DIFFUSION IN BINARY SOLUTIONS Variation of DzfLsion Cocficient with Composition A L A I N VIGNES
Ecole .Vationale Suptrieure de la .liiitallurgie et de I’lndustrie dps .Mines, C’niaersity
of
.Vanci., .Vuncy, France
The coefficient o f interdiffusion, experimentally determined from Fick’s second law, i s the product of an activity-corrected diffusion coefficient and a thermodynamic factor which represents the departure from ideality of the solution: DAB= DABcxAB = D A B ( 1 d In f,/d In N.4). Analysis of all the experimental results concerning the nonassociated solutions, ideal or nonideal, liquid or solid, leads to the following general empirical correlation: The activity-corrected coefficient of interdiffusion, DAB, varies exponentially with the mole fraction of one of the components of the solution between the two extreme values: D A B = (DAB0).”B(DBAo).’.*, In the case of diffusion within associated solutions, this result remains valid whenever the average degree of molecular association of the associated compound can b e considered constant.
+
HE
phenomenological theory of diffusion. based on the
Tthermodynamics of irreversible processes (70,27,50) briefly summarized below, leads to the following definition of the diffusion coefficient. Let $A and @B be the fluxes of components A and B with respect to stationary coordinate axes. u be the molar average velocity defined by
ch =
$‘A
+
$‘B
(1)
and J A and J B be the molar diffusion fluxes relative to this velocity, defined by
T h e phenomenological theory of diffusion establishes linear relations between the diffusion fluxes, J , and the driving forces -i.e., the gradient of (chemical potential of each component. These relations can be written:
(3) From the classical expression of the chemical potential p = /LAO
+ RT In fA.VA
(4)
where iVA is the mole fraction and fA is the activity coefficient, and from the Gibbs-Duhem relation, a simple transformation leads to the following expression for the molar diffusion fluxes:
JA
=
-D*Boc,&
b.VA = -J B bZ ~
(5)
during the mixing of the diffusing species-it can be shown easily that Fick‘s second law can be Lrritten in its usual form:
When the partial molar volumes d o not remain constant, knowledge of their variations is required in order to solve the problem. This procedure has never been used and the diffusion coefficient experimentally determined is the coefficient defined in Fick‘s second laiv (8). Thus, the diffusion coefficient experimentally determined, Da+g,is the product of an activity-corrected diffusion coefficient, DAB.and a thermodynamic factor. aAB:
Kinetic Theories of Diffusion
T h e phenomenological theory does not indicate the kinetic aspect of diffusion. Numerous reviews and discussions of the kinetic theories of diffusion have been published (36, 39, 49, 6 2 ) . Only the main results of these theories are presented here. Equation 10, proposed by Adamson (2) for gases, by Hartley and Crank (37) for liquids, and by Darken (79) for solids, is characteristic of one of these theories This kind of theory assumes a n intrinsic mobility or a n intrinsic diffusion coefficient for each component, DA and Dg. these coefficients being coupled in the interdiffusion process: DAB
where DAB is related to the phenomenological coefficient, 1,
and a A B is a thermodynamic factor which represents the solution departure from ideality:
(7) When the partial molar volume of each component remains constant over the whole composition range, which is often a good first approximation for both liquid and solid binary solutions (Vegard’s law)-i.e., when no volume change occurs
(9)
D A B = DABff4B
=
DA.VB
+ DB.VA
(10)
Furthermore, this theory assumes that these coefficients can be identified with the corresponding coefficients of self-diffusion, DA* and DB*. which leads to the relation: D A B = (DA*.VB
f
DB*ATA)ffAB
(11)
Experimental results ( 9 , 29, 45) for diffusion in the liquid phase invalidate Darken‘s relation, as shown in Figure 1, where the ordinate of the curves represents the logarithm of the ratio of the experimental diffusion coefficient to that predicted by Darken’s equation. T h e values differ by as much as a factor of 8 ( 6 6 ) . I n the solid phase, no decisive test of the validity of this relation has yet been given. Another kind of kinetic theory (27, 37, 67), based on the VOL. 5
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189
T
0
0.5
1
