Guggenheim's Rule and the Enthalpy of Vaporization of Simple and

J. Phys. Chem. B 2010, 114, 9183–9194

9183

Guggenheim’s Rule and the Enthalpy of Vaporization of Simple and Polar Fluids, Molten Salts, and Room Temperature Ionic Liquids Volker C. Weiss* Bremen Center for Computational Materials Science, UniVersita¨t Bremen, Am Fallturm 1, 28359 Bremen, Germany ReceiVed: March 24, 2010; ReVised Manuscript ReceiVed: May 31, 2010

One of Guggenheim’s many corresponding-states rules for simple fluids implies that the molar enthalpy of vaporization (determined at the temperature at which the pressure reaches 1/50th of its critical value, which approximately coincides with the normal boiling point) divided by the critical temperature has a value of roughly 5.2R, where R is the universal gas constant. For more complex fluids, such as strongly polar and ionic fluids, one must expect deviations from Guggenheim’s rule. Such a deviation has far-reaching consequences for other empirical rules related to the vaporization of fluids, namely Guldberg’s rule and Trouton’s rule. We evaluate these characteristic quantities for simple fluids, polar fluids, hydrogen-bonding fluids, simple inorganic molten salts, and room temperature ionic liquids (RTILs). For the ionic fluids, the critical parameters are not accessible to direct experimental observation; therefore, suitable extrapolation schemes have to be applied. For the RTILs [1-n-alkyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imides, where the alkyl chain is ethyl, butyl, hexyl, or octyl], the critical temperature is estimated by extrapolating the surface tension to zero using Guggenheim’s and Eo¨tvo¨s’ rules; the critical density is obtained using the linear-diameter rule. It is shown that the RTILs adhere to Guggenheim’s master curve for the reduced surface tension of simple and moderately polar fluids, but that they deviate significantly from his rule for the reduced enthalpy of vaporization of simple fluids. Consequences for evaluating the Trouton constant of RTILs, the value of which has been discussed controversially in the literature, are indicated. 1. Introduction The principle of corresponding states has proved extremely useful and illuminating in the attempt to develop a unified description of the thermodynamic behavior of fluids.1,2 In addition to identifying the generic behavior of the comparatively simple fluids that were studied first, it allows one to classify different fluids according to the predominant type of interactions that determine their respective properties.3,4 Simple fluids are composed of almost spherical particles that interact exclusively via dispersion forces (London forces). In polar fluids, the molecules possess a permanent dipole moment, and thus, Keesom and Debye forces come into play. Due to their exceptional properties, it has turned out to be useful to divide the group of polar fluids into those that are capable of forming hydrogen bonds and those that are not (aprotic polar fluids). In ionic fluids, the Coulomb interaction usually dominates over other types of intermolecular forces.5 Guggenheim formulated his principle of corresponding states for simple fluids, such as noble cases and homonuclear diatomic species.1 Among the many thermodynamic properties he addressed were the critical compressibility factor, the enthalpy and entropy of vaporization, the normal boiling point, and the surface tension. If these quantities are properly reduced by the critical parameters, all simple fluids show the same behavior. The origin of this uniformity can be traced back to the similarity of the underlying interaction potentials, which forms the statistical-mechanical basis of the corresponding-states principle.6 * Corresponding author. Tel.: +49 421 218 7762. Fax: +49 421 218 4764. E-mail: [email protected].

In detail, Guggenheim reported the generic simple fluid to have a critical compressibility factor of Zc ) PcVc/(RTc) ) 0.29, where Tc is the critical temperature, Pc the critical pressure, Vc the critical molar volume, and R the universal gas constant. Much earlier, Guldberg had found the empirical rule that the ratio of the normal boiling temperature Tb and the critical temperature, Tb/Tc ≈ 0.6, was universal, while Trouton had noted that the molar entropy of vaporization at the normal boiling point is ∆vapS(Tb) ) 80 ( 8 J K-1 mol-1 for many fluids. Guggenheim pointed out that relating properties at atmospheric pressure did not do justice to the corresponding-states idea and suggested to compare the above properties at a pressure P that amounts to 1/50th of the critical pressure Pc of the respective fluid; the temperature at which P(T) ) Pc/50 he denoted as Ts. Since for many simple fluids Pc ≈ 5 ( 1 MPa [due to the logarithmic dependence of T on P along the liquid-vapor phase boundary (see below), a 20% difference in Pc is hardly relevant in this context], Tb and Ts have very similar values. It was, however, seen that Ts/Tc ≈ 0.58 was more nearly a constant for different fluids than Tb/Tc. Similar observations hold for ∆vapS(Ts) and ∆vapS(Tb).1 The temperature dependence of the vapor pressure is described by the Clapeyron equation in one of the following three forms:

∆vapH dP ) dT T∆vapV

10.1021/jp102653a  2010 American Chemical Society Published on Web 06/24/2010

(1)

9184

J. Phys. Chem. B, Vol. 114, No. 28, 2010

∆vapH dln P ) dT PT∆vapV

Weiss

(2)

T∆vapH(T) dln P )d(1/T) P∆vapV(T)

(3)

where ∆vapH is the molar enthalpy of vaporization and ∆vapV ) Vv - Vl is the difference of molar volumes in the vapor phase (v)andintheliquidphase(l).TheapproximativeClausius-Clapeyron equation is derived from eq 3 by assuming that ∆vapV ) Vv Vl ≈ Vv ≈ RT/P, that is, by realizing that the molar volume of the vapor phase is much larger than that of the liquid phase and by assuming the ideal gas law to hold for the vapor. The Clausius-Clapeyron equation in the form

∆vapH(T) dln P )d(1/T) R

(4)

can be used to determine ∆vapH from vapor-pressure data taken at different temperatures if this range of temperatures is sufficiently narrow, so that ∆vapH is approximately constant. The assumptions made above to obtain the Clausius-Clapeyron equation from the rigorous Clapeyron equation are valid at low temperatures, but must be expected to fail at higher temperatures, especially near the critical point. It is, however, often seen that a plot of ln P vs 1/T is linear all the way from the triple point to the critical point. Therefore, it is useful to define an apparent enthalpy of vaporization, ∆vapHapp, which is independent of temperature and whose value is a characteristic quantity for a given fluid app

∆vapH dln P ) const ) d(1/T) R

(5)

In the corresponding-states approach, the pressure and temperature are reduced by their critical values and a new universal constant emerges, namely ∆vapHapp/(RTc)

dln(P/Pc) ∆vapHapp )d(Tc /T) RTc

(6)

Since ∆vapS(Ts) ) ∆vapH(Ts)/Ts ≈ 75 J K-1 mol-1 and Ts/Tc ) 0.58 for simple fluids,1 one finds ∆vapHapp/(RTc) ≈ 5.2; this relation is known as Guggenheim’s rule for simple fluids. Comparing eqs 3 and 5, it becomes clear that

∆vapH(T) ) ∆vapHapp

P∆vapV(T) RT

∆vapH(T) ) ∆vapHapp∆Z(T)

(7)

(8)

where ∆Z ) Zv - Zl ) PVv/(RT) - PVl/(RT) ) P∆vapV/(RT). If eq 5 holds, then ∆Z(T) may be used to calculate ∆vapH(T) at different temperatures without the need to refer to heat capacities. The reduced surface tension defined by

γred ) c0Mr2/3γ/(Fc2/3Tc)

(9)

where γ denotes the temperature-dependent surface tension, c0 a constant, Mr the molar mass, and Fc the critical density, shows corresponding-states behavior as well.1 In reduced form, the relation between the reduced surface tension γred and the reduced temperature T/Tc reads γred ) γ0(1 - T/Tc)11/9, where the amplitude γ0 has a universal value of γ0 ) 4.3.1,7 In the above definition of γred, the value of the constant is c0 ) 1/(kBNA2/3) ≈ 1.016 × 10 7 J -1 K mol 2/3 if γ is measured in N m-1, Mr in g mol-1, Tc in K, and Fc in g m-3; here, kB denotes Boltzmann’s constant and NA Avogadro’s number. The above expression involving the exponent 11/9 is capable of describing the surface tension over a very wide range of temperatures, essentially all the way from the triple point to the critical region. Far away from Tc, it reproduces a nearly linear relation between γ and T, while closer to Tc, it correctly introduces a pronounced curvature into γ(T). Asymptotically, the surface tension vanishes according to γ ∼(Tc - T)µ, where the critical exponent µ has a value of 1.26,8 which is not too different from Guggenheim’s exponent of 11/9. Despite his impressive success in unifying the properties of different simple fluids, Guggenheim warns about the cases for which the principle of corresponding states cannot be expected to hold.1 Rather than regarding these deviations as a failure of his approach, he stresses that they offer additional information on the interaction among particles in the respective fluid, which must go beyond dispersion interactions. Keeping his warning in mind, one must be prepared for different patterns of behavior in the cases of polar and ionic fluids as well as for nonpolar fluids in which the particles deviate from spherical shape. The fact that polar fluids do not adhere to all the empirical rules found for simple fluids is now textbook knowledge. The corresponding-states behavior of molten salts has also been studied, but for this class of fluids, the approach outlined above faces serious challenges due to the difficulty to determine critical parameters with sufficient accuracy. This problem can be appreciated in light of the critical temperatures of alkali halides being roughly at 3000 K.9 Instead, the corresponding-states approach is either based on the charges and the diameters of the ions to set the characteristic energy and length scales of the fluid,10-12 or the melting temperature at atmospheric pressure is used as an alternative reference point.13 In recent years, there has been an increasing interest in the thermodynamic properties of room temperature ionic liquids (RTILs). RTILs have been defined as molten salts displaying normal melting points below 100 °C, while the normal melting point of alkali halides is well above 800 K.9 It can therefore be expected that the critical temperatures of RTILs are also lower (and, therefore, more readily accessible) than those of molten inorganic salts by a considerable margin. From a more practical point of view, the interest in RTILs is warranted by the fact that their physicochemical properties make them promising candidates as solvents and catalysts in new syntheses, extraction processes, and electrochemical applications.14-16 In addition, their chemical stability and their extremely low vapor pressure raise the hope that RTILs can soon replace volatile organic solvents in many processes. By combining different cations and anions and by modifying side chains on the ions, the properties of the RTIL can be adjusted to one’s needs.17 The variety of different RTILs that can be synthesized in this way, however, prohibits the experimental determination of all their relevant properties. Also for this reason, it is important to study the generic behavior of RTILs. The corre-

