b_virial

Fluid Phase Equilibria 211 (2003) 3549

Second virial coefficients of polar haloalkanes2002

Constantine Tsonopoulos18 Dorothy Drive, Morristown, NJ 07960, USA

Received 24 January 2003; accepted 25 March 2003

Abstract

The 1975 investigation of the second virial coefficients of polar haloalkanes [AIChE J. 21 (1975) 827] demon-strated an apparent difference between polar haloalkanes and non-hydrogen bonding fluids such as ketones andethers that had been examined earlier [AIChE J. 20 (1974) 263]. The difference was in the dependence of the polarcontribution to B (second virial coefficient) on r (reduced dipole moment). However, the 1975 observation wasbased on data for only three polar haloalkanes. The large number of new measurements that have been made since1990 on chlorofluoroalkanes and, especially, the environmentally-benign hydrofluoroalkanes has made it possibleto re-examine the B of polar haloalkanes. This re-examination has included the correlation of Weber [Int. J. Ther-mophys. 15 (1994) 461], which differs both in the non-polar terms and in the dependence of the polar contributionto B on r.

A brief examination of three non-polar compounds (Ar, n-butane, n-octane) has helped clarify the strengthsand weaknesses of three non-polar correlations (PitzerCurl, Tsonopoulos, Weber). A definitive study of the polarcontribution to B may require an improvement of the non-polar B correlation. The two different approaches to theB of polar haloalkanes (Tsonopoulos, Weber) were investigated primarily for haloalkanes with r > 100, and theconclusion was that the two approaches are roughly equivalent, with Weber possibly having an advantage whenr < 50, but not for chloroalkanes. The 1975 recommendation for the dependence of the polar contribution to Bon r was confirmed by the new data. A more definitive conclusion on this dependence may be realized by usingmolecular modeling techniques. 2003 Elsevier B.V. All rights reserved.

Keywords: Second virial coefficients; Non-polar compounds; Polar haloalkanes; Reduced dipole moment

1. Introduction

The 1975 investigation of haloalkanes [1] was hampered by the limited databaseboth in quantityand quality. These limitations led to the erroneous conclusion that only monohaloalkanes (alkyl halides) This paper was presented at the Halocarbon Workshop at the 17th IUPAC Conference on Chemical Thermodynamics (ICCT

2002), Rostock, Germany, 28 July2 August 2002. Tel.: +1-973-540-9229.

E-mail address: [email protected] (C. Tsonopoulos).

0378-3812/03/$ see front matter 2003 Elsevier B.V. All rights reserved.doi:10.1016/S0378-3812(03)00112-2

36 C. Tsonopoulos / Fluid Phase Equilibria 211 (2003) 3549

required a polar contribution to their B (second virial coefficient). That was because the non-polar part ofthe B correlation appeared to fit satisfactorily the B of the moderately polar dichloromethane (r = 57.6)and the strongly polar 1,1-difluoroethane (r = 153). Thus, the polar contribution to B and its dependenceon r were based on just three compounds: chloromethane, fluoromethane, and chloroethane. But evenwith these limited data it became clear that the dependence of the polar contribution on r was differentfrom that found earlier for non-hydrogen bonding CHO and CHN compounds [2]. For r < 200,the polar contribution to the B of haloalkanes was much weaker than that for the organic compoundscontaining O or N. However, for r > 200, the polar contribution appeared to approach the same limitfor all compounds.

This difference is generally attributed to the fact that at low r values the location and direction of thedipole moment are important, but at much larger values (r > 200) the magnitude of the reduced dipolemoment overshadows the contributions of location and direction, and thus magnitude alone affects thepolar contribution to B.

The large number of new B data that have become available since around 1990, many of them men-tioned in Dymonds 2000 review [3], can now be examined with the goal of testing and, if needed,updating the 1975 correlation for polar haloalkanes; as well as comparing it with Webers 1994 corre-lation [4]. Most of the new B data are on chlorofluoroalkanes and, especially, hydrofluoroalkanes. Theenvironmentally-benign hydrofluoroalkanes have already started replacing the ozone-depleting chloroflu-orocarbons that are currently being used as refrigerants.