NA Figure 1 . Binary diffusion coefficients (66) Test of Darken's relation 1. Acetone (A)-CCId (9) 2. Ethanol (A)-CCld (9) 3. Nitromethane (A)-CCla (B) 4. Nitromethane (A)-benzene (9) 5. Benzene (A)-CCId (9) 6. Acetone (A)-chloroform (9)
''I
Stokes-Einstein relation, explicitly considers friction between the two diffusing species. Hartley and Crank (37) have proposed the following relation :
If CYAand O ( B , the friction factors, are assumed constant, they can be expressed in terms of the extreme values of the diffusion coefficient and Equation 12 can be rewritten: 0.5
0
NA According to this equation, D A B ~ A B / c Y A B should vary linearly with the mole fraction of the solution. Equation 13 is not verified experimentally, as appears in Figure 2. For nonideal solutions, Dullien and Shemilt (23) and Lisnyanskii (42) have computed through Equation 9 the activity-corrected diffusion coefficient, DAB, for the ethanol-wafer and acetone-water systems and have shown that it varies linearly in a first approximation. No other authors have tried to generalize this conclusion. Thus, no satisfactory solution has yet been proposed for predicting the variation of the diffusion coefficient with composition. Proposed Correlation between Interdiffusion Coefficient and Composition
T h e proposed empirical correlation is obtained from all experimental results published so far, for both liquid solutions of nonelectrolytes and solid solutions (binary alloys). The binary solutions can be classified in three main categories: Ideal solutions Nonideal solutions Associated solutions, among which one can consider: solutions in which there is association of the molecules of the same kind (alcohols-benzene) and Solutions in which molecules of different species associate (chloroform-acetone, chloroformether).
Ideal Binary Solutions. For the ideal solutions, the diffu190
I&EC
FUNDAMENTALS
Figure 2.
Binary diffusion coefficients ( 6 )
Upper. lower.
Acetone (A)-water (9) Acetone (A)-chloroform (9)
sion coefficient experimentally determined, D A B , is equal to DAB. As appears in Figure 3, DABdoes not vary linearly with the composition (expressed in mole fraction). T h e departure from linearity increases as the difference between the limiting values of the diffusion coefficient increases; the same observation applies to the product D A B ~ A Bas, Bidlack and Anderson (9) showed. Since the diffusion isotherms, D A B ( N A ) , usually exhibit a negative departure from the straight line joining the extremities of the curve, one is led to represent the trend of these curves by the relation log DABIDAB' =
LYA'A
(14)
As appears in Figure 4, all the experimental results are satisfactorily represented by such a relation, which can be rewritten as a n exponential relation including the extreme values of DAB: DAB = (DAB')"B (DBA'Y-4
(15 )
Nonideal Binary Solutions. In nonideal solutions, the isotherms DAB(NA) generally exhibit a minimum. DAB decreases as the concentrations of A in B and of B in A increase. I n this paper the activity-corrected diffusion coefficient
0
0..5
1
NA
Very few experimental results are available for diffusion within alloys; however. for the systems Cu-Ni, Ag-Au, and Fe-Ni the previous conclusion is verified (Figure 7 ) . So for all these binary systems, the variation of the diffusion coefficient can be predicted from the limiting values of these coefficients and from data on thermodynamic activities. Associated Solutions. In these systems, the thermodynamic factor has no meaning, since its numerical value varies with the nature of the species. T h e first kind of associated solutions-for example, solutions of methanol or ethanol in benzene or in carbon tetrachlorideare characterized by a n average degree of association of the alcohol molecule. I t then becomes impossible to speak of diffusion in a binary system, since each species diffuses a t its own velocity. These associated solutions depart considerably from the ideal solution law. According to Prigogine and Defay (57), most of this departure can be explained through the interactions leading to the formation of associated complexes. Furthermore, the resulting system of single-molecule complexes must approach a n ideal system. This model has been tested in the case of solutions of alcohol in benzene or carbon tetrachloride. T h e experimental results (Figure 8) show that < 1, the fraction within the concentration range 0.3 < AV(nlcollol) of alcohol molecules which exists as single molecules is almost equal to zero, and the average degree of association, X , remains almost constant within this concentration range (Figure 9).
