J . Phys. Chem. 1990, 94, 6099-6105
between intergranular coupling and microwave penetration. For slab samples the LFMA response is proportional to the surface area when the demagnetization factor is taken into account. The results suggest that another possible factor responsible for the observed anisotropy is grain orientation and ordering within the slab samples to form local crystallike regions.
6099
Acknowledgment. The authors thank Professor Maurizio Romanelli from the Institute of Chemistry, University of Basilicata, Potenza, Italy, for helpful discussions. This research was supported by the Texas Center for Superconductivity at the University of Houston by prime grant MDA 972-88-G-0002 from the Defense Advanced Projects Agency and by the State of Texas.
Theory and Measurement of the Concentration Dependence of the Dtfferential Diffuslon Coefficient Using a Diaphragm Cell with Compartments of Unequal Volume John C. Clunie, Ning Li, Merle T. Emerson, and James K. Baird* Department of Chemistry, Unioersity of Alabama in Huntsville, Huntsuille, Alabama 35899 (Received: July 1 1 , 1989; In Final Form: March 5, 1990)
A diaphragm cell consists of two well-stirred solution compartments on opposite sides of a porous membrane, which is usually a sintered glass disk. When the compartments are filled with different concentrations of a solution, diffusion sets in across the membrane. We have integrated exaqtly the equation of motion for the concentration difference across the membrane for the important general case where the compartment volumes are unequal and the differential diffusion coefficient is an arbitrary function of the concentration. Our integral takes the form of an infinite series involving the concentration difference. We have tested this integral experimentally in the case of aqueous solutions of HC1 using diaphragm cells of the Stokes design as well as of a new design which permits the continuous determination of the electrical conductivity of the top compartment. By combining the data according to our new result, we have computed the differential diffusion coefficient and its first two derivatives with respect to concentration for HCI at 25 OC and 1 M.
1. Introduction Diffusion describes the ensemble average motions of a system of molecules measured with respect to an internal coordinate frame. For a system consisting of two chemical components, the theory of nonequilibrium thermodynamics specifies that the coefficients for interdiffusion of the two components are related, the form of the relation depending upon the frame of reference chosen. Because most solutions are thermodynamically nonideal, the coefficients for interdiffusion are in general a function of concentration.' However, in the case of trace diffusion (interdiffusion in the limit of infinite dilution), the concentration dependence is often negligible.*J Likewise, in the case of selfdiffusion (interdiffusion of isotopically labeled forms of the same chemical component), there is no detectable concentration effect, although there is sometimes a measurable dependence upon the isotopic m a s e 6 The diaphragm cell method for the determination of binary diffusion coefficients was introduced by Northrop and Anson' and developed into its present form by McBain,* Mouquin and Cathcart: and Stokes.loJ1 The cell consists of two magnetically stirred solution compartments on opposite sides of a porous membrane, which in practice is usually a sintered glass frit. The stirring guarantees that the contents of each compartment are uniform, so that mixing is limited to transport through the frit. If the stirring bars are rotated at an angular speed in excess of 1 Hz, the stirring in the compartments extends a negligible and reproducible distance into the frit, and transport through it is by diffusion." In operation, the cell is oriented so that the normal to the plane of the frit is parallel to the gravity (1 ) Fitts, D. Nonequilibrium Thermodynamics; McGraw-Hill: New York, 1962. (2) Dymond, J. H.; Easteal, A. J.; Woolf, L. A. Chem. Phys. 1988.99, 39. (3) Weingartner, H.; Braun, 8. M. Ber. Bunsen-Ges. Phys. Chem. 1985, 89, 906. (4) Weingartner, H. 2.Phys. Chem. (Munich) 1982, 132, 129. (5) Easteal, A. J.; Woolf, L. A. J . Phys. Chem. 1985, 89, 1064. (6) Easteal. A. J.; Edge, A. V.; Woolf, L. A. J. Phys. Chem. 1985.89.106. (7) Northrop, J. H.; Anson, M. L. J . Gen. Physiol. 1929, 12, 543. (8) McBain, J. W.; Liu, T. H. J . Am. Chem. SOC.1931, 53, 59. (9) Mouquin, H.; Cathcart, W. H. J . Am. Chem. Soc. 1935, 57, 1791. (IO)Stokes, R. H. J . Am. Chem. SOC.1950, 72, 2243. (1 1) Stokes, R. H. J . Am. Chem. SOC.1950, 72,763.
