Correlation and prediction of the surface tensions of mixtures

Correlation and prediction of the surface tensions of mixtures

The Chemical Engineering Journal, 38 (1988) 205 - 208 205 Short Communication u* = A(T*)B Correlation and Prediction of the Surface Tensions of ...

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The Chemical Engineering

Journal,

38 (1988)

205 - 208

205

Short Communication u* = A(T*)B

Correlation and Prediction of the Surface Tensions of Mixtures RALPH DIGUILIO

and AMYN

S. TEJA

School of Chemical Engineering, Georgia Institute Technology, Atlanta, GA 30332-0100 (U.S.A.) (Received September 25, 1987; February 15,1988)

of

in final form

ABSTRACT

The recently proposed Lielmezs-Her-rick equation for the surface tension of pure saturated liquids has been extended to mixtures using three sets of mixing rules. The results are compared with those obtained using a modified Guggenheim equation for 21 binary and 4 ternary systems. Our results show that the simple Kay’s rules can be used to calculate the surface tensions of mixtures with surprising accuracy. 1. INTRODUCTION

Lielmezs and Herrick [l] have recently proposed a semiempirical equation for the surface tension of saturated pure liquids between the triple point and the critical point. They tested their equation for 34 liquids and found that their results compared favorably with the results obtained by using the methods of Brock and Bird [2], Jasper [ 31 and Sprow and Prausnitz [4]. We have extended the Lielmezs-Herrick equation to mixtures using mixing rules that are analogous to those used in corresponding states correlations. Three sets of mixing rules as well as a semiempirical equation proposed by Guggenheim [5] were tested on 21 binary and 4 ternary systems and the results are presented below. 2. EXTENSION EQUATION

OF THE LIELMEZS-HERRICK

TO MIXTURES

Lielmezs and Herrick proposed the following functional relation for the surface tension of pure liquids: 0300-9467/88/$3.50

(1)

where A and B are universal constants. The values of A and B were determined by regression of surface tension data on 34 liquids and were found to be A = 1.002 855

B = 1.118 091

(2)

The reduced surface tension u* and the reduced temperature T* were introduced on the basis of phenomenological scaling and renormalization group theory and were defined as follows: (T* =

T*

UlT UNBITNB

(3)

(TcP”) - 1

_

(TJTNB)

-

1

(4)

where u is the surface tension at any temperature T, UNB is the surface tension at the normal boiling point TNB and T, is the critical temperature. A knowledge of UNB, TNB and T, is therefore required in order to calculate the surface tension of the pure liquid at any temperature T between the triple and critical points. Values of UNB, TNB and T, for 34 liquids were reported by Lielmezs and Herrick in their work. A simple way to extend the Lielmezs Herrick equation to mixtures is by extension of Kay’s mixing rules. Thus T,= zXiT,i i TNB

(5)

=>iTNBi i

UNB = BiuNBi i

(7)

where xi is the mole fraction of component i in the mixture. Equation (5) is the original Kay’s rule for the pseudocritical temperature of the mixture. Mixing rules for the pseudoboiling temperature and pseudoboiling surface tension are proposed here by analogy. Equations (5) - (7) are termed “mixing rules A” in the work described below. @ Elsevier Sequoia/Printed

in The Netherlands

206

It is well known [6] that Kay’s rules are only valid for mixtures of spherical molecules and become increasingly inadequate as the components of a mixture differ in size, shape or polarity. A more theoretically justified mixing rule for the pseudocritical temperature was proposed by van der Waals and may be written

Tc= C~XiXj(T,iT~)1'2 i

(8)

j

The van der Waals mixing rule works well for non-polar mixtures of components which differ moderately in size and shape. It must often [6] be modified to include some experimental information as follows:

TABLE 1 Summary of results in average absolute per cent deviations between calculation and experiment for the four mixing rules Mixtures

Cryogenics Nitrogen-argon [ 41 Nitrogen-carbon monoxide Nitrogen-methane [ 4 ] Argon-methane [ 41 Carbon monoxide-methane Average