Guggenheim’s Rule sponding-states ideas mentioned above1 offer an efficient way of identifying general trends in the properties of a class of substances. These approaches, however, rely on the critical parameters, which are not always easy to obtain for ionic fluids. For molten inorganic salts, the critical temperature is simply too high to allow accurate measurements. RTILs, on the other hand, face the problem of decomposition at elevated temperatures, which render experiments beyond 500 K impossible.18,19 The critical properties are thus not only difficult to obtain but purely hypothetical. Nevertheless, the hypothetical critical point will still govern the behavior of the fluid at lower temperatures, which makes the effort to determine its location by extrapolation worthwhile.20 The properties on which we focus in this work are all related to the equilibrium of liquid and vapor phase: the reduced apparent enthalpy of vaporization ∆vapHapp/(RTc), the normal boiling temperature in relation to the critical temperature, Tb/Tc (Guldberg ratio), and the entropy of vaporization at the normal boiling point, ∆vapS(Tb) (Trouton constant); as byproducts in the attempt to deduce the critical parameters of the RTILs, the critical compressibility factor Zc and the reduced surface tension γred are studied. Despite their reputation of being essentially nonvolatile at ambient temperatures, RTILs have been proved to have measurable vapor pressures at elevated temperatures, that is, in the range of 400-500 K.21 The demonstration of the presence of ions in the gas phase, mostly in the form of ion pairs,22,23 opens up the possibility to distill RTILs20,24 and naturally raises the question of their normal boiling points and their enthalpies of vaporization. The latter is not only a fundamental thermodynamic property, but also offers a good target quantity for the parametrization of empirical force fields for molecular simulations.17,25 While early estimates of the enthalpy of vaporization turned out to be unreliable, different methods to determine ∆vapH are now available,21,22,26,27 and their results generally agree to within the indicated uncertainties of a few percent. In our discussion, we focus on 1-n-alkyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imides, denoted as [xmim] [NTf2], for which the largest number of experiments were carried out. In this notation scheme, “ x ” stands for a general alkyl side chain at the 1-position of the 3-methylimidazolium ion [mim], which may be ethyl (x ) e), butyl (x ) b), hexyl (x ) h), or octyl (x ) o). For these four RTILs, vapor pressures were measured using the Knudsen-effusion method,21 and ∆vapH was deduced from these data using the Clausius-Clapeyron equation (eq 4); direct measurements of ∆vapH were carried out using mass spectrometry,22 microcalorimetry,28 thermogravimetry,27 and the transpiration method.26 For [emim] [NTf2], there is good agreement among all measured data;25,29 for longer alkyl chains, the microcalorimetry data of Santos et al.28 indicate somewhat larger values than the other methods, which may be related to the high temperatures (580 K) at which the experiments were carried out and at which the RTILs may already start to decompose. The results of the first method (Knudsen effusion), namely the vapor pressure and the enthalpy of vaporization in a certain range of temperatures, allows the extrapolation of the vapor pressure to atmospheric pressure via the Clausius-Clapeyron equation, thereby providing us with an estimate of the normal boiling point Tb.21 If the critical temperature Tc is known, the critical pressure Pc may, in principle, be obtained as well by this method. The first attempt to estimate the critical temperature and the normal boiling points of RTILs was made by Rebelo et al.20

J. Phys. Chem. B, Vol. 114, No. 28, 2010 9185 They extrapolated experimental surface-tension data to the temperature at which γ reaches zero using the empirical expressions of Guggenheim and of Eo¨tvo¨s (see below); this temperature was identified as the critical one. Equipped with this estimate of Tc, Rebelo et al. proceeded to obtain the normal boiling temperature Tb via a modification of Guldberg’s rule by assuming that Tb ) 0.6Tc. Guldberg’s original relation states that Tb/Tc ) 2/3, but already Guggenheim realized that, for simple fluids, the numerical value of this ratio is closer to 0.6.1 From their experimental vapor-pressure data, Zaitsau et al. deduced the enthalpy of vaporization and used it in conjuction with eq 4 to extrapolate the pressure to the temperature at which it would reach atmospheric pressure.21 The so-obtained estimates of the hypothetical boiling point differ significantly from the values reported by Rebelo et al.20 Due to the limited availability of vapor-pressure data, the comparison includes only four RTILs, namely [emim] [NTf2], [bmim] [NTf2], [hmim] [NTf2], and [omim] [NTf2]. While Rebelo et al.20 gave ranges for the normal boiling point of the RTILs in the order indicated above of 660-725, 607-646, 559-580, and 520-530 K, Zaitsau et al.21 obtained normal boiling temperatures of 907, 933, 885, and 857 K, respectively. Zaitsau et al. attributed these discrepancies of 200-300 K in the estimates of Tb to the inapplicability of Guggenheim’s and Eo¨tvo¨s’ empirical expressions for the surface tension, which had been developed for molecular fluids, to RTILs. In addition, they stressed that the surface-tension data which Rebelo et al. extrapolated to about 1000 K20 had been measured in a narrow range of temperatures near 300 K; such a long extrapolation must be suspected to suffer from comparatively large uncertainties. Recently, Weiss et al.30 simulated the liquid-vapor interface of the RTIL [bmim] [PF6] using the force field of Bhargava and Balasubramanian31 for temperatures up to 800 K and confirmed the estimate of the critical temperature (Tc ) 1100 K), which Rebelo et al.20 had deduced for this substance from experimental data taken at much lower temperatures. In addition, no unusual behavior, such as the appearance of an inflection point in γ(T), which is known to occur for hydrogen-bonding fluids8 and which invalidates the applicability of Guggenheim’s and Eo¨tvo¨s’ approaches to these fluids, has been observed. The methodology of Rebelo et al.,20 thus, seems justified also for RTILs; the remark on the uncertainty of the estimates of Tc caused by the wide extrapolation range is certainly in order, but there is little that can be done about it because the substances start to decompose at 500 K. The goal of the present contribution is an attempt to reconcile the approaches and the numerical estimates of Tb put forward by Rebelo et al.20 and by Zaitsau et al.,21 as well as to reinterpret some of their findings in the framework of a correspondingstates approach. To set the stage for the study of ionic fluids, we first gather some of the (mostly well-known) facts about simple and polar fluids. We then address molten inorganic salts, such as NaCl and KCl, and repeat the analyses carried out by Kirshenbaum et al.,32 but reconsider the assigned uncertainties. Last but not least, we turn to the RTILs and discuss which of the relevant properties can be stated with reasonable certainty and which must be regarded as being essentially unknown based on the available experimental evidence. The remainder of the paper is organized as follows. The next section briefly shows how the critical parameters Tc and Fc are deduced from the available experimental data for the surface tension of RTILs and for the density of the liquid phase, as these parameters are prerequisites for the corresponding-states analysis. This procedure is not only applied to the [xmim] [NTf2]

9186


Weiss

RTILs, but, for comparison, also to the molten inorganic salts NaCl and KCl. In section 3, we repeat the analysis carried out by Zaitsau et al.21 to deduce the enthalpy of vaporization and the normal boiling point of the four RTILs from their measurements of the vapor pressure; the same approach is applied to the vaporpressure data of NaCl and KCl obtained by Barton and Bloom.33 The first part of section 4 is devoted to the correspondingstates behavior of the surface tension of the four RTILs and to a comparison with the known patterns of behavior of simple fluids, aprotic polar fluids, and inorganic molten salts.3,4 The second part of section 4 compiles and discusses the characteristic properties related to the vaporization of the different classes of fluids. In this part, we will also compare with generic model fluids for each class, the properties of which are determined from simulation data (taken from the literature) and by employing simple mean-field theories. The main conclusions of the study are summarized in section 5, which also gives a few warnings and recommendations for future work.

