Fluid Phase Equilibria 230 (2005) 15–20

An improved method to predict second cross virial coefficients

Alessandro Vetere∗

Viale Gran Sasso 20, Milan 20131, Italy

Received 1 June 2004; received in revised form 19 October 2004; accepted 19 October 2004

Abstract

A previous method of the author to calculate the second cross virial coefficients is re-considered to improve the reliability of the predictionsand to reduce the number of empirical rules. The method is based on the reduced second cross coefficient at the normal boiling temperature,B∗

b, whose value is always assumed equal to unity. This value is then extrapolated to the experimental temperatures using only two empiricalconstants:K1, a corrective multiplying factor ofB∗

b andK, in an exponential term as a multiplying factor of temperature. To improve thereliability of the method, literature experimental data are grouped in three binary classes:

- non-polar or slightly polar fluids;--

low3©

K

1

aot

wr

u

e ap-per-ul influ-

ules.d to

be-notu-ter-

hisbitd toxy-

edic-

ting

0d

at least one strongly polar fluid;strong interactions of the acid–base type.

Only the critical constants and the normal boiling temperature are required as input parameters.Deviations of calculated results from experimental one are in the range 25–40 cm3 mol−1 for the first and the second class and be

00 cm3 mol−1 for the third class.2004 Elsevier B.V. All rights reserved.

eywords: Second cross virial coefficients; Prediction; Polar mixtures; Non-polar mixtures; Associated mixtures

. Introduction

Methods to calculate second virial coefficients rely on thecentric factor concept,ω, put forward by Pitzer[1]. The sec-nd virial coefficient of a pure compound can be calculated

hrough a relation having the following form:

BPc

RTc= F0(Tr) + ωF1(Tr) (1)

here bothF0 and F1 are the power series of the inverseeduced temperature.

Following efforts of several researchers, Eq.(1) has beenpdated[2]:

BPc

RTc= 0.083−0.422

T 1.6r

+ω

(0.139− 0.172

T 4.2r

)+ A

T 6r

− C

T 8r(2)

∗ Tel.: +39 02 295 24380; fax: +39 02 520 57657.E-mail address:a vetere@tin.it.

The last two terms, suggested by Tsonopoulos, enablplication to polar compounds characterized by strongmanent dipoles. The Pitzer–Tsonopoulos is successfpredicting the second virial coefficients of numerousids, but is less reliable for strongly associated molecDifficulties arise when Pitzer-type methods are extendethe calculation of the second cross coefficientBij . Com-bining rules proposed to take into account interactionstween different molecules are often oversimplified orwell justified. Any attempt to link the properties of pure flids to those of a mixture often entails some binary inaction parameters,Kij , which are not easily evaluated. Tis particularly evident for mixtures of fluids which exhistrong hydrogen bonds, like alcohols or water, or tenform acid–base complexes, like chloroform with some ogenated fluids. As a result, no proposed method is prtive [3–8]. Only the Hayden and O’Connell correlation[3]is able to describe mixtures of strongly associating/solvaspecies.

378-3812/$ – see front matter © 2004 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2004.10.034

16 A. Vetere / Fluid Phase Equilibria 230 (2005) 15–20

This paper demonstrates that prediction of the secondcross virial coefficient can be obtained for any type of mixtureusing simple rules, similar to but better than those previouslyproposed by the author.

2. The proposed model

Vetere[9–12] showed that the following equation for thesecond virial coefficients of pure fluids can be derived froman eight-parameter equation of state:

B =( a

1.987T

)exp(b − KT )

−(

a

1.987TBo

)exp(b − KTBo) (3)

where the second term on the right-hand side enables thecalculation of the absolute value ofB up to the Boyle temper-ature,TBo, and beyond. At temperatures appreciably belowTBo, roughlyT≤ 0.85–0.9TBo, Eq.(3) reduces to

B = a

1.987Texp(b − KT ) (4)

without loss of reliability.Eq. (4) can correlate with good accuracy all the experi-

mental range ofB for the vast majority of fluids, both polara

3

or-m

B

a

B

w( Eq.(

K

B

B

w ts atae esst sB f-fiT

according to Martin[13]. Written in reduced form, Eq.(8)becomes

BPc

RTc= TbB

∗b

(exp[K(Tb − T )]

T

)(9)

The non-dimensional parameterB∗b is often near unity also

for polar compounds. Calculation of cross second virial co-efficients according to Eq.(9) requires the following simplemixing rules:

(Tb)ij = K2(Tbi × Tbj)0.5 (10)

(Tc)ij = (Tci × Tcj)0.5 (11)

(Pc)ij = (Pci × Pcj)0.5 (12)

(B∗b)ij = K1[(B∗

b)i × (B∗b)j]

0.5 (13)

Eq.(13)can be approximated by

(B∗b)ij = K1 (14)

if we assume (B∗b)

i= (B∗

b)j

= 1, as suggested by Martin. As aresult, the working relation used in this work for the secondcross virial coefficients is

Bij(Pc)ij(

exp[K((Tb)ij − T )])

w ts apac h intc thatK alb ffer-e .A da

E oylet gooda

T

f

T

fa

nd non-polar, using only two empirical parameters.

