SOME ASPECTSOF DIFFUSION IN TERNARY SYSTEMS
2949
and the effect of this attraction is directly opposite to that of surface heterogeneity: heterogeneity leads to a decrease in the heat of adsorption while adsorbateadsorbate attraction causes its increase. The adsorption properties depend on the geometry and chem-
istry of the surfaces. I n particular instances, when the effect of this attraction balances or overcomes the effect due to the heterogeneity of the surface, the surface behaves more or less like an energetically homogeneous surface.
Some Aspects of Diffusion in Ternary Systems’ by V. Vitagliano and R. Sartorio Istituto Chimico dell’ Universitlc di Napoli, Naples, Italy
(Received December 17, 1969)
+
Diffusionequations of three model ternary systems are discussed: (1) ideal ternary solution: solvent (0) A B; (2) ideal ternary solution with a chemical equilibrium: 0 A B eAB (= Z); (3) ternary soluand Dunlop4equations tion with a salting out effect of component A (or B) on component B (or A). Laity2$* are chosen to describe the flow process, X k = & R k 2 c ; ( ~ k - vi) (I). If no interactions exist between the flows (Le., the thermodynamic frictional coefficients relating the solute flows are zero and the only nonzero frictional coefficients are those relating solvent and solutes flows) the thermodynamic flow equations reduce to X i= RtOcO(u, - uo) (11),where the index 0 refers to the solvent. It is shown that when eq I1 describes the flow process, the cross-diffusion coefficients Dl2 and DZl of the extended Fick’s equations, J, = - ZDik X (dcklbs), are not zero and they may even reach orders of magnitude equal to those of the main terms DU and D22. They are generally negative for ideal solutions and for solutions with positive interactions between solutes, such as a chemical reaction (systems 1 and 2). They are positive when repulsive interactions (salting out effects) are present between the solutes (system 3). Numerical and graphical examples are given.
+ +
+
Diffusion in multicomponent systems has been the subject of increasing interest in recent years. The theoretical hydrodynamic and thermodynamic background describing this transport process has been already fully discussed and papers from several authors are readily available in the Iiteraturea6-l0 We think it might be of some interest to write out explicitly the expressions for diffusion in some model systems and show the numerical values that transport coefficients can assume. It is known that diffusion in ternary systems may be described by an extended Fick’s law of the type
Jz
=
bc1 bc2 -Dzi ax - Dzz dX
where the Dlk are the four experimentally measurable diffusion coefficients and indices 1 and 2 (and 0) refer to components 1 and 2 ( and to the solvent). Generally if D12 = Dzl = 0 one speaks of noninteracting flows. Actually a definition of “noninteracting flows” applies better to phenomenological transport equations in terms of thermodynamic flows and forces. Assuming that interactions among flows should imply
the existence of some relative friction between species whose flows interact, we think that the best way to define diffusion with no interacting flows is through the use of the phenomenological equations proposed by la it^^,^ and by Dunlop4
where pi = chemical potential of component i, cI = concentration of component i, Rki = frictional coefficient relating components i and IC, vi = velocity (1) This work has been carried out with the financial support of the Italian C. N. R. (2) R.W.Laity, J. P h ~ s Chem., . 63, 80 (1959). (3) R.W.Laity, J. Chem. Phys., 30, 682 (1959). (4) P.J. Dunlop, J. Phys. Chem., 68, 26 (1964). (5) D. G. Miller, Chem. Rev., 60, 15 (1960). (6) D.G.Miller, J. Phys. Chem., 6 3 , 570 (1959). (7) D . G. Miller, ibid., 71, 616 (1967). (8) D. G. Miller, ibid., 71, 3588 (1967). (9) T. K.Kett and D. K. Anderson, ibid.,73, 1262 (1969). (10) An extensive literature on this subiect can be found in the “Annual Reports on Mass Transport,” published by Ind. Eng. Chem.; for example, A. Gomezplata and T. M. Regan, Ind. Eng. Chem., 60, No. 12, 53 (1969); K.B. Bischoff and D. M. Himmelblau, ibid., 60, No. 1, 66 (1969); K.B. Bischoff and D. M. Himmelblau, ibid., 58, No. 12,32 (1966). The Journal of Physical Chemistry, Vol. 74?No. 16,1970
