Variation of Liquid Diffusion Coefficients with Composition Binary Systems John Leffler a n d Harry T. Cullinan, Jr. Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, N . Y . 14214
The predictive capability of the Vignes equation is limited, with considerable deviations encountered in a number of systems. A modification of this expression is developed accounting for solution viscosity. Comparison with data for a large number of known systems shows that this modified version improves the over-all predictability. The concept of segmental diffusion is used to explain the results for mixtures of n-alkanes.
of the variation with concentration of the mutual diffusion coefficient of binary liquid systems is a problem of continuing practical and theoretical significance. On the practical side, the desired result of all effort in this area is a generally useful predictive relation which would give the mutual diffusion coefficient a t any concentration in terms of easily accessible mixture properties. On the theoretical side, a complete and self-consistent theory capable of accounting for the concentration dependence of the mutual diffusion coefficient would represent a significant advance. Unfortunately, no entirely satisfactory answer to either side of the problem has emerged despite a considerable effort in both directions. Theoretical efforts have ranged from attempts a t the construction of models based on hydrodynamic or kinetic considerations to rigorous statistical mechanical formulations. None of these have been successful as yet in the present context. However, some theoretical models have provided bases for empirical approaches to the problem. Most of these have been restricted to dilute solutions, and, even in this case, success has been only moderate. A recent empirical effort (Vignes, 1966), directed a t the concentration variation of the binary diffusion coefficient, is of interest here for several reasons. It was apparent that the results had some theoretical implications (Cullinan, 1966), thus opening up the possibility for extension to the multicomponent diffusion problem. The Vignes work (1966) is, however, purely correlative in nature, showing that for a large number of liquid mixtures the logarithm of the activity-corrected diffusion coefficient tended to vary linearly Tyith mole fraction. If such a correlation is used in a predictive sense to give a mixture diffusion coefficient in terms of the limiting values, some danger exists if the limiting values actually do not lie on the best correlating line. I n fact, the Vignes result leads to large errors when considered as a predictive theory. In this nork an improved version of the Vignes result is derived and compared with existing data. T H E cH.mAcTmIzATIox
84
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VOL. 9 NO. 1 FEBRUARY 1970
Previous Theory
For a binary system, isothermal, isobaric diffusion in the absence of external forces may be described by an equation of the form (Bearman, 1961) --PI
=
FlPCP(V1
- V,)
(1)
which can be taken as a defining equation for the friction coefficient, F12.This coefficient has fundamental significance and may be interpreted in various mays, by the various theories of liquid diffusion. I n the hydrodynamic theory, Flz is interpreted in terms of a flow resistance. I n the statistical mechanical theory, F12is related to intermolecular potential and pair correlation functions. According to the theory of absolute rates, the friction coefficient is given in terms of it frictional activation energy and a characteristic diffusion distance (Cullinan, 1966) :
The relation between the friction coefficient and the binary mutual diffusion coefficient is
RT a, - - a - CFI2
(3)
where a=l+-
d In ~1 d In x1
(4)
Combination of Equations 2 and 3 gives the absolute rate theory result:
(5) For a binary system with given thermodynamic properties the problem of the composition variation of the diffusion coefficient can be resolved if the composition dependence of
the parameters on the right side of Equation 5 is determined. For example, if the activation energy, AGI2, is taken to be linear in mole fraction (Cullinan, 1966), the Vignes (1966) result follows immediately, provided A D is assumed constant :
ture rather than being peculiar to a given process. On this basis, if all the distances are taken as the same,
CY
312
--
CY
= (D*lo)y3120)z2
(6)
However, it can also be argued (Cullinan, 196813) that the activation energy, AGlz, should actually be identified with a free energy of formation of a diffusional activated complex, the equilibrium constant for which ought t o depend only on temperature. If this is the case, a geometric mean diffusion distance
