Ind. Eng. Chem. Process Des. Dev. lS85, 24, 1306-1308
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Dependence of Mutual DMusivHies on Composition in Regular Solutions: A Rationale for a New Equation A predictive equation for the dependence of mutual diffusivitles on concentration in binary regular solutions has been obtained. The proposed equation proved to be superior to exlsting predictive equations. An attempt to rationalize the proposed equation on the basis of Eyring’s absolute rate theories of diffusion and viscous flow is
presented.
Correlations that can accurately describe the dependence of mutual diffusivities on concentration are indispensible mathematical tools for mass-transfer equipment design. However, up until now, no general correlation has been obtained that can successfully predict, with reasonable accuracy, the dependence of mutual diffusivities on concentration for all classes of binary liquid mixtures. The main reason for this is our almost complete ignorance of liquid structure. Many attempts have been made to obtain empirical correlations, e.g., Rathbun and Babb (1966) and Vignes (1966), and semiempirical correlations, e.g., Leffler and Cullinan (1970). However, these attempts have not met with satisfactory success. Whereas such correlations succeed in predicting the dependence of diffusivity on concentration for a certain class of binary liquid systems, they fail badly in their predictions of another class and break down completely for a third class. Asfour (1979) and Dullien and Asfour (1985), realizing this predicament, reported a correlation that gave better predictions in cases of binary regular solutions than those due to Vignes (1966) and due to Leffler and Cullinan (1970). The correlation due to Asfour (1979) and Dullien and Asfour (1985) has the following advantages over the existing correlations: (i) It does not require the tedious and somewhat uncertain correction for thermodynamic nonideality. This represents a definite advantage from a design standpoint. (ii) It gives better predictions; moreover, it has the capability to track concentration variations for most regular systems where other correlations fail. This capability was clearly manifested in the diagrams reported by Dullien and Asfour (1985) on 16 regular binary solutions. A curious feature of the AsfowDullien correlated is that it uses the quotient of diffusivity and viscosity, rather than the product of the two quantities as conventional wisdom suggests. The objective of this communication is to provide a rationale for a predictive equation that was found to be superior to existing correlations, on the basis of Eyring’s absolute rate theories of diffusion and viscous flow. A modified form of the equation reported in this communication has been obtained. This modified form was found to give exceptional predictions for binary systems consisting of n-alkanes. The modified form consists of the equation reported here multiplied by a correction factor, which can be determined from molecular properties. The results of this investigation will be published shortly. Another modified form of the equation reported in this communication is being currently developed for the case of associated systems. Preliminary results obtained so far seem to be promising. Development of the Equation Eyring and co-workers (Kincaid et al., 1941; Powell et al., 1941) reported the following expression for mutual diffusivity 0196-4305/85/1124-1306$01.50/0
where X is the distance traveled by the “jumping“ molecule, V, is the free volume, k is Boltzmann’s constant, m is the mass of the diffusing molecule, T is the absolute temperature, and AG$ is the activation energy of diffusion. Envisaging that viscous flow is a rate process like diffusion and applying the same arguments of the theory of reaction rates, Eyring arrived a t the following expression for the viscosity, q , A1
77=X2X2X3
( 2 ~ m k r ) l / ~ Vexp(AGt/RT) f1/~
(2)
where X1 is the distance between two adjacent layers, X2 and X3 are the distances between molecules within the same layer, and AGg is the activation energy for viscous flow. Assuming that the form of Eyring’s expressions for AG$ and AG#, is correct, then these expressions may be added (rather than subtracted as assumed by Leffler and Cullinan, 1970). It is possible that the sum of AG$ and AG#, is a quantity that varies more linearly with the mole fraction of the mixture than either AGf, or AG! alone (Asfour, 1979). Consider the quantity AG# = AG$
+ AG!
(3)
The expressions of AG#,can be found by rearranging eq 1 and 2 , respectively, as
and
where N is Avogadro’s number and M is the molecular weight. Substitution of eq 4 and 5 into eq 3 and rearrangement yields
Now, we will designate AGQ as the value of AG# at X A = 0 and AG#, as the value of AG# a t xA = 1. Further, we will assume a linear dependence of AG# on composition. This assumption can be shown to be practically true on the basis of the results of, e.g., Sanni (1976). Thus, AG#(x) = AG$3CB = AG#,xA
-
-
(7)
Using eq 6 and 7 , realizing that when X A 0.0, Dm is DiB, and a t xA 1.0, D A B is DZB, and assuming that A, = X2 0 1985 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
Table I. Percent Average Absolute Error for the Three Mutual-Diffusivity Predictive Equations percent av abs errora Leffler- Asfourvignes’ Cullinan, Dullien system eq eq eq 9.6 10.5 3.9 OMCTS-benzene 3.7 4.4 5.2 bromobenzene-benzene 7.0 3.4 5.1 n-hexane-toluene 0.33 0.7 0.7 chlorobenzene-bromobenzene 17.1 8.2 5.3 diethyl ether-chloroform 4.9 4.0 2.1 benzene-toluene 8.2 8.8 1.5 chloroform-benzene 8.7 5.7 3.8 chloroform-toluene 3.5 4.0 0.8 benzene-carbon tetrachloride 2.1 6.2 3.1 OMCTS-carbon tetrachloride 2.2 0.6 2.3 ryclohexane-carbon tetrachloride 6.6 2.2 7.4 n-hexane-carbon tetrachloride 1.01 1.1 0.9 toluene-carbon tetrachloride 2.3 3.9 2.1 chloroform-carbon tetrachloride 11.7 8.2 2.3 n-hexane-benzene 11.1 3.8 4.0 cyclohexane-benzene 6.3 4.7 3.2 % overall av aPercent average absolute error = l / n 100.