2950
V. VITAQLIANO AND R. SARTORIO
of component i measured with respect to any reference frame." Interaction at least with the soIvent must obviously exist, so we define a system with no interacting flows as one in which Rki = 0 if both i and k do not refer to the solvent (i and k ti 0). This assumption implies that there are no frictional effects between components i and k ; (eq 3 is given belowll). We shall show that on this assumption experimental cross-diffusion coefficients may be measured whose values may be of the same order of magnitude as the main terms. Three model systems will be discussed: (1) ideal ternary solution (solvent A B); (2) ideal ternary solution with a chemical equilibrium
+ +
A + B ~ A B = Z (4) B Z ) ; (3) a ternary system where
+ + +
where the fi are the rational activity coefficients. To change from moIe fractions to concentrations, the following expressions can be used COVO
+ ClVl + c2v2 = 1
(10) where the V , are the molar volumes (constant because VA, V2 VB, of the ideality of the solution, VI Vl V2 5 VZ)
+
1 cz = 2-(cl
1 + + -) KV c2
-
V(C1 f
C2
f 1 / K v ) 2 - 4CiC2 (11)
V being the total volume V = Novo
+ N A V ~+ N B V ~+ Nz( Vi + V2)
(12)
(solvent A the A (or B) component promotes a salting out effect on the B (or A) component. The first two systems show negative values of the D12 and D21 cross coefficients, the third one shows positive values. I n the following, index 1 refers to the stoichiometric component A irrespective of its actual form in sohtion and index 2 refers to the component B. Capital letters (A, B, or Z) refer to the actual species existing in solution. The index 0 always refers to the solvent.
System 3. We assume the following expressions for the activity coefficients of components 1 and 2
Thermodynamic Background
- = yo
Systems 1 and 2. The various chemical potentials of an ideal system are PA
=
PA'
PB = PB'
pz = fiz'
p o = PO'
+ R T In N A + R T In NB + R T 111Nz + R T In NO
(5)
In f1 = X Y ~ Y ~ ~ / Y O ~ Inf2 = XYI~YZ/YO~ where, of course, y l = NA,y2 = NB, and yo No. The S > 0 coefficient accounts for the salting out effect. Equations 13 obey the Gibbs-Duhem expression, the total free energy of mixing being AF RT
In yo
+ y1 In y1 + y2 In y2 + 21
y12y22
__
yo3
(13a) This model system exhibits a phase separation (see Figure 1). The Flow Equations. For systems 1 and 2 eq 2 may be written
where the Ni are the mole fractions of the various species. From the equilibrium eq 4 we have PZ = P A
+
PB
(6)
also the equilibrium constant K relates the mole fractions of the solutes
Nz = KNANB
(7)
Therefore the stoichiometric mole fractions will be
+ KNANB) vi = ( N A + K N A N B ) / ( + ~ KNANB) = (NB + K N A N B ) / ( + ~ KNANB) go = No/(l
(8)
y2
The stoichiometric chemical potentials of the three components remain 10, PA, and PB, NO,N A , and NB being the activities uo = No = foyo ai = N A = fiyi a2 = N B = f2y2 The Journal of Physical Chemistry, Vol. 74, No. 16,1970
(9)
where the cross-frictional terms RAB, RAZ,and RBZ are absent in accordance with our definition of non(11) The flow process described by eq 2 is independent of reference frames because only relative velocities ( v k - v i ) appear in eq 2. The Onsager reciprocal relations hold among the frictional coefficients of oq 2; Rib = Rki (3).
2951
SOME ASPECTSOF DIFFUSION IN TERNARY SYSTEMS
where
H = -
1
0
Figure 1. Phase diagram of a ternary system with activity coefficients and free energy of mixing given by eq 13 and 13a.
