never negligible. For example, in the lattice frame, which we assumed identical with the volume frame for our calculations on the toluene-chlorobenzene-bromobenzene system, there is no formal contribution to coupling. The most encouraging aspect of our calculations is the agreement with experiment in the general form of the concentration dependence of the theoretical phenomenological coefficients. This dependence, as given by (34) and (35), appears to be similar t o that obtained as a n empirical rule (Leffler and Cullinan, 1970), but is now obtained as a consequence of the model and theoretical arguments. The most discouraging aspect is the failure of the parameters a and b to have the theoretically predicted relative sizes when obtained by fitting data. This would indicate that more work on the kinds of two-body processes included in calculations and the assumed kinds of lattices would be justified, especially if more and better data become available. For example, a better test of the theory would be possible if one had data for large mole fractions of the diffusing species. literature Cited
Bennett, C. H., Alder, B. J., J . Chem. Phys. 54, 4796 (1971). Burchard, J. K., Toor, H. L., J . Phys. C h e p 66,2015 (1962). Burton, J. J., Jura, G., J . Phys. Chem. Solzds 28,705 (1967). Cullinan, H. T., IND.ENG.CHEM.,FUNDAM. 7, 331 (1968).
Cullinan, H. T., Cusick, N. R., IND.ENG.CHEM.,FUNDAM. 6, 72 (1967). Fehder, P.’L., Emeis, C. A,, Futrelle, R. P., J . Chem. Phys. 54, 4921 (1971). Glasstone, S Laidler, K. J. Erying, H., “The Theory of Rate Processes,” 1st ed, pp 477-522, McGraw-Hill, New York, N. Y., 1941. Green, M. S., J . Chem. Phys. 20, 1281 (1952). Green, M. S., J . Chem. Phys. 22, 398 (1954). Holmes, C. F., Mortimer, R. G., J . Chem. Phys. 52,4345 (1970a). Holmes, C. F., Mortimer, R. G., J . Chem. Phys. 52,4355 (1970b). Kirkdood, J. G., J . Chem. Phys. 14, 180 (1946). Kirkwood, J. G., Baldwin, R. L., Dunlap, P. J., Gosting, L. J., Kegeles, G., J . Chem. Phys. 33, 1505 (1960). Kirkwood, J. G., Fitts, D. D., J . Chem. Phys. 33, 1317 (1960). Kubp, R., J . Phys. SOC.J a p . 12,570 (1957). Lane, J. E., Kirkaldy, J. S., Can. J . Phys. 42,1643 (1964). Leffler, J., Cullinan, H. T., IND.ENG.CHEM.,FUNDAM. 9, 88 11970). Miller, D. G., J . Phys. Chem. 63, 570 (1959). Mortimer, R. G., J . Chem. Phys. 46, 309 (1967). Mortimer, R. G., J . Chem. Phys. 48, 1023 (1968a). Mortimer, R. G., IND.ENG.CHEM.,FUNDAM. 7,330 (196813). Mprtimer; R. G.; J . Chem. Phys. 52,4334 (1970). Rice, S. A., Allnatt, A. R., J . Chem. Phys. 34,2144 (1961). Vignes A., IND.ENG.CHEM.,FUNDAM. 5 , 189 (1966). Yon, 6. hL, Toor, H. L., IND.ENG.CHEM.FUNDAM. 7, 319 (1968). RECEIVED for review August 11, 1969 RESUBMITTED July 28, 1971 ACCEPTED July 28, 1971 Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research.
Molecular Binary Diffusion for Nonideal Liquid Systems Jerry 1, Haluska and C. Phillip Colver” School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, Okla. 73069
A double Savart plate birefringent interferometer was used to measure molecular diffusion coefficients as a function of concentration for the systems toluene-methylcyclohexane and toluene-aniline at 25, 45, and 6OoC and aniline-methylcyclohexane at 60°C.These systems were chosen for their structural similarity and their nonideality. The diffusion coefficients at infinite dilution were compared to several correlations, and it was found that no one predictive equation yielded satisfactory results. An equation for the prediction of the concentration dependence of the diffusion coefficient was developed. This expression successfully correlates the data obtained in this study.