3
< xA < 4 xA
-
for methanol in CC1, for ethanol in CC14
4
Within this range of composition, log D A B varies linearly with the alcohol mole fraction (Figure IO). So, for large alcohol concentration, it appears as if complexes made of three to four alcohol molecules diffuse through carbon tetrachloride. This conclusion appears justified, since the extrapolated straight line, log D A B ( S A ) , gives for very small values of ,VA a diffusion coefficient one third or one fourth of the diffusion coefficient of single molecules of alcohol in benzene or carbon tetrachloride (Figure 10, lower). For low alcohol concentrations, where the degree of association varies rapidly, a mixture of complexes exists and the diffusion coefficient experimentally determined is only a n average coefficient. I n the second kind of associated solution. a n association between molecules of different species takes place, as in mixtures of acetone and chloroform: CH3
\ 0.5
I 1
0:s
0
NA Figure 3. Upper. lower.
Binary diffusion coefficients Dodecane (A)-hexane (6) (8) Hexadecane (A)-heptane (B) ( 9 )
DAB, has been computed from activity values reported in the literature. For all the systems studied the activity-corrected diffusion coefficient, D A B , appears, in Figures 5 and 6, to vary exponentially with the composition (in mole fraction) of the solution as predicted by Equation '15.
CO-H-C-C1
c1 / \ c1
/
CH3
T h e diffusion isotherms, D A B ( N A ) , shown in Figure 11 exhibit positive departure from the straight line joining their extremities. For these systems the thermodynamic factor has no meaning either. I t was impossible in this case to find a n interpretation, even limited, of the shape of these curves. Conclusions. Except for the associated solutions where a complex of molecules of different species exists, analysis has shown that the variation of the interdiffusion coefficient with the composition of the solution can be satisfactorily represented by an exponential expression of the form VOL. 5
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191
T
041
T
0.6-L
0
Figure 4.
0 Bromobenzene (A)-benzene (B) ( 4 5 ) 0 Bromobenzene (B)-toluene (B) ( I 4 )
Upper left.
A Upper righf.
Lower left.
lower right.
192
l&EC FUNDAMENTALS
Binary diffusion coefficients
H Bromobenzene (9)-chlorobenzene (A) (17) E Chlorobenzene (B)-toluene (A) (17) 0 Hexane (9)-dodecane (A) (9) 0 n-Octane (B)-dodecane (A) ( 6 8 ) A Hexane (B)-hexadecane (A) (8) m Heptane (B)-hexadecane (A) (9) H Propanol (B)-methanol (A) ( 6 1 ) A 2-Butanol-methanol (A) (61) Propanol (B)-2-butanol (A) (61) H Hexane (B)-CC14 (A) (9) 0 Benzene (B)-CC14 (A) (17) A Cyclohexane (B)-CCIh (A) ( 4 0 )
In t
n 0
0
.
0.J
0.4
0
I 0
1
015
1
NA
NA
0.5
0.4
0.J
0.2 r\
+
,3
.-
0.1
0
0.2 -0.1
0.q t
n .-0.2
-0.5
0
a -
+
? -
0.5
NA
Figure 5. Upper left. Upper right. Lower left. Center right. lower right.
0
-
0.4
Binary diffusion coefficients Acetone (A)-benzene (8) ( 6 ) Acetone (A)-CCId (B) ( 6 ) Acetone (A)-water (B) (61 Ethanol (A)-water (8) ( 2 8 ) Methanol (A)-water (B) ( 4 1 )
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193
-0.1 10
0.5
NA
t
4
0.4
0.2
T Figure 6.
Upper left. Upper righf. Cenfer left. Center righf. lower left. Lower right.