+-
CACB
-
cOc1c2
Rio
=
Rzo
=
R12
(15)
=
and since
c1 CA
+
Rio
'CZ
CZ(V2
- vo)
cz
= -X
A
CORZO
+
where the flows are measured with respect to a solvent fixed reference, (Jo)s = 0. Equations 17 are in the form of thermodynamic flow equations. The entropy production is given by TU
=
- ( J I ) ~ X A-
(J2)sxB
(18)
and the Onsager reciprocal relations hold between the cross coefficients cz
Liz = - - LZl
coRzo
(19)
Equations 17 may be written in the form of eq 2 by solving for X A as a function of ( V I - vo) and (vi - v d , and for XB as a function of (v2 - uo) and (v2
-
2/11
+m
(23) eq 21 becomeI2
> c2 =
RAO
R12
- RAO = cORAO/(Ci - c2)
R12
=
Rzo eq 14 becomes
RAO
RBO Rzl 0
in the opposite limit, where K
=
(22)
Equations 20 and 21 show that the equilibrium 4 gives rise to apparent interactions between the flows (R12# 0 ) if they are measured as stoichiometric flows of components 1 and 2. In the limit where K = 0, eq 21 reduce to
interacting flows. The stoichiometric flows of components 1 and 2 are
Ct
CACZ
[RA~RB~ RAORZO
=
Rzo
-
C&BO/(CZ
~ i )
System 3. System 3 needs no further comments; eq 14a and 14b hold for it, since K = 0 eq 23 are obt ained . (12) If
CI =
cz = c we obtain
+ (RAO+ RBO- R z o ) ] R ~= o R z ~ R B ~ / [+ R z ~ (RAOf RBO- R z o ) ]
Rlo = R z o R ~ o / [ R z o
(25)
CORAORZOCZ
+
R1z= ( c c ~ ) [ c R z o CZ(RAO -t RBO- Rzb) as K goes to infinity Rn .-* m . It must be pointed out that in the absence of free A and B the only physically meaningful driving force is X z which is formally written ( X A XB),and eq 20 with the aid of 25 then becomes
+
-
X z = RZOCO(U 00)
(26)
As we expect, the only measurable coefficient is Rzo which refers to the only existing species, Le., Z ; the cross coefficient Rtz disappears in (26). The Journal of Physical Chemistry, Vol. 7 4 , No. 16, 1970
2952
V. VITAGLIANO AND R. SARTORIO
The Diffusion Coefficients The flow eq 1 are measured with respect to a volume fixed frame n
c VJJ,
i=l
=
0
they are related to the thermodynamic forces by equations l 3
where14
lim Dtk = - co ci
+
Ck+O
(37) on the other hand16 lim Dkt = 0
(38)
Cb+0
While the main coefficients are always positive, the sign of the cross coefficients depends on the sign of the second factor in the right side of eq 36. A rearrangement of that equation gives the following inequality (relation 39). Table I: Diffusion Coefficients in a Ternary Ideal Solution with No Interaction between the Solute Flows and Constant Frictional Coefficients, V O= 0.020, VA = 0.040, VB = 0.070, RAO= 0.004, RBO= 0.006
On the other hand, the driving forces can be written as
where
at being the activities, eq 9 and 13. Solving eq 28 with the aid of eq 31, the Fick's eq 1 are obtained with the following values for the four diffusion coefficients
It is interesting to discuss briefly eq 35 and 36, which show that even in the absence of interactions between components A and B ( K = 0) the cross coefficients Die are not zero. In the limit of C, + 0, eq 35 reduces to the value of the diffusion coefficient of component i in a binary solution;15 surprisingly the D,, coefficient does not disappear The Journal of Physical Chemistru, Vol. 74- N o . 16,1970
DZI
Dzz
c2 = 0.0 0.000 0.0120 0.0136 0.0024 - 0 0242
0.000 0.000 0.000 0.000 0 * 000
3,3333 3.4722 3.6232 3.7879 3.9682
2.0 3.0 4.0
5.3763 5.5036 5,6376 5.7789 5.9282
c2 = 1 . 0 0.000 0.0137 0.0155 0,0028 - 0.0279
-0.1449 -0.1846 -0.1655 -0.1776 -0,1911
3.5088 3.6583 3.8216 4.0006 4.1976
0.0 1.0 2.0 3.0 4.0
5,8140 5,9783 6.1439 6,3256 6.5203
e2 = 2 . 0 0.000 0.0157 0.0179 0.0032 -0.0325
- 0,3170 - 0,3400 - 0,3657 - 0.3947 - 0.4274
3.7037 3,8655 4.0429 4.2385 4.4553
0.0 1.0 2.0 3.0 4.0
6,3291 6.5317 6.7501 6,9866 7.2439
cz 3.0 0.000 0.0182 0.0209 0.0038 - 0.0385
- 0.5242 - 0.5655 -0.6121 - 0.6650 - 0.7256
3.9216 4.0974 4,2914 4,5066 4.7468
0.0 1.o 2.0 3.0 4.0
6.9444 7,2046 7,4890 7 .8018 8.1481
= 4.0 0.000 0.0214 0.0247 0.0045 - 0,0463
-0.7778 0.8446 -0.9210 - 1,0090
4,1667 4.3590 4,8724 4,8108 5.0794
c1
Dii
0.0 1.0 2.0 3.0 4.0
5 000 5.1020 5.2083 5.3191 5,4348
0.0 1.o
I
Dia
I
-
-1,1111
(13) Reference 10 eq 16 and 17. (14) The R ~ oand Rzb are the stoichiometric frictional coefficients defined by eq 21, 22, and 23. (15) Reference 10 eq 40. (16) L. Sundelof, Ark. Kemi,20, 369 (1963). (17) These values are arbitrary; they have been chosen to give diffusion coefficients in the range of unity, that is the order of magnitude of experimental diffusion coefficients if measured in cm2 sec-l X 106.
