# A simple modification of the Pitzer method to predict second virial coefficients.

INTRODUCTION

The literature on the second virial coefficient prediction over the last twenty years appears stagnant, as compared with the new methods widely proposed in the sixties and seventies. Few papers have been published since 1980 showing only marginal contributions to the matter, being only intended to improve the performances of the existing methods for a narrow class of compounds not previously considered.

There is a general consensus that after the relevant improvement of the Pitzer relation due to Tsonopoulos (Tsonopoulos and Prausnitz, 1970; Tsonopoulos, 1974, 1975, 1978; Tsonopoulos and Heidman, 1990), there is not much to do, instead waiting for the "true method" that new statistical thermodynamics promoted by Panagiotopoulos and other theoretical physicists will give in the next ten years (Prausnitz, 2005).

However, in spite of its undeniable success, the Tsonopoulos method has a weak point: the dipole moment value, [mu], on which the method hinges, is not always well known. Further, [mu] is not significant enough to improve the Bi predictions for some classes of compounds. Therefore, there is room for some improvements based on two constraints: first, to improve the Pitzer equation without modifying its analytical form; second, to restrict the input parameters required to apply the new Pitzer relation to the normal boiling temperature and critical values only.

THE PROPOSED MODEL

The Pitzer equation for the calculation of the second virial coefficients of pure compounds, written in its more updated form according to Tsonopoulos (Prausnitz et al., 1999) is:

[BP.sub.c]/[RT.sub.c] = 0.1445 - 0.330/[T.sub.r] - 0.1385/[T.sup.2.sub.r] 0.0121/[T.sup.3.sub.r] - 0.000607/[T.sup.8.sub.r] + [omega] (0.0637 + 0.331/[T.sup.2.sub.r] - 0.423/[T.sup.3.sub.r] - 0.008/[T.sup.8.sub.r]

Relation (1) gives reliable results for normal and slightly polar fluids. Tsonopoulos greatly improved the performance of the Pitzer relation for a wide class of polar compounds by adding a simple function of [T.sub.r] to Equation (1):

F([T.sub.r]) = + a/[T.sup.6.sub.r] - b/[T.sup.8.sub.r] (2)

in which both a and b are simple functions of the reduced dipole moment, defined as:

[[mu].sub.R] = 98690[[mu].sup.2][P.sub.c]/[T.sup.sub.c] (3)

Both a and b require different empirical relations for each class of chemical species.

However, the unmodified Pitzer relation reveals a surprising feature: it is very reliable in predicting the second virial coefficient of some strongly polar inorganic compounds which have an appreciable dipole moment, like water ([mu] = 1.8), ammonia ([mu] = 1.5), hydrogen chloride ([mu] = 1.1), sulphur dioxide ([mu] = 1.6) and phosphine ([mu] = 0.6). As a result, at least in some cases, [mu] should not be considered an appropriate parameter to evaluate the polarity degree of a molecule. These relevant exceptions to the assumptions of the Tsonopoulos method suggest the possibility of introducing a new parameter, [beta], which embodies two contributions: the first related to the acentricity, [alpha], and the second to the polarity, [[alpha].sub.p], of the considered fluid. In other words, if we introduce the [beta] parameter defined as:

[beta] = [alpha] + [[alpha].sub.p] (4)

instead of [omega] into Equation (1) it becomes:

[BP.sub.c]/[RT.sub.c] = 0.1445 - 0.330/[T.sub.r] - 0.1385/[T.sup.2.sub.r] - 0.0121/[T.sup.3.sub.r] - 0.000607/[T.sup.8.sub.r] + [beta](0.0637 + 0.331/[T.sup.2.sub.r] - 0.423/[T.sup.3.sub.r] - 0.008/[T.sup.8.sub.r] (5)

The analytical form of the Pitzer equation is preserved and an appreciable gain in simplicity with respect to the Tsonopoulos method is achieved. According to the second constraint of this work, [beta] must be generalized for classes of fluids by correlating a wide range of experimental data for each class of fluids. This can be achieved by using relations involving [T.sub.b], M or critical constants only.

Accordingly, we assume that [beta] can be generalized for classes of compounds by using a linear relation of the Hildebrand solubility parameter:

[beta] = A + B [[delta].sub.i] (6)

or a linear relation of the polarity factor, [W.sub.p]:

[beta] = [A.sub.1] + [B.sub.1] [W.sub.p] (7)

being:

[W.sub.p] = [T.sub.b] 1.72/M - 263 (8)

The calculation of the Hildebrand solubility parameter in Equation (6) does not require the knowledge of the experimental enthalpy of vaporization, since this datum can be calculated with good accuracy by applying the Vetere method (Vetere, 1999), based on [T.sub.b] and M only.