[IDexptl- DpJ/
Dezpt11 X
= A3 = A, as was shown by Leffler and Cullinan (1970), one obtains
and since
therefore,
Arithmetic averages of molecular weights of pure components are conventionally used to calculate molecular weights of mixtures of those components. But since geometric means are more mathematically tractable than arithmetic averages and since molecular weights and free
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volumes of the systems being considered are of roughly the same order of magnitude, one can utilize the relationships M = MAX~MBX~ (11) It should be noted here that mathematically speaking, the geometric mean of two numbers A and B is very close to the arithmetic average of the same two numbers as long as the ratio greater number/lesser number < 2. In our case, of eq 11, the largest deviation between the arithmetic average of A and B and the geometric mean of the same two numbers occurs a t xA = XB = 0.5. Evidently, the arithmetic average and the geometric mean will have the same values a t the terminal points, XA = XB = 1. Now, for all the systems listed in Table I, one can notice that the ratio of the molecular weights of the components is less than 2 except in the case of the OMCTS-benzene system where such a ratio is about 3.8. Notwithstanding this relatively high ratio of molecular weights, the percent average absolute error is only 3.9%, which is impressive when compared with the results obtained by other existing correlations, e.g., Vignes 9.6% and Leffler and Cullinan 10.5%. However, the case of the OMCTS-benzene system is an extraordinary case which is not likely to be encountered when one is dealing with regular solutions for which the proposed equation is intended. Consequently, the condition of the ratio of molecular weights will be satisfied in the great majority of the cases encountered. The relationship given by eq 12 is not directly amenable to test. However, should one attempt to test it by using Hildebrand’s fluidity equation (19711, then extensive viscosity-composition-temperature data are needed. Unfortunately, such data are very scarce. Using data for the system benzene-carbon tetrachloride, the only regular solution for which relatively extensive viscosity-composition-temperature data are available, Asfour (1979) was able to show that the average absolute deviation between the experimental free volumes of the mixture and those calculated by eq 12 is about 6%. Moreover, using similar arguments as those used by Leffler and Cullinan (1970), one can assume that k3 = (xA3)xA ( x B 3 ) x B (13) The assumptions made in eq 11,12 and 13 reduce eq 10 to the form
where DAB has been written instead of D&. D g has been often identified with DAB, the activity corrected diffusion coefficient. It turned out that eq 14
Table 11. Sources of Data system OMCTS-benzene bromobenzene-benzene n-hexane-toluene chlorobenzene-bromobenzene diethyl ether-chloroform benzene-toluene chloroform-benzene chloroform-toluene benzene-carbon tetrachloride OMCTS-carbon tetrachloride cyclohexane-carbon tetrachloride n-hexane-carbon tetrachloride toluene-carbon tetrachloride chloroform-carbon tetrachloride n-hexane-benzene cyclohexane-benzene
viscos diffus Marsh, 1968 Marsh, 1968 International Critical Tables, 1929 Miller and Carman, 1959 Ghai. 1973 Gahi, 1973 Caldwell and Babb, 1956 Caldwell and Babb, 1956 Anderson and Babb, 1961 Anderson and Babb, 1961 Asfour, 1979 Asfour, 1979 Asfour, 1979 Sanni and Hutchison, 1973 Asfour, 1979 Asfour, 1979 Caldwell and Babb, 1956 Grunberg, 1954 Marsh, 1968 Marsh, 1968 Kulkarni et al., 1965 Kulkarni et al., 1965 Bidlack and Anderson, 1964 Bidlack and Anderson, 1964 Ghai, 1973 Ghai, 1973 Kelly et al., 1971 Kelly et al., 1971 Harris et al., 1970 Asfour, 1979 Sanni, 1973 Grunberg, 1954
vap-liq equilib Marsh, 1968 McGlashan and Wingrove, 1956 Funk and Prausnitz, 1970 Bourrely and Chevalier, 1968 Dolezalek and Schulze, 1913 Bell and Wright, 1927 Campbell et al., 1966 Rao et al., 1956 Christian et al., 1960 Marsh, 1968 Scatchard et al., 1939 Christian et al., 1960 Wang et al., 1970 McGlashan et al., 1954 Funk and Prausnitz, 1970 Funk and Prausnitz, 1970
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Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
gives excellent predictions for regular solutions without the activity correction. Equation 14 was tested by using literature data on 16 regular binary solutions. The average absolute error for each system is given in Table I and is compared with the average absolute errors obtained when the correlations due to Vignes (1966) and due to Leffler and Cullinan (1970) are used. Table I1 lists all the literature sources of the data used in testing eq 14. It is clear that eq 14 gives better predictions than the equations due to Vignes and due to Leffler and Cullinan. Moreover, the diagrams reported by Dullien and Asfour (1985) show clearly the superior capability of eq 14 in tracking mutual diffusivity variations with composition. Acknowledgment I acknowledge with thanks the financial support of the Natural Sciences and the Engineering Council of Canada (NSERC). Notation = mutual diffusion coefficient AG = activation energy for diffusive flow DA% AGv = activation energy for viscous flow k = Boltzmann constant m = mass of the diffusing molecule M = molecular weight N = Avogadro’s number R = universal gas constant T = absolute temperature Vf = free volume xi = mole fraction of component i, where i = A, B Literature Cited Anderson, D. K.; Babb, A. L. J. Phys. Chem. 1961, 6 5 , 1281. Asfour, A. A. Ph.D. Thesls, Unlverslty of Waterloo, Waterloo, Ontarlo, Canada, 1979. Bell, T.; Wright, R. J. Phys. Chem. 1927, 3 1 , 1885.