194
T
l&EC FUNDAMENTALS
Binary diffusion coefficients
Propanol (A)-water (B) ( 4 1 ) Methyl ethyl ketone (A)-CC14 (6) ( 4 ) Nitromethane (Akbenzene (B) ( 4 5 ) Ethanol (A)-chloroform (B) ( 4 1 ) Diphenyl (A)-benzene (8) ( 5 5 ) Cyclohexane (A)-benzene (8) (30, 47)
This applies to both liquid and solid solutions. The variation of the diffusion coefficient can then be predicted from the limiting values of the coefficient and from data on thermodynamic activities. Analysis of Experimental Results
Nonassociated Liquid Solutions. ZESE
0 5
0
NA
t
-'I4 0
0 5
NA
t
4
-1 2 IO
-1 3 L 10 0
Figure 7. Upper.
0r5
I
NA Binary diffusion coefficients
Copper (A)-nickel (6)
---a
I
AND
left). These ideal systems have been studied by Caldwell and Babb (77) a t lo", 27", and 40" C. T h e isotherms D,4Bexhibit a negative departure from the straight line joining their extremities and log D A B varies linearly with AVA. TOLUE~E-BROMOBENZENE (Figure 4, upper left). This ideal system has been studied by Burchard and Toor (74) a t 29" C. The diffusion isotherm drawn by these authors is linear, but the difference between the straight line and the exponential curve is smaller than 3% for a n equimolar composition. BENZENE(B)-BROMOBENZENE (A) (Figure 4, upper left). This system has been studied by Miller and Carman (45)a t 25' C. Experimental values are shoi+n in Figure 12. T h e dashed line is the author's line. T h e extreme value of the diffusion coefficient of bromobenzene in benzene, shown in this figure, is given by Kamal and Canjar (38). T h e average curve starting from this point and going through the experimental points of Carman has been drawn. T h e thermodynamic factor (25, 45) varies between 1 and 1.02, and log DABvaries linearly with 'YA. ~-OCTASE-~-DODECASE (Figure 4, upper right). This ideal system has been studied by Van Geet and Adamson (68) a t 25" and 60" C. T h e diffusion isotherms show a negative departure from linearity. Log DABvaries linearly with the composition. HEXANE (A)-HEXADECASE (Figure 4, upper right). This system has been studied by Bidlack and Anderson (8). The diffusion isotherm shows a marked negative departure from linearity. T h e activities have been determined by Bronsted and Koefoed (17) and
Log D A B varies linearly with .VAS TZ-HEXANE--~-DODECASE AND HEPTAKE-HEXADECANE (Figure 4, upper right). These two systems have been studied by Bidlack and Anderson ( 9 ) a t 25" C. T h e diffusion isotherms show a marked negative departure from linearity (Figure 3). T h e thermodynamic activities have been determined by Bronsted and Koefoed ( 7 7 ) a t 20' C. CYAB
= 1
aAB= 1
+ 0.082 .\r~LV~
+ 0 . 1 8 .VA.YB
for hexane-dodecane for heptane-hexadecane
T h e first system can be considered as ideal; log D A B varies linearly with ITA. For the second system, log D A B varies linearly with A V A .
METHANOL-PROPANOL,METHANOL-BUTANOL,AKDBUTANOLPROPANOL (Figure 4, lower left). These three systems have been studied by Shuck and Toor (67) a t 30' C. The difl usion isotherms show a negative departure from linearity. T h e methanol-propanol system shows a n appreciable departure from ideality. T h e activities determined from vapor pressure measurements of Hill and Van IVinkle (32) lead to the following values of the thermodynamic factor:
(20, 6 5 )
Cenfer. Gold (A)-silver (B) at 900' C. (60) lower. Nickel (A)-iron (6) at 1000' C. ( 2 6 )
TOLUENE-CHLOROBEN-
CHLOROBESZENE-BROMOBENZESE (Figure 4, upper
A'biethanol aAB
0 1
0.2 0.92
0 .4 0.885 VOL. 5
0.6 0.885
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0.92
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1 1 195
Log DAB varies linearly with the composition. For the two other ideal systems, log DAB varies linearly with the mole fraction. H E X A X E - C C(Figure ~~ 4, lower right). This system has been studied a t 20' C. by Bidlack and Anderson ( 9 ) . The diffusion isotherm shows a marked negative departure from linearity. Log D A B varies linearly with iVA. The thermodynamic activities have been determined by Christian (78).
