TRACER AND MUTUAL DIFFUBIVITIES IN CHLOROFORM-CARBON TETRACHLORIDE current values, once it will be known which kind of corrections to make. I n any case, independently of this revision, the association sequence found in the present work for the alkali perchlorates can be explained by taking into account the effect of solvation on ion pairing.11r27-29 If we consider, in fact, the extreme alkali perchlorates, the following considerations may be made. Sodium, as well as lithium ions, on account of their small radii and high charge densities, strongly orientate the water molecules. Due to the sheath of solvent molecules around the ions, ion pairing almost should not occur for them. The largest cesium cation, on the other hand, is almost un-
3293
able to orientate the water molecules and must be considered as a bare ion in aqueous solution. Cesium perchlorate should be, therefore, much more associated in water than the other alkali perchlorates.
Acknowledgment. The author wishes to acknowledge the valuable aid of Professor R. M. Fuoss in the present research. (27) A.D’Aprano and R. M. Fuoss, J . Phys. Chem., 67, 1704, 1877 (1963). (28) W.R. Gilkerson and E. Eaell, J . Amer. Chem. Soc., 87, 3812 (1965); 88,3484 (1966). (29) H. K. Bodenseh and J. B. Ramsey, J . Phys. Chem., 69, 543 (1965).
Tracer and Mutual Diffusivities in the System Chloroform-Carbon Tetrachloride at 25 by
O
C. M. Kelly,*l G. B. Wirth, and D. K. Anderson
Department of Chemical Engineering, Michigan State University, East Lansing, Michigan (Received November 12, 1970)
48883
Publication costs borne completely by The Journal of Physical Chemistry
Experimental measurements were made of tracer diffusivities, mutual diffusivities, viscosities, and densities at 25’ in the binary liquid system chloroform-carbon tetrachloride. The diffusivities are compared with those predicted by the Hartley-Crank equation for nonideal nonassociated liquid systems. The thermodynamic correction factor in the Hartley-Crank equation undercorrects the mutual diffusivity data. This undercorrection is an unusual result. Introduction and Theory A comprehensive model of diffusion must be able to predict both mutual and tracer diffusivities. It should be able to correlate diffusion coefficients to molecular size and shape, to intermolecular interactions, and to temperature and pressure. It must also describe the relationship between tracer and mutual diffusivities. One moderately successful model is the hydrodynamic model, which treats a diffusing molecule as a particle flowing through a continuous medium. I n its earliest form, as set forth by Sutherland2and E i n ~ t e i nit, ~considers the diffusing molecule as a sphere and predicts the diffusivity as a function of temperature, viscosity, and molecular radius. This has been a fairly good approximation for large molecules in dilute solutions. More recent developments of the hydrodynamic model have differentiated between mutual and tracer diffusion and have included the effectsof intermolecular associations, molecular size and shape (as distinct from molecular radius) , and solution nonideality.
It has been shown4 that according to the hydrodynamic model the tracer diffusivity of a component which is not associated in solution should be given by Di*= RT vat
(1)
where T is the temperature, v is the viscosity of the solution, R is the gas law constant, and ut is the “friction coefficient” of the component, assumed constant. The mutual diffusivity in such a system is related to the tracer diffusivities of the components by (2) (1) To whom correspondence should be addressed a t Department of Chemical Engineering, Villanova University, Villanova, Pa. (2) W. Sutherland, Phil. Mag., 9, 784 (1905). (3) A. Einstein, Ann. Phgs. Chem., 17, 549 (1905). (4) L. S. Darken, Trans. Faraday SOC.,45, 801 (1949). The Journal of Physical Chemistry, Vol. 76, No. 81, 1971
C. M. KELLY,G. B. WIRTH, AND D. K. ANDERSON
3294 This is the familiar Hartley-Crank equation. It was initially derived6 for the case of constant volumes. It was later shown that this is not a necessary restrictionS6 The Hartley-Crank equation has been found to predict mutual diffusivities quite accurately in ideal liquid solution^,^ where the thermodynamic correction factor is unity. I n nonideal systems, however, it has not been so successful. I n almost all cases the thermodynamic correction factor has been found to overcorrect. That is, the predicted difference between actual mutual diffusivities and the mutual diffusivities if the solution were ideal is usually too large. I n most cases the predicted diffusivities do at least qualitatively agree with experimental results, though. The shape of the predicted graph of mutual dsusivity os. composition is generally verified by measurements. The present paper presents a binary liquid system, chloroform-carbon tetrachloride, in which the HartleyCrank equation appears to undercorrect for solution nonideality.