. Procedures

In our previous method, the first modification is only fal. Eqs.(3) and(4) are rewritten in a equivalent form as

= K1

[ a

1.987Texp(− KT )

](5)

nd

= K1

[a

1.987Texp(−KT ) − a

1.987TBoexp(−KTBo)

](6)

hereK1 is a substitute for the exp(b) term in Eqs.(3) and4). The other modification is substantial. By applying5) atTb, we have

1a = BbRTb exp(KTb) (7)

y substituting(7) in (5), we obtain

(T ) = TbBb

(exp[K(Tb − T )]

T

)(8)

hich enables calculation of the second virial coefficienny temperature using two unknown terms,Bb andK. How-ver, sinceBb is not easily predicted, it is useful to exprhis quantity through the reduced parameterB∗

b, defined a∗b = BbPc/RTc. It is well known that the reduced virial coecients of pure compounds can be easily generalized atTb andc at least for non-polar compounds:B∗

b = 1 andB∗c = 1/3

R(Tc)ij= K1(Tb)ij

T(15)

hich contains only two empirical constants. There exisossible justification for the apparently too simple rules(11)nd(12): if we introduce a corrective constant,Kij , to cal-ulate (Pc)ij and (Tc)ij , these constants would appear bothe nominator and in the denominator of Eq.(15), providingompensation. Further, because it was empirically found2 in Eq. (10) is different from unity only when the normoiling temperatures of two components are markedly dint, deficiency in the geometric mean rises as�Tb increasest temperatures beyond 0.9TBo, Eq. (15) must be enlarges follows:

Bij(Pc)ijR(Tc)ij

= K1(Tb)ij

{exp[((Tb)ij − T )]

T

−exp[K((Tb)ij − TBo)

ij]

(TBo)ij

}(16)

q.(16)requires no new empirical constants, since the Bemperature for pure compounds can be predicted withccuracy by

Bo = 10+ 4.4Tb (17)

or fluids whoseTb is below 120 K and

Bo = 100+ 2Tc (18)

or all the other fluids. (TBo)ij is obtained from Eqs.(17)nd/or(18)by applying the geometric mixing rule.

A. Vetere / Fluid Phase Equilibria 230 (2005) 15–20 17

Table 1Prediction ofBij for non-polar or slightly polar mixtures

Mixtures K K1 AAD (cm3 mol−1)

This work Ref.[10]

Ar–ethanea 0.0035 0.80 2.3 7.0Ar–propanea 0.0035 0.80 10.7 11.4Ar–pentanea 0.0035 0.80 4.9 13.9Ar–m-xylene 0.0035 0.80 13.4b 37.7Ar–SF6

a 0.0035 0.90 4.3 19.2N2–methanea 0.0035 0.80 3.5 7.0N2–benzenea 0.0035 0.80 13.4 14.9Methane–ethane 0.0035 0.80 1.6 10.0Methane–ethylenea 0.0035 0.80 4.2 2.1Methane–propane 0.0035 0.80 3.3 11.9Methane–n-pentane 0.0035 0.80 8.5 12.9Methane–n-hexane 0.0035 0.80 12.1 7.4Methane–n-heptane 0.0035 0.80 9.7 14.1Methane–n-decanea 0.0035 0.80 16.2b 51.7Methane–benzene 0.0035 0.80 18.1 33.0Methane–CCl4 0.0035 0.80 15.6 36.4Ethylene–CCl4 0.0035 0.80 16.2 69.3Ethane–naphthalene 0.0035 0.80 22.7 45.2Propane–n-hexane 0.0035 0.80 65.8 28.6Propane–n-octane 0.0035 0.80 31.4 90.6Propane–CO2a 0.0055 0.80 13.3 17.3Propane–CH3Br 0.0035 0.90 9.3 21.0n-Butane–n-octane 0.0035 0.8 74.5 35.7n-Butane–CO2a 0.0055 0.80 14.5 22.1n-Butane–CH3Br 0.0035 0.9 31.9 48.6Benzene–cyclohexane 0.0035 0.9 87.3 24.1Benzene–toluene 0.0035 0.9 85.3 7.9Benzene–carbon monoxidea 0.0035 0.9 3.7 7.4Benzene–CHCl4 0.0035 0.9 126.5 170.2CH3Cl–CS2 0.0035 1.0 33.9 9.6CH3Cl–CH3Br 0.0035 1.0 24.5 28.9Mean ADD 25.2 29.6Max. AAD 126.5 170.0

a Systems predicted by applying Eq.(16) instead of Eq.(15).b Systems for which (Tb)ij = 0.8[(Tb)i (Tb)j ] instead of (Tb)ij = [(Tb)i (Tb)j ].