2. Correlating equations for non-polar gases

The reduced second virial coefficient (BPc/RTc) can be expressed as a sum of three functions of reducedtemperature, Tr(= T/Tc):

BPcRTc

= f (0) + f (1) + f (2) (1)

The first function, f (0), gives the reduced B for compounds with (acentric factor) = 0, while f (1)corrects the value of the reduced B for compounds with = 0 (that is, for deviations from sphericalsymmetry). For polar compounds, that is, compounds that have a non-zero dipole moment, f (2) gives thepolar contribution to the reduced B (which will be examined in the following section).

Three correlations for non-polar compounds are considered. The first one and the best known wasproposed by Pitzer and Curl in 1957 [5]:

f(0)PC = 0.1445

0.33Tr

0.1385T 2r

0.0121T 3r

(2)

f(1)PC = 0.073 +

0.46Tr

0.5T 2r

0.097T 3r

0.0073T 8r

(3)

Pitzer and Curl determined the coefficients of f (0)PC by fitting the B data for Ar, Kr, and Xe, which have zeroacentric factor. In 1974, Tsonopoulos [2] examined data for Ar and Kr that were reported in 19671969and found that Eq. (2) was unsatisfactory for Tr < 0.75; at Tr = 0.53, the error was 10%. This is illustratedin Fig. 1 that was taken from [2]. Tsonopoulos proposed a slight modification of Eq. (2) that made it

C. Tsonopoulos / Fluid Phase Equilibria 211 (2003) 3549 37

Fig. 1. Second virial coefficient of gases with 0 (from [2]).

possible to fit the data included in Fig. 1 to within 1%:

f(0)T = f (0)PC

0.000607T 8r

(4)

Tsonopoulos also modified Eq. (3) by fitting the 1969 recommendations of Dymond and Smith [6] forn-butane ( = 0.200) and the experimental data of Connolly and Kandalic [7] and of McGlashan andPotter [8] for n-octane ( = 0.398):

f(1)T = 0.0637 +

0.331T 2r

0.423T 3r

0.008T 8r

(5)

Differences from Eq. (3) are relatively small, but persist up to high reduced temperatures (see examplefor CO2 in Fig. 3 of [2]).

In 1994, Weber proposed a new correlation for polar haloalkanes [4]. In his introduction, Weber notedthat he used the modifications by Tsonopoulos, that is, Eqs. (4) and (5) for non-polar gases, but actuallyhe removed the T8r terms from both equations. Thus, Webers correlation can be summarized as

f(0)W = f (0)PC (6)

f(1)W = f (1)T +

0.008T 8r

(7)

Eqs. (1)(7) will be compared with selected data for Ar, n-butane, n-octane, and a haloalkane that isessentially non-polar.


2.1. Argon

In recent years it has been noted, for example by Dymond [3], that many of the older PvT data,which historically have been the source of the B values in the literature, were not corrected for physicaladsorption effects, and therefore the resulting Bs are too negative. This is especially the problem atsubcritical temperatures. Although it is possible to correct for these effects, apparently this was not donefor the Ar and Kr data shown in Fig. 1, which today are generally considered to be too negative.

Eqs. (2) and (4) were examined against the 1994 data of Gilgen et al. [9] for Ar. (In an accompanyingpaper [10], Gilgen et al. also measured the critical properties and the vapor pressure of Ar, which wereused to calculate the acentric factor: = 0.0022.)