Experimental Section I n this study tracer diffusivities were measured by the microcapillary method. In this method, a Teflon capillary approximately 8 cm in length and 2 mm i.d. is filled with a solution, one of whose components is tagged with 14C. The capillary is immersed in a large volume of a bulk solution of the same chemical composition, but with no tagged molecules. After a 3- to 5-day period, the capillary is removed from the bulk solution, emptied, and counted. The capillary is then filled with the same tagged solution and immediately emptied and counted. The ratio of these two count rates is then compared to a tabulated solution of the associated boundary value problem to determine the diff usivity. To eliminate convective transfer from the open end of the capillary into the bulk solution (a major source of error in this technique) a porous glass disk covers the open end during the experiment. I n this case the boundary condition for diffusion is not a zero concentration at the open end. Rather, the concentration gradient is proportional to the concentration at the end of the capillary. The constant of proportionality involves a resistance term for transfer through the disk, which depends on pore size and structure within the disk. This resistance constant appears in the solution of the boundary value problem. It may be determined experimentally by performing the experiment on a substance whose diffusivity is known. This was done using carbon tetrachloride, with a self-diffusivity of 1.32 X 10-5cm2/sec (as suggested by Rathbun and Babb7). The absolute precision of this method depends on the accuracy of the calibration procedure, hence upon the reported value of the self-diffusivity of carbon tetrachloride. The Journal of Physical Chemistry, Vol. '76,N o . 81, 1971
The precision of this method was determined by repetitive measurements and comparison with known mutual diffusivities extrapolated to infinite dilution. The precision level to be expected in a given experiment was generally found to be *2%. Mutual diffusivities were measured by means of a Mach-Zehnder interferometerss Bidlacks gives details of construction and operation. This method consists of following diffusion into a pseudo-infinite medium from a free boundary by means of optical interference fringes. The interference fringes represent a plot of refractive index vs. location in the diffusion cell, which can be related to concentration vs. location. These are photographed a t various times during the experiment, and measurements taken from the photographs are compared to a solution of the boundary value problem to determine the diffusivity. This procedure yields the integral diffusion coefficient over the concentration difference between the two solutions on either side of the interface. I n this investigation the refractive indices of the components were such that differences in concentration of the two solutions were generally on the order of 0.01 mole fraction or less. Under these conditions the integral diffusion coefficient obtained from solution of the boundary-value problem should be a good approximation to the differential diffusion coefficient at the average composition. Since the overall change in the diffusion coefficient is approximately 0.7 X cm2/sec, the change in diffusivity during one experimental determination should be on the order of 0.007 X cm2/sec, which is relatively small. The precision of the method was determined by comparison of experimental determinations of mutual diffusivities to published results in the sucrose-water system.1° Agreement for this system was within *0.5%. Experimental difficulties in handling the volatile chloroform-carbon tetrachloride system decrease the precision somewhat. Repetitive measurements in this system indicate that a reasonable estimate of the experimental precision is 1.501,. Viscosities were measured using a Cannon-Fenske capillary viscometer. Repetitive measurements and comparison to various accepted literature values indicate a precision of *0.5%. All experiments were conducted in a water thermostat with a miximum temperature cycle of *0.1".
*
Results and Discussion Experimentally determined values of tracer diffusivi(5) (6) (7) (8) 28, (9)
G. 9. Hartley and K. Crank, Trans. Faruduy SOC.,52, 781 (1956). C. 9. Caldwell and A. L. Babb, J . Phys. Chem., 60, 61 (1956). R. E. Rathbun and A. L. Babb, ibid., 65, 1072 (1961). C. 5. Caldwell, J. R . Hall, and A. L. Babb, Rev. Sci. Instrum., 816 (1967).
D. L. Bidlack, Ph.D. Thesis, Michigan State University, 1964. (10) L. J. Gosting and M. S. Morris, J . Amer. Chem. Soc., 71, 1998 (1949).
3295
TRACER AND MUTUAL DIFFUSIVITIES IN CHLOROFORM-CARBON TETRACHLORIDE 1.4
-
1.s
-
0
0
0
1.2 H
0
0
0
0
0
2.2
-
2.1
-
2.0
-
0
E
P 1.1
-
u9
c 0 I
I P P
1.0
0.9
-
P.8
-
0.7
1
I
I
I
1
0
0.2
OA
0.6
0.0
MOLE
FRACTION
I
1.o
CHCIS
Figure 1. Tracer diffusivity-viscosity product in the system CHCla-CCla a t 25": 0, VDCHCIZ; 0,VDC014*.
0
I
I
I
I
0.2
0.4
0.6
0.8
MOLE
ties, mutual diffusivities, viscosities, and densities a t 25" are given in Tables I, 11, and 111, respectively. The variation, of diffusivity and diff usivity-viscosity product with composition are illustrated in Figures 1 and 2.
FRACTION
1.o
CHCI,
Figure 2. Mutual diffusivity in the system CHCla-CC14 at 2 5 0 : -----, 4; -, 5 ; 0, experimental.
Table 11: Mutual Diffusivities in the System Chloroform-Carbon Tetrachloride a t 25" 'Fable I : Tracer Diff usivities in the System Chloroform-Carbon Tetrachloride a t 25"
DAB X 108, XCHCli
DCECI~* X I@, XCHCls
0.00 0.08 0.18 0.29 0.42 0.50 0.75
0.76
DCCl4*
x
I@,
cma/sec
cmz/sec
...