2953
SOMEASPECTSOF DIFFUSION IN TERNARY SYSTEMS
(39)
At higher values of ci the Dak coefficients are always negative, but if
Rio
Vo
KO+%-
le0
there will be one value of ct, independent of ck, for which
Dtk
=
0
At lower civalues the Dik will be positive (see Figure 2).
Numerical Examples Some numerical examples may help to clarify the implications of the expressions obtained so far. All data will refer to a model system with the following values of the molar volumes and frictional coefficients : Vo = 20 cm/mol, VA E V1 = 40, VB 5 V2 = 70,
-
-091
0.1
-D, -
C*'O
I
-0.1
I
-
'
.:'.' 0,.
I 0.02
ki 1)
-
0.0 2
V2
I 0.00
,
I 1
+
- a03
RA= ~ 0.004, RBO
1 I
Figure 3. Cross diffusion coefficients in a ternary system with a chemical reaction (A B = Z) between the solutes as a function of the equilibrium constant K (eq 7). The component 2 concentrations are: (1) c2 = 4.0; (2) = 3.0; ( 3 ) = 2.5; (4) = 2.1; (5) = 2.0; (6) = 0.2; (7) = 1.0; (8) .= 1.7; (9) = 1.5; (10) = 0 . 5 ; (11) = 1.8.
C,2 CZ
-0.04-
0.10
Figure 2. Cross diffusion coefficients in an ideal ternary system (see data of Table I) as a function of the two solutes concentrations.
= 0.006, RZO= 0.050.17 Concentrations are in moles per liter. Computations have been made with the help of an IBM 1620 computer. The four diffusion coefficients tabulated as functions of ci and c k in the concentration range 0 to 4 M are given in Table I for K = 0. Figure 2 shows the DIz and D21 coefficients as a function of c1 and cz in a Zimm-like-plot graph. The values of the diffusion coefficients when all of A or of B react to give Z ( K -+ a) are given in Table 11). The behavior of the diffusion coefficients and of the frictional coefficients in the region of equilibrium 4 is shown in Figures 3, 4,and 5 . Figure 3 shows the behavior of D12 and D21 at c1 = 2 M as a function of K for different c2 values. Figures 4 and 5 show similar plots for Rloand R12 ( = Rzl),eq 21. Finally, the behavior of a system with a salting out effect according to eq 13 is shown in Table 111, The Journal of Phyaical Chemistry, Vol. 74, No. 16, 1970
2954
V. VITAGLIANO AND R. SARTORIO ~
~~
~~
~
~~~
~
~~~
Table 11: Diffusion Coefficients in a Ternary System Containing Component A and B which React Completely to Form a Compound Z ( K 4 m, eq 7), VO= 0.020, VA = 0.040, VB = 0.070, RAO= 0.004, REO= 0.006, Rea = 0.050 Dii
CL
DLZ
Da
Dzi Ca + 0
0.0"
...
...
1.0 2.0 3.0 4.0
5,102 5.208 5.319 5.435
-4.272 -3.932 -3.578 -3.212
0.0 1.O" 2.0 3.0 4.0
0.4301
,..
...
0.000 0.000 0.000 0.000
0.4167 0.4348 0.4546 0.4762
Deo = 0.4000
ca = 1.0
0.000
...
...
-2.810
3.509
-0.2312 - 0.2482 - 0.2672
0.6686 0.6820 0.6960
Dz
5,542 5.689 5.846
-4.590 -4.200 -3.792
0.4651 0.7256
0.000 - 0.2758
- 0.2674 -2.935
3.704 3.721
6,113 6.324
-5.000 -4.546
-0.5330 0.5779
0.9886 0.9946
= 0.4301
c2 = 2 . 0
0.0 1.0 2.0" 3.0 4.0
...