3.0
I
0.4 0.2
CYAB =
1 - 0 , 3 6 :VAA7B a t 20' C 0
T h e activity correction leads to values of log DABwhich exhibit a slight positive difference from linearity. BENZEKE-CCI~ (Figure 4, lower right). This system has been studied by Caldwell and Babb (77) a t lo'? 25', and40'C. T h e diffusion isotherms present marked negative departures from linearity and log DABvaries linearly with iVA. T h e thermodynamic activities have been determined by Scatchard (59)and Christian (78): CYAB
Figure 8. Fraction of alcohol molecules existing as monomolecules (5 1 ) CCI4 (8)-ethanol (A)
t
= 1
1
= 1 - 0 . 2 8 lTA2VB
CYAB =
1
- 0 . 2 NAh'B
- 0 . 8 8 1VAA'B -
1.0
3.0
5.0
4.0
x,
Figure 9. Average association degrees, of methanol ( 1 ) and ethanol (2,' in CC14 (B) ( 5 1 )
METHANOL-WATER (Figure 5, lower right) and PROPANOLWATER (Figure 6, upper left). These systems have been studied by Lemonde (47), the first a t 15' C., the second a t 11° C. T h e values of aABhave been computed graphically from the values of the activity coefficients determined by Butler, Thomson, and McLennan (75) and Ewert (24). METHANOL-WATER SYSTEM
0.314 .VAdT~(iVB- iVA) a t 45' C.
Acetone(A)-CC14(B)
2.0
CA
a t 20' C.
T h e activity correction leads equally to values of log DAB presenting a slight positive difference from linearity. ACETONE-BEXZEKE, ACETONE-CC14,and ACETONE-WATER (Figure 5 ) . These systems have been studied a t 25' C. by Anderson, Hall, and Babb (6). Each diffusion isotherm shows a strongly marked minimum. T h e activity coefficients for the two first systems have been determined by Brown and Smith (73)at 45' C. Acetone(A)-benzene(B) cvAB
1
NA
The activity correction leads equally to values of log DAB presenting a slight positive difference from linearity. CYCLOHEXASE-CC14 (Figure 4, lower right). This system has been studied by Hammond and Stokes ( 2 8 ) ,and recently by Kulkarni, Allen, and Lyons (40) a t 25" C. using the Gouy interferometric technique. T h e diffusion isotherm obtained by these last authors presents a negative departure from linearity and log D A B varies linearly with iV.4. The thermodynamic activities have been determined by Scatchard (58). l n f A = 0 . 1 iVB2
0.5.
0
NMethanol ffAB
0 1
0.2 0.72
iVprop,,panol
0
ffAB
1
0.1 0.48
0.4 0.67
0.5 0.69
0.6 0.74
0.8 0.88
1 1
0.8 0.53
1 1
PROPANOL--\VATER SYSTEM
T h e activity coefficients of the acetone-water system have been determined by Beare, McVicar and Ferguson (7). The values of the thermodynamic factor, a A B , have been computed graphically. They differ slightly from the values computed from the Van Laar equations. AIACetone
c(AB
0 0.1 0.2 1 0.59 0 . 4
0.3 0.3
0.4 0.3
0.5 0.31
0.6 0.8 0.35 0.62
1 1
system, the extreme value of DABofor iVqcetane= 0 is larger than the experimental value. ETHANOL-~YATER (Figure 5, center right). This system has been studied and discussed by several authors (23, 28, 47,42, 62). The most recent values of Hammond and Stokes ( 2 8 ) , obtained a t 25' C., have been used with the thermodynamic factor values given by these authors. Log DAB varies linearly with KA. 196
IhEC FUNDAMENTALS
0.4 0.12
0.6 0.23
Log D A Bvaries linearly with iVA for the two systems. The extreme value of the diffusion coefficient of methanol in water measured by Lemonde appears too high. From the values obtained by Rossi (54) a t 20' and Johnson and Babb (37) a t 27' C., the value obtained a t 15' C., assuming that the product D p / T remains constant, is equal to 1 . 0 9 X 10-6 sq. cm. per second. This agrees with the value obtained by extrapolation of the straight line: log DAB for N b f e t h a n o l
Figure 5 (upper left and right and lower left) shows that log
DABvaries linearly for each system. For the acetone-CC14
0.2 0.22
+
0
METHYL ETHYLK E T O N E - C C(Figure ~~ 6, upper right). For this system, studied a t 25' C. by Anderson and Babb ( 4 ) ,the diffusion isotherm shows a minimum. The values of LYAB have been computed graphically from the values of the activity coefficients determined by Fowler and Norris (25). NKetone
CYAB
Log
0 1
DAB
0.2 0.82
0.4 0.785
0.5 0,845
varies linearly with N A .
