1262
TERENCE K. KETT AND DONALDK. ANDERSON
Multicomponent Diffusion in Nonassociating, Nonelectrolyte Solutions by Terence K. Kett and Donald K. Anderson Department of Chemical Engineering, Michigan State University, East Lansing, Michigan
(Received July 6 9 , 1 9 6 8 )
Hydrodynamic theory is used to describe multicomponent diffusion in nonassociating, nonelectrolyte solutions. Generalized expressions are developed for the fluxes of each component relative to the volume-average velocity, for the diffusion coefficients,Daj, and for the phenomenological coefficients,Lij. The Onsager reciprocal relations are derived from the model and it is shown that the phenomenological coefficients should be independent of activity data.
Introduction The Onsager reciprocal relations have been of considerable interest during the past decade, especially in the case of isothermal diffusion in liquids. has derived a sufficient condition for which the Onsager reciprocal relation is valid. This equation is derived for the case of isothermal ternary diffusion with the experimental condition of no volume flow. Only in the last few years have reasonable ternary data become available largely through the contributions of Gosting, et U Z . , ~and ~ Toor, et U Z . ~ - ~ They, along with Miller,1*21e have made attempts to verify the Onsager reciprocal relations using experimental data and the condition derived by Miller.1 Considering the limits of their experimental accuracy, the overall conclusions have been that their data indicate the validity of the reciprocal relations. In this paper, hydrodynamic theory is applied to multicomponent diffusion in nonassociating nonelectrolyte systems. The theory shows that the Onsager reciprocal relations are valid for nonassociating systems. Experimental evidence supporting this statement is presented in a companion paper. Other work concerning associating systems is currently in progress. Hydrodynamic Theory According to the hydrodynamic theory of Hartley and Crank,l0 the flux of a species results from purely molecular motion of the species and bulk motion of the medium. The former is generally referred to as intrinsic diffusion and results when a particular molecule acquires enough energy to escape the potential barrier of its surrounding molecules. This motion occurs regardless of concentration gradients which might exist. The name bulk motion is somewhat misleading since it seems to imply a flowing liquid solution through a container with open ends, Actually bulk motion referred to here results even in a closed system in which the volume is constant. For example, when a component of type i migrates across a coordinate-fixed reference plane in a closed volume container, the increase of material resulting in the one section must be compensated for by a bulk flow in the opposite direction. Thus, the total diffusion that is studied experimentally consists of both the bulk motion and intrinsic diffusion. The Journal of Physical Chemistry
The Flux Equations The driving force for diffusion is generally accepted as the negative gradient of chemical potential, thus (driving force for one-dimensional diffusion of i in the x direction) = - api/ax Although the forces resisting molecular diffusion are not fully understood, experimental evidence indicates that the resisting force is proportional to the viscosity of the medium. However, there has been considerable discrepancy regarding its dependence on the size and shape of the diffusing molecule and the effects of the other components in the medium. Therefore, let (resisting force for diffusion of molecule i) = -uiqui where q is the viscosity of the medium, ui is the velocity of species i relative to the velocity of the medium (defined later), and ui is a proportionality constant which depends on the molecular size and shape of the diffusing molecule. The negative sign is a result of the velocity of i being in the opposite direction as the resisting force. Summing the forces acting on the diffusing molecule
multiplying both sides by the concentration of i, Ci, and rearranging yield
(1) D.G. Miller, Chem. Rev.. 6 0 , 15 (1960). (2) D.G. Miller, J. Phys. Chem., 63, 570 (1959). (3) P. J. Dunlop and L. J. Gosting, J. Amer. Chem. Soc., 77, 5238 (1955). (4) H. Fujita and L. J. Gosting, ibid., 7 8 , 1099 (1956). (5) H.Fujita and L. J. Gosting, J. Phys. Chem., 64, 1256 (1960). (6) J. K.Burchard and H. L. Toor, ibid., 6 6 , 2015 (1962). (7) F. 0.Shuck and H. L. Toor, {bid., 67, 540 (1963). (8) H. T. Cullinan, Jr., and H. L.Toor, ibid., 69, 3941 (1965). (9) D. G. Miller, ibid.. 69, 3374 (1965). (10) G. 9. Hartley and J. Crank, Trans. Faraday Soc., 45, 801 (1949).