This equation describes all available data in the temperature range 290-415 K to within 0.2% (or better), including the sets of Fredlake et al.40 and Tariq et al.,41 which were not used for the correlation because the values of the former are slightly higher than those of the above-mentioned sets, whereas the latter data are somewhat lower. In general, the selection of data sets to be used for the representative fit was based on internal consistency and/or agreement with other sets. In the following paragraphs, the reason for not using a particular data set in the correlation will be given in parentheses. The representative equation for [bmim] [NTf2] has been adjusted to describe the data of Krummen et al.,35 Gomes de Azevedo et al.,42 Jacquemin et al.,37,43 and Harris et al.44 It also describes the data reported by Fredlake et al.40 (visible curvature, densities too high), Tokuda et al.36 (densities too high), Troncoso et al.45 (densities too low), Wandschneider et al.39 (densities too low), and Tariq et al.41 (densities too low), which were not considered for the correlation. The linear fit reads

Fl ) 1.719 g cm-3 - 0.0009447 g cm-3 K-1T

(13)

2. Critical Parameters of the Ionic Fluids The determination of the critical parameters of the four RTILs and the inorganic molten salts NaCl and KCl is bound to rely on extrapolation of the available experimental data. For the alkali halides, the critical temperature of about 3000 K is simply too high to carry out accurate measurements; for the RTILs, the decomposition of the substances prevents such measurements even in principle. Instead, the critical temperature may be estimated by fitting the coexistence curve to a nearly cubic shape if data for the liquid and vapor densities are available32 or by extrapolating surface-tension data to the temperature at which the surface tension would reach zero.20 The former approach does not seem viable for RTILs due to the very limited temperature range in which density data may be collected; the latter strategy was adopted by Rebelo et al.20 to deduce hypothetical critical temperatures of several RTILs. We will follow the same approach here, but repeat the analysis using all data currently available in the literature for the respective substances. The surface-tension data γ (T) are fitted to the empirical expressions proposed by Guggenheim (Gug)1 and by Eo¨tvo¨s (Eot),34 which read

γ(T) ) a(Tc - T)11/9

(10)

(Gug)

γ(T)[Fl(T)]-2/3 ) b(Tc - T)

(11)

(Eot)

where a and b are constants that play no role in determining Tc. Whereas in Guggenheim’s approach only surface-tension data are needed, Eo¨tvo¨s’ expression requires the density of the liquid phase, Fl, measured at the same temperature at which the surface tension was recorded. For this reason, we also determine a representative equation for the liquid density as a function of temperature from the available experimental data for each of the four RTILs. For [emim] [NTf2], the correlation of the density of the liquid phase is based on the data reported by Krummen et al.,35 Tokuda et al.,36 Jacquemin et al.,37 Gardas et al.,38 and Wandschneider et al.,39 which are in excellent agreement with each other. It is represented by -3

Fl ) 1.817 g cm

-3

- 0.001 g cm

-1

K

For [hmim] [NTf2], the correlation relies on data by Widegren and Magee,46 Kandil et al.,47 Esperança et al.,48 Muhammad et al.,49 and Marsh et al.50 and reads

Fl ) 1.6409 g cm-3 - 0.0009023 g cm-3 K-1 T

(14) This equation also describes the data by Tokuda et al.36 (markedly different slope, densities too low), Kato et al.51 (densities too low), and Gomes de Azevedo et al.42 (densities too low), which were not used for the correlation. The correlation for [omim] [NTf2] was adjusted to the data sets of Kato et al.,51 Gardas et al.,38 and Tariq et al.;41 it reads

Fl ) 1.5850 g cm-3 - 0.00089161 g cm-3 K-1 T

(15) The data of Tokuda et al.36 were not used for the fit because they indicate a significantly more negative slope than the other sets. There are fewer data sets for the surface tension of the [xmim] [NTf2] RTILs than for the liquid density. We have not considered literature data that consist of just one point because there is no way of deducing Tc from them. The experimental data and the representative equations are shown for [emim] [NTf2] and [bmim] [NTf2] in Figure 1 and for [hmim] [NTf2] and [omim] [NTf2] in Figure 2. The data for [emim] [NTf2] in Figure 1 (filled symbols) are seen to agree on the slope, but they differ significantly with respect to the intercept. The data of Kilaru et al.52 (filled diamonds) are about 6 mN/m larger than the ones of Carvalho et al.53 (filled circles) and of Wandschneider et al.39 (filled squares); the latter display a visible kink and show the presumably correct slope only at higher temperatures. The correlation is based on all three sets (discarding the low temperature data of Wandschneider et al.)

γ ) 53.434 mN m-1 - 0.048957 mN m-1 K-1 T T

(12)

(16)

Guggenheim’s Rule

J. Phys. Chem. B, Vol. 114, No. 28, 2010 9187

γ ) 48.83 mN m-1 - 0.05282 mN m-1 K-1 T

(17)

Figure 1. Experimental surface tension of the RTILs [emim] [NTf2] (filled symbols) and [bmim] [NTf2] (open symbols). Shown are the data for [emim] [NTf2] measured by Kilaru et al.52 (filled diamonds), by Carvalho et al.53 (filled circles), and by Wandschneider et al.39 (filled squares) as well as those for [bmim] [NTf2] obtained by Carvalho et al.53 (open circles) and by Wandschneider et al.39 (open squares). The symbols are connected by dashed lines to guide the eye. The upper continuous line represents the linear fit, eq 16, for [emim] [NTf2], the lower continuous line the one for [bmim] [NTf2], eq 17.

In view of the above discussion about the consistency of different data sets, it should be noted at this point that the single datum reported by Huddleston et al.19 at 298 K is 4 mN/m higher than the values found from the above sets. The data for [hmim] [NTf2] shown by filled symbols in Figure 2 suffer from the same problem as the ones for [emim] [NTf2] in Figure 1. Once again, the data of the set reported by Kilaru et al.52 (filled squares) are significantly (4 mN/m) higher than the ones of Carvalho et al.53 (filled circles). Without any obvious reason to prefer one set over the other, we simply average the two and obtain the following fit (upper continuous line), which represents the data to within 2 mN/m:

γ ) 49.586 mN m-1 - 0.052364 mN m-1 K-1 T

(18) The data set measured by Muhammad et al.49 (filled diamonds) has been excluded from the fit because it shows a completely different slope than the other two sets. For [omim] [NTf2], surface-tension data are particularly scarce. Only the set reported by Carvalho et al.53 (open circles) is suited for our purpose, and so the fit (lower continuous line) is based solely on these data

γ ) 50.011 mN m-1 - 0.061829 mN m-1 K-1 T

(19) Figure 2. Experimental surface tension of the RTILs [hmim] [NTf2] (filled symbols) and [omim] [NTf2] (open symbols). Shown are the data for [hmim] [NTf2] measured by Kilaru et al.52 (filled squares), by Carvalho et al.53 (filled circles), and by Muhammad et al.49 (filled diamonds) as well as those for [omim] [NTf2] obtained by Carvalho et al.53 (open circles) and by Zaitsau et al.21 (open squares). The symbols are connected by dashed lines to guide the eye. The upper continuous line represents the linear fit, eq 18, for [hmim] [NTf2], the lower continuous line the one for [omim] [NTf2], eq 19.

This fit (upper continuous line) represents all data to within 3 mN/m, which is the uncertainty we assign to all experimental surface-tension data here. This estimate is not only based on the error margins of the individual measurements, but takes into account that, as in the above case, the reported values differ by up to 6 mN/m for the same substance and at the same temperature. These discrepancies are attributable to different methods to determine the surface tension, different samples, and different impurities and water contents. A similar situation was encountered in the study of the experimental surface-tension data for [bmim] [PF6],30 for which two of the four available data sets were consistent with each other, but indicated surface tensions that were about 6 mN/m higher than those of the other two sets, which also were in good mutual agreement. In the absence of criteria and, therefore, reasons to disregard any of the available data sets, we decided to base the correlation on all three sets here, but we assign larger uncertainties to obtain a representative equation which is compatible with all experimental data. The two available sets for [bmim] [NTf2] by Carvalho et al.53 (open circles) and by Wandschneider et al.39 (open squares) agree very well; cf. Figure 1. The linear fit to both sets (lower continuous line) yields

The set of Zaitsau et al.21 (open squares) consists of only two points, making it impossible to assign an error to the slope, which, in addition, is seen to differ from the one in the set of Carvalho et al. The estimates of the critical temperatures resulting from the individual sets as well as from the respresentative equations within Guggenheim’s and Eo¨tvo¨s’ approaches are compiled in Table 1. The scatter is seen to be considerable and the uncertainty of the “best estimate” for Tc from each set is typically 100-150 K; for comparison, the critical temperatures quoted by Rebelo et al.20 are given, too; they are seen to be lower than the new values in all four cases. In Table 2, the best estimates of the critical temperatures are compiled together with their assigned uncertainties. An estimate for the respective critical density is obtained from the liquid densities applying the linear-diameter rule, which states that Fdia ) (Fl + Fv)/2 is a linear function of temperature. In the temperature range in which liquid densities have been measured, the vapor density is essentially zero, making Fdia ≈ Fl/2 a reasonable approximation. Neglecting critical fluctuations,54 the linear-diameter rule implies that Fdia(T) establishes also a connection between Tc and Fc.30 Thus, using the values of Tc obtained from the surface-tension data, an estimate of Fc is given by Fc ) h + mTc; the values of the coefficients h and m are listed in Table 2. For future reference, the same analysis is used to derive estimates of the critical parameters of the inorganic molten salts NaCl and KCl. From the available experimental surface-tension data (std),9,55-57 the critical temperatures marked by “std” in Table 2 are obtained, and the critical densities are estimated using the coexistence data reported by Kirshenbaum et al.32 The recommended critical parameters (based on fitting the coexistence curve, not by extrapolating the surface-tension data) of

9188


Weiss

TABLE 1: Estimates of the Critical Temperature Tc from Different Data Sets of the Surface Tension of [emim] [NTf2], [bmim] [NTf2], [hmim] [NTf2], and [omim] [NTf2] Based on Guggenheim’s (Gug) and Eo¨tvo¨s’ (Eot) Methods data set