...
De = 0.4651
CZ
= 3.0
0.000
0.0 1.o 2.0 3.0" 4.0
0.5063 0.7785 1.108
- 0,3202
0.0 1.o 2.0 3.0 4.0"
- 0,6429
-2.524 -2.789 -3.100
3.922 3.967 3.999
6,884
-5.549
- 0.9432
1.410
0.5556 0.8400 1* 190 1.624
0.000 - 0.3764 - 0.7627 -1.154
...
...
...
...
...
Cg
..,
De
= 0.5063
= 4.0
-2.356 -2.625 -2.944 -3.326
...
4.167 4.248 4 319 4.376 I
...
De = 0.5556
In solution only the species Z is present.
1
I
I
I
I
10
100
I
I
K io3
I
I
1 o4
1 o5
Figure 4. Main frictional coefficient Rlo (eq 21) as a function of the equilibrium constant K (eq 7) a t fixed cI = 2.0 concentration and varying cz concentra.tion. The c2 values are given on each curve. The Journal of Physical Chemistry, Vol, 74, N o . 16, 1970
Figure 5 . Cross-frictional coefficient R1z (eq 21) as a function of the equilibrium constant K (eq 7) a t fixed c1 = 2.0 concentration and varying cZ concentration. The cz values are given on each curve.
2955
SOMEASPECTSOF DIFFUSION IN TERNARY SYSTEMS in which we have collected the four diffusion coefficients as functions of the S parameter for some systems at different concentrations, including systems at the critical mixing composition (see Figure 1).
Table III : Diffusion Coefficients in a Ternary Solution with Salting Out Effect on Components A and B According to E q 13, Vo = 0.020, VA = 0.040, VB = 0.070, RAO= 0.004, RBO= 0.006 S
DII
DZI
Diz
Dza
IDik/
= 2.0 cz = 4.0 c1
20 50 100 200 300 340 350 360
20 50 100 200 300 340 350 360
7.574 7.700 7.912 8.335 8.758 8.927 8 969 9.011 I
8.682 9.484 10.821 13.49
16.17 17.24 17.50 17.77
0.222 0.519 1.014 2.003 2.992 3.387 3.486 3.585
-0.764 -0.528 -0.136 0.650 1.435 I .749 1.828 1.906
~1
= 4.0
c2
= 4.0
1.241 3.173 6.392 12.83 19.27 21.84 22.49 23,13 c1 =
-0.552 0.287 1.685 4,482 7.278 8.397 5.676 S .96
4.664 4.803 5.033 5.494 5.954 6.138 6.184 6.230
Figure 6. The four Dik diffusion coefficients in a system containing A -/ AB as a function of the Rzo frictional coefficient a t constant RAO= 0.004.
(3) Repulsive interactions between species A and B, promoting a salting out, have an opposite effect on the cross-diffusion terms; these shift to positive
5.412 5.912 6.745 8.411 10.077 10.744 10.91 11.08
+47.9 +47.0 +62 .O $57.0 +23.0 +1.4 -0.4" -1.Ob
6.300 8.092 9.670
+45.8 +31.3 -0.5'
3.0
cz = 3 . 0
500 1000 1440
9.660 12.33 14.68
6.210 12.42 17.88
2.422 5.508 8.224
a Approximate critical mixing composition. Unstable system, composition corresponding to a two-phase region.
values and may become even greater than the main terms. As may be expected,18 in the region of critical mixing
DiiDzz
=
DizDzi
(42)
For a three-component system eq 42 corresponds in fact to the relation D = 0 at a critical point of a binary mixture.
Acknowledgments. The authors wish to thank Dr. H. Schonert for reading the paper and for discussions and useful suggestions.