1.32
1.57
...
1.82
...
1.49
,..
...
1.73
2.02 2.23
,..
*
.
I
0.98
...
1.00
2.44
... 2.04 2.16
...
Equation 1 predicts that the product for a nonassociated component is constant a t a given temperature. It can be seen from Figure 1 that this product is constant in this system. This is more or less expected, since the outer electron shells of the chlorine atoms do
0.027 0.234 0 234 0.431 0.690 0.739 0.774 0.774 0.907 0,942 0.946 0.977 I
cmz/sec
1.505 1.557 1,572 1 644 1.680 1 752 1.757 1.779 1.881 1,972 I. 976 2 007 I
I
I
not support hydrogen bonding with the hydrogen of chloroform. Any other intermolecular attractions are likely to be too weak to maintain a stable associated complex. Furthermore, absence of a temperature effect upon The Journal of Physical Chemistry, Vol. 76, No. 81,1971
C.M. KELLY,G. E. WIRTH,AND D. K. ANDERSON
3296 Table 111: Densities and Viscosities in the System Chloroform-Carbon Tetrachloride a t 25” 9,
XCHClg
OP
0.000 0.112 0.232 0.492 0.845 1I000
0.889 0.826 0.761 0,661 0 568 0.540 I
1.5945 1.5744 1.5800 1.5322 1.4955 1.4798
the heat of mixing supports the conclusion that this system is not ass0ciated.l’ The Hartley-Crank equation has been modified to explain the usual overcorrection by the thermodynamic factor in terms of simultaneous diffusion of monomers and associated complexes (dimers, trimers, etc.), These have been reasonably s u c c e ~ s f u l ~in~predicting -~~ the direction of the overcorrection. They usually contain an extra parameter (such as the equilibrium constant for the association) which quantitatively removes the overcorrection for reasonable values of the bond strength of the associated complex. For nonassociated systems such as the present system, however, the Hartley-Crank equation should apply. If the solution were ideal, the Hartley-Crank equation would have the form
DAB =
-[z+]X;g RT X A 17
(3)
or, after some substitutions and rearrangement (4)
The terms qDA* and ~ D Bare * constant and may be obtained from Figure 1. Equation 4 is plotted in Figure 2 for comparison with the experimental measurements. The thermodynamic correction factor may be obtained from isothermal vapor-liquid equilibrium data. Hala, et aZ.,16have fit published vapor-liquid data for a variety of systems to various descriptive equations (such as Van Laar, Margules, etc.) relating activity coeficients to solution composition.15 The constants they obtained for the present system from the data of McGlashan, Prue, and Sainsbury16 were used to calculate thermodynamic correction factors for use in the equation
The Jozirnal of Physical Chemistry, Vol. 76,No. $1, 1071
It may be noted that this system does not deviate too badly from ideality, so that the correction factor is rather small. Equation 5 is also plotted in Figure 2. Comparing the experimental data to the predictions of eq 4 and 5 shows that the Hartley-Crank equation predicts qualitatively the shape of the DAB us. mole fraction curve. It does not, however, correct properly for solution nonideality. In fact it undercorrects, which is unusual. I n the past, such deviations have often been attributed to intermolecular associations, but this cannot be the case here. It is possible that the vapor-liquid equilibrium data are in error, since small errors in this sort of data are amplified when determining activity coefficients. This seems somewhat unlikely, for the experimental data fit both Van Laar and three-term Margules equations very well. I t is hoped that further refinements in diffusional models and theory will explain the undercorrection in this system. At present, however, the most probable reason is a slight error in determining solution nonideality (ie., obtaining activity coefficients from vaporliquid equilibrium data). Although the Hartley-Crank equation is ordinarily at least qualitatively applicable to nonideal solutions, care must be taken in applying it predictively. I n this system at least, there are some anomalous effects which are as yet unexplained. Perhaps future experimental or theoretical work will help t o clarify this situation. Appendix. Nomenclature a D* DAB R T X
activity tracer diffusivity, cm2/sec mutual diffusivity, cm2/sec ga’slaw constant temperature mole fraction viscosity, cP ?I densit’y, g/cm3 P U friction factor, cm/g-mol subscript A refers to component A subscript B refers t o component B subscript i refers to component i (11) J. H. Hildebrand and R. L. Scott, “The Solubility of Nonelectrolytes,” 3rd ed, Reinhold, New York, N. Y., 1950. (12) P. C . Carman, J . Phus. Chem., 71, 2565 (1967). (13) D. K. Anderson and A. L. Babb, ibid., 66, 1281 (1962). (14) D. K. Anderson and A. L. Babb, ibid., 66, 899 (1962). (15) E . Hala, I. Wichterle, J. PolLk, and T. Boublik, “Vapor-Liquid Equilibrium Data a t Normal Pressures,” Pergamon Press, Elmsford, N. Y., and Oxford, 1968. (16) M. L. McGlashan, J. E. Prue, and J. E. J. Sainsbury, Trans. Faraday Soc., 50, 1284 (1954).