TcGug (K)

TcEot (K)

TABLE 3: Estimates of the Apparent Enthalpy of Vaporization ∆vapHapp and of the Normal Boiling Point Tb for the Four Room Temperature Ionic Liquids and the Two Inorganic Molten Saltsa substance

∆vapHapp (kJ mol-1)

Tb (K)

[emin] [NTf2]

118.9 ( 11.9 114.7 ( 11.5c 118.1 ( 11.8b 123.5 ( 12.4b 131.3 ( 13.1b 174.0 ( 17.4d 169.4 ( 16.9d

906 (830-1020) 966 (900-1060) 933 (840-1060) 884 (815-990) 862 (800-960) 1736 (1724-1750) 1680 (1669-1694)

b

Carvalho et al.a Wandschneider et al.b Kilaru et al.c lin. fit, eq 16d Rebelo et al.e

[emim] [NTf2] 1235 ( 165 1190 ( 160 1300 ( 180 1265 ( 175 1100

1443 ( 135 1365 ( 165 1570 ( 130 1490 ( 90 1209

Carvalho et al.a Wandschneider et al.b lin. fit, eq 17d Rebelo et al.e

[bmim] [NTf2] 1032 ( 130 1092 ( 140 1060 ( 135 1012

1107 ( 105 1203 ( 115 1154 ( 75 1077

a In addition to the best estimate of Tb, the range of conceivable values of Tb is given in parentheses. b Reference 21. c Reference 26. d Reference 33.

Carvalho et al.a Kilaru et al.c lin. fit, eq 18d Rebelo et al.e

[hmim] [NTf2] 1020 ( 130 1156 ( 150 1087 ( 140 932

1090 ( 105 1303 ( 90 1190 ( 75 967

Carvalho et al.a Rebelo et al.e

[omim] [NTf2] 920 ( 110 882f

950 ( 85 866f

data of Emel’yanenko et al.26 for [emim] [NTf2], and from the vapor pressures reported for NaCl and KCl by Barton and Bloom.33 The values agree with those quoted by the respective authors, but, for the purpose of extrapolation to the normal boiling point, we assign a larger uncertainty of 10% to ∆vapHapp. We stress that we do not suspect the values of ∆vapHapp to be in error by this margin in the temperature range in which the pressure data were recorded. Rather, it is the only approximate constancy of ∆vapHapp over a wider range of temperatures which forces us to assume a larger uncertainty. For benign cases (noble gases), ∆vapHapp is constant to within 5% from the triple point to the critical point; in more problematic cases (water), ∆vapHapp may vary by up to 15% in this temperature range (figures based on experimental vapor-pressure data58,59). Typically, the dependence of ∆vapHapp is such that it displays relatively large values near the triple point and near the critical point and a minimum in the region 0.6 e T/Tc e 0.7, that is, near the normal boiling point for most fluids. More erratic behavior is, however, also observed in some cases. It therefore seems reasonable to assume an uncertainty of 10% for ∆vapHapp of the ionic fluids for the current purpose. Based on the range of conceivable values of ∆vapHapp, the normal boiling temperature is estimated by extrapolating the vapor pressures to atmospheric pressure using the ClausiusClapeyron equation. The results are shown in Table 3, with the best estimates agreeing with the figures quoted by Zaitsau et al.21 for the RTILs and with those of Kirshenbaum et al.32 for the inorganic molten salts. For the RTILs, the uncertainty in ∆vapHapp results in a range of boiling temperatures of about 60-100 K around the best estimate. For NaCl and KCl, the range is much smaller because the extrapolation is carried out over a temperature range of only 200 K, whereas for the RTILs, there is a difference of 500 K between the estimated normal boiling point and the temperatures at which the vapor pressures were measured.

a Reference 53. b Reference 39. c Reference 52. d This work. Reference 20. f Values were linearly interpolated between those for [hexyl-mim] [NTf2] and [decyl-mim] [NTf2]. e

TABLE 2: Estimates of the Critical Parameters Tc and Gc and Coefficients of the Correlation for the Diameter of the Coexistence Curve, Gdia ) h + mT, for the Ionic Fluids substance

Tc (K)

[emim] [NTf2] [bmim] [NTf2] [hmim] [NTf2] [omim] [NTf2] NaCla NaCl (std) b KCla KCl (std) b

1300 ( 150 1100 ( 100 1100 ( 150 950 ( 100 3400 ( 200 2890 ( 110 3200 ( 200 2820 ( 150

Fc (g cm-3) h (g cm-3) m (g cm-3 K-1) 0.259 ( 0.075 0.340 ( 0.047 0.324 ( 0.068 0.369 ( 0.045 0.222 ( 0.048 0.343 ( 0.026 0.175 ( 0.054 0.256 ( 0.040

0.9085 0.8595 0.8205 0.7925 1.0293 1.0293 1.0308 1.0308

-5.0000 × 10-4 -4.7235 × 10-4 -4.5115 × 10-4 -4.4581 × 10-4 -2.3736 × 10-4 -2.3736 × 10-4 -2.6741 × 10-4 -2.6741 × 10-4

a Reference 32. b Estimates based on extrapolation of surface-tension data (std).

Kirshenbaum et al.32 are also given. There is a marked difference in the estimates of Tc, which must be suspected to be caused by the inapplicability of Guggenheim’s and Eo¨tvo¨s’ relations to inorganic molten salts.30 In this context, it is noteworthy that the estimates of Tc based on the surface-tension data are lower than the ones derived from the coexistence data; if the approaches of Guggenheim and Eo¨tvo¨s failed due to the occurrence of an inflection point in the function γ(T), one would expect the surface-tension-based estimates to be too high. For the hydrogen-bonding fluids water, methanol, and ethanol, Guggenheim’s method overestimates Tc by 30-60 K. Despite this unexplained discrepancy for the inorganic salts, it will be useful to compare the results of both approaches for Tc in the corresponding-states analysis carried out in section 4. 3. Enthalpy of Vaporization and Normal Boiling Point of Ionic Fluids In Table 3, the apparent enthalpy of vaporization as deduced from vapor-pressure data taken at different temperatures for the four RTILs and the inorganic molten salts are compiled. The Clausius-Clapeyron equation, eq 4, was used to obtain ∆vapHapp from the data of Zaitsau et al.21 for all four RTILs, from the

[bmim] [NTf2] [hmim] [NTf2] [omim] [NTf2] NaCl KCl

4. Corresponding-States Properties In this section, the estimates of the critical parameters of the ionic fluids are used to analyze thermodynamic properties in the framework of a corresponding-states approach, which enables us to obtain characteristic values for each class of substances and to compare them to typical data for simple and polar fluids. We start with the surface-tension data and analyze the quantities related to vaporization afterward. 4.1. Reduced Surface Tension. The corresponding-states analysis of the surface tension is based on the relation proposed by Guggenheim,1 who suggested that the reduced surface tension γred as defined in eq 9 should be a universal function of the

Guggenheim’s Rule


reduced temperature T/Tc for all fluids adhering to the principle of corresponding states. As Guggenheim showed,1,7 this expectation is indeed fulfilled for simple fluids, such as noble gases and homonuclear diatomic molecules. He found the amplitude γ0 as defined by

(

γred ) γ0 1 -

T Tc

)

11/9

(20)

to have a value of γ0 ) 4.3 for strictly spherical molecules, such as noble gases and methane. A distortion of the spherical shape to a dumbbell (nitrogen and ethane) causes a slight increase of γ0 to about 4.5.7 Aprotic polar fluids have been shown to display somewhat higher values of γred with γ0 increasing to 5.3.3,4 Polar fluids in which the molecules are capable of forming hydrogen bonds are known to display a completely different kind of behavior.3,8 The association of molecules leads to an inflection point in the function γ(T), which usually occurs at 0.8 e T/Tc e 0.85.3 Due to the inflection point, the values of γred for this class of fluids at low reduced temperatures are much lower than for simple and aprotic polar fluids.3 Furthermore, their behavior cannot be captured by Guggenheim’s (or Eo¨tvo¨s’) formula. That the occurrence of an inflection point in γ(T) is indeed caused by association phenomena was convincingly demonstrated by Blas et al.60 and by Gloor et al.,61 who studied fluids in which the particles could interact through specific binding sites in addition to a spherically symmetric attractive interaction. It was shown that dimerization (one binding site) was insufficient to produce an inflection point in γ(T), but that two or more binding sites, producing particle chains or more complex networks, gave rise to an inflection point. Inorganic molten salts, such as NaCl and KCl, also display lower values of γred3,4 and, consequently, of γ0. Experimental values are usually only available at relatively low temperatures 0.3 e T/Tc e 0.4; it is, therefore, not known if these ionic fluids show an inflection point in γ(T) at higher temperatures. Simulations of simple model fluids, which combine the Coulomb interaction either with a hard-sphere potential62 or with a soft repulsion,63,64 suggest the occurrence of such an inflection point. As in the case of hydrogen-bonding polar fluids, one may rationalize such a behavior of γ(T) with the association pattern of ions being different at lower temperatures than at higher ones, at which aggregates of ions tend to dissociate. The concept of ion pairing is, however, of very limited use in a dense molten salt, in which an ion is in close contact with many counterions at the same time. As far as room temperature ionic liquids are concerned, only the reduced surface tension of [bmim] [PF6] has been analyzed.30 It was found that this particular RTIL displays a behavior which is practically identical to that of aprotic polar fluids. As already mentioned in the introduction, this study extended the accessible temperature range to 800 K by simulating the RTIL using a classical force field, but no indication of an inflection point in γ(T) was observed. Figure 3 shows the data for [xmim] [NTf2] based on the critical parameters deduced in the previous sections. Again, the values are compatible with those of polar fluids (exemplified by data for CHClF2,65 continuous line), but not with the values of inorganic molten salts. For the latter, data based on the two sets of critical parameters, one obtained from extrapolating the coexistence curve32 (filled triangles) and the other from the surface tension (open triangles), are shown for NaCl (triangles left) and for KCl (triangles right). It is reassuring to see that,