The data extend from 110 (Tr = 0.73) to 340 K (Tr = 2.26), and the comparison of the deviations ofEqs. (2) and (4) from the data in Fig. 2 clearly illustrates that these data from Wagners group do notsupport the correction made to f (0)PC in 1974 (to fit the data for Ar and Kr that are plotted in Fig. 1). TheBs from the PvT measurements of Gilgen et al. [9] are in excellent agreement with those derived fromthe speed-of-sound measurements of Estrada-Alexanders and Trusler [11] at 110450 K; the maximumdifference is 0.56 cm3 mol1 (0.53%) at 135 K.

2.2. n-Butane and n-octane

These two compounds were used in 1974 to establish the new f (1). The best data for n-butane are nowconsidered to be [12] the isochoric Burnett measurements of Gupta and Eubank [13] at 264.9450 K.

Fig. 2. B of argon: deviation from data of Gilgen et al. [9]; (- - -) PitzerCurl [5]; () Tsonopoulos [2].


Fig. 3. B of n-butane: deviation from data of Gupta and Eubank [13]; (- - -) PitzerCurl [5]; () Tsonopoulos [2]; ( )Weber [4].

All three correlations are compared in Fig. 3. The 1974 modification of the PitzerCurl correlation givesthe smallest deviation, except for Tr < 0.7, where the original PitzerCurl correlation is superior. On theother hand, Webers correlation is everywhere too positive, and the deviation increases considerably asTr goes from 0.8 to 0.6.

The B values used for n-octane were those calculated at 375575 K with an equation determined byAl-Bizreh and Wormald [14] from fitting the data used in the 1974 investigation [7,8] along with those

Fig. 4. B of n-octane: deviation from equation of Al-Bizreh and Wormald [14], which fits their data and those of Connolly andKandalic [7] and of McGlashan and Potter [8]; (- - -) PitzerCurl [5]; () Tsonopoulos [2]; ( ) Weber [4].


Fig. 5. B of CCl2F2: deviation from smoothed experimental data [1519]; (- - -) PitzerCurl [5]; () Tsonopoulos [2]; ( )Weber [4].

they obtained from their JouleThomson measurements. As Fig. 4 demonstrates, at = 0.398 the over-prediction of B with Webers correlation becomes much more pronounced, while the 1974 modificationappears to give the best results over the entire range. Thus, as increases, the removal of the T8r termfrom f (1)W makes this correlation progressively worse, while the addition of the T8r term to f

(0)T becomes

rapidly less of a factor.

2.3. Dichlorodifluoromethane

CCl2F2 (R-12) is essentially non-polar ( = 0.51 D; r = 7.2), and so it is included in this section. ItsB was examined by Dymond, who found that Webers correlation was significantly better than the 1974modification of the PitzerCurl correlation (see Fig. 5 in [3]). The same data are examined here, and thedeviation of the three non-polar correlations from the smoothed data [1519] is plotted in the new Fig. 5.

Table 1RMSD of non-polar correlations from experimental B data

Compound (; r) RMSD (cm3 mol1)PitzerCurl Tsonopoulos Weber

Ar (0.0022; 0) 0.7 1.1 0.7n-Butane (0.200; 0) 8.6 10.8 29.2n-Octane (0.398; 0) 32.1 6.1 87.2CCl2F2 (0.180; 7.2)a 15.7 17.2 8.6

a and from [20].


Although Webers correlation becomes too positive at Tr < 0.7, resulting from the elimination of the T8rterm, overall it provides the best fit.

This is also shown in Table 1, which presents the RMSD (root-mean-square deviation) of the threecorrelations from the smoothed experimental data for the four non-polar gases examined in this section.It is perhaps surprising that Weber does better than PitzerCurl for CCl2F2 ( = 0.180), even though itis significantly worse for n-butane ( = 0.200). It may be that there is no exact correspondence betweenalkanes and even weakly polar haloalkanes in the dependence of B on .

