NOTES
1591
Analysis of Multicomponent Diaphragm Cell Data by George €3. DeLancey’ Institute for Basic Standards, National Bureau of Standards, Washington, D. C. 20234 (Received September 3 , 1 9 6 8 )
The diaphragm cell has been commonly used for determining the values of multicomponent diffusion coefficients in isothermal liquid systems. In this technique, solutions of different concentration and equal volume are placed on each side of the sintered diaphragm. Diffusion is allowed to proceed through the pores of the diaphragm under isothermal conditions. Assuming that (1) a quasi-steady state is reached, (2) diffusion occurs only within the fluid, (3) no convective currents exist within the diaphragm, and (4) the solutions on either side of the diaphragm are completely mixed, t8he following equations have been derived to describe the transient behavior of the system2na
(d/dt)Ac
=
-0DAc
(1)
and AC =
ACo
(2)
( t = 0)
The constant, p, is a function of the diaphragm and is obtained by calibration of the apparatus with a known system. The concentration of component j , cj, is expressed in mol/cm3, and A is used to denote differences between conditions on either side of the diaphragm. The concentration differences are collected in the vector Ac on which a superscript 0 is used to denote initial conditions ( t = 0). The matrix, D, is composed of average values of the local diffusion coefficients, i.e. (3)
The integral on the right is to be interpreted as a line integral whose path, I’, extends from the conditions on one side of the diaphragm to those on the other and passes through the states of the system within the diaphragm. The solution to eq 1 and 2 for the widely studied ternary case has been written as4 A c ~=
(Dii - az) Ac~’
+ DijAQ
I
+
[
a2
- g1 (i,j
gi
=
Act = exp( -pDtj) Aco
(6)
where the number of chemical species in the mixture is 1. Hence, the dimension of the considered to be N system of equations in eq 6 is N . The duration of the experiment is denoted by tj. The matrix exponential is defined by exactly the same series as in the scalar case and is always nonsingular. Since the matrix of diffusion coefficients is N-dimensional, N independent experiments must be performed.
+
e-fir1 t
- a2 (Dii - a 1 ) A ~ i ’+ DijAcj’ g1
calculation of the ternary diffusion coefficients from concentration differences measured a t specific times. The nonlinear character of the diffusion coefficients in this form of the solution has motivated various workers to devise additional mathematical and experimental restrictions or to utilize relatively complex numerical procedures to evaluate the transport coefficients. For example, Burchard and Toor*”and have replaced the exponential factors in eq 4 with truncated linear expansions which places a restriction on the systems that can be studied and the duration of an experiment with the diaphragm cell. Kelly and Stokes? limited their experiments to cases when the concentration of one component was the same on each side of the diaphragm initially. This would appear to be generally inadequate for a thorough study of coupled behavior. Cussler and Dunlop4 used an iterative technique that did not contain a number of restrictions associated with a method of successive approximations presented earlier by Kim.* All of these methods impose additional assumptions and grow more complex with the number of species in the mixture being studied. It is the purpose of this note to provide general analytical techniques for the evaluation of multicomponent diffusion coefficients from standard diaphragm cell data. No experimental or mathematical restrictions will be required to apply the techniques, with the exception of those inherent in eq 1, and the number of species in the system will be of no particular analytical concern. In addition, the error in the diffusion coefficients arising from errors in the concentrations and cell constant is estimated. In preference to component form, as for the ternary case in eq 4, the unique solution to eq 1 may be written as9
=
1
e-Bc2 1
1, 2; i # j )
(4)
3CDu 4- Dzz
+ ( -1) ‘+‘d (Dii - Dzz) + ~DIZDZI] 2
(i
=
1, 2)
(5)
These equations have been the starting point for the
(1) NRC-NBS Postdoctoral Research Associate. Address correspondence to Department of Chemistry and Chemical Engineering, Stevens Institute of Technology, Hoboken, N. J. 07030. (2) A . R . Gordon, Ann. N . Y . Acad. Sci., 46, 285 (1945). (3) (a) H. L. Toor, J. Phys. Chem., 64, 1580 (1960); (b) J. K. Burchard and H. L . Toor. ibid., 66, 2015 (1962). (4) E . L. Cussler and P. J . Dunlop, ibid., 7 0 , 1880 (1966). (5) E’. 0. Shuck and H. L. Toor, i b i d . , 67, 540 (1963). (6) H . T. Cullinan and H. L. Toor, ibid., 69, 3941 (1965). (7) F. J. Kelly and R. H . Stokes in “Electrolytes,” B. Pesce, Ed., Permagon Press, Oxford, 1962, pp 96-100. (8) H. Kim, J. Phys. Chem., 7 0 , 562 (1966). (9) F. R . Gantmacher, “Theory of Matrices,” Vol. I, Chelsea Publishing Co., New York, N. Y., 1959, p 118. Volume YS, Number 6 M a y 1969