Figure 3. Reduced surface tension γred as a function of the reduced temperature T/Tc for the RTILs [emim] [NTf2] (filled circles), [bmim] [NTf2] (filled squares), [hmim] [NTf2] (open diamonds), and [omim] [NTf2] (open circles). The continuous line represents data for polar fluids, exemplified by CHClF2, whereas the dotted line for methane and the dash-dotted line for ethane are more typical of simple fluids. For comparison, data for inorganic molten salts, such as NaCl (triangles left) and KCl (triangles right), are shown; for data points represented by filled triangles, the best estimates of the critical parameters (cf. Table 2) are used, while for open triangles, the alternative sets of critical parameters (marked by “std” in Table 2) are employed.

although the latter approach leads to significantly lower estimates of the critical temperature (the values marked by “std” in Table 2), the data of γred lie on the same curve for both methods and are just shifted to higher reduced temperatures. For the RTILs with short alkyl chains on the cation, [emim] [NTf2] (filled circles) and [bmim] [NTf2] (filled squares), the agreement with the continuation of γred(T/Tc) for CHClF2 is almost perfect. For [hmim] [NTf2] (open diamonds) and [omim] [NTf2] (open circles), there is a visible increase of γred, which is at the border of being significant, but may as well be related to the uncertainty of the critical parameters. For comparison, typical data of simple (nonpolar) fluids are shown, namely for methane (dotted line) and for ethane (dashdotted line).65 4.2. Reduced Enthalpy and Entropy of Vaporization. Having discussed the differences and similarities of the reduced surface tension γred of the different classes of fluids, we now turn to the properties related to vaporization. Table 4 shows the critical compressibility factor Zc, the reduced apparent enthalpy of vaporization ∆vapHapp/(RTc), the ratio of the normal boiling temperature and the critical temperature, Tb/Tc (Guldberg ratio), as well as the entropy of vaporization at the normal boiling point ∆vapS(Tb) (Trouton constant) for selected simple, aprotic polar, and hydrogenbonding fluids. In Table 5, the same quantities are compiled for the ionic fluids, that is, for inorganic molten salts and for the RTILs. For simple fluids composed of spherical particles, Zc is close to 0.29;1 deviations from the spherical shape decrease the value of Zc slightly. As stated indirectly by Guggenheim, one finds ∆vapHapp/(RTc) ≈ 5.2 for simple fluids, while a deviation from the spherical shape increases this value significantly. The Guldberg ratio (Tb/Tc) is seen to be near 0.58 for noble gases and methane, but increases to 0.70 for octane.66 ∆vapHapp/(RTc) increasing more rapidly than Tb/Tc in this sequence of nonpolar fluids leads to a slight increase of the Trouton constant from 73 J K-1 mol-1 for argon and methane to 86 J K-1 mol-1 for octane. Polar fluids are known to feature somewhat lower values of Zc due to dipole-dipole interactions and/or hydrogen bonds. As a rule of thumb, the more polar a molecule, the lower Zc. Guggenheim’s ratio ∆vapHapp/(RTc) increases with polarity,

9190


Weiss

TABLE 4: Critical Compressibility Factor, Zc, Reduced Apparent Enthalpy of Vaporization, ∆vapHapp/(RTc), Ratio of Normal Boiling to Critical Temperature, Tb/Tc, Entropy of Vaporization at the Normal Boiling Point, ∆vapS(Tb), and ∆Z(Tb) ) ∆vapH(Tb)/∆vapHapp for Simple Fluids, Aprotic and Hydrogen-Bonding Polar Fluidsa ∆vapHapp/(RTc)

Tb/Tc

∆vapS(Tb) (J K-1 mol-1)

substance

Zc

∆Z(Tb)

Ar CH4 C2H6 C3H8 n -octane

0.293 0.286 0.279 0.280 0.259

Nonpolar Simple Fluids 5.3 0.58 5.4 0.59 5.9 0.60 6.5 0.62 8.9 0.70

73.7 73.3 79.6 82.4 86.3

0.97 0.96 0.97 0.95 0.82

CH3Cl CHClF2 acetone

0.268 0.269 0.237

Aprotic Polar Fluids 6.6 0.60 7.2 0.63 7.8 0.65

85.9 86.9 88.4

0.94 0.91 0.89

NH3 H2O CH3OH C2H5OH

0.235 0.230 0.222 0.241

Hydrogen-Bonding Fluids 7.1 0.59 97.3 7.6 0.58 108.9 9.2 0.66 104.2 9.8 0.68 109.7

0.97 1.00 0.90 0.92

a All entries were derived using data taken from CRC handbooks.58,59

which, thus, has a similar effect as deviations from the spherical molecular shape. Guldberg’s ratio is seen to increase as well. Polar molecules also display larger Trouton constants of ∆vapS(Tb) g 86 J K-1 mol-1. Hydrogen-bonded fluids usually combine a relatively large dipole moment with the specific effects of the highly directional hydrogen bond. Consequently, Zc is often even lower than for aprotic polar fluids, and ∆vapHapp/(RTc) is seen to be larger, reaching a value of almost 10 for ethanol. The ratio Tb/Tc is in the same range found for nonpolar and aprotic polar fluids. As is well-known, the Trouton constant of hydrogen-bonded fluids is characteristically larger (∆vapS(Tb) g 100 J K-1 mol-1) than for simple fluids, reflecting the association of particles in the liquid phase. The respective values for ionic fluids are listed in Table 5. To obtain Zc, the critical pressure must be determined first. The same value of Pc as the one quoted by Kirshenbaum et al.32 was found by extrapolating the vapor-pressure data of Barton and Bloom33 to the critical temperature, the value of which is uncertain by 200 K. This uncertainty combined with the variation of ∆vapHapp (which Kirshenbaum et al. ignored) leads to an error margin that is almost as large as the value of Pc itself. Zc must, therefore, be considered to be essentially unknown. The same applies to the estimates of Pc and Zc for the RTILs given in Table 5. The Guggenheim ratio ∆vapHapp/ (RTc) for the inorganic molten salts is similar to the values for polar fluids, whereas the Guldberg ratio Tb/Tc is seen to be somewhat lower. Only if the estimate of Tc is derived from the surface-tension data (std), Tb/Tc is found in the range for simple and polar fluids. The entropy of vaporization at the boiling point is known32 to be about ∆vapS ≈ 100 J K-1 mol-1, encompassing values of hydrogen-bonding fluids. For the RTILs, ∆vapHapp/(RTc) is seen to be significantly larger than for the other fluids. Apart from uncertainties in ∆vapHapp and Tc, these large values may be due to the considerable contribution of dispersion interactions in addition to the Coulomb forces acting among the ions. For the inorganic salts NaCl and KCl, the ratio of the Coulomb part and the dispersive contribution to the interaction energy in an ion pair at contact

can be estimated using crystallographic radii as well as data for the polarizability volumes and the ionization potentials of the ions67 in conjunction with London’s formula. The results indicate that the Coulomb term is at least 200 times more important than the dispersive part. For RTILs, this ratio decreases to values between 2 and 5.68-70 Of particular interest is the ratio of Tb/Tc, which according to the estimates of Rebelo et al.20 for Tc and by Zaitsau et al.21 for Tb, must have been expected to be exceptionally high (0.75-1). Using our revised estimates of Tc, the values in Table 5 show that a quite normal ratio of Tb/Tc ) 0.7 (cf. the value for octane in Table 4) is conceivable, but significantly larger values are equally possible. For [omim] [NTf2], in particular, it is within the range of the uncertainties that Tb > Tc, which means that the critical pressure is below 1 atm and the fluid has no normal boiling point, not even a hypothetical one. As for inorganic molten salts, Zc of the RTILs is largely unknown due to the uncertainties in the estimates of Pc resulting from the extrapolation of vapor-pressure data to Tc. To determine Pc, it seems more reasonable to assume that Zc is somewhere in the range 0.05-0.25, which allows us to estimate at least the order of magnitude of Pc, which should be 0.5-2.5 MPa, that is, lower than for most simple fluids (5 MPa). The possibility of Zc being as low as 0.05 for ionic fluids is based on the finding that the critical parameters of the restricted primitive model (RPM, a model that features just hard-sphere repulsion in addition to Coulomb interactions, but in which attractive dispersion forces are absent) yield Zc ) 0.024.71 We expect that the presence of dispersive interactions in real ionic fluids will increase this value somewhat. Due to the relatively large values of Tb/Tc for RTILs, it is not straightforward to estimate their Trouton constant ∆vapS(Tb). For Tb/Tc e 0.6, the true enthalpy of vaporization at the normal boiling point ∆vapH(Tb) is almost the same as its apparent value, ∆vapHapp, because ∆Z does not deviate significantly from unity in this temperature range, and the Clausius-Clapeyron equation (cf. eq 4) holds. If, however, T/Tc is very much larger than 0.6, one must expect the true value of ∆vapH(T) to be lower than its apparent value ∆vapHapp, which remains nearly constant all the way to the critical point, whereas the true ∆vapH(T) must approach zero as T f Tc. In order to be able to obtain an estimate of ∆vapH(Tb) for the high-boiling RTILs according to eq 8, one needs information on the variation of ∆Z with temperature for these fluids. This kind of information is unavailable from experiments, but could, in principle, be supplied by molecular simulations of a suitable classical force field, in which the substance cannot decompose. Here, we choose another, more generic approach and study the behavior of ∆Z(T/Tc) within mean-field theories of simple, polar, and ionic fluids.72 For comparison, we also analyze the behavior of model fluids using simulation data taken from the literature. The model systems considered for this purpose are the Lennard-Jones (LJ) and square-well (SqW) fluids as representatives of simple fluids and the restricted primitive model (RPM) of ionic fluids.72 In addition to modeling simple fluids, the SqW fluid allows one to vary the interaction range systematically, and the influence of this parameter can be studied. For the corresponding simplest model of a polar fluid, dipolar hard spheres (DHS), it is not certain whether it displays liquid-vapor equilibrium at all; rather, it seems that a minimum of dispersion interaction is needed to obtain a regular phase diagram.73 As all real polar fluids display liquid-vapor equilibrium, we deduce the behavior of ∆Z for this class of fluids from Onsager’s theory. For simple fluids, a van der Waals(vdW)-type of equation is

