learning and very importantly on the quality, quantity and nature of experimental (or calculated) data used for the learning process.
Equations of state are well known for determining the relationship between pressure, temperature, volume and composition of components providing a theoretical way to calculate some physical properties such as density. The main advantage of this method is the simplicity in their use but a possible drawback is that to obtain the density of the desired compound usually some previous experimental (or calculated) data of certain critical properties (temperature, pressure, volume,…) are needed.
In the GCM methods, the desired property is obtained as individual contribution of each of the components of the final compound. Then, predictions are obtained using available simple equations. The main drawback of these methods is that their predictive capability depends on previous experimental (or calculated) data. With some exceptions, it is impossible to predict density if no data are available for both of the components of an ionic liquid. On the other hand, usually good predictions can be obtained very easily for any combination of known counterions in a few minutes.
Estimating the density of ILs by correlations between this property and other physical properties is especially interesting if the measurement of the property used is easier or cheaper than the measurement of the density like in the case of refractive index. Nevertheless, the main drawback is that no prediction can be obtained if previous correlations have not been properly established.
Our method provides a fast and simple way to predict the density of ILs with alkyl imidazolium cations and with different anions (and very likely combinations of ions of similar chemical structure) based on the molecular volume in a wide range of temperatures from simple theoretical calculations of the structure of the proposed ions.
The correlation between density and the refraction index is usually described in the literature by equations which are assumed to be temperature-independent; we have shown that a substantial improvement can be obtained if the dependence on temperature is included.
Prediction of refractive index of ILs has not been so widely studied in the literature like in the case of density because the relationships between this property and the ion constituents of the ILs are not so direct and accessible. For this reason, more studies in this field are required.
We support the idea that ILs can be designed with adjustable properties (at least density and refractive index, but very probably some others) based on the structure of the cation and anion chosen. Our method may provide valuable contributions for the design and study of present and future ILs.
Acknowledgments
This work was partially supported by FEDER/ERDF funds from the European Commission, the Spanish Ministry of Economy and Competitiveness (MINECO) (Ref. CTQ2014-57467-R) and the research support programme of the Seneca Foundation of Science and Technology of Murcia, Spain (Ref. 19499/PI/14). Mercedes G. Montalbán acknowledges support from MINECO (FPI grant, BES-2012-053267).
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