4. Experimental data

Experimental data forB are from the monograph by Dy-mond and Brian Smith[14]. When possible, the choice amongseveral literature sources is limited to the recommended data.Otherwise, the choice is enlarged to include less reliable datato examine a wider family of fluids that differ in molecularstructure and polarity. To compare the new method with theprevious one,Tables 1–3report also the results obtained formixtures studied in Ref.[10].

Critical constants and normal boiling temperatures arefrom Prausnitz et al.[2].

5. Rules and results

By definition, the second virial coefficient arises from var-ious types of forces acting between molecules (typically, os-cillating or permanent dipoles). For second cross virial coeffi-cients, we must consider specific interactions of the acid–basetype which strongly depend on the electronic structures of the

two components. Because a universal procedure is not feasi-ble, we propose different rules for three binary families:

- non-polar or slightly polar fluids;- at least one strongly polar fluid;- strong interactions of the acid–base type.

5.1. Mixtures of the first family

Bij data for mixtures of this family, characterized byweak interaction forces, can be predicted with good accu-racy according to simple rules: the empirical parameters areK = 0.0035 for all examined mixtures, whileK1 assumes threevalues, namely: 0.8 for the vast majority of cases, 0.9 for mix-tures which contain a polar fluid or benzene, 1.0 for mixturesof two slightly polar fluids. Only when the difference be-tweenTbi andTbj is exceedingly high we assumeK2 = 0.8instead of unity to correct the mixing rule expressed by Eq.(10). This occurs for mixtures of CH4, N2, Ar and CO withhigh-molecular-weight fluids, as indicated inTable 1. Exceptfor a few cases, Eq.(15)was used instead of Eq.(16).

18 A. Vetere / Fluid Phase Equilibria 230 (2005) 15–20

Table 2Prediction ofBij for polar mixtures

Mixtures K K2 AAD (cm3 mol−1)

This work Ref.[10]

Ar–methanola 0.008 1.0 14.4 15.5Ar–ethanola 0.008 1.0 7.7 28.0Ar–H2Oa 0.008 0.75 4.1 4.6N2–methanola 0.008 1.0 10.8 5.9N2–n-propanola 0.008 1.0 15.3 16.1N2–H2O 0.008 0.75 1.7 12.5CH4–methanola 0.008 1.0 6.2 28.4CH4–ethanola 0.008 1.0 8.9 28.9Ethane–methanol 0.008 1.0 5.7 22.1H2O–ethanea 0.008 0.75 8.8 10.8n-Butane–acetone 0.008 1.0 247.8 129.9n-Pentane–methanol 0.008 1.0 73.9 132.0Methane–H2Oa 0.008 0.75 1.5 8.3Benzene–acetone 0.008 1.0 95.9 136.4Benzene–methanol 0.008 0.75 43.8 287.4Ethanol–benzene 0.008 0.75 91.5 163.5N2O–methanol 0.008 1.0 11.6 5.1CS2–acetone 0.008 1.0 67.3 27.5Acetonitrile–cyclohexane 0.008 0.75 77.1 184.5Benzene–nitromethane 0.008 1.0 24.6 79.0Cyclohexane–diethylamine 0.008 1.0 2.5 0.6CO2–H2Oa 0.008 1.0 28.3 14.7N2O–H2O 0.008 1.0 7.3 8.3Mean AAD 40.9 65.8Max. AAD 247.8 287.4

a Systems predicted by applying Eq.(16) instead of Eq.(15). K1 is equal to unity for all the above mixtures.

5.2. Mixtures of the second family

Table 2shows that for these mixtures the applied rulesare also very simple. The constants are:K = 0.008 andK1 = 1 for all systems. Only for mixtures containing verystrong polar fluids, like water and acetonitrile, the reli-ability of the method increases by usingK2 = 0.75 in-stead of unity. The lower value is used also for the

Table 3Prediction ofBij for strongly associated mixtures

Mixtures K1 K2 AAD (cm3 mol−1)

This work Ref. [10]

CHCl3–methylformate 1.137 1.2 106.2 210.5CHCl3–methylacetate 1.081 1.2 170.7 72.6CHCl3–ethylacetate 0.964 1.2 128.0 164.6CHCl3–acetone 1.197 1.2 131.6 163.1CHCl3–diethylenamine 1.085 1.2 84.7 427.4CHCl3–diethylether 0.800 1.2 123.5 170.9CHCl3–propylformiate 1.018 1.2 413.9 100.6Acetone–nitromethane 1.239 1.2 471.7 850.1NH3–ethylene 1.322 1.2 44.4 34.9NH3–acetylene 1.05 1.75 57.6 14.9Methanol–methylamine 1.05 1.75 755.0 252.4Methanol–ethylamine 1.05 1.75 572.6 728.1Methanol–propylamine 1.05 1.75 479.4 981.0Mean AAD 272.2 320.8M