Guggenheim’s Rule


TABLE 5: Critical Compressibility Factor, Zc, Reduced Apparent Enthalpy of Vaporization, ∆vapHapp/(RTc), Ratio of Normal Boiling to Critical Temperature, Tb/Tc, and Entropy of Vaporization at the Normal Boiling Point, ∆vapS(Tb) for the Molten Inorganic Salts and the Room Temperature Ionic Liquidsa substance

Tb/Tc

∆vapS(Tb) (J K-1 mol-1)

0.51 (0.48-0.55) 0.52 (0.49-0.56) 0.60 (0.57-0.63) 0.60 (0.56-0.63)

97 (88-106) 97 (88-106) 92 (82-102) 93 (83-103)

Room Temperature Ionic Liquids 11.0 (8.9-13.7) 0.70 (0.57-0.89) 12.9 (10.6-15.6) 0.85 (0.70-1.06c) 13.5 (10.3-16.4) 0.80 (0.63-0.99) 16.6 (13.5-20.4) 0.91 (0.76- 1.13c)

110 (55-156) 76 (0-131) 95 (8-157) 69 (0-140)

∆vapHapp/(RTc)

Zc

NaCl KCl NaCl (std)b KCl (std)b

0.34 (0.12-1.07) 0.52 (0.16-1.88) 0.09 (0.04-0.20) 0.17 (0.07-0.47)

[emim] [NTf2] [bmim] [NTf2] [hmim] [NTf2] [omim] [NTf2]

1.72 (0.06-48) 0.13 (0.007-1.80) 0.41 (0.008-17.3) 0.09 (0.003-2.82)

Molten Salts 6.2 (5.2-7.2) 6.4 (5.4-7.5) 7.2 (6.3-8.3) 7.2 (6.2-8.4)

a The numbers in parentheses after the “best estimate” give the conceivable range of values for the respective quantity. b Estimates of the critical parameters based on extrapolation of surface-tension data (std). c A value of Tb/Tc > 1 implies that the substance has no normal boiling point because Pc < 1 atm.

TABLE 6: Critical Compressibility Factor Zc and Reduced Apparent Enthalpy of Vaporization ∆vapHapp/(RTc) for Simple, Polar, and Ionic Model Fluids As Obtained from Simulations and Mean-Field Theories substance/method Lennard- Jonesa square well (l ) square well (l ) square well (l ) square well (l ) van der Waalsd

1.5)b 1.5)c 2.0)b 3.0)c

Zc Simple Fluids 0.32 0.256 0.252 0.295 0.331 0.359

∆vapHapp/(RTc) 5.1 6.2 5.9 5.6 5.0 5.0

Onsagerd

Polar Fluid 0.281

4.8-5.0

RPMc Debye-Hückeld

Ionic Fluid 0.024 0.096

7.5-9.5 8-10

a Reference 74. b Reference 75. c Reference 71. d In conjunction with the Carnahan-Starling hard-sphere term, cf. ref 72.

chosen, with the hard-sphere part described by the CarnahanStarling expression instead of van der Waals’ original excluded volume term.72 For the RPM, predictions of the Debye-Hu¨ckel (DH) theory are evaluated. To allow a comparison with their real counterparts, Table 6 shows the values of Zc and of ∆vapHapp/ (RTc) obtained from simulations and from mean-field theoretical calculations for these model fluids. It is seen that Zc of the Lennard-Jones fluid74 is slightly larger than for real simple fluids, whereas the vdW equation overestimates Zc considerably. For the SqW fluid, an interaction range of l ) 2 (twice the hardsphere radius)75 yields a value of Zc which is comparable to the one of noble gases, whereas a shorter interaction range leads to lower values of Zc. For larger interaction ranges, the vdW limit is approached. The Onsager theory for polar fluids predicts Zc to be only slightly smaller than the value for simple fluids. Particularly interesting is the value of Zc for the ionic model fluid; simulations indicate an exceptionally small value of Zc ) 0.024,71 and also the DH theory yields Zc < 0.1, which is smaller than the values found for molecular fluids. Properties related to the normal boiling point cannot be studied for model fluids, as the latter have no way of experiencing atmospheric pressure, but the Guggenheim ratio for the apparent reduced enthalpy of vaporization ∆vapHapp/(RTc) is nevertheless accessible. LJ and SqW fluids show the values found for real simple fluids (cf. Table 6); the trend with varying interaction range is that ∆vapHapp/(RTc) decreases with increasing interaction range and approaches the vdW limit of 5.0 asymp-

Figure 4. Difference of the compressibility factors in the vapor phase and in the liquid phase, ∆Z, as a function of the reduced temperature T/Tc. Shown are the data for the Lennard-Jones fluid (open circles) obtained from simulations by Errington,74 for the restricted primitive model of ionic fluids (open squares) by Orkoulas and Panagiotopoulos71 as well as results from mean-field theories: van der Waals (continuous line) for simple fluids, Onsager (dotted line) for polar fluids, and Debye-Hu¨ckel (dashed line) for ionic fluids.

totically. The Onsager theory indicates no difference between the behavior of polar fluids and simple fluids in this respect. The data found from simulations of the RPM and from DH theory show significantly larger values of ∆vapHapp/(RTc) for ionic fluids, which is in good agreement with the trend indicated by the data for RTILs. It is, however, not clear why these larger values are not found for inorganic molten salts which should be described more accurately by the RPM than the RTILs. Finally being able to return to the behavior of ∆Z(T/Tc), this function is shown for simple, polar, and ionic model fluids in Figure 4. The data deduced from simulations of the LJ fluid74 (circles) are seen to agree very well with the predictions of the vdW equation (continuous line). We, therefore, expect this curve to show the behavior of real simple fluids. A comparison of experimentally determined values of ∆vapH(Tb)/∆vapHapp ) ∆Z(Tb) compiled in Table 4 to the calculated data of ∆Z(T) for model fluids at the same respective value of Tb/Tc in Figure 4 shows that this approach indeed gives very reasonable results. It is also seen that if Tb/Tc e 0.6, as is usual for simple fluids, then ∆Z(Tb) g 0.96 and ∆vapHapp ≈ ∆vapH(Tb) to within a few percent. For the RPM, simulation data71 are available only at relatively high temperatures, T/Tc g 0.85. The data in the range 0.85 e T/Tc e 0.9 (squares) are seen to agree well with the predictions of DH theory (dashed line), but they deviate for temperatures closer to Tc. One possible explanation for this disagreement is that, close to Tc, the asymptotic behavior of ∆Z is determined by the critical exponent β for the order parameter. A mean-field theory predicts β ) 1/2 and, therefore,

9192


Figure 5. Difference of compressibility factors in the vapor phase and in the liquid phase, ∆Z, as a function of the reduced temperature T/Tc for the square-well fluid with variable interaction range l. Data for l ) 1.5 obtained from simulations by del Rıó et al.75 (circles) and by Orkoulas and Panagiotopoulos71 (squares) connected by dashed lines are shown along with data for l ) 2.0 by del Rio et al.75 (diamonds, connected by a dash-dotted line) and for l ) 3.0 by Orkoulas and Panagiotopoulos71 (triangles, connected by a dotted line). For comparison, the predictions of the van der Waals equation of state, the limiting case for l f ∞, are shown as a continuous line.