3. Polar contribution to B

In Eq. (1), f (2) is the polar contribution to B. It was shown in 1974 [2] that for non-hydrogen bondingpolar fluids (such as ketones, ethers, as well as acetonitrile) f (2) can be given by a simple function of Tr:

f (2) = aT 6r

(8)

where the parameter a can be correlated with the reduced dipole moment of the fluid.

r = 2Pc

1.01325 T 2c(9)

In Eq. (9), , the dipole moment of the compound (in the gas phase), is in Debyes (1 D = 3.33564 1030 C m), Pc is in pascals, and Tc is in kelvins. (The factor 1.01325 is used to make r numericallyidentical to that used in [1,2], where it was defined as 1052Pc/T 2c , but Pc was in atmospheres; 1 atm =1.01325 105 Pa.) The relationship between a and r was determined to be

a = 2.140 104r 4.308 10218r (10)In the 1975 investigation of the B of polar haloalkanes [1], it was observed that Eq. (10) worked well

only at high values of the reduced dipole moment (r > 200). However, for r < 200, the absolute valueof a decreased more steeply with decreasing r than Eq. (10) predicted. Thus, on the basis of the datafor chloromethane, fluoromethane, and chloroethane, a different dependence of a on r was proposed forpolar haloalkanes:

a = 2.188 10114r 7.831 10218r (11)(As noted in the introduction, it was erroneously concluded in 1975 that this correction would be requiredonly for monohaloalkanes.)

The difference between Eqs. (10) and (11) for r < 200 is clearly illustrated in Fig. 6. Thus, althoughthe polar parameter a depends very strongly on the reduced dipole moment, the functional dependenceis not universal, unless the reduced dipole moment is very large, say, r > 200. At lower values of r, asnoted earlier, the location and direction of the dipole moment apparently must be taken into account.

Weber also used Eq. (8) for the polar contribution to B, but found a different dependence of a on rthat is given by Eq. (12) which is plotted in Fig. 6.

a = 9 1072r (12)


Fig. 6. Dependence of polar parameter a on reduced dipole moment,r ; (- - -) CHO, CHN [2]; () haloalkanes [1]; ( )Weber [4].

Weber, by removing the T8r terms from his f (0) and f (1), increased the contribution of f (2)W to the B ofpolar halolkanes for r < 200 so that it is non-negligible even for r < 50 (where the a parameter in the1975 correlation can be neglected).

Webers correlation (Eqs. (1), (68) and (12)) is compared with the 1975 correlation for polar haloalka-nes (Eqs. (1), (4), (5), (8) and (11)) in the following section. Although Eq. (11) is used to determine thevalue of a for r < 100, for higher values the new B data are regressed to obtain the optimum value fora in the 1975 correlation.

4. Polar haloalkanes

4.1. Halomethanes

Dymond noted in his recent review [3] that the most extensively investigated polar halomethanes inthe 1990s were CHClF2 (R-22) and the much more environmentally friendly CH2F2 (R-32). The firstcompound has a r < 100, and thus its parameter a was obtained from Eq. (11), but the data for the morepolar CH2F2 were regressed to determine the optimum value for a. Figs. 7 and 8 show the deviation ofthe two correlations from the smoothed experimental data. Both figures show the characteristic increasein the deviation of Webers correlation as Tr decreases.

CHClF2, CH2F2, and three other halomethanes with r > 100 are listed in Table 2 along with theirvalues for and r, the data sources, the optimum value for a in the 1975 correlation, and the RMSDs forthe two polar correlations. (Unless otherwise noted, Tc, Pc, , and were taken from the DIPPR datacompilation [20] .) Generally, the two correlations are equivalent, except for CH3Cl, where the RMSDwith Webers correlation approaches 26 cm3 mol1.


Fig. 7. B of CHClF2: deviation from smoothed experimental data [1619,2123]; () Tsonopoulos [2]; ( ) Weber [4].

4.2. Haloethanes

According to Dymond [3], two-thirds of the measurements on haloethanes were made in the 1990s,and the majority of them were on hydrofluoroethanes with 5 (R-125), 4 (R-134a), 3 (R-143a), and 2(R-152a) fluorine atoms. All four compounds were investigated, along with three other haloethanes withr > 100, and the results are listed in Table 3. As in the case of the halomethanes, the two correlationsare roughly equivalent, even for C2H5Cl, where they show the largest deviations (2729 cm3 mol1).