K

benzene–alcohol mixtures. The new method appears tobe appreciably better and simpler than the previous one,based on four complex relations to calculate the re-quired parametersB∗

b and B∗c and several ad hoc empir-

ical parameters to improve the prediction[10]. On thewhole, deviations from the experimental data appear ac-ceptable, with the butane/acetone system as the only excep-tion.

ax. AAD

= 0.01 for all mixtures above.

755.0 981.0

A. Vetere / Fluid Phase Equilibria 230 (2005) 15–20 19

5.3. Mixtures of the third family

These mixtures comprise the most difficult cases; forthese systems, no literature method is predictive. Within thisclass, it is useful to make a further subdivision: mixtureswhich interact through strong hydrogen bonds to give weakcomplexes, like CHCl3 with acetone and other oxygenatedcompounds and mixtures which exhibit interactions of theacid–base type, like amines with alcohols or ammonia withacetylene. These systems show the highest absolute values ofBij .

The rules for the first sub-class are:K = 0.003, whileK1varies according to the relation:

K1 = 0.8 + 0.085ωp (19)

Eq.(19)contains the so-called “polar factor”,ωp, defined as

ωp = T 1.72b

M− 263 (20)

Eq. (19) is applied only to data for the oxygenated flu-ids. The best value forK2 is 1.2. The second subclass ispredicted by usingK = 0.01, K1 = 1.05 andK2 = 1.75 forall the systems considered. The results obtained are re-ported in Table 3 for both sub-classes. Deviations fromthe experimental values are often high, particularly fort stemss ausetv

6

nlyt sim-p l co-e ewmP lit-e d, ex-pi re insv waya lityo omec en-t ad-v mwo is oft om-p ities( owne

also for the correspondingBij versusT curve. This experi-mental trend enables reliable extrapolation of the experimen-tal data.

The new method presented here offers a useful alternativeto the literature equations based on Pitzer’s acentric factorfor the calculation of second cross virial coefficients.

List of symbolsa empirical parameter in Eqs.(3) and (7)

(cal cm3 mol−1)A empirical constant in Eq.(2)AAD absolute average error (cm3 mol−1)b empirical constant in Eqs.(3) and(4)B∗

b absolute reduced value of the second virial coeffi-cient of pure compounds atTb, defined asBbPc/RTc

B∗c absolute reduced value of the second virial coeffi-

cient of pure compounds atTc, defined asBcPc/RTcBi absolute second virial coefficient of pure compound

i (cm3 mol−1)(B∗

b)ij

absolute reduced value of the second cross virialcoefficient atTb

C empirical constant in Eq.(2)K empirical constant in Eqs.(3) and(9)M molecular weightPc critical pressure (bar)R 3 −1 −1

TTTTTV

Gω

ω

A

R

.G.,all,

ases0–

v. 14

Dev.

Pro-

he methanol/methylamine system, because these syhow negative deviations from the Raoult’s law and bechey are typically characterized by anomalously highBij

alues.

. Conclusions

Tables 1–3show that a new simple equation, with owo adjustable parameters easily evaluated according tole rules, enables prediction of the second cross viriafficients for a large variety of interacting fluids. The nethod requires only very limited input data (Tb, Tc and

c only), providing an advantage not offered by otherrature methods. Regarding the accuracy of the methoressed by the absolute deviations reported inTables 1–3,

s important to remember that the experimental data aome instances doubtful, primarily for two reasons: theBij

alues are not experimentally accessible in a directnd, further, it is difficult to measure of the non-ideaf a gas mixture at very low densities. Therefore, in sases, the deviation of calculated results from experimal results is within the experimental uncertainty. Theantage of Eqs.(15) and (16) lies in the exponential terhich provides the correct dependence of bothBi and Bij

n temperature. This exponent arises from an analyshe experimental behaviour of the isometrics of pure counds, which are linear in a wide range of fluid denssee Ref.[12]). Therefore, the same linear behaviour shlsewhere for theBi versusT curve (see Ref.[11]) is found

universal gas constant (bar cmbar K mol )absolute temperature (K)

b normal boiling temperature (K)Bo Boyle temperature (K)c critical temperature (K)r reduced temperaturelb liquid molar volume atTb (cm3 mol−1)

reek lettersacentric factor

p polar factor, defined by Eq.(20)

cknowledgements

Musa, cantai la lode della mia strofe lunga!

eferences

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