has to deviate from the curves for systems that belong to the Ising universality class, in which β ≈ 0.326. Below T/Tc ) 0.9, however, we expect DH theory to yield reasonable estimates of ∆Z for fluids whose behavior is dominated by Coulomb interactions. The curve following from the Onsager theory for polar fluids (dotted line) is seen to lie inbetween the ones for simple and for ionic fluids. For completeness, the behavior of ∆Z for the SqW fluid with variable interaction range is shown in Figure 5. It is seen from the simulation data71,75 that, for a given reduced temperature T/Tc, ∆Z decreases with increasing interaction range and asymptotically approaches the limit set by the vdW equation (continuous curve). It becomes clear from Figure 4 that, if the reduced normal boiling point of RTILs is really at Tb/Tc g 0.7, then the variation of ∆Z(T) has to be taken into account to deduce ∆vapH(Tb) from ∆vapHapp reliably. For example, at T/Tc ) 0.7, ∆vapH will be about 80% of the experimentally determined ∆vapHapp, at T/Tc ) 0.8 only 60-70%, while if Tb/Tc is as high as 0.9, then ∆vapH will be only half of ∆vapHapp ! Taking the combined uncertainties of ∆vapHapp, Tb, Tc, and ∆Z(Tb/Tc) into account, it becomes very difficult to state anything about the entropy of vaporization, ∆vapS(Tb), of RTILs at their normal boiling point with reasonable certainty. Table 5 lists the “best estimates” of the Trouton constants for the four different RTILs, but the scatter in the data for this set of supposedly very similar fluids and the range of conceivable values of ∆vapS(Tb) given in parentheses make clear that nothing reliable can be said. At best, it can be concluded that RTILs display entropies of vaporization near their normal boiling points that are somewhere in the range covered by molecular fluids. A discussion of the exact value of Trouton’s constant on the basis of the currently available experimental data, however, does not seem particularly fruitful. 5. Summary and Conclusion Various corresponding-states properties, which were originally proposed by Guggenheim,1 have been investigated for simple fluids, aprotic polar fluids, hydrogen-bonding fluids, inorganic molten salts, and room temperature ionic liquids (RTILs). Mostly experimental data taken from the literature have been used, but it also turned out to be valuable to include

Weiss simulation data (from the literature) and predictions of simple mean-field theories for model fluids, which allow us to isolate one of the characteristic types of interactions (dispersion forces, dipolar interactions, or the Coulomb potential). The properties investigated for the different classes of fluids are the reduced surface tension γred, the critical compressibility factor Zc, the reduced apparent enthalpy of vaporization ∆vapHapp/(RTc), the reduced normal boiling temperature Tb/Tc (Guldberg ratio), and the entropy of vaporization at the normal boiling point ∆vapS(Tb) (Trouton constant). The latter two quantities refer to atmospheric pressure and are, strictly speaking, not actual correspondingstates properties. The behavior of γred for the different fluids is shown in Figure 3, whereas the results for the other properties are compiled in Tables 4-6. The reduced surface tension of the four RTILs considered here is seen to be comparable to the one characteristic of moderately polar fluids,3 which is in line with their experimentally determined dielectric permittivities.17 Molten inorganic salts are known to display much lower values of γred at a given reduced temperature; the same is true for hydrogen-bonding fluids.3,4 The critical compressibility factor decreases from Guggenheim’s value of Zc ) 0.29 for simple nonpolar fluids with increasing polarity of the fluid. Deviations from a perfectly spherical molecular shape have a similar effect. After careful re-evaluation of the available experimental data, the value of Zc of ionic fluids must be considered to be unknown; for a generic model of ionic fluids, simulation71 and theory indicate that Zc may be significantly lower than for simple and polar fluids. Guggenheim’s ratio ∆vapHapp/(RTc) is about 5.2 for simple fluids, but this value increases in a very similar way in which Zc decreases, that is, with increasing polarity and/or deviation from the spherical shape. For hydrogen-bonding fluids, values twice as high as the ones for simple fluids can be found, but the ones for RTILs are even higher and truly exceptional (cf. Table 5). Since inorganic molten salts and generic models of ionic fluids show values that are comparable to those of moderately polar fluids, one must suspect the results for RTILs to be a combined effect of Coulomb interactions and significant contributions from dispersion interactions.69 Guldberg’s ratio has a value of Tb/Tc ) 0.58 for simple fluids. Polarity of the molecule and deviation from spherical shape can increase this value to at least 0.7, maybe even beyond.66 Whereas the molten inorganic salts display an anomalously small value of Tb/Tc ) 0.5, the RTILs show very large values of this ratio ranging from 0.7 to (hypothetical) values greater than one, which would imply that the critical pressure is below one atmosphere and the fluid, therefore, has no normal boiling point. It must be stressed at this point that the extrapolation of vapor-pressure data for the ionic fluids to atmospheric pressure introduces a considerable uncertainty into our best estimates of the normal boiling points (cf. Table 5). The behavior of Trouton’s constant for molecular fluids is well-known: For simple fluids, one finds ∆vapS(Tb) ≈ 73 - 80 J K-1 mol-1; aprotic polar fluids show slightly larger values, up to 88 J K-1 mol-1. Higher values mark a significant association in the liquid phase and are found for hydrogenbonding fluids and for simple inorganic molten salts (cf. Tables 4 and 5). For the RTILs, estimating ∆vapS(Tb) is even more difficult than determining their normal boiling point. In addition to the uncertainties in ∆vapHapp, Tc, and Tb, one must also take into account that the true value of ∆vapH(T) differs from the apparent value ∆vapHapp by a factor of ∆Z(T), which is not

Guggenheim’s Rule known for the RTILs. Instead we adopt values of ∆Z(T/Tc) for simple and ionic model fluids derived from simulations and theoretical calculations (cf. Figure 4), which have been validated to be accurate for molecular fluids (cf. Table 4). The combined uncertainty of ∆vapS(Tb) for RTILs is, however, so large that nothing definite can be said. Due to the decomposition of the real substances, experiments will not be carried out near Tb; the only hope of deriving ∆vapS(Tb) for RTILs with greater accuracy seems to be by means of computer simulations employing accurate classical force fields at elevated temperatures, as it was done for the surface tension of [bmim] [PF6].30 The corresponding-states behavior of the four RTILs studied here is difficult to classify. Their reduced surface tension γred is typical of moderately polar aprotic fluids, as it was observed for [bmim] [PF6].30 The vaporization-related quantities, however, either differ significantly from what is usually found for molecular fluids or are impossible to quantify on the basis of the currently available data (Zc and ∆vapS). For Guldberg’s ratio, a value of Tb/Tc ) 0.7 is at the lower end of the conceivable range for RTILs, whereas ∆vapHapp/(RTc) > 11 is markedly higher than the values of Guggenheim’s ratio found for molecular fluids. A possible reason for these different patterns of behavior may be that we still underestimate the critical temperature. As demonstrated for NaCl and KCl (cf. Figure 3), γred is not very sensitive to the estimate of Tc because its effect is partially balanced by the corresponding estimate of Fc. For the other properties, there is no such counter-balancing mechanism. If Tc was higher, both Guldberg’s and Guggenheim’s ratios would assume less exceptional values for the RTILs. An upper limit to Tc seems to be set by the values at which the average value of the liquid and vapor densities, Fdia(T), reaches zero, which is near 1800 K for all four RTILs studied here. If the surface tension γ(T) of the RTILs showed anomalous behavior that would invalidate Guggenheim’s and Eo¨tvo¨s’ approaches, higher values of Tc would be conceivable and are, in fact, predicted by the group-contribution method applied by Valderrama and Robles.76 In contrast, the occurrence of an inflection point in the temperature-dependent surface tension, as it is found for hydrogen-bonding fluids, would lead to somewhat lower estimates of Tc. At present, however, there is no indication of any anomaly in the function γ(T) for the RTILs studied,30 and the values of Tc derived from surface-tension data remain the only estimates based on actual experiments. Acknowledgment. The author thanks Athanassios Panagiotopoulos for making the simulation data presented graphically in ref 71 available in numerical form. References and Notes (1) Guggenheim, E. A. J. Chem. Phys. 1945, 13, 253. (2) Pitzer, K. S. J. Chem. Phys. 1939, 7, 583. (3) Weiss, V. C.; Schroër, W. J. Chem. Phys. 2005, 122, 084705. (4) Weiss, V. C.; Schroër, W. Int. J. Thermophys. 2007, 28, 506. (5) Israelachvili, J. N. Intermolecular and Surface Forces; Academic: London, 1985. (6) See, e.g., Hill, T. L. An Introduction to Statistical Thermodynamics; Addison-Wesley: Reading, MA, 1960. (7) Guggenheim, E. A. Proc. Phys. Soc. London 1965, 85, 811. (8) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon: Oxford, 1982. (9) Janz, G. J. Molten Salts Handbook; Academic: New York, 1967. (10) Friedman, H. L.; Larsen, B. J. Chem. Phys. 1979, 70, 92. (11) Pitzer, K. S. J. Phys. Chem. 1984, 88, 2689. (12) McGahay, V.; Tomozawa, M. J. Chem. Phys. 1992, 97, 2609. (13) Reiss, H.; Mayer, S. W.; Katz, J. L. J. Chem. Phys. 1961, 35, 820. (14) Marsh, K. N.; Boxall, J. A.; Lichtenthaler, R. Fluid Phase Equilib. 2004, 219, 93. (15) Earle, M. J.; Seddon, K. R. Pure Appl. Chem. 2000, 72, 1391. (16) Wasserscheid, P.; Keim, W. Angew. Chem., Int. Ed. 2000, 39, 3772.