leads equally to Equation 6. This seems to be the case for several systems obeying Equation 6 (Cullinan, 196813). Although the Vignes (1966) proposal correlated existing binary data well for a large number of systems, use of Equation 6 as a predictive device can lead to substantial erroreven though existing data could be fitted with a straight line with In D12/ous. zl, there exists sufficient deviation from the correlation in some cases to cause significant error when limiting values are used in Equation 6 to predict values of D12in a mixture. Proposed Theory
I n spite of its prediction limitations, the Vignes result indicates some trends toward universal composition variation which, with proper interpretation, might still be useful as part of a predictive scheme. Equation 6, although accounting for the solution thermodynamics in an appropriate way, does not explicitly incorporate the effect of solution viscosity. This implies that either diffusion and viscous flow are independent processes or viscosity has a composition variation of a n analogous type (Cullinan, 1968a). The notion that diffusion and viscous flow are independent phenomena directly contradicts hydrodynamic theories of diffusion. Relationships between diffusion coefficients and viscosity emerge from most usable statistical mechanical results (Bearman, 1961). Interpretations of absolute rate theory descriptions also usually result in such relations. T o correct Equation 6 for the effect of viscosity, the absolute rate theory interpretation (Cullinan, 1966) can be adjusted accordingly. The fundamental expression in this theory for the mixture viscosity is (Glasstone et al., 1941)
e(AGq-AGlr)/RT
h
The activation energies are associated with the specific rates of the two processes. If the rates are assumed proportional, the difference in activation energies is constant. I n this case the product of activity-corrected diffusion coefficient and viscosity varies inversely with the distance, A, at constant T. For several systems for which the Vignes result (Equation 6) yields adequate predictions, the geometric mean variation of h seems to apply (Cullinan, 1968b). If this is a general characteristic, the use of Equation 7 with Equation 10, assuming the difference in activation energies is independent of composition, yields
This is the desired modification of Equation 6 incorporating the effect of solution viscosity. Comparison of Results
Typical comparisons of the predictions of Equation 6 and Equation 11 with experimental data are illustrated in Figures 1 to 3. For these comparisons, as well as the subsequent ones, the thermodynamic data for each system are the same as those used by Vignes (1966). The infinitely dilute values of the diffusion coefficients used in the predictive equations are either experimental or estimated values extrapolated from available data. The sources for the viscosity data used in Equation 11 are listed in Table I. The sources for the diffusion coefficient data used to test the equations are listed in Table
11. For the systems hexane-carbon tetrachloride and cyclohexane-benzene, the Vignes equation predicts significantly low values. Equation 11 improves the predictions substan-
4.00 I
3.50 in
ex
$ Q
3.00 2.50
\ N
22.00 There is little basis upon which the various distances appearing in Equation 8 can be differentiated. The reasonable approach usually taken is the assumption that they are all the same and related to the mean molecular volume. If Equation 8 is multiplied by Equation 5 , the result is
1.50
EXPERIMENThLIBlDLACK 8 ANDERSON,I964) (THIS W O R K )
-PREDICTED PREDICTED I n a structural interpretation of these equations, A, and are very similar, being essentially the distance a molecule involved in the transport process moves during the unit step. Esthetically a t least, i t would be satisfying if such distances were associated mith a state property of the mixhD
0.00 0.00
.20
.40
iYIGNES,19661
.60
.80
1.00
X" Figure 1. Measured and predicted binary mutual diffusion coefficients for system hexane-carbon tetrachloride VOL 9 NO. 1 FEBRUARY 1970
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85
Table 1.
Reference
Cyclohexane-carbon tetrachloride Hexane-carbon tetrachloride
(Kulkarni et al., 1965) (Bidlack and Anderson, 1964b) (Fischler, 1913) (Grunberg, 1954) (Grunberg, 1954) (Anderson and Babb, 1962) (Van Geet and Adamson, 1964) (Bidlack and Anderson, 1964b) (Yajnik et al., 1925) (Toropov and Galakov, 1956) (Caldwell and Babb, 1956) (Herz, 1918) Toropov and Galakov, 1956) (Bidlack and Anderson, 1964a) (Bidlack and Anderson, 1964b) (Yajnik et al., 1925)
Acetone-benzene Cyclohexane-benzene Benzene-carbon tetrachloride Methyl ethyl ketone-carbon tetrachloride Octane-dodecane Hexane-dodecane Benzene-bromobenzene Toluene-chlorobenzene
Chlorobenzene-bromobenzene Methanol-propanol Butanol-propanol Hexane-hexadecane Heptane-hexadecane Acetone-carbon tetrachloride Table
II.