Bidlack, D. L.; Anderson, D. K. J. Phys. Chem. 1964, 6 8 , 3790. Bourrely, J.; Chevaller. V. J. C t ” . Phys. 1968. 65,1961. Caldwell, C. S.;Bab, A. L. J. Phys. Chem. 1956, 6 0 , 51. Campbell, A. N.; Kartzmark, E. M.; Chatterjee, R. M. Can. J . Chem. 1966, 44, 1183. Chrlstian, S. D.; Naparko, E.; Affsprung, H. E. J. Phys. Chem. 1960, 6 4 , 442. Dolezalek, F.; Schulze, T. 2.Phys. Chem. 1913, 83, 45. Dulllen, F. A. L.; Asfour, A. A. Ind. Eng. Chem. Fundam. 1985, 2 4 , 1. Funk, E. W.; Prausnitz, J. M. Ind. f n g . Chem. 1970, 62 (9), 8. Ghai, A. K. Ph.D. Thesis, University of Waterloo, Waterloo, Ontario, Canada, 1973. Grunberg, L. Trans. Faraday SOC. 1954, 5 0 , 1293. Harrls, K. R.; Pua, C. K.; Dunlop, P. J. J. Phys. Chem. 1970, 74, 3518. Hlldebrand, J. H. Science (Washington,D.C.) 1971, 774, 490. “International Critical Tables”; McGraw-Hill: New York, 1929; Vol. 5, p 43. Kelly, C. M.; Wirth, G. B.; Anderson, D. K. J. Phys. Chem. 1971, 75,3293. Kincald, J. F.; Eyrlng, H.; Stearn, A. S. Chem. Rev. 1941, 8 , 301. Kukarni, M. V.; Allen, G. F.; Lyons, P. A. J. Phys. Chem. 1965. 69, 2491. Leffler, J.; Cullinan, H. T. Ind. Eng. Chem. Fundam. 1970, 9 , 84. Marsh, K. N. Trans. Faraday SOC. 1968, 6 4 , 894. Marsh, K. N. Trans. Fara&y SOC. 1988, 64, 883. McGlashan, M. L.; Wlngrove, M. Trans. Faraday SOC. 1956, 52,470. McGlashan, M. L.; Prue, J. E.; Sainsbury. I. E. J. Trans. Faraday SOC.1954, 50, 1284. Miller, L.; Carmen, P. C. Trans. Faraday SOC. 1959, 55, 1831. Powell, R. E.; Roseveare, W. E.; Eyring, H. h e . f n g . Chem. 1941, 3 3 , 430. Rao. R. M.; Sitapathy, R.; Anjaneyulu, N. S. R.; Raju, G. Jj, V.; Rao, C. V. J. Scl. Ind. Res., Sect. 8 1956, 75B,556. Rathbun. R. E.; Babb, A. L. Ind. Eng. Chem. Process Des. Dev. 1966, 5 , 273. Sannl. S. A. Chem. Eng. J. 1976, 72,119. Sanni, S. A.; Hutchison, H. P. J. Chem. Eng. Data 1973, 78, 317. Scatchard, G.; Wood, S. E.; Mochel, J. M. J. A m . Chem. SOC. 1939, 67. 3206. Vignes, A. Ind. Eng. Chem. Fundam. 1966, 5 , 189. Wang, J. L. H.; Boubllkova, L.; Lu, B. C. Y. J. Appl. Chem. 1970, 20, 172.
Department o f Chemical Engineering University of Windsor Windsor, Ontario, Canada N9B 3P4
Abdul-Fattah A. Asfour
Received for review July 5, 1984 Revised manuscript received April 17, 1985 Accepted May 2, 1985