T h i s paper presents the experimental and theoretical results of a study of binary molecular diffusion in nonideal liquid systems of cyclic compounds. Accurate experimental data were obtained and used to test a predictive equation which was developed to relate the concentration dependence of the diffusion coefficient. Binary diffusion coefficients were measured a s a function of concentration for the miscible systems toluene-methylcyclohexane and toluene-aniline a t temperatures of 25, 45, and 6OoC and for the system aniline-methylcyclohexane at 60OC. Measurements on the latter system were restricted t o a single temperature due t o the formation 610 Ind. Eng. Chem. Fundom., Vol. 10, No. 4, 1971
of two phases near 45OC. The choice of systems studied was based on several factors. Experimentally, it was required that the systems be transparent to laser light and that their boiling points be above 8OoC. For investigative purposes, systems were selected for which thermodynamic data, necessary to evaluate predictive equations, existed. Further, these systems were thermodynamically nonideal and the components were of structural similarity. The need for an accurate relationship describing the concentration dependence of binary liquid diffusion is well defined in the literature. Presently, existing correlations have
had only limited success in predicting the concentration dependence of the diffusion coefficient for a binary nonideal system. An equation based on an empirical expression for the concentration dependence of the friction coefficient was developed. The correlation is compared with data obtained in this work as well as the data of Wirth (1968) and Rao (1968). Apparatus and Procedure
The experimental equipment consisted of a double Savart plate birefringent interferometer, a constant-temperature air bath, a flowing junction diffusion cell, a 35-mm still camera, and a time measurement system. A complete description of this equipment has been detailed elsewhere by Merliss and Colver (1969), Haluska (1970), and Haluska and Colver (1970), so only a brief summary will be given here. Principally, the birefringent interferometer consisted of an optical bench, a lens system, and a laser light source. The flowing junction diffusion cell was installed in a constanttemperature air bath equipped with internal control heaters, baffles, and a n air circulating fan. Temperatures of the bath were controlled to within 0.01OC with a Hallikainen Thermatrol controller. The test cell cavity was l / 4 in. wide by 31/2 in. high by 3 in. thick, giving a geometrical light path of 3 in. through the cell. Horizontal slits, used to draw off liquid a t the flowing junction, were centered vertically on each side wall and adjustable up t o a maximum width of 0.006 in. An experiment was conducted by forming a horizontal interface between two solutions in the flowing junction test cell. With the more dense solution on the bottom, a sharp boundary was formed between the two test solutions by withdrawing both fluids from horizontal exit slits centered along each side of the cell. For all diffusion measurements the concentration difference between diffusing solutions was within 0.4 wt yo.After a satisfactory interface was established, fluid withdrawal was stopped and the experiment was initiated. The decay of the interface w a k followed by taking approximately 30 photographs of the interference patterns a t known times. Photographs were taken with a Kikon F 35-mm camera at specified times measured with a 0.1-see timer. The photographs shoued symmetric Gaussian-shaped curves with increases in the degree of dispersion of the bell shape with time. A Sherr micro-projector equipped with a 20X magnifying lens and a micrometer table capable of measurements to 5/100,000 in. along both axes was used to measure the changing width of the profiles. Data Interpretation