Fig. 8. B of CH2F2: deviation from smoothed experimental data [2628]; () Tsonopoulos [2]; ( ) Weber [4].


Table 2RMSD of polar correlations from experimental B data for halomethanes

Compound (; r)a Data sources Optimum a for 1975 correlations RMSD (cm3 mol1)Tsonopoulos Weber

CHClF2 (0.219; 76.5)b [1619,2123] (0.00076)c 8.5 11.9CHF3 (0.264; 144.6) [16,2426] 0.0152 5.4 5.0CH2F2 (0.277; 181.0)b [2628] 0.0236 5.0 10.2CH3Cl (0.153; 133.1) [2931]d 0.00489 11.9 25.9CH3F (0.198; 197.2) [23,26,29] 0.0451 3.3 7.0

a and from [20], unless otherwise noted.b from [32].c a obtained from Eq. (11).d Did not use recommended values in [29] at 500600 K.

Two examples are given. Fig. 9 shows the deviations from the smoothed data for 1,1,1,2-tetrafluoroethane(R-134a). Once again we see the large increase in the deviation with Webers correlation as Tr goes below0.7. But in the case of 1,1-difluoroethane (R-152a) in Fig. 10, there is no such increase, even at Tr = 0.6,and as a result Webers correlation is slightly better than the 1975 correlation with an optimum valuefor a.

4.3. Dependence of a on r

The data analysis summarized in Sections 4.1 and 4.2 led to 10 optimum values for the a parameterin the 1975 correlation. These values were expected to give a more definitive relationship between a andr for haloalkanes that would improve upon Eq. (11)which had been based on relatively limited dataavailable in 1975 for just three monohalolkanes: CH3Cl, CH3F, and C2H5Cl.

Table 3RMSD of polar correlations from experimental B data for haloethanes

Compound (; r)a Data sources Optimum a for 1975 correlations RMSD (cm3 mol1)Tsonopoulos Weber

CHF2CF3 (0.305; 75.9)b [3335] (0.00073)c 4.2 4.3CH2FCF3 (0.327; 121.1)b [18,33,3638] 0.00667 3.8 6.1CH3CClF2 (0.231; 108.5) [18,22] 0.00244 8.6 19.1CH3CF3 (0.261; 170.0)b [33,39] 0.0191 4.5 5.9CH3CHF2 (0.275; 152.8)b [18,22,33,38] 0.0143 8.6 7.0CH3CH2Cl (0.190; 103.1) [16,31,40,41] 0.00323 27.4 29.3CH3CH2F (0.220; 132.6) [33] 0.00476 0.7 17.9

a and from [20], unless otherwise noted.b from [42].c a obtained from Eq. (11).


Fig. 9. B of CH2FCF3: deviation from smoothed experimental data [18,33,3638]; ()Tsonopoulos [2]; ( ) Weber [4].

The result was somewhat surprising. When the optimum a values for the 10 haloalkanes were plottedversus r in Fig. 11, the 1975 relationship, Eq. (11), turned out to give a most satisfactory representationof the 2002 results. Thus, although a slight improvement in the fit of the 2002 optimum a values is possiblewith some adjustment in the coefficients of Eq. (11), it was decided to retain the original Eq. (11) forrepresenting the dependence of a on r for polar haloalkanes.

Fig. 10. B of CH3CHF2: deviation from smoothed experimental data [18,22,33,38]; () Tsonopoulos [2]; ( ) Weber [4].


Fig. 11. Dependence of 2002 results for polar parameter a on r : () 1975 correlation [1]; () 2002 optimum values.