MULTICOMPONENT DIFFUSION
1263
where J i m is the flux of i relative to the medium in mol/cm2 sec, Ci is the concentration of i in mol/cma, and ui is the velocity of i relative to the velocity of the medium in cm/sec. Ji is the flux of i resulting from intrinsic diffusion only since the bulk contribution has not been considered. Notice that since ui is the velocity of i relative to the velocity of the medium, J i m is the flux of i relative to a coordinate plane moving with the velocity of the medium. Expressions for the velocity of the medium, which is the bulk motion referred to earlier, can be obtained by looking a t the volume flux across a coordinate-fixed plane. The volume flux across this plane due to the intrinsic diffusion of N components is given by
The same result is obtained in the case of constant molar volumes since here v ~ =, 0 ~and vm,c is given by eq 3. Substitution of eq 2 into eq 7 gives the desired result
I n the remainder of this discussion, only constantvolume systems will be considered. For the ternary case, eq 8a with the superscript V dropped becomes J1=---
c1aPl +c1-BlclaNl
Qlq ax
Qlq
ax
N
i-1 J2=
This is a volume flux relative to the velocity of the medium. If the system is closed, then this accumulation on the one side of the plane must be compensated for by a flow back across the plane to relieve the hydrostatic pressure that would be built up. Let us ~ , volume flux call this compensating flow, v ~ , the resulting from the flow of the medium relative to fixed coordinates. Hence
BlCl 8/41 - -c2 -aP2 +c2--
QZV
Ja=
ax
alq
ax
Plclakl c 3 8/43 --+c3-u311ax
clq
ax
(net volume flux relative to a fixed-coordinate plane) N
+
PiJim
=
Um,o
From the Gibbs-Duhem relation
i-1
For the case of constant molar volumes, the net volume flux relative to fixed coordinates is zero. Therefore Therefore
N ~ m , o=
-
BiJim
(3)
i-1
However, for the case of varying molar volumes, the net volume flux is not zero but rather vV+,. That is
JlPl
N
C PiJim+ Vm,a i-1
vv,o
(4)
The flux of a component with respect to fixed coordinates is given by
Jjc = J j m
+
~m,oCj
(5)
However, most equations describing diffusion are written with respect to the volume-average plane, the plane across which the net volume flux is zero. This . plane moves with velocity v ~ , ~Therefore Jjv
= Jjm
+
(vm,a
By the definition of Ji, it also follows that
- vv,a)Cj
+
J2P2
+
J378
=
0
(11)
and thus only two of the fluxes are independent. Utilizing eq 10 in eq 8, the two independent fluxes of components 1 and 2 are
Q1
(12)
(6)
Su.bstitution of eq 4 into eq 6 gives N
JjV
= J . 3m
- Cj C PjJim i-1
(7) Volume 73,Number 6 May 1069
TERENCE K. KETT AND DONALD K. ANDERSON
1264 For the case of N components, the Gibbs-Duhern relation gives
where
therefore By equating like coefficients of eq 12 and 19, four linear equations are obtained which can be solved for L11, L22, ,512, and L21. These are given below.
Also
Thus there are only N - 1 independent fluxes, J i , and - 1 api/dx terms in each flux equation. Performing the same algebraic operations as was done for the ternary case gives the generalized flux equations Ji=
P2CIC2 + LIZ6 = -a39 P8ClC2
Lllp
N
Q271
Q29
-!?((' 9
Qi
L2lp
a19
P3C2' + L226 = - (1 - P2C2) + c 2
a29
US?