For the small concentration differences used between the diffusing solutions, Fick's second law with a composition independent diffusion coefficient was considered valid. The interference patterns produced by the interferometer represent optical path gradients where the optical path is defined as the product of the refractive index of the solution and the geometrical length through the diffusion cell. Over small composition intervals the refractive index varies linearly with concentration. I n this case, the optical path gradient is equated to the concentration gradient which may be represented mathematically by solution of Fick's second law. Determination of the diffusion coefficient is accomplished from a measurement of the interference patterns photographed during the course of a diffusion run. These measurements are then related to the solution of Fick's second law for determination of the diffusion coefficient. The exact data reduction procedure, including the final forms of the equa-
tions used, are given in a previous article by the authors (Haluska and Colver, 1970). It is important to point out that whereas it was not possible experimentally to establish an infinitely sharp interfacial boundary between test solutions, the imperfect initial condition was corrected mathematically by determining a zero time correction. Moreover, the determination of the diffusion coefficient, as well as the zero time correction, was accomplished by a regression analysis utilizing a Gauss-Newton nonlinear least-squares technique. Predictive Equation
From statistical mechanics, Bearman (1961) has derived the following expression for the diffusion coefficient
PERT d In a DAB= --
EAB
d In CA
Hartley and Crank (1949) have shown that d1nXA = - XB dlnC~ ~BCB Using this relationship and the fact that
in combination with eq 1 gives DAB
= RT -
EAB
+
(VAXA VBXB)d In a d In X ~
(4)
Loflin and AlcLaughlin (1969) point out that when the friction coefficient tAB is represented by the geometric mixing rule (5) eq 4 reduces to the form
A similar equation for the concentration dependence of the diffusion coefficient has also been presented by Bearman (1961), Darken (1948), and Rathbun (1965). I n general, eq 6 does not satisfactorily represent the available data for diffusion in liquids. It has been suggested by Loflin and McLaughlin (1969) and Rathbun (1965) that the thermodynamic factor overcorrects the effects of the ( X A D B A ' X B D A B ' ) term. On the other hand, it is quite possible that the activity term in eq 6 does not, in reality, overcorrect, but that the friction coefficient is in need of better description. Accordingly, a suitable description of the concentration dependence of the friction coefficient might be sought. Implicit in the form of eq 6 is that the friction coefficient is constant with composition. A logical step in the development of a n expression for the concentration dependence of diffusion coefficients would involve description of the concentration dependence of the friction coefficient. To accomplish this end, a linear dependency of the friction coefficient with concentration of the folloving form was tested.
+
(7) Use of eq 7 with eq 4 represents an improvement over eq 6. Still, there is significant disagreement on testing with highly nonideal systems. Carmen and Stein (1956) and Leffler and Cullinan (1970) have derived predictive expressions with viscosity correction terms applied to the two infinite diffusion coefficients. I t seems more reasonable, however, to apply this type of corInd. Eng. Chem. Fundom., Vol. 10, No.
4, 1971 61 1
DAB =
Table 1. Molecular Diffusion Coefficients Mass fraction of
Mole fraction of
toluene
toluene
Diffusion coefficient DAB X los, cmZ/sec
Toluene-Methylcyclohexane System T = 25°C 0.0025 0.0026 1.65 0.3333 0.3477 1.61 0.6673 0.6813 1.74 0.9967 0.9970 2.21
T 0.0025 0.3333 0.6673 0.9967
T 0.0025 0.3333 0.6673 0.9967
0.0034 0.1535 0.3336 0.5012 0.6687 0.8469 0.9971
Toluene-Aniline System T = 25°C 0.0035 0.1548 0,3394 0,5042 0.6710 0.8483 0.9975
T
T
2.18 2.16 2.40 3.09
= 60°C
0.0026 0.3477 0.6813 0.9970
0.0034 0.1535 0.3336 0.5012 0.6687 0,8469 0.9971 0.0034 0.1535 0.3336 0.5012 0.6687 0.8469 0.9971
= 45°C 0.0026 0.3477 0.6813 0.9970
2.73 2.63 2.94 3.66
0.478 0.366 0.279 0.296 0.568 1.040 2.100
= 45OC 0.0035 0.1548 0.3394 0.5042 0.6910 0.8483 0.9975
0.880 0.758 0.588 0.605 0.889 1.63 2.78
= 60°C 0.0035 0.1548 0.3394 0.5942 0.6910 0.8463 0.9975
1.27 1.13 0,997 1.09 1.57 2.48 3.60
+
~ J A ~ J B R T ( V A XVAB X B ) d In a ~ J ~ ( X A ~ B [ BXAB' V A ~ A Bd' )In X A
+
(9)
Infinite dilution friction coefficients are determined in the usual manner by letting X A + 0, DAB DAB' so eq 9 becomes -+