5. Where are we?

For non-polar gases, f (0)T appears to be too negative at subcritical temperatures (Fig. 2), but thisdeficiency becomes smaller as increases, because f (1)T becomes progressively more important. On theother hand, the removal of the T8r term from f

(1)W makes Webers correlation too positive at low Tr, and

this deficiency increases as increases to 0.2 (Fig. 3) and 0.398 (Fig. 4). However, Webers correlationis surprisingly good for CCl2F2, = 0.18 (Fig. 5).

For polar haloalkanes, the Tsonopoulos [1] and Weber [4] correlations are roughly equivalent, especiallyconsidering that the results with Webers correlation can be improved by using component-specificparameters in place of Eq. (12). In the case of the Tsonopoulos correlation, results with Eq. (11) aresignificantly worse than with the optimum a value only for CH2F2. This is also the only case where adirect fit of the new a values (Tables 2 and 3) would be a significant improvement over Eq. (11); otherwise,the 2002 investigation supports the 1975 results for the dependence of a on rbut not the erroneous1975 conclusion that only monohaloalkanes require a polar correction to their B.

The elimination of the T8r term from f(1)W improves the predictions with Webers correlation for mildly

polar haloalkanes (r < 50), but not for chloroalkanes (with or without fluorine). Although there is nolonger much interest in chloroalkanes, at least as refrigerants, it would be of interest to measure their Bat low Tr, because some recent data may be too positive. More experimental data may also be needed forC2H5F, for which only three B values have been determined [33].

6. What more do we need for haloalkanes?

If there is an interest in improving the generalized correlations for polar haloalkanes, it may benecessary, as Dymond recommended in his review [3], to go back to the non-polar correlation and


improve both f (0) and f (1) over a wide Tr range. There are limitations both at Tr < 0.7 andTr > 2.

Once f (0) and f (1) are improved, f (2) and the relationship between a and r can be tested and putin some final form. Molecular modeling techniques should be very helpful in establishing the properdependence of the polar contribution to B onr, at both low and high values. Such techniques should makepossible the quantification of dipole location and direction effects, which presumably are important atlow r values ( 200) only the magnitude of the dipole momentmatters. If that turns out to be true, as suspected at present, then all strongly polar, but non-hydrogenbonding, gases would have the same dependence of a on r.

Nothing has been said here about mixtures of haloalkanes. The data available in 1975 were examinedin [1], and a similar approach can be taken in analyzing the new data. On the basis of what has beenfound for other classes of compounds, when the two components of a binary are very dissimilar insize or chemical nature (or both), then a binary interaction constant may be needed to correct for thedeviation of the characteristic temperature, Tcij, from the geometric mean of the two critical temperatures[1,2]:

Tcij = (TciTcj)0.5(1 kij) (13)Finally, what about calculations at high densities, where B alone will not suffice? The traditional

approach is to use the virial equation of state with the less well-known third virial coefficients and, atstill higher densities, the virtually unknown fourth virial coefficients. Or an alternative, simpler approachcan be taken that may work just as well: use the VirialRedlichKwong equation of state [43], whichcombines the B correlation with the original RedlichKwong equation of state.

List of symbolsa parameter of polar contribution to B, f (2)B second virial coefficientf (0), f (1), f (2) dimensionless functions of Tr in B correlationskij characteristic binary constant; see Eq. (13)P pressurePsat vapor pressureR gas constantT absolute temperature (in kelvins)Tcij characteristic temperature used in B correlation for mixtures; see Eq. (13)v molar volume

Greek letters dipole moment, in Debyes (1 D = 3.33564 1030 C m)r reduced dipole moment {r = 2Pc/(1.01325 T 2c ); where Pc is in Pa} acentric factor { = log10 (P satr ) 1}

Subscriptsc critical propertyi, j property of component i, jPC functions in PitzerCurl correlation


r reduced property (Mr = M/Mc, for M = P or T, but not )T functions in Tsonopoulos correlationW functions in Weber correlation

Acknowledgements

The author is grateful to Alexi Tsonopoulos, for assistance in preparation of Figs. 211.

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