Utilizing the relation
(i = 1, . . .,N - 1) (16)
--1 c*P3
yp-as=
The Phenomenological Coefficients MillerlB2 showed that for the Onsager reciprocal relations to be valid, the independent fluxes and forces for the ternary case should be as follows
and the fact that for a ternary system
CIPl+
c2v2
+ c.373 = 1
(22)
one obtains the following expression for LIZand L12
=
L21
ClC2Pl
L21 = - -(1 - PIC,) Ql9
where
6 d j is the Kronecker 6, and
are the phenomenological coefficients. Because the J i as well as the Yi are independent, the Onsager reciprocal relation Ll2
Lij
=
L2l
(18)
should be valid. Substituting for the Yi)s in eq 17 and rearranging gives
A point of interest here is that no activity data are required in order to calculate the phenomenological coefficients. For the case of N components, expressions for the phenomenological coefficients can be obtained in a similar procedure to that for the ternary. Extending Miller'slv2 analysis to a system of N components and using eq 14 and 15, one obtains N-1
Ji =
LijYj
(i = 1, . . ., N
- 1)
(24)
j-1
where
and The JOzlTnal of Phgsical Chemistry
6jk
is the Kronecker 6. Substituting eq 25 into 1
1265
MULTICOMPONENT DIFFUSION eq 24 and collecting coefficients of the aP/axls yields
It should be noted that these equations are written in
+ G)
terms of concentration gradients which are more meaningful in terms of experiment. I n order to derive expressions for the ternary diffusion coefficients, Dij, use will be made of the identity
N-I N-1
Ji =
Lij k-1
(&
j-1
apk
VjCk
(i = 1, I . ., N - 1)
(26)
R’ow equating coefficientsof eq 16 and 26 gives ( N - 1) linear equations with ( N - 1)2 unknowns, the Lij. These equations are
where f = f ( x ~ x2, , . I , zn) and x i = x i ( t 1 , t 2 , . . ., tm). Since eq 1-2 are writien in terms of the gradients of chemical -potential, it is obvious that these must be changed to gradients of concentration in order to equate corresponding coefficients. At constant T and P, for the ternary case I
N-1 j-1
(i,k
=
1,
..., N - 1)
(27)
Solving these equations for the cross-phenomenological coefficients, one obtains
Lij = Lji =
PI = f(C1, C2)
C,Cj --
(33)
t
P2
= f(C1, C*)
The pi’s are not a function of all three concentrations because only two of the concentrations are independent as a result of eq 22. Applying eq 32 to eq 33 gives (i,j= 1,
. . ., N - 1 ; i Zj)
(28)
Also
(34)
+
where the superscripts M and M 1 refer to a system of M components and to the same system with one additional component. Thus the hydrodynamic theory, for the case of a nonassociating system with constant molar volumes of its components, predicts that the Onsager reciprocal relations should be valid. Experimental evidence has been obtained by these same authors for a threecomponent system to substantiate this statement and is given in a subsequent article.
The chemical potential of the ith constituent can be written as p i = piO
+ RT In ai
(35)
where pio is a function of T and P, R is the gas constant, and ai is the activity. Substituting eq 34 into eq 12, introducing eq 35, rearranging, and then equating corresponding coefficients with eq 31 yield the desired expressions for Dij.
The Diffusion Coefficients Dij Baldwin, Dunlop, and Gosting” generalized Fick’s law for a system of N components as follows N- 1 Ji
-
(i = 1,
Dijr3)
8%
1-1
. ., N - 1)
(30)
t
+ ClC2
(2- :)(%)c,l
It is interesting to note that the diffusion coefficients defined by eq 30 can be obtained in terms of the ui’s. For the ternary case eq 30 becomes
J2
=
acl - ljZ2acz - D21ax ax
(11) E. L. Baldwin, P. L. Dunlop, and L. J. Gosting, J. Amer. Chem. SOC.,77, 5235 (1955). Volume Y3*Number 6 ‘Mau 1969
TERENCE K. KETTAND DONALD K . ANDERSON
1266
Substituting eq 42 into eq 16, collecting the coefficients of the a C j / d x , and equating coefficients with those in eq 30 give the desired generalized expression for Dij
(1
+ c2(
- 72C2)
+
P;)(a;;.>,,l
(i,j= 1,
ff2
Miller* has obtained expressions for the phenomenological coefficients in terms of the ternary diffusion coefficients, Dij. These are Ll2
=
aD12 ad
- cD11. - bc ’
L2l
=
dD21 ad
- bD22 - bc
(37)
where
. . ., N - 1)
(43)
A check on the hydrodynamic approach is provided. The diffusion coefficients Dij can be obtained experimentally by the methods of Fujita and Gosting416and Burchard and Toor.6 From them, phenomenological coefficients can be obtained. Also from eq 36, the ternary diffusion coefficients Dij can be determined, which serves as a check in itself, and from eq 23, the phenomenological coefficients Lij can be determined. Hence the hydrodynamic theory can be checked by comparing the diffusion coefficients Dij and by comparing the phenomenological coefficients Lij. Determination of Friction Coefficients, ui
