COMPOSITION DEPENDENCE OF BINARY DIFFUSION COEFFICIENT A geometric mean rule for the lattice spacing is proposed as an alternate basis for the variation of the binary diffusion coefficient with composition. The implications of this viewpoint are discussed, and a comparison with some available data is offered in support. effort has been expended on the problem of the composition dependence of the bindi-y diffusion coefficient in completely miscible liquid systems (Arnold, 1930; Bearman, 1961; Carmen and Stein, 1956; Cullinan, 1966; Hartley and Crank, 1949; Vignes, 1966). From a practical viewpoint the important objective of any work of this kind is the prediction of rates of diffusion in concentrated binary systems. From a more fundamental viewpoint the major objective is a more detailed understanding of the mechanism of the diffusion process in liquid systems and, with such understanding, a firmer characterization of the nature of the liquid state. The two viewpoints are, of course, strongly interrelated. The less theoretical the input to a predictive attempt, the more empirical it must become and thus the more unreliable when extrapolated from known to unknown. On the other hand, a purely theoretical approach without regard for what is already known or for what possibly can be known is of little immediate value. The best of the predictive theories combines the two kinds of emphasis to yield workable predictions along with fundamental understanding. Such theories, however, are rare. This note deals with a recent effort which began with an empirical correlation of data for a variety of systems (Vignes, 1966). The surprisingly simple form of the resulting relationship for the binary diffusion coefficient, in terms of the infinite dilution values and a thermodynamic factor, led to an attempt a t a partial theoretical explanation of the results (Cullinan, 1966). The Vignes expression (Vignes, 1966) : ONSIDERABLE
laboring the issue, because the lattice model is not a very satisfactory molecular representation of a liquid. The present pioblem then devolves to one of differentiation between the two candidates for an empirical rule, Equation 3 or Equation 4. The relation (Bearman, 1961) (5) can be used with Equation 2 to give the activity-corrected diffusion coefficient:
It-
..
d In x i
If the lattice spacing and the activation energy are both insensitive to temperature changes, the problem is easily resolved, provided that diffusion coefficient-temperature data are available. I n this case a semilog plot of D;/T against l / T with mole ftaction as a parameter should yield a set of straight lines of varying slope but equal intercept if activation energy varies and lattice spacing is constant. O n the other hand, a set of parallel lines indicates a constant activation energy and varying lattice spacing. If either the activation energy or the lattice spacing is strongly dependent on temperature, the problem is not as straightforward. Only three binary systems of the type which follows Equation
10
-
9-
is consistent with a frictional activation energy defined by
Fij =
hill -
ca2
8 -
exp ( A G i j / R T )
which varies linearly with mole fractions-Le. AGi, = xiAGjio
+ x, AG,,'
7 -
(Cullinan, 1966)
(3)
This result, which must be regarded as essentially empirical in nature, offers no real kinetic or molecular picture of the diffusion process despite the apparent connection with the theory of absolute rates. I n fact, if a structural view toward Equation 2 is adopted, it is seen that the implicit assumption of constant lattice spacing leading to Equation 3 as a basis for Equation 1 is probably not a very good one. I t also might be argued that the activation energy, AGtJ, should really be a free energy of formation of the i-j activated complex, the equilibrium constant for which ought to depend only on temperature. These arguments lead to a geometric mean lattice spacing a = (a,)". (a5)'3
6 -
(4)
as a more appropriate choice. If it were not for the hope of extending these results to systems of more than two constituents, there would be little use in be-
5-
40
-
XT
0.013 0.250 0.499 0.749 0.986
3-
A
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I
I
I
I
I
3.2
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3.3
3.4
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Figure 1 .
x
lo3,
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Test of theory for chlorobenzene-toluene VOL. 7
NO. 3
AUGUST 1968
519
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*'u 0
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3.2
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Figure 2. Test of bromobenzene
theory
for
chlorobenzene-
Figure 3. Test tetrachloride
of
x
3.4 lo3,
theory
for
3.5
3.6
OK-'
benzene-carbon
1 have been measured at more than two temperatures (Caldwell and Babb, 1956). I n Figure 1, a semilog plot of D,*/Tvs. 1/T is presented for the system, chlorobenzene-toluene (Caldwell and Babb, 1956). This system is ideal, so no thermodynamic correction is necessary. The lines are definitely curved: indicating a strong dependence of activation energy on temperature. However, the curves are nearly parallel; the slope tends to be independent of composition a t a given temperature. According to Equation 6 :
in slope from the curves nearer the composition extremes Additional thermodynamic data, as well as diffusion coefficienttemperature data, are needed for a definitive test. I n this context the preliminary evidence tends to support the view that the observed variation of the binary diffusion coefficient with composition is attributable to the solution thermodynamics and to the geometric variation of the lattice spacing (or the average distance a diffusing entity travels in a single step). Additional implications of these results in their application to multicomponent systems will be the subject of a later paper.
If the lattice spacing is not strongly temperature-dependent, AGZjis indicated to be a function of temperature only, since the slope depends only on temperature. This supports the contention that the activation energy is independent of composition, and the lattice spacing varies geometrically with mole fraction. Figure 2 is a similar plot for the ideal system, chlorobenzenebromobenzene (Caldwell and Babb, 1956). The activation energy is strongly dependent on temperature and the curves tend to be parallel. According to the analysis above, this indicates an activation energy independent of composition and a lattice spacing varying geometrically with mole fraction. Figure 3 is a plot of the data for the regular solution, benzenecarbon tetrachloride (Caldwell and Babb, 1956). These results are somewhat inconclusive. The problem may be in the use of a single activity correction over the entire temperature range. The correction used here is the same as that used previously (Vignes, 1366). The effect of this correction is strongly felt only a t the middle compositions, where the curves deviate
Acknowledgment
520
I&EC F U N D A M E N T A L S
The author expresses appreciation to the National Science Foundation for its support under NSF GK 1747. literature Cited Arnold, J. H., J . Am. Chem. Sod. 52, 3937 (1930). Bearman, R. J., J . Phys. Chem. 65, 1961 (1961). Caldwell, C. S., Babb, A. L., J . Phys. Chem. 60, 51 (1956). Carmen, P. C., Stein, L. H., Trans. Faraday SOC.52, 619 (1956). 5 , 281 (1966). Cullinan, H. T., IND.ENG.CHEM.FUNDAMENTALS Hartley, G. S., Crank, J., Trans. Faraday SOC.45, 801 (1949). Vignes, A., IND.ENG.CHEM.FUNDAMENTALS 5 , 189 (1966).
HARRY T. CULLINAN, JR.
State University of ,Vew York Buffalo,>V. Y . RECEIVED for review September 20, 1967 RESUBMITTED March 11, 1968 ACCEPTED March 28, 1968