EXPERIMENTAL DATA

The source of experimental data for [B.sub.i] used in this paper is the monograph by Dymond and Smith (1980). When possible, the choice among several literature sources was limited to the recommended data only. Less reliable data have been included in the data base only to examine a wider family of fluids of dissimilar structure and polarity. Critical constants and the normal boiling temperatures were taken from Poling et al. (1999a). The pure component properties required to calculate the Hildebrand solubility parameter were evaluated by applying the Vetere relations (Vetere, 1999) for the enthalpies of vaporization, which are based on [T.sub.b] and molecular weight only, and the Rackett equation for molar volumes.

The relations used to calculate the Tsonopoulos parameters a and b were taken from Poling et al. (1999a), corrected for a printing error reported in Table 4.5 of this source for the class of alkyl halides, mercaptans, sulphides and disulphides.

RULES AND RESULTS

Table 1 reports the generalized relations used in this work to calculate [beta] according to the modified Pitzer method for several chemical species. They were derived by regressing the experimental data of some compounds pertaining to a same family and labelled with an asterisk in Tables 2-4. The optimized [beta] values of these compounds are generalized as a linear functions of the Hildebrand solubility parameter, [[delta].sub.i], or the polar parameter [W.sub.p]. Table 2 reports comparisons between the proposed modification of the Pitzer method and the Tsonopoulos method for alkyl halides, mercaptans, sulphides, disulphides and ethers representing the class of slightly polar compounds. It must be stressed that perfluoro-hydrocarbons require ad hoc parameters for the modified Pitzer methods. A possible justification of this rule is the following: the molecules of perfluoro-compounds are surrounded by a cloud of negatively charged fluorine atoms which partly compensate the attractive part of the interaction potential. Table 3 reports the results obtained in predicting the [B.sub.i] values of the strongly polar fluids: esters, ketones, aldehydes and alcohols. It appears that the modified Pitzer relation is slightly better than the Tsonopoulos method. Table 4 reports several data pertaining to amines and heterocyclics. It must be stressed that for these compounds Tsonopoulos does not give any rules. This fact, perhaps, confirms that the reduced polar moment is not a significant parameter for all types of polar interactions. On the contrary, the use of the new parameter, [beta], appreciably improves the Pitzer relation performances.

Overall, the AAD factor is generally acceptable, except in few cases. However, when the deviations are very high (> 90-100 [cm.sub.3] [mol.sub.-1]) with different method, we can share the statement of O'Connell: "Often, all methods are poor, suggesting that the data may be incorrect" (Poling et al., 1999b).

CONCLUSIONS

The well known motto "sick simplicity and wring its neck" is doubtful in many cases. This work shows that a drastic simplification of an old method leads to an improvement of its reliability in predicting second virial coefficients for a large number of polar and slightly polar fluids. Above all, the relevant advantage of the new method is that it removes from the second virial coefficient calculations the need for experimental data which are in some cases unknown or not reliable, such as the dipole moment. The new parameter, [beta], which substitutes for both [omega] and [mu], can be evaluated using the simple relations (6)-(8) that are related to the normal boiling temperature and the critical constants only, as required by the assumed constraints.
```NOMENCLATURE

a empirical parameter in Equation (2)
b empirical parameter in Equation (2)
A and B empirical parameters in Equation (6)
[A.sub.1] and [B.sub.1] empirical constant in Equation (7)
[B.sub.i] absolute second virial coefficient of pure
compound i, [cm.sup.3]/mol
F function defined by Equation (2)
M molecular weight
[P.sub.c] critical pressure, bar
R universal gas constant, bar [cm.sup.3]/K
T absolute temperature, K
[T.sub.b] normal boiling temperature, K
[T.sub.c] critical temperature, K
[T.sub.r] reduced temperature
[W.sub.p] polar factor, defined by Equation (8),
[K.sup.1.72]

Greek Symbols

[alpha] non-polar contribution to the acentric
factor in Equation (4)
[[alpha].sub.p] polar contribution to the acentric factor in
Equation (4)
[beta] empirical parameter defined by Equations (4)
and (7)
[[delta].sub.i] Hildebrand solubility parameter,
[cal.sup.0.5]/[cm.sup.1.5]
[mu] dipole moment, debye
[[mu].sub.R] reduced dipole moment, defined by Equation
(3)
[omega] Pitzer acentric factor
```

Manuscript received January 19, 2006; revised manuscript received April 25, 2006; accepted for publication August 28, 2006.

REFERENCES

Dymond, H. and E. B. Smith, "The Virial Coefficients of Pure Gases and Mixtures," Clarendon Press, Oxford (1980).

Poling, B. E., J. P. O'Connell and J. M. Prausnitz, "The Properties of Gases and Liquids," 5th ed., McGraw-Hill, New York (1999a).

Poling, B. E., J. P. O'Connell and J. M. Prausnitz, "The Properties of Gases and Liquids," 5th ed., McGraw-Hill, New York (1999b), pp. 40-41.

Prausnitz, J. M., R. N. Lichtenthaler and E. G. Gomes de Azevedo, "Molecular Thermodynamics of Fluid Phase Equilibria," Prentice Hall, New Jersey (1999), pp.165.

Prausnitz, J. M., private communication, CALTECH, Berkeley, U.S.A. (2005).