and c and d are the same, respectively, as a and b except that (a/aC1)c2is replaced by (a/aC2)c,. From eq 37, it is easy to show that the necessary and sufficient condition for the Onsager reciprocal relations to be valid is that aDi2
+ bD22 =
CDii
The crux of the hydrodynamic method is obtaining accurate values of the friction coefficients ~ i . For the nonassociative case, these can be obtained by tracer techniques. I n tracer diffusion, the concentration is uniform throughout the system and, as a result, the velocities of all reference planes relative to fixed coordinates are zero. That is
+ dD21
Vm,c
(39) ad - bc f 0
It can be shown by substituting eq 36 and 38 into eq 37 that L12and LZlreduce exactly to those given by eq 23. Expressions for the diffusion coefficients of an Ncomponent system can be obtained by carrying out the s8me procedure followed for the ternary. For the N-component case, eq 22 and 33 become
= Vv,a = 0
(44)
Thus eq 6 for unidirectional diffusion of the tagged species reduces to
(45) Substituting for chemical potential from eq 35, eq 45 becomes Ji*
=
ci* a In ai* - -RT- a x Ui*V
i-1
which can be rearranged to give
and Ji*
a In ai* aCi* ._ RT -____ ui*v a In Ci* ax
(47)
The tracer diffusion coefficient which is measured is defined by
Therefore, by comparing eq 47 and 48, the measured The Journal of Physical Chemistry
MULTICOMPONENT DIFFUSION
1267
tracer diffusion coefficient Di* is
RT a In ai* Di* = --ai*~ a In Cr*
(49)
If we make the reasonable assumption that the physical properties of the labeled species are the same as the unlabeled species, then ai* =
aj
(50)
For the case of tracer diffusion
a In ai* -6, In
Ci*
binary at infinite dilution of A and VB is the viscosity of pure B. In a ternary system, two such values of b~ can be obtained from the binaries AB and AC-similarly for UB and ac. Another value of the friction coefficient can be obtained from the self-diffusion coefficient, Di. For example (57) In summary, if self-diffusion data and binary data at infinite dilution in each component are available, several values for each ai can be obtained. These values will be reasonably constant if the friction coefficient is independent of the other species present. A reasonable value for ai is therefore a weighted average of all the ai values based on mole fraction. That is
1
Therefore from eq 49 and 50
D;* RT air]
and hence
(53) Thus by tagging each component separately and keeping the concentrations the same as those at which the diffusion coefficients, Dij, are measured, all of the friction coefficients can be determined. If there are N components, N tracer runs would of course be required to obtain the N ai's. If tagged species are not available, reasonable values of the friction coefficients can be obtained from the infinitely dilute mutual-diffusion coefficients of the binaries and the self-diffusion coefficients of the pure components. Hartley and Cranklo and others have shown that alnaA DAB=-RT -XA + - XB ___ 7 [UB @AlalnXA
(54)
where DABis the mutual-diffusion coefficient of the A-B binary. At infinite dilution of component A, eq 54 becomes
Independent Dji vs. Independent ui The hydrodynamic theory presented here says that for an N-component, nonassociating system only N independent coefficients, ai, are necessary to describe the diffusion in the system. If diffusion is described in terms of the diffusion coefficients, Dij, and the Onsager reciprocal relationships, LO = Lji, N ( N - 1)/ 2 independent coefficients are necessary. Thus in a ternary system there are as many ai as there are independent Dij but for a system of more than three components there are fewer ai. Hydrodynamic theory, as presented here, therefore says that there must be other relationships among the phenomenological coefficients in addition to the Onsager relations. These additional relationships can be obtained from eq 28 and are of the form
Ci
Lij = - L,j ck
7, + CiCj __ ; ( - ?) 1 (iZj#k;i,j,k=l,
..., N - 1 )
(59)
Thus
> 3) component system, only ( N ( N - 1 ) / 2 ) - N of the above equations are independent.
where DAB'is the mutual-diffusion coefficientof the AB
Acknowledgment. This work was supported by a grant from the Petroleum Research Fund, administered by the American Chemical Society. Grateful acknowledgement is hereby made to the donors of the fund.
(55)
It should be noted that for an N ( N
Volume 73, Number 6 May 1969