0022-3654/90/2094-6099$02.50/0
vector. The heavier solution is placed in the lower compartment, while the lighter solution is placed in the upper compartment; this arrangement precludes gravitational convection within the frit.12 The cross-sectional area, A , and the average interstitial thickness, I , of the frit may be combined with the volumes of the compartments below and above the frit, VI and V2,respectively, to define a geometric constant
-( $)
p= I 1 +
A VI for the cell. In general, /3 is not known a priori, and the diaphragm cell must first be calibrated with a solution whose interdiffusion coefficient is already available. In standard practice, this is accomplished at 25 O C with 0.5 M aqueous KC1 below the frit and water above, the concentration difference across the frit being followed as a function of the time, t . If c , ( t ) and c z ( t ) are the KCI concentrations below and above the frit, respectively, and ~ ' ( 0 )and cz(0) are the initial values, then the cell constant, /3, can be calculated from the formula
where D = 1.847 X cm2/s is the "integral" diffusion coefficient for aqueous KC1 over the concentration range of the calibration. 2-1 In the absence of a volume change upon the total volume of liquid in the cell is a constant during a given experiment. Further, since the volume of solution in the lower compartment is fixed by gravitational stabilization, then when a quantity of (1 2) Cussler, E. L. Dif/usion and Mass Transfer in Fluid Systems; Cambridge University Press: New York, 1984. (13) Robinson, R. A.; Stokes, R. H. Electrolyte Solufions; Butterworths: London, 1959; pp 253-261. (14) Mills, R.; Woolf, L. A. The Diaphragm Cell; Australian National University Press: Canberra, 1968. (15) Tyrrell, H . J. V.; Harris, K . R. Dij&sion in Liquids; Butterworths: Boston, 1984; pp 105-120. (16) Kirkwood, J. G.; Baldwin, R. L.; Dunlop, P. J.; Gosting, L. J.; Kegeles, G. J . Chem. Phys. 1960, 33, 1505. (17) Robinson, Jr., R. L.; Edminster, W. C.; Dullien, F. A. L. J . Phys. Chem. 1965, 69, 258.
0 1990 American Chemical Society
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Clunie et al.
The Journal of Physical Chemistry, Vol. 94, No. 15, 1990
solute departs the lower compartment, it must be replaced by an equal volume of solvent from the upper compartment. Consequently, during the diffusion, the center of volume of the liquid remains at rest with respect to the cell. (The center of mass, by contrast, moves upward.) An experiment under these conditions determines the interdiffusion coefficient with respect to the cell and, hence, with respect to the center of volume as origin of coordinates. For a two-component system, the interdiffusion coefficients in this frame are identical, and only one coefficient is needed to describe the diffusion.] The theory of the diaphragm cell in the case of two components with a concentration-independent interdiffusion coefficient was initiated by BarnesI8 and completed by Mills et aI.l9 Taken together, their work showed that, if the interstitial volume of the frit were small as compared with the volumes of the compartments, the initial concentration distribution within the frit decayed rapidly to a steady state. Given a short enough relaxation time, this result permits the cell to be operated in the steady state. The steady state may be established in the laboratory through a process called "prediffusion".12-ls In one of the several versions of this procedure, which are covered by the theory,I8,I9VI and the frit are filled with solution, while V2 is filled with solvent. The cell is then allowed to run for several hours to establish a steady concentration gradient within the frit. At the end of this prediffusion period, VI and V2 are emptied and refilled with solution and solvent, respectively, and the experiment is restarted. The concentrations c,(t) and c2(t) are followed, and the results are substituted into eq 1.2 to compute the unknown value of D for the given binary system. GordonZoproposed that D was related to the differential diffusion coefficient, D(c), which appears in Fick's first law, through the equation (1.3)
possessing an arbitrary dependence upon c, we have solved exactly the equation of motion for a diaphragm cell where VI = V,. In this paper, we have extended our previous theoretical work to encompass the general case where VI # V, and have carried out careful experiments using aqueous HCI as a test of our results. To facilitate this effort, we have constructed a cell which permits c 2 ( t ) to be monitored continuously. This new cell, as well as the cell of standard Stokes design, which we have used is described, and the data we have accumulated on aqueous HCI with both cells are reported below. By combining these data according to our new method, we have computed the differential diffusion coefficient and its first two derivatives with respect to concentration for HCI at 25 "C and 1 M. 2. Theory In this section, we provide a summary of our theoretical results. The complete derivations, which constitute a generalization of our previous methods,30 may be found in the Appendix. The concentrations cl(t) and c,(t) are determined by two coupled first-order ordinary differential equations of motion, which are solved by two integrals. The first of these is