Tsonopoulos, C. and J. M. Prausnitz, "Fugacity Coefficients in Vapor-Phase Mixtures of Water and Carboxylic Acids," Chem. Eng. J. 1, 273-276 (1970).

Tsonopoulos, C., "An Empirical Correlation for Second Virial Coefficients," AIChE J. 20, 263-272 (1974).

Tsonopoulos, C., "Second Virial Coefficients of Polar Aloalkanes," AIChE J. 21, 827-829 (1975).

Tsonopoulos, C., "Second Virial Coefficients of Water Pollutants," AIChE J. 24, 1112-1115 (1978).

Tsonopoulos, C. and J. L. Heidman, "From the Virial to the Cubic Equations of State," Fluid Phase Equilib. 57, 261-276 (1990)

Vetere, A., "An Improved Method to Predict the Second Virial Coefficients of Pure Compounds," Fluid Phase Equilib. 164, 49-59 (1999).

Alessandro Vetere

Viale Gran Sasso, 20, 20131 Milan, Italy

```Table 1. Rules to calculate the parameter for the modified
Pitzer method

Classes of compounds [beta]

Heterocyclics 0.14 + 0.0023 [W.sub.p]

Perfluoro-compounds 0.55

Ethers, alkyhlalides, mercaptans, 0.25
sulphides, disulphides

Esters 0.44 + 0.0037 [W.sub.p]

Ketones 0.44 + 0.0037 [W.sub.p]

Aldehydes 0.44 + 0.0037 [W.sub.p]

Amines 0.285 + 0.0022 [W.sub.p]

Alcohols -0.94 + 0.16 [[delta].sub.i]

Table 2. Prediction of [B.sub.i] for slightly polar compounds
with two methods

Compounds [beta] Mod Pitzer Tsonopoulos

C[Cl.sub.4] 0.25 75.7 99.2
CH[Cl.sup.*.sub.3] 0.25 9.9 28.0
[CH.sub.3]Br 0.25 29.8 57.2
1,2 Dichloroethane * 0.25 10.1 8.8
1,1 Difluoroethane * 0.25 9.9 14.0
Fluoro-benzene * 0.25 15.6 16.3
Hexafluorobenzene 0.55 20.6 108.0
Perfluoro-2-methylpentane 0.55 61.4 73.0
Dichlorofluoromethane 0.25 111.2 93.6
Perfluoropropane 0.55 68.7 15.0
Methylethylsulphide 0.25 97.0 91.2
Diethylsulphide 0.25 97.0 58.8
Ethylmercaptane 0.25 40.9 26.6
Dimethylether 0.25 32.3 43.8
Diethylether 0.25 87.6 77.1
Mean 50.9 57.6

* Compounds whose experimental data were used to derive
the relation reported in Table 1 for this class of molecules

Table 3. Prediction of [B.sub.i] of polar compounds with two methods

Compounds [beta] Mod Pitzer Tsonopoulos

Methylformate * 0.551 21.7 29.6
Ethylformate 0.395 74.7 56.6
Methylacetate 0.511 39.3 20.7
Ethylacetate 0.630 65.2 103.8
Propylformate 0.406 87.0 65.5
Methylpropionate 0.545 67.8 71.9
Acetone * 0.820 17.7 25.6
Methylethylketone 0.620 97.0 234.5
Methylpropylketone 0.630 85.9 141.2
Diethylketone * 0.630 16.6 26.4
Acetaldehyde 1.020 40.4 91.5
Propanal 0.830 90.0 110.8

Mean 70.3 80.9

Alcohols

Methanol 1.260 74.9 80.9
Ethanol 1.166 163.7 68.6
1-Propanol 0.598 68.7 20.4
2-Propanol * 0.840 50.2 85.5
1-Butanol 0.473 81.9 32.3
2-Butanol * 0.630 43.0 175.8
Isobutanol * 0.630 38.0 153.2
Ter-Butanol 0.630 59.3 106.4
Mean 72.5 90.4

* Compounds whose experimental data were used to
derive the relation reported in Table 1 for
this class of molecules

Table 4. Prediction of [B.sub.i] for amines and heterocyclics

Compounds [beta] Pitzer Mod Pitzer

Methylamine 0.761 33.6 44.4
Dimethylamine * 0.496 49.2 14.4
Ethylamine * 0.544 76.8 18.9
Diethylamine 0.348 67.8 36.4
Trimethylamine * 0.294 30.4 7.7
Triethylamine 0.256 191.5 48.6
Pyridine 0.361 47.4 159.1
A-Picoline 0.281 25.2 47.8
Pyrrolidine 0.340 54.0 10.4
Pyrrole * 0.573 464.5 8.8
2,6 Dimethylpyridine 0.224 129.0 312.1
Furane 0.167 25.7 31.7
Methylfurane 0.162 137.0 48.6
Thiophene 0.208 54.8 62.4
Mean 99.1 63.9

* Compounds whose experimental data were used to derive the
relation reported in Table 1 for this class of molecules
```
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