NOTES
1592
It will be required that each of the experiments have the same duration or, more generally, the same value of Ptf. This is not restrictive. Also, the average composition in each of the experiments should be identical in order to maintain, approximately, the invariance of the diffusion coefficientsbetween the experiments. This has been the program of past experiments with the diaphragm cellabJQ except for the condition that Ptf be constant. The same analysis will follow if one is able to measure the concentration differences in one experiment at equal intervals in time without disturbing the parameters of the system. However, this latter method may impose experimental difficulties and will not be explicitly considered further. The concentration measurements from N experiments may be summarized in matrix form as Cf =
[AClf, ACzf,
. . ., ACnrf]
(7)
. . ., ACNO]
(8)
Hence, the xi are usually near unity and the logarithm may be expanded in powers of (X - I ) . It should be noted that this method may only be preferred over the former when the norm of (X - I) is near zero, in which case the logarithmic series converges rapidly. The choice of either method must be made on an individual basis. An analysis of the error in the diffusion coefficients arising from experimental errors in the concentrations and cell constant may be carried out after first writing eq 11 as exp[n(D
Acz',
Cf = exp ( -aD) Co
+ E,)
(15)
+
=
+ + €)ED
ED (1
(16)
As a first step toward relating ED to E, and matrix exponential in eq 15 may be factored as'l eXp[Z(D
(9)
where
%(I
+
ED'
It follows directly from eq 6 that
=
The error in X is introduced by errors in the concentration measurements and may be accounted for by the form given in eq 15. The parameter a is written as ~ ( le) due to errors in the cell constant, and the transport coefficients are written as D ED, so that
and
co= [ACi',
+ ED')]
E,
+ ED')] = exp(nD)Qol[E]
the
(17)
where a = Ptf
E
(10)
Since the matrix exponential is always nonsingular and the initial value matrix, Co, is nonsingular, eq 9 may be inverted to yield
x = CO(Cf)-I
(12)
Note that X is dimensionless and nonsingular and that the concentrations may be considered as mole fractions when the molar density is constant. The matrix D is known to be diagonalizable and to have positive real eigenvalues (di).'' The modal matrix of X is then the modal matrix of D and the eigenvalues of the two matrices are related by
(i = 1, 2,
. . ., N )
(13)
where xi are the eigenvalues of X. Hence D may be constructed from the eigenvalues and eigenvectors of X directly. Alternatively, an explicit solution for the diffusion coefficients may be constructed as = (l/a)In
X
(14)
EQot[E] dt
(19)
This result contains ED implicitly and may be used for estimating the required precision in the concentration measurements for fixed errors in the diffusion coefficients and cell constant. The converse may be approximated by expanding the integrand in a Taylor series and integrating to give E,
=
SED'
+ +@(ED'D - I)&') + . . .
(20)
where. . , indicates terms of a lower order of magnitude. These terms may be neglected when the errors are of, a t most, the order of for the diffusion coefficients and loe2 for e, i.e., when the error in X is of the order of loF2or less. Assuming this to be the case, the first two terms in eq 20 givela ED'
Here, In X is defined as the matrix equivalent of a scalar logarithmic series whose interval of convergence contains the eigenvalues of X and excludes the rig in.^ Since the xi are strictly positive, a suitable expansion may always be found. In most cases, the parameter a is of the order of lo4 and the d i are of the order of The Journal of Physical Chemieirzl
1 1
E, =
where
D
(18)
and !Jot is the matricant.12 Considering D and X as the quantities D and X treated earlier, it follows from eq 15 and 17 and the definition of the matricant that
eaD = X
di = (l/a) lnxi
= 8 exp ( -nD,t) ED' exp (EDt)
=
2 " e-2t exp(aDjt)E, exp( -nDt) ;
dt
This result is valid provided that the eigenvalues
(21)
& of
(10) H.T. Cullinan, I n d . Eng. Chem., Fundam., 4, 133 (1966). (11) See ref 9, Vol. 11, p 128. (12) See ref 9,Vol. 11, p 127. (13) R. Bellman, "Introduction to Matrix Analysis," McGraw-Hill Book Do.. Inc., New York. N . Y., 1960,p 231.