[AB'
and X B
-+
=
RTVB DAB'
0, DBA--c DBA' so eq 9 becomes &Ao
RTVA DBA'
= -
Use of eq 10 and 11 with eq 9 permits calculation of diffusion coefficients over the entire concentration range. Comparisons of eq 9 with experimental data are presented later. Experimental Results and Discussion
Diffusion data were obtained for each of the following systems a t the temperatures given: toluene-methylcyclohexane a t 25, 45, and 60°C; toluene-aniline at 25, 45, and 60°C; methylcyclohexane-aniline a t 60°C. The data are tabulated in Table I. Concentrations listed in the table are average values between the two diffusing test solutions. The test solutions had maximum concentration differences in nearly all runs of less than 0.003 solute mole fraction. These results are also plotted in Figures 1, 2, and 3. The curves in the figures represent eq 9. Indeed, the figures clearly illustrate the nonideal behavior of these systems. Each system exhibits a negative departure from linearity which is characteristic of mixtures which deviate positively from Raoult's law. As a n indication of experimental precision, data from triplicate runs for one set of diffusing test solutions gave diffusion coefficients having an average absolute deviation of 0.17% from their mean value. 4.0
I
I
I
I
I
Met~hylcyclohexane-AnilineSystem T = 6OoC 0 0022 0.0018 0.865 0.2513 0.2415 0.388 0.4904 0.363 0.5032 0.7755 1.02 0.7828 0.9981 2.69 0.9983 t
rection to the friction coefficient. Hence, the following empirical relationship is proposed 1.4 I
0
Combining this expression with eq 4 and approximating the partial molar volumes by pure component molar volumes gives 612
Ind. Eng. Chem. Fundam., Vol. 10, No. 4, 1971
I
0.2
I
I
I
0.4 0.6 0.8 MOLE 'FRACTION TOLUENE
I 1.0
Figure 1 . Diffusion coefficients for toluene-methylcyclohexane system compared to eq 9
c
{I
nr
3.0
I
2.0
u)
0 x
n
t
-
-1.0 -
- 2.0
0 x
m
U
0
e
-
I .o
-
0
I
0.2
0
MOLE
I
0.4 FRACTION
I
I
0.6
0.8
1.0
METHYLCYCLOHEXANE
Figure 3. Diffusion coefficients for methylcyclohexaneaniline system compared to eq 9 0
0
I
I
I
I
0.2
0.4
0.6
0.0
MOLE FRACTION
1
I.o
TOLUENE
Figure 2. Diffusion coefficients for toluene-aniline system compared to eq 9
The system toluene-methylcyclohexane is more nearly ideal than the other two systems studied. This system is composed of two hydrocubons--the first a n aromatic and the latter a saturated cycloparaffin. Although methylcyclohexane is similar in structure, it does not exist in a planar conformation as does toluene. Perhaps a more important difference between these compounds is the presence of a conjugated ?r structure in the aromatic. The methyl group on the aromatic ring causes a small dipole moment, making toluene a mildly polar molecule, and, as such, it can be expected t o contribute t o some degree of nonideal behavior in solution. The systems toluene-aniline and methylcyclohexaneaniline are highly nonideal. Aniline is a highly polar molecule and has a tendency toward association. It is of interest to note that though toluene is miscible with aniline in dl1 proportions a t room temperature, methylcyclohexane is only very slightly soluble in aniline. The enhanced miscibility of toluene may be attributed to the formation of charge-transfer complexes, whjrh result from mild electronic interactions between the x structure of the aromatic and the strongly polar solvent. A majority of the presently available infinite diffusion data represent a n extrapolation from higher concentrations. One of the advantages of using a birefringent interferometer i, that extremely small concentration differences may be used. Consequently, values of the diffusion coefficients a t infinite dilution may be obtained without extrapolation. Recall that the average concentration difference at the compositional extremes is generally less than 0.003 solute mole fraction. The ability to perform measurements a t these extremes of composition IS of particular significance in that all existing correlations of the concentration dependence of diffusion coefficients are highly dependent upon accurate values of infinite dilution coefficients. I n this work, the infinite dilution diffusion coefficients are taken to be these experimentally determined values.