0.6 0.88
0.8 0.92
1 1
0.6
t I
- 0.2I O
O 1S
1 NA
I
I
0.5
0
NA O.'
6
Figure 10.
Binary diffusion coefficients
Upper left. Methanol (A)-benzene (8) 0 (6)
w (35)
Upper right. A w (6)
Ethanol (A)-benzene (B)
A (35) lower left.
0 H CCI4 (B)-ethanol (A) (5,2 8 ) 0 CC14 (B)-methanol (A) (5)
NITROMETHANE (A)-BENZENE(B) (Figure 6, center left). This system has been studied by Miller and Carman (45). T h e diffusion isotherm exhibits a marked minimum. T h e activity coefficients have been determined a t 45' C. by Brown and Smith (72) and
log DAB varies linearly with A A . T h e limiting value of the diffusion coefficient of benzene in nitromethane, DBao,obtained by these authors appears to be too small. ETHANOL-CHLOROFORM (Figure 6, center right). This system has been studied by Lemonde (47) a t 15' C. T h e diffusion isotherm shows a minimum. T h e values of a A B have been computed graphically from the values of the activities determined by Scatchard and Raymond (57). '$'Ethanol ffAB
0 1
0.2
0.50
0.4 0.47 VOL. 5
0.6 0.65
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0.8 0.96 1966
1 1 197
DIPHESYL-BENZE~E (Figure 6, lower left). This system has been studied by Sandquist and Lyons (55) for a diphenyl concentration range A V ~ l p h e 0.4. For this composition range, the average degree of association remains constant. METHANOL-cC14 (Figure 10, lower left). This system has been studied a t 25' C. by Anderson and Babb (5). The diffusion isotherm shows a sharp minimum, as does the curve of the thermodynamic factor variations. Log DABvaries linearly with the alcohol mole fraction for A'A > 0 3. ETHANOL-CC14 (Figure 10, lower left). This system has been studied by Hammond and Stokes (28) and Anderson and Babb (5) for small ethanol concentrations. T h e values re198
l&EC FUNDAMENTALS
2 0.5
I
NA Figure 1 1.
Binary diffusion coefficients
Upper.
Diethyl ether (AI-CHCh (B)
lower.
Acetone (A)-chloroform (6) ( 6 )
0 (3)
ported by these last authors are much larger than Hammond and Stokes' values. T h e thermodynamic factor shows, as the diffusion isotherm, a sharp minimum. Log DAB varies linearly with the alcohol > 0.3. mole fraction for
n 0
r
4 l.OL--
0
0.5
1
NA Figure 12.
Binary diffusion coefficients
Brornobenzene (A)-benzene (B)
--- (‘45) 0 (30)
NITROMETHANE-CCl?,. This system has been studied by Miller (45). T h e diffusion isotherm, D A B ( N ~shows ) , a sharp minimum, as does the curve of the thermodynamic factor variations. (The nitromethane dimerizes in CC14.) DIETHYLETHER-CHLOROFORM (Figure 11, upper). This system, studied by Anderson and Babb (3),is characterized by the production of a complex. T h e diffusion isotherm shows a positive departure from the straight line joining the extreme points. ACETOSE-CHLOROFOF:M (Figure 11 lower). This system has been studied by Anderson, Hall, and Babb ( 6 ) a t 26’ and 40’ C . T h e diffusion isotherm shows a positive departure from the straight line joining the extreme points. T h e thermodynamic factor also goe,sthrough a maximum. For this system, as for the chloroform-ethyl ether system, Powell, Roseveare, and Eyring (49) have shown that the product D p / a A Bvaries linearly. However, more recent computations made by Anderson, Hall, and Babb ( 6 ) disprove this result (Figure 2, upper). SUCROSE-WATER ASD SACCHAROSE-WATER. These two systems have been studied by Irani and Adamson (2, 34) a t 25’ C. for low sucrose and saccharose concentrations. I n both systems, n molecules of water are tied u p with each molecule of sugar. T h e diffusion coefficient decreases, but the thermodynamic factor increases, as the sugar concentration increases. ~
References (1) Adamson, A.