No. of points

19 10 12 11 11

[ 41

[ 41

Non-polar Isooctane-n-dodecane [7 ] Isooctane-cyclohexane [7 ] Isooctane-benzene [7 ] Cyclohexane-n-hexane [9 ] Benzene-n-dodecane [lo] Benzene-n-hexane [ 10 ] Average

9 5 9 19 23 23

Polar Acetone-1-butanol [ 111 Carbon tetrachloride-1-propanol Ethanol-1,4dioxane [ll] Methanol-1,4dioxane [ll] Average

T (K)

83.32 83.32 90.67 90.67 90.67

303.16 303.15 303.16 298.16 - 308.16 298.16 - 313.16 298.16 - 313.16

Mixing rules

Eqn. (11) Q

A

B

C

D

6.62 2.62 21.61 5.61 10.23 9.34

6.61 2.62 21.71 5.67 10.32 9.39

5.18 2.10 17.84 2.30 6.35 6.75

0.96 2.48 1.15 0.61 2.78 1.59

0.0501 0.0228 0.0514 0.0406 0.0663

1.36 6.74 10.96 3.11 7.65 7.20 6.17

1.32 6.75 10.96 3.13 7.75 7.22 6.19

1.28 1.79 1.60 1.87 2.64 1.25 1.74

1.37 1.38 1.46 1.68 0.80 1.03 1.29

-0.0625 0.0079 0.0114 0.0193 0.0028 0.0141

8 6 6 6

303.16 303.16 303.16 303.16

3.96 5.48 3.67 4.59 4.42

3.93 5.48 3.67 4.56 4.41

1.79 1.80 3.62 2.14 2.34

2.30 2.59 3.64 2.88 2.85

-0.0013 0.0007 -0.3586 -0.0176

Binaries used in ternaries Hexane-octane [ 121 Hexane-tetradecane [ 121 Octane-tetradecane [ 12 ] Hexane-octane [ 121 Hexane-tetradecane [ 121 Octane-tetradecane [ 12 ] Hexane-octane [ 12 ] Hexane-tetradecane [12] Octane-tetradecane [12] Hexane-tetradecane [13, 141 Hexane-hexadecane [ 15 ] Decane- hexadecane [ 13 ] Average

8 5 7 8 5 7 8 5 7 11 7 4

313.16 313.16 313.16 333.16 333.16 333.16 353.16 353.16 353.16 303.16 293.16 303.16

1.75 2.96 1.33 0.99 3.86 1.67 1.40 3.86 1.74 2.69 3.17 3.31 2.39

1.15 3.48 1.56 0.97 4.32 1.89 1.39 4.26 1.94 2.49 3.12 4.06 2.55

0.46 0.94 0.88 0.39 1.66 0.87 0.84 1.57 0.78 1.20 1.72 0.68 1.00

0.80 0.75 0.95 0.68 1.58 0.99 0.76 1.47 0.96 1.88 0.86 1.24 1.03

-0.0083 -0.1320 -0.0513 -0.0104 -0.1438 -0.1526 -0.0093 -0.1625 -0.4594 -0.0083 -0.0807 -0.0095

Ternaries Hexane-octane-tetradecane Hexane-octane-tetradecane Hexane-octane-tetradecane Hexane-decane-hexadecane Average

24 24 24 24

313.16 333.16 353.16 303.16

1.83 2.23 4.29 4.81 3.29

6.73 6.71 8.03 19.13 10.15

5.06 5.13 6.52 21.61 9.58

4.62

5.58

3.35

Overall average

[ 161 [ 161 [ 16 ] [ 13 ]

[ 111

1.48

where 6ij is a binary interaction parameter which is obtained from experimental data on the binary system. The advantage of eqn. (9) is that au is often close to zero and that the extension to ternary and higher systems requires experimental information only for the constituent binaries. Equations (6) - (8) are termed “mixing rules B” and eqns. (6), (7) and (9) are termed “mixing rules C” in the work described below. Finally, Guggenheim [5] proposed a singleparameter equation for the surface tension of a binary mixture of molecules of the same size as follows: exp(-