Table
II. Comparison of Diffusion Coefficients at Infinite Dilution for a Common Solvent Solute
Solvent
Yiethylcyclohexane (A) ;iniline (B)
Toluene (C)
Toluene (A) Aniline (B) Toluene (A) Methylcyclohexane (B)
Temp,
'c
25
DACo/DBCo
60
1.05 1 11 1.02
hlethylcyclohexane (C)
60
1 01
Aniline (C)
25
1 47
45
Toluene (A) Aniline (C) 25 Benzene. (B) Data from Kao (1968); DBC = 0.540 cm*/sec.
0 89
Comparisons
Values of infinite dilution coefficients for these systems were calculated by the correlations of Wilke and Chang (1955), Othmer and Thakar (1953), Lusis and Ratcliff (1968), and Reddy and Doraiswamy (1967). Comparison of these with the values determined experimentally are presented by Haluska (1970); however, it can be stated that, in general, these correlations provide a poor overall description of diffusion in the systems studied here. The correlations give the proper temperature dependence of the diffusion coefficient, but as stated previously quantitative comparison with the data is generally poor and shows only order of magnitude agreement. Comparison is especially poor in cases where aniline is the solvent. I n most of these instances, the predictive equations yield lower values than those obtained experimentally. Similar comparisons wheil aniline is the solvent have been reported previously by Rao (1968). The results of the statistical mechanical analysis of Bearman (1961) and the modified absolute rate theory equations of Gainer (1966) indicate that a t infinite dilution, the diffusion coefficients of two solutes which are similar in chemical Ind. Eng. Chem. Fundam., Vol. 10, No.
4, 1971 613
~~
Table 111. Comparisons of Experimental Diffusion Coefficients with Eq 9 and Correlation of Vignes-Cullinan Reference
This work Wirth (1968) Wirth (1968) Rao (1968) Rao (1968)
Amount of data tested
System
Toluene-meth ylcyclohexane a t 25, 45, 60°C Toluene-aniline a t 25, 45, 6OoC Methylcyclohexane-aniline a t 60°C Benzene-carbon tetrachloride a t 25°C Chloroform-carbon tetrachloride a t 25OC Aniline-benzene a t 25°C Aniline-carbon tetrachloride a t 25°C
Average absolute % dev Vignes-Cullinan Eq 9 correlation
6
2.57
2.13
15 3
4.01 32.1
5
2.28
3.11
6
4 29
5.34
10 10
14.0 8.30
8.92 11.2
15.4 44.3
structure are identical for a common solvent; Le., DAcO/DBCO = 1. I n order to test this hypothesis, ratios of the infinite dilution coefficients for structurally similar compounds in a common solvent were calculated from the data of this work and that of Rao (1968). These results are summarized in Table 11. I n considering these comparisons, it must be remembered that the theories were developed for structurally similar molecules which exhibited no polarity effects. Nevertheless, the calculated ratios are generally in good agreement with the arguments of Bearman and Gainer. The effect, of polarity is more pronounced in considering changes of solvent for a given solute. For toluene or methylcyclohexane as solute, a change from a polar to a nonpolar or slightly polar solvent resulted in approximately a threefold increase in the infinite diffusion coefficient. Conversely, with aniline as solute, there was very little change in the infinite diffusion coefficient for a change in solvent. The experimental results a t each temperature were compared with eq 9 and the Vignes (1966)-Cullinan (1966) correlation. Table I11 presents average per cent deviations between the correlations and the data for each system studied, plus the data of Wirth (1968) and Rao (1968). I n addition, Figures 1, 2, and 3 give a graphical comparison of eq 9 with the data. The necessary thermodynamic values used in the correlations were calculated by the method proposed by Wilson (1964). For the system toluene-methylcyclohexane, the vapor-liquid equilibrium data of Schneider (1961), Weber (1955), and Ellis and Contractor (1964) were used. The data of Billes and Varsanyi (1963) and Schneider (1960) were analyzed for the toluene-aniline system. The activity gradients were determined for the rnethylcyclohexaneaniline system from the data of Schneider (1961) and Rock and Seig (1955). Viscosities and densities used in eq 9 were determined experimentally and &re tabulated by Haluska (1970). From Table I11 it can be seen that except in two instances, eq 9 on the average more closely approximates the behavior of the data than the Vignes-Cullinan relationship. I n any case, for the diffusion data tested, eq 9 generally provides an excellent description of the concentration dependence of the diffusion coefficients.