7
Sources of Viscosity Data
System
4*00
n
0 .L
f 2.50 c In
'.OO
0.00 0.00
Summary of Comparison of Equations 6 and 11 with Experimental Data Maximum % Deviation (Predicted-Experimental) Equation 6 Equation 1 1
System
86
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-4.1 -12.7 -6.0 -12.0 -7.0 -3.0 -4.1 -5.0 1.5 1.5 -18.5
.20
.40
.EO
.60
1.00
1
4.00 r
Reference
Bidlack and Anderson, 196413 Acetone-benEene Anderson et al., 1958 Cyclohexane-benzene Harned, 1957 Benzene-carbon tetrachloride Caldwell and Babb, 1956 Methyl ethyl ketone-carbon Anderson and Babb, tetrachloride 1962 Cyclohexane-carbon tetrachloride Kulkarni et al., 1965 Octane-dodecane Van Geet and Adamson, 1964 Hexane-dodecane Bidlack and Anderson, 196413 Benzene-bromobenzene Miller and Carman, 1959 Toluene-chlorobenzene Caldwell and Babb, 1956 Chlorobenzene-bromobenzene Caldwell and Babb, 1956 Methanol-propanol Shuck and Toor, 1963 Butanol-propanol Shuck and Toor, 1963 Hexane-hexadecane Bidlack and Anderson, 1964a Hept ane-hexadecane Bidlack and Anderson, 196413 Anderson et al., 1958 Acetone-carbon tetrachloride
Cyclohexane-carbon tetrachloride Hexane-carbon tetrachloride Acetone-benzene Cyclohexane-benzene Benzene-carbon tetrachloride Toluene-chlorobenzene Chlorobenzene-bromobenzene Methanol-propanol Butanol-propanol Methyl ethyl ketone-carbon tetrachloride Acetone-carbon tetrachloride
EXPERIMENTAL,(HARNED, 1957 1 PREDICTED,(THIS WORK1 PREDICTED ,( VIGNES, I966 I
Figure 2. Measured and predicted binary mutual diffusion coefficients for system cyclohexane-benzene
Hexane-carbon tetrachloride
Table 111.
_-----
XB
Sources of Diffusion Data
System
t
1.5 -1.9 -1.5 4.0 -7.5 -4.0 -4.1 -6.5 2.0 -1 .o -19.6
VOL. 9 NO. 1 FEBRUARY 1970
3.50
3.00
=
c,
L
2.50
x
ci
a? : 2.00
N
E u
1.50
t0
I
EXPERIMENTAL,(ANOERSON 8 BABE, 1962)
-PREDICTED, ITHIS WORK)
PREDICTEO, (VIGNES, 1966)
0.00 0.00
.20
.40
.60
.EO
1.00
XM Figure 3. Measured and predicted binary mutual diffusion coefficients for system methyl ethyl ketone-carbon tetrachloride
tially. As indicated below, similar comparisons are observed for the systems cyclohexane-carbon tetrachloride and acetone-benzene. The results illustrated in Figure 3 for the system methyl ethyl ketone-carbon tetrachloride are typical of the results for systems for which the Vignes equation predicts the concentration dependence fairly well. For these systems, Equation 11 predicts equally well on the average. Considered as a modification of the Vignes equation, the proposed Equation
I i improves the agreement with data in cases where the unmodified equation yields large deviations while preserving the agreement in cases where the unmodified equation yields good agreement. There are, however, some exceptions. Comparisons of Equation 6 and Equation 11 with experimental data are summarized in Table 111. For each system the maximum deviation of the smoothed equation prediction curve from the smoothed experimental curve is reported. For most systems this maximum occurs near the midpoint between the binary composition extremes, but the precise location depends on the particular characteristics of the given system. The system acetone-carbon tetrachloride is a notable exception to the trends indicated above. This system is one of the most thermodynamically nonideal of the apparently nonassociating systems of the present type. The diffusion coefficient data (Anderson et al., 1958) appear consistent and accurate. There exists a distinct minimum in the diffusion coefficient data a t about 20 mole % acetone, but each equation predicts a deeper minimum than actually reported. The thermodynamic factor, a, is large, but it is unlikely that errors in this quantity are sufficient to explain the discrepancy. Mixtures of n-alkanes also appear to be exceptions. For the systems hexane-hexadecane, heptane-hexadecane, octanedodecane, and hexane-dodecane, the Vignes equation predicts high values (as high as +9.0Y0 for hexane-hexadecane) while Equation 11 severely overcorrects (a maximum deviation of - 18.070 occurring for hexane-hexadecane). Van Geet and Adamson (1964) suggested that in mixtures of n-alkanes the diffusion process, unlike that for a system of nonelongated molecules, is characterized by a rate-determining step consisting of the activated displacement of a segment of the hydrocarbon chain. If this is the case, these deviations can be interpreted within the framework of the underlying theory. Suppose that the larger chain molecule in a binary mixture diffuses segmentally. Then the diffusion distance, A D , a t any relative composition should be equivalent to the diffusion distance in a hypothetical mixture of the shorter chain molecule with an appropriately shorter second molecule, this hypothetical system assumed to undergo "normalJ' diffusion (no segmental motion). In other words, Equation 7 should be replaced by the form