Appendix Some explicit expressions are given here to be used in the text equations
Conclusions From the expressions obtained in the previous pages and from inspection of the numerical results we may suggest the following conclusions. (1) I n an ideal ternary system with constant frictional coefficients and without interaction between the solute flows, the experimental cross-diffusion coefficients are not zero. They are likely to be negative and they may even be of the same order of magnitude of the main terms. If relation 40 holds, the Dlkt erm may become positive on dilution of component i, and it goes to zero from the positive side as ca goes to zero (see Figure 2). (2) The appearance of a chemical equilibrium between species A and B, with the existence of a third species Z = AB, increases the negative values of the cross-diffusion coefficients which may probably reach values close to those of the main terms.'*
pi1 =
b In N A b In yz +----b In yz b s b In b In y1 b In N A b In yz +----b In yl bc2 b In y2 bcz b In N A b In y1 __ b In y1 b c ~
~
N A
p12
= ___
_ I _
(44)
(444
b In_ NA _ _I
b In y1
(18) The signs of the cross-diffusion coefficients do depend, of course, on the relative values of the three frictional coefficients. I n the case that Rzo is lower than RAOor RBOthe D,k may become positive as shown in Figure 6. This case appears t o be physically unreasonable and it looks like a simple algebraic speculation.
The Journal of Physical Chemistry, Vol. 74, No. 16,1970
2956
KOICHIRO NAKANISHI AND TERUKO OZASA Expressions for NB,p21, and ~ 2 2 , and for the derivatives can be obtained by interchanging indices A and B, 1 and 2. From eq 13 we have
b- In - a, In Yi
-.b
In YI b In c1
-
TI
+l
/ ~
(46a)
-1+s
YiYk2(Y0
+ 3y,)
yo4
(47)
Equations 47 and 47a substitute the (b In N / b In y) in eq 44 and 44a to compute the pik.
Diffusion in Mixed Solvents. I. Iodine in Ethanol-Water and t-Butyl Alcohol-Water Solutions by Koichiro Nakanishi and Teruko Ozasa Department of Industrial Chemistry, Kyoto University, Kyoto, Japan
(Received Februarg 16, 2970)
The diffusion coefficientsof iodine diluted in ethanol-water and t-butyl alcohol-water solutions at 25.00' have been measured by the capillary-cell method. It is found that the diffusion coefficients do not vary linearly with the molar composition of mixed solvents and that the product Dllot/T (D1l0is the diffusion coefficient of iodine, 7 is the viscosity of solvent, T is the temperature) has a pronounced maximum at lower mole fraction (-0.05-0.1) of each alcohol. This fact is interpreted in terms of the structural anomaly in alcoholwater solutions.
Introduction Although the liquid-phase diffusion in binary systems has received much attention in the past decades, very little work has been carried out in ternary systems. This is due to our incomplete understanding of the structure and properties of multicomponent liquid mixtures that is pertinent to the interpretation of experimental data and also to the relative difficulty in making precise determinations of diffusion coefficients. There are only a few reports on diffusion in ternary systems. I n their pioneering works, Dunlop, Fujita, and Gosting' have measured the diffusion coefficients of two electrolytes or nonelectrolytes in water. Later, Toor, et U Z . , ~ - ~ and Anderson, et aE.,5 have studied the diffusivity in ternary systems of nonelectrolyte solutions such as alcohol mixtures. The present study is confined to establishing the limiting values of the mutual diffusion coefficient of a single solute in mixed solvents where the concentration of diffusant is practically equal to zero. This quantity, denoted as D1+' hereafter, is not identified with any The Journal of Physical Chemistrg, Val. 74, N o , 16,1970
binary diffusion coefficients. However, since the fluxes are uncoupled at infinite dilution, Dllocan be treated as a pseudobinary coefficient and our aim is to investigate how DIIo depends on the molar compositions in mixed solvents. Although this case of ternary diffusion has been studied experimentally by Toor, el Wilke, et U Z . , ~ Himmelblau, et ul.,' and Perkins, et aZ.,* their experimental work and theoretical analysis are confined almost exclusively to the cases of nonassociating solutions (1) F o r quick reference, see, for example, D. D. Fitts, "Non-Equilibrium Thermodynamics," McGraw-Hill, New York, N. Y., 1962, Chapter 10. (2) J . K. Burchard and H . L. Toor, J . Phys. Chem., 66, 2015 (1962). (3) F. 0. Shuck and H. L. Toor, ibid., 67, 540 (1963). (4) 1%. T. Cullinan, Jr., and H. L. Toor, ibid., 69, 3941 (1965). (5) T. K. Kett and D. K. Anderson, ibid., 73, 1268 (1969). (6) J. T. Holmes, C. R. Wilke, and D. R. Olander, AIChE J., 8 646 (1962). (7) Y. P. Tang and D. M. Himmelblau, ibid., 1 1 , 54 (1965). (8) L. R. Perkins and C. J. Geankoplis, Chem. Eng. Sci., 24, 1036 (1969).