J. Phys. Chem. B, Vol. 114, No. 28, 2010 9193 (17) Weinga¨rtner, H. Angew. Chem., Int. Ed. 2008, 47, 654. (18) Tokuda, H.; Hayamizu, K.; Ishii, K.; Susan, M. A. B. H.; Watanabe, M. J. Phys. Chem. B 2004, 108, 16593. (19) Huddleston, J. G.; Visser, A. E.; Reichert, W. M.; Willauer, H. D.; Broker, G. A.; Rogers, R. D. Green Chem. 2001, 3, 156. (20) Rebelo, L. P. N.; Canongia Lopes, J. N.; Esperança, J. M. S. S.; Filipe, E. J. Phys. Chem. B 2005, 109, 6040. (21) Zaitsau, D. H.; Kabo, G. J.; Strechan, A. A.; Paulechka, Y. U.; Tschersich, A.; Verevkin, S. P.; Heintz, A. J. Phys. Chem. A 2006, 110, 7303. (22) Armstrong, J. P.; Hurst, C.; Jones, R. G.; Licence, P.; Lovelock, K. R. J.; Satterley, C. J.; Villar-Garcia, I. J. Phys. Chem. Chem. Phys. 2007, 9, 982. (23) Kelkar, M. S.; Maginn, E. J. J. Phys. Chem. B 2007, 111, 9424. (24) Earle, M. J.; Esperança, J. M. S. S.; Gilea, M. A.; Canongia Lopes, J. N.; Rebelo, L. P. N.; Magee, J. W.; Seddon, K. R.; Widegren, J. A. Nature 2006, 439, 831. (25) Ludwig, R.; Kragl, U. Angew. Chem., Int. Ed. 2007, 46, 6582. (26) Emel’yanenko, V. N.; Verevkin, S. P.; Heintz, A. J. Am. Chem. Soc. 2007, 129, 3930. (27) Luo, H.; Baker, G. A.; Dai, S. J. Phys. Chem. B 2008, 112, 10077. (28) Santos, L. M. N. B. F.; Canongia Lopes, J. N.; Coutinho, J. A. P.; Esperança, J. M. S. S.; Gomes, L. R.; Marrucho, I. M.; Rebelo, L. P. N. J. Am. Chem. Soc. 2007, 129, 284. (29) Esperança, J. M. S. S.; Canongia Lopes, J. N.; Tariq, M.; Santos, L. M. N. B. F.; Magee, J. W.; Rebelo, L. P. N. J. Chem. Eng. Data 2010, 55, 3. (30) Weiss, V. C.; Heggen, B.; Mu¨ller-Plathe, F. J. Phys. Chem. C 2010, 114, 3599. (31) Bhargava, B. L.; Balasubramanian, S. J. Chem. Phys. 2007, 127, 114510. (32) Kirshenbaum, A. D.; Cahill, J. A.; McGonigal, P. J.; Grosse, A. V. J. Inorg. Nucl. Chem. 1962, 24, 1287. (33) Barton, J. L.; Bloom, H. J. Phys. Chem. 1956, 60, 1413. (34) Eo¨tvo¨s, R. Wied. Ann. 1886, 27, 452. see also. Shereshefsky, J. L. J. Phys. Chem. 1931, 35, 1712. (35) Krummen, M.; Wasserscheid, P.; Gmehling, J. J. Chem. Eng. Data 2002, 47, 1411. (36) Tokuda, H.; Hayamizu, K.; Ishii, K.; Susan, M. A. B. H.; Watanabe, M. J. Chem. Phys. B 2005, 109, 6103. (37) Jacquemin, J.; Husson, P.; Mayer, V.; Cibulka, I. J. Chem. Eng. Data 2007, 52, 2204. (38) Gardas, R. L.; Freire, M. G.; Carvalho, P. J.; Marrucho, I. M.; Fonseca, I. M. A.; Ferreira, A. G. M.; Countinho, J. A. P. J. Chem. Eng. Data 2007, 52, 1881. (39) Wandschneider, A.; Lehmann, J. K.; Heintz, A. J. Chem. Eng. Data 2008, 53, 596. (40) Fredlake, C. P.; Crosthwaite, J. M.; Hert, D. G.; Aki, S. N. V. K.; Brennecke, J. F. J. Chem. Eng. Data 2004, 49, 954. (41) Tariq, M.; Forte, P. A. S.; Costa Gomes, M. F.; Canongia Lopes, J. N.; Rebelo, L. P. N. J. Chem. Thermodyn. 2009, 41, 790. (42) Gomes de Azevedo, R.; Esperança, J. M. S. S.; Szydlowski, J.; Visak, Z. P.; Pires, P. F.; Guedes, H. J. R.; Rebelo, L. P. N. J. Chem. Thermodyn. 2005, 37, 888. (43) Jacquemin, J.; Husson, P.; Pa´dua, A. A. H.; Majer, V. Green Chem. 2006, 8, 172. (44) Harris, K. R.; Kanakubo, M.; Woolf, L. A. J. Chem. Eng. Data 2007, 52, 1080. (45) Troncoso, J.; Cerdeirinã, C. A.; Sanmamed, Y. A.; Romanı´, L.; Rebelo, L. P. N. J. Chem. Eng. Data 2006, 51, 1856. (46) Widegren, J. A.; Magee, J. W. J. Chem. Eng. Data 2007, 52, 2331. (47) Kandil, M. E.; Marsh, K. N.; Goodwin, A. R. H. J. Chem. Eng. Data 2007, 52, 2382. (48) Esperança, J. M. S. S.; Guedes, H. J. R.; Canongia Lopes, J. N.; Rebelo, L. P. N. J. Chem. Eng. Data 2008, 53, 867. (49) Muhammad, A.; Mutalib, M. I. A.; Wilfred, C. D.; Murugesan, T.; Shafeeq, A. J. Chem. Thermodyn. 2008, 40, 1433. (50) Marsh, K. N.; Brennecke, J. F.; Chirico, R. D.; Frenkel, M.; Heintz, A.; Magee, J. W.; Peters, C. J.; Rebelo, L. P. N.; Seddon, K. R. Pure Appl. Chem. 2009, 81, 781. (51) Kato, R.; Gmehling, J. J. Chem. Thermodyn. 2005, 37, 603. (52) Kilaru, P.; Baker, G. A.; Scovazzo, P. J. Chem. Eng. Data 2007, 52, 2306. (53) Carvalho, P. J.; Freire, M. G.; Marrucho, I. M.; Queimada, A. J.; Coutinho, J. A. P. J. Chem. Eng. Data 2008, 53, 1346. (54) Kim, Y. C.; Fisher, M. E.; Orkoulas, G. Phys. ReV. Lett. 2003, 67, 061506. (55) Janz, G. J. J. Phys. Chem. Ref. Data 1980, 9, 791. (56) Pandey, J. D.; Chaturvedi, B. R.; Pandey, R. P. J. Phys. Chem. 1981, 85, 1750. (57) Bloom, H.; Davis, F. G.; James, D. W. Trans. Faraday Soc. 1960, 56, 1179.

9194


(58) Lide, D. R., Ed.; CRC Handbook of Chemistry and Physics, 82nd ed.; CRC Press: Boca Raton, FL, 2001. (59) Lide, D. R., Ed.; CRC Handbook of Chemistry and Physics, 89th ed.; CRC Press: Boca Raton, FL, 2008. (60) Blas, F. J.; Martıń del Rıó, E.; de Miguel, E.; Jackson, G. Mol. Phys. 2001, 99, 1851. (61) Gloor, G. J.; Jackson, G.; Blas, F. J.; Martıń del Rıó, E.; de Miguel, E. J. Chem. Phys. 2004, 121, 12740. (62) Gonza´lez-Melchor, M.; Alejandre, J.; Bresme, F. Phys. ReV. Lett. 2003, 90, 135506. (63) Alejandre, J.; Bresme, F.; Gonza´lez-Melchor, M. Mol. Phys. 2009, 107, 357. (64) Gonza´lez-Melchor, M.; Bresme, F.; Alejandre, J. J. Chem. Phys. 2005, 122, 104710. (65) Wohlfarth, C.; Wohlfarth, B. In Surface Tension of Pure Liquids and Binary Liquid Mixtures. Landolt-Bo¨rnstein, New Series, Group IV; Lechner, M. D., Ed.; Springer: Berlin, 1997; Vol. 16. (66) Bowden, S. T. Nature 1954, 174, 613.

Weiss (67) Serr, A.; Netz, R. R. Int. J. Quantum Chem. 2006, 106, 2960. (68) Del Po´polo, M. G.; Voth, G. A. J. Phys. Chem. B 2004, 108, 1744. (69) Rebelo, L. P. N.; Canongia Lopes, J. N.; Esperança, J. M. S. S.; Guedes, H. J. R.; Lachwa, J.; Najdanovic-Visak, V.; Visak, Z. P. Acc. Chem. Res. 2007, 40, 1114. (70) Morrow, T. I.; Maginn, E. J. J. Phys. Chem. B 2002, 106, 12807. (71) Orkoulas, G.; Panagiotopoulos, A. Z. J. Chem. Phys. 1999, 110, 1581. (72) Weiss, V. C.; Schroër, W. J. Stat. Mech. 2008, P04020 and references therein. (73) van Leeuwen, M. E.; Smit, B. Phys. ReV. Lett. 1993, 71, 3991. (74) Errington, J. R. J. Chem. Phys. 2003, 118, 9915. ´ valos, E.; Espıńdola, R.; Rull, L. F.; Jackson, G.; (75) del Rıó, F.; A Lago, S. Mol. Phys. 2002, 100, 2531. (76) Valderrama, J. O.; Robles, P. A. Ind. Eng. Chem. Res. 2007, 46, 1338.

JP102653A

Guggenheim's Rule and the Enthalpy of Vaporization of Simple and

Recommend Documents