where 1 refers to a shorter-chain-length molecule relative to 2. The factor, f, is interpreted as the fraction of the total chain length of the longer molecule involved in segmental motion. This fraction certainly depends on the relative amounts of the two hydrocarbons, approaching unity as the shorter molecule becomes infinitely dilute and some unspecified (but presumably lower) value a t the other composition extreme. If Equation 12 is used in Equation 10, the following modified version of Equation 11 is obtained for straight-chain hydrocarbon mixtures: (13) I t is of interest to use the available data for the four nalkane mixtures mentioned above to calculate f from the measured diffusion coefficients according to Equation 13. These results are presented in Figure 4.
1.c .9
.8
.7
LL
.6
.5
.4
.3
XI
A HEXANE 0 HEPTANE 0 HEXANE x OCTANE
x2 HEXADECANE HEXADECANE DODECANE DODECANE
\
.2
0.0
.2
.4
.6
.8
I.o
XI Figure 4. Segmental motion factor for various binary nalkane systems
The fraction of the total chain length of the larger molecule involved in the segmental motion is seen to be proportional t o the relative number of longer molecules. The strength of this proportion is also greater the larger the difference in chain length between the two constituents. These notions are somewhat analogous to the concept of congruence in nalkane mixtures (Van Geet and Adamson, 1964). However, no a priori means for the determination of f is available even if the segmental theory is valid. At present, then, neither the proposed Equation 11 nor the Vigries result appears applicable in unmodified form to mixtures of nalkanes.
Conclusions The predictive capability of the Vignes equation (Vignes, 1966) appears to be limited, with rather large deviations encountered in a number of known systems. With binary nalkane systems excepted, the proposed modification (Equation 11) improves the results in all cases (except acetonecarbon tetrachloride, as noted above) in which substantial disagreement existed and preserves the previous good agreement in the remainder. The failure of the proposed Equation 11 in the case of +alkanes can be accounted for in terms of the segmental diffusion concept (Van Geet and Adamson, 1964). This necessarily diminishes the utility of the result in a predictive context because of the introduction of a segmentation parameter not amenable to a priori determination. From a more fundamental viewpoint the results presented here afford a better characterization of the composition variation of the binary mutual diffusion coefficient for a large class of systems than previously available. This should have some implications within the framework of more widely applicable theories of liquid-phase transport, notably the statistical mechanical theories. Finally, the results presented here afford a basis for extension to the multicomponent diffusion case. VOL. 9
NO. 1 FEBRUARY 1970
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87
Cited
Nomenclature
literature
C C,
Anderson, D. K., Babb, A. L., J. Phys. Chem. 66, 899 (1962). Anderson, D.K., Hall, J. R., Babb, A. L., J . Phys. Chem. 62,
? F,, AG,, AG7l
h
IC
N R T V, 21
V
= total molar concentration
molar concentration of species i binary mutual diffusion coefficient fraction of total chain length of larger molecule involved in segmental motion = friction coefficient = activation energy for diffusion process = activation energy for viscous process = Planck constant = Boltzmann constant = Avogadro’s number = gas constant = absolute temperature = velocity of species i = mole fraction of species i = gradient = = =
GREEKLETTERS CY = thermodynamic factor defined by Equation 4 = activity coefficient of species i Yz = characteristic distance for diffusion process AD A,, X A , AB, A 0 = characteristic distance for viscous process x = characteristic distance applicable to diffusion or viscous process = characteristic distance for pure species i A, = chemical potential of species i w* 7 = solution viscosity = viscosity of species i 7L SUBSCRIPTS = component 1, 2, 3 1, 2, 3 SUPERSCRIPT 0 = infinite dilution of indicated species
404 (19.554). \--_-,.