W., J . Phys. Chem. 58, 514 (1954). (2) Adamson, A. W., Irani, R., J . Chim. Phys. 55, 102 (1958). (3) Anderson, D. K., Babb, A . I,.,J . Phys. Chem. 65, 1281 (1961). (4) Ibid., 66, 899 (1962). (5) Ibid., 67, 1363 (1963). (6) Anderson, D. K., H:all, J. R., Babb, A. L., Ibid., 62, 404 (1958). (7) Beare, W. G., McVicar, G. A., Ferguson, J. B., Ibid., 34, 1310 11930). ( 8 j Bidlack, D. L., Anderson, D. K., Ibid., 68, 206 (1964). ( 9 ) Ibid., p. 3790. (10) Bird, R. B., Stewart, N. E., Lightfoot, E. N., “Transport Phrnomena,” M’iley, New York, 1962. (11) Bronsted, J. N., Koefoed, J. K., Danske Vidensk Selsk 17, 22 (1946). (12) Brown, I., Smith, F., Australzan J . Chem. 8, 501 (1955). (13) Ibtd., 10, 423 (1957:. (14) Burchard, J. K., Toor, H. L., J . Phys. Chem. 66,2015 (1962). (15) Butler, J. A . V., Thomson, D. W., McLennan, W. J., J . Chem. Sot. 1933, p. (174. (16) Caldwell, C. S., Babb, A . L., J . Phys. Chem. 59, 1113 (1955). (17) Ibzd., 60, 51 (1956). (18) Christian, S . D., Neparko, E., Affsprung, H. E., Ibzd., 64, 442 (1960).
(19) Darken, L. S., Trans. A I M E 175, 184 (1948). (20) Da Silva, L. C., Mehl, R. F., Ibid., 191, 155 (1951). (21) De Groot, S. R., Mazur, P., “Nonequilibrium Thermodynamics,” North-Holland, Amsterdam, 1961. (22) Dolezalek, F., Schulze, A , . Z. Physzk. Chem. 83, 45 (1913). (23) Dullien. F. A . L., Shemilt, L. I\’., ,Vature 190, 526 (1961). (24) Ewert, H.. Bull. Sot. Chim. Belge 45, 493 (1936). 125) Fowler. R. T.. Norris. G. S.. J . AMI. Chem. 5. 266 11955). (26) Goldstein, J . I., Hanneman, R:‘E., Ogilvk, R. E., Trans. A I M E 233,812 (1965). (27) Gordon. A. R.. J . Chem. Phys. 5,522 (1937). (28) Hammond, B. R., Stokes, R. H., Trans. Faraday SOC. 52, 781
, --.,.
( 1~, 05AI
(29) Hardt, A. P., Anderson, D. K., Rathburn, R., Amar, B. I\’., Babb. A. L., J . Phys. Chem. 63, 2059 (1959). (30) Harned, H. S., Discusstons Faraday SOL.24, 7 (1957). (31) Hartley, G. S., Crank, J., Trans. Faraday Sot. 45, 801 (1948) (32) Hill, \V. O., Van iYinkle, M., Ind. Eng. Chem. 44, 2450 (1952). (33) Hultgren, R., Orr, R. L., Anderson, P. D., Kelley, K. K., “Selected Values of Thermodynamic Properties of Metals and Alloys.” IViley, New York, 1963. (34) Irani, R. R., Adamson, A. W., J . Phys. Chem. 62, 1517 (1958).
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RECEIVED for review July 15, 1965 ACCEPTED January 13, 1966
VOL. 5
NO. 2
MAY 1966
199