NOTES
1593
D satisfy di
- ttj < (2/a)
(i,j
= 1, 2,
. . ., N )
(22)
which is consistent with the estimates of these quantities given earlier. By defining kDI =
Q-lED’Q
(23)
where Q is th,e modal matrix of X, and with a similar definition of E,, the result of the integration in eq 21 may be written as
In summary, the two proposed analytical techniques may be applied without modification to mixtures containing any number of components and do not impose additional experimental or mathematical restrictions. The estimate of the experimental error in the diffusion coefficients is valid within a specified range of conditions that is believed to apply to most meaningful experiments.
Acknowledgment. The author is grateful to the National Research Council and the National Bureau of Standards for a Postdoctoral Research Associateship which supported this work:
(24) Expanding the denominator o,f eq 24 inna power series and using the definition of ED’ and E,, the matrix formulation of the result is
+ &(DEx- ExD) + . . .
ED‘ = (l/S)Ex
(25)
The first two illustrated terms in the expansion are sufficient since eq 21 is itself an approximation. The true error in the diffusion coefficients may now be approximated by
+ Ea
ED = Eo
for Electroosmosis by R. P. Rastogi, Kehar Singh, and Shri Nath Singh Chemistry Department, Gorakhpur University, Gorakhpur, India (Receiwed September 30, 1 9 6 8 )
(26)
where
+ &(DE, - ExD)
(27)
+ +(DE, - E,D) + D]
(28)
E, = (l/E)E,
Nonlinear Phenomenological Equation
For a Markoffian system, the flux Ji can be expressed as a function of forces XI, Xz, . . . and the structural factor G , as1n2
and Ea = -eC(l/n)E,
Ji
Terms of the order of e2 have been neglected in comparison to unity and E and the total error separated into a component due to concentration errors, E,, and a component due to errors in the cell constant, E,. As a hypothetical illustration of the error contributions, suppose that 1.00 0.50
D
=
[
]
cm2/sec
0.25 2.00 X
]
(29)
1.00 1-00
*[
Ex =
X
(30)
1.00 1.00
E = 1X
lo4 sec/cm2
(31)
E = fO.O1 (32) Using eq 27 and 28, the resulting errors in the diffusion coefficients are given by
1.01 0.95 1.05 0.99
and 1.10 0.59
Ea =
*[
]
0.35 2.10
X lo-’ cm2/sec
(34)
=
f(xi,Xz, . . . , G )
(1)
If the structural factor G does not alter, the flux J i may be expressed as
Ji
=
CLijXj j
+ 4C XLijkXjXk + . . i
I
(2)
k
where Lij and Lijk, etc., are phenomenological coeficients. Expressions similar to eq 2 have been found to be valid in the case of chemical reactions3 and thermoosmosis of ideal gases.4 Recent measurements on electroosmosis in Pyrex-acetone systems show that the transport processes involved are nonlinear.6 Precise measurements on electroosmotic velocity in Pyrexacetone systems show that mass flux J is given by eq 2 retaining the terms up to third ordersG It was thought desirable to test this conclusion by examining the electroosmosis of differential liquid through a plug of different material and different geometry. Accordingly, electroosmosis of methanol through a quartz plug was studied. In the present study, electroosmotic (1) J. C. M.Li, J . Chem. Phys., 29, 747 (1958). (2) R. P. Rastogi and K. Singh, Trans. Faraday Soc., 6 3 , 2917 (1967). (3) R. P. Rastogi, K. Singh, and R. 0. Srivastava, ibid., 61, 854 (1965). (4) H.J. M.Hanley and W. A. Steele, ibid., 6 1 , 2661 (1965). (5) R. P.Rastogi and K. M. Jha, ibid., 6 2 , 585 (1966). (6) R. P. Rastogi, K. Singh, and M. L. Srivastava, J . Phys. Chem., 73, 46 (1969). Volume YS, Number 6 May 1969