Beaiman R. J., J. Phys. Chem. 65, 1961 (1961). Bidlack, D.L., Anderson, D. K., J . Phys. Chem. 68,206 (1964s). Bidlack, D.L. Anderson, D. K., J . Phys. Chem. 68,3790 (1964b). Caldwell, C. d.,Babb, A. L., J . Phys. Chem. 60, 51 (1956). Cullinan, H. T., IND. ENG.CHEM.FUNDAMENTALS 5, 281 (1966). ENG.CHEM.FUNDAMENTALS 7,177 (1968a). Cullinan, H.T., IND. ENO.CHEM.FUNDAMENTALS 7,519 (1968b). Cullinan, H.T. IND. Fischler, J., Z.hlektrochem. 19, 126 (1913). Glasstone, S. K.,Lardler, K. J., Eyring, H., “Theory of Rate Processes,” McGraw-Hill, New York, 1941. Grunberg, L., Trans. Faraday SOC.50, 1293 (1954). Harned, H.S., Discussions Faraday SOC.24, 7 (1957). Hera, W., Z.Anorg. Chem. 104,251 (1918). Kulkarni, M.V., Allen, G. F., Lyons, P. A., J . Phys. Chem. 69, 2491 (1965). - -, Miller, L., Carman, P. C., Trans. Faraday SOC.55, 1831 (1959). Shuck, F. O.,Toor, H. L., J . Phys. Chem. 67,540 (1963). Toropov, -__ N. A., Galakhov, F. V., Bull. Akad. Sci. USSR 1956, \
158.
Van Geet, A. L., Adamson, A. W., J. Phys. Chem. 68,238 (1964). ENG.CHEM.FUNDAMENTALS 5, 189 (1966). Vignes, A.,IND. Yajnik, N. A.,Bhalla, M. D., Talwar, R. C., Soofi, A., 2.Phys. Chem. A118, 305 (1925).
RECEIVED for review April 1, 1969 ACCEPTED October 2, 1969
Work supported by the National Science Foundation under NSF GK-1747. The first author (J. L.) was the recipient of a NASA Traineeship.
Variation of Liquid Diffusion Coefficients with Composition Dilute Ternary Systems John Leffler and Harry T. Cullinan, Jr. Department of Chemical Engineering, State University of New York ut Buffalo, Buffalo,N . Y . 14214
The modification of the Vignes equation, proposed and demonstrated for binary liquid systems by Leffler and Cullinan, is extended to the case of a dilute species diffusing in a mixture of two solvents. Experimental results are reported for the three dilute limits of the systems hexane-benzene-cyclohexane, acetonebenzene-cyclohexane, and acetone-hexane-carbon tetrachloride. These data and previous results are used to test the proposed predictive equation. Agreement is generally good.
THE of diffusion in liquid systems consisting more than two components is complicated by two CHARACTERIZATION
of distinct factors. First, interactions between chemical species cause coupling of the diffusive fluxes. These effects are more pronounced the more nonideal the system; and this involves the second difficulty, the adequate description of the thermodynamic properties of multicomponent liquid systems. 88
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VOL. 9 NO. 1 FEBRUARY 1970
Any consistent theory of diffusion in such systems must allow for coupling of the fluxes, phenomenologically or otherwise, and also account for the solution thermodynamics. Although the thermodynamics of irreversible processes has provided a convenient basis, the development of such theories has been slow. Not the least among the reasons for this situation has been a general lack of reliable diffusion coefficient