DAB= binary diffusion coefficient, cm2/sec Do = binary diffusion coefficient a t infinite dilution, cmZ/sec R = universal gas constant, l./(atm deg mole) T = temperature V = molal volume, l./mole F = partial molal volume, l./mole X = mole fraction, moles/mole
Nomenclature
Presented at the 161st National hleeting of the American Chemical Society, Los Angeles, Calif., Mar 1971. Financial support was provided by the National Science Foundation under Research Grant N o . GK-1023. Graduate Fellowship support was provided by the National Aeronautics and Space Administration.
a A
B
= activity = component A = component B
C
=
concentration, moles/l.
614 Ind. Eng. Chem. Fundam., Vol. 10, No. 4, 1971
GREEKLETTERS = viscosity, rnoles/l. sec = mixture viscosity, moles/l. sec = friction coefficient = friction coefficient a t infinite dilution
q qm
literature Cited
Bearman, R. J., J . Phys. Chem. 65, 1961 (1961). Billes, F., Varsanyi, Gy., Acta Chim. (Budapest) 35, 147 (1963). Carmen, P. C., Stein, L. A., Trans. Faraday SOC. 52, 619 (1956). Cullinan, H. T., IND.ENG.CHEM.,FCNDAM. 5, 281 (1966). Darken. L. S.. Trans. Am. Inst. Minina Met. Ena. 175. 184 (1948). Ellis, S. R. M., Contractor, R. M., Birmingham Cniv. Chem. Eng. 15,lO (1964). Gainer, J. L., IND.ENG.CHEY.,FUNDAM. 5 , 436 (1966). Haluska, J. H., Ph.D. Thesis, University of Oklahoma, Norman, Okla., 1970. Haluska, J. H., Colver, C. P., A.I.Ch.E. J . 16, 691 (1970). Hartley, G. S., Crank, J., Trans. Faraday SOC.45, 801 (1949). Leffler. J.. Cullinan. H. T.. IXD.ENG.CHEM.,FUNDAM. 9. 84 (1970). ’ Loflin. J.. McLauehlin. E.. J . Phvs. Chem. 73. 186 (1969). Lusis,’M: A., Ratocliff, ‘G. A., Cui. J . Chem. gng. 46, 385 (1968). Merliss, F. E., Colver, C. P., J . Chem. Eng. Data 14, 149 (1969). Othmer, D. F., Thakar, M .S., I n d . Eng. Chem. 45, 589 (1953). Rao, S. S.,Ph.D. Thesis, University of Connecticut, Storrs, Conn., 1968. Rathbun, R. E., Ph.D. Thesis, University of Washington, Seattle, Wash., 1965. Reddy, K. A., Doraiswamy, L. K., IND. ENG.CHI:M., F U N D A M . 6 , 77 (1967). Rock, H., Seig, L., Z . Phys. Chem. (Frankfurt am M u m ) 3, 355 (1955). Schneider, G., Z . Phys. Chem. (Frankfurt am Main) 24, 12 (1960). Schneider, G., Z . Phys. Chem. (Frankfurt am Main) 27, 171 (1961). 5, 189 (1966). Vignes, A., IND.ENG.CHEM.,FUNDAM. Weber, J. H., Ind. Eng. Chem. 47, 454 (1955). Wilke, C. R., Chang, P., A.I.Ch.E. J . 1, 264 (1955). Wilson, G. Rl., J . Amer. Chenl. SOC.86, 127 (1964); Wirth, G. B., Ph.D. Thesis, Michigan State University, East Lansing, Mich., 1968. RECEIVED for review December 28, 1970 ACCEPTED July 14, 1971