$)

=r,exp(-

$)

(10) where u, u1 and u2 are surface tensions of the mixture, pure component 1 and pure component 2 respectively; a is the average surface area per molecule, kT is the thermal energy and x1, 3t2 are mole fractions of components 1 and 2. Evans and Clever [7] have shown that the Guggenheim equation can be used for the surface tensions of moderately non-ideal mixtures if an empirical value is employed for the surface area a. We have used the following form of eqn. (10): exp(-

-$)

=x1 exp[-

1. The calculations included the pure species at either end of the composition range. Values of TNsi and Tci were taken from the compilation of Reid et al. [6] when they were not reported by Lielmezs and Herrick [ 11. Similarly, the surface tensions of the pure components at the normal boiling point uNB~were obtained from the compilation by Jasper [3] when not given by Lielmezs and Herrick. A total of 21 binary systems and 4 ternary systems were evaluated. They included cryogenic mixtures and non-polar mixtures as well as mixtures containing polar molecules. The Lielmezs-Herrick equation reproduces the surface tensions of mixtures with average absolute deviations of 5% when combined with mixing rules A and B, 3.5% when combined with C and 1.5% when combined with mixing rules D. Comparison of methods A and B shows that the simple Kay’s rules work as well as the van der Waals rules - in agreement with the findings of Brucks and Murad [8] who used a different method for the calculation of pure component surface tensions. Methods C and D require experimental data and are, therefore, of limited utility. Although method D (modified Guggenheim equation) represents the data best, it is limited to binary systems and cannot be used for predictive purposes. We therefore recommend that the extended Kay’s rules (mixing rules A) be used for the calculation of the surface tensions of mixtures with the LielmezsHerrick equation.

2) REFERENCES

+x,exp

(

-

2

CIT1

(11)

where the empirical parameter (Yis obtained from experimental data on the binary system. Equation (11) is termed “mixing rules D” in the work.described below. Since the parameter (Yhas no theoretical meaning, eqn. (11) cannot easily be extended to ternary and higher mixtures.

3. RESULTS

AND CONCLUSIONS

The results of our calculations for the four sets of “mixing rules.” are presented in Table

J. Lielmezs and T. A. Herrick, Chem. Eng. J., 32 (1986) 165. J. R. Brock and R. B. Bird, AZCh.E J., 1 (1955) 174. J. J. Jasper, J. Phys. Chem. Ref. Data, 1 (1972) 174. F. B. Sprow and J. M. Prausnitz, Trans. Faraday Sot., 62 (1966) 291. E. A. Guggenheim, Mixtures, Clarendon, Oxford, 1952. R. C. Reid, J. M. Prausnitz and T. K. Sherwood, The Properties of Gases and Liquids, McGrawHill, New York, 1977, 3rd edn. H. B. Evans and H. L. Clever, J. Phys. Chem., 68 (1964) 3433. M. G. Brucks and S. Murad, Chem. Erg. Commun., 40 (1986) 345.

9 H. L. Clever and W. E. Chase, J. Chem. Eng. Data, 8 (1963) 291. 10 R. L. Schmidt, J. C. Randall and H. L. CIever, J. Phys. Chem., 70 (1966) 3912. 11 D. T. Ling and M. Van Winkle, Znd. Eng. Chem., 3 (1958) 88. 12 P. P. Pugachevich and A. I. Cherkasskaya, Russ. J. Phys. Chem., 54 (1980) 1330.

13 J. D. Pandey and N. Pant, J. Am. Chem. Sot., 104 (1982) 3299. 14 P. P. Pugachevich and E. M. Beglyarov, Kolloidn. Zh., 32 (1970) 895. 15 A. Aveyard, Trans. Faraday Sot., 63 (1967) 2778. 16 P. P. Pugachevich and A. I. Cherkasskaya, Russ. J. Phys. Chem., 54 (1980) 1328.