V,CI(t) + Vzcz(t) - Vl'lCl(0) + VZCZ(0) (2.1) VI + v2 VI + VZ which serves to define the volume average mean concentration, i?. The second integral is t =
Bo + BL In Ac + Bl(Ac) + B2(Ac)Z+ B,(Ac)~+ ... (2.2)
where AC = c , ( t ) - c2(t)
(2.3) The leading coefficient, Bo,is determined by the initial values, cl(0) and c2(0). Expressions for the next several coefficients are the following:
where c is the concentration variable and Pi =
f/z[Cf(t)
+ Ci(O)],
i = I, 2
(2.4) (1.4)
The Gordon theory has been generalized to multicomponent diffusion by Cussler and Dunlopz1 and by Cullinan and collabor a t o r ~ and ~ ~ has . ~ been ~ used to interpret experimental data on integral diffusion coefficients in several system^.^^-^^ Instead of D, nonequilibrium thermodynamics suggests that the differential diffusion coefficient, D(c), is of fundamental importance. Stokes developed a graphical procedure based upon eq 1.3 for extracting the functional form of D(c) from data involving D. He noted, however, that eq 1.3 lacked a rigorous mathematical basis.I0 Since then, Miller et al. have stressed this point.29 They suggested that, although the integral diffusion coefficients measured in a diaphragm cell were adequately reproducible, combining them according to the Stokes-Gordon procedure was likely to introduce large errors in the calculated values of D(c). They concluded that use of the diaphragm cell should be avoided except in cases of negligible concentration dependence of D(c) (as, for example, in trace or self-diffusion). In order to place the diaphragm cell and the determination of D(c) on a rigorous basis, we have recently approached this problem from a theoretical point of view.w Specifically, in the case of two components interdiffusing with a differential diffusion coefficient (18) Barnes, C. Physics 1934, 5, 4. (19) Mills, R.; Woolf, L. A.; Watts, R. 0. AIChE J. 1968, 14, 671 (20) Gordon, A. R. Ann. N.Y.Acad. Sci. 1945, 46, 285. (21) Cussler, E. L.: Dunlop, P. J. J . Phys. Chem. 1966, 70, 1880. (22) Kosanovich, G.; Cullinan, Jr., H. T. Can. J . Chem. Eng. 1971, 49, 75. (23) (24) (25) (26) (27) (28) (29)
Rai, G. P.; Cullinan, Jr., H. T. J . Chem. Eng. Data 1973, 18, 213. Leaist, D.G . J. Solution Chem. 1985, 14, 709. Leaist, D.G. Aust. J . Chem. 1985, 38, 249. Leaist, D.G. J . Phys. Chem. 1985,89, 1486. Leaist, D.G.; Wiens, B. Con. J. Chem. 1986, 64, 1007. Noulty, R. A.; Leaist, D.G. J. Phys. Chem. 1987, 91, 1655. Miller, D.G.;R a d , J. A.; Epstein, L. B.: Robinson, R. A. J. Solution
Chem. 1980, 9. 467.
By differentiating a linear combination of the equations of motion, one finds the added result
which links the differential diffusion coefficient evaluated at the top and bottom concentrations at some time, t , to the first and second derivatives of Ac evaluated at the same time. 3. Experiment 3.1. P h n . We have carried out diaphragm cell diffusion experiments using aqueous HCI in order to test eqs 2.2 and 2.7. In the case of a differential diffusion coefficient depending weakly upon concentration, we can expect the constant and logarithmic terms in eq 2.2 to determine most of the variation o f t with Ac, while the terms Bl(Ac) and B2(Ac),, etc., serve as corrections. Since the largest of these correction terms is likely to be Bl(Ac) (which according to eq 2.5 is nonzero only when V, # V2), our experiments were designed with unequal volumes in order to detect this term. Our experimental strategy was as follows: We calibrated our cell with the volumes equal using aqueous KCI in the standard (30) Baird. J. K.; Frieden, R. W. J . Phys. Chem. 1987, 9 / , 3920
The Journal of Physical Chemistry, Vol. 94, No. 15, 1990 6101
Differential Diffusion Coefficient
TABLE 1: HCI Data from Cell 1“ 1, s cd0, M 25 200 46 800 82 920 103 560 149 400 196 800
1.386 1.346 1.281 1.249 1.202 1.154
c*(t), M
0.1157 0.2027 0.3290 0.3934 0.5157 0.6136
P, M 0.9927 0.9920 0.9863 0.9841 0.9895 0.9867
IIP
1’1
a VI = 1 1 I .5 cm’, V, = 50.0 cm3, A / / = 5.179 cm, j3 = 0.150cm-2, cl(0) = 1.432 M,~ ( 0 =) 0,E = 0.989M,temperature = 25.00f 0.01 OC.
way described in the Introduction. The concentration of KCI was determined with a calibrated conductivity meter available to us. To test our equations with volumes unequal, we chose aqueous HCI, not only because its concentration could be followed with the same meter but also because its differential diffusion coefficient has been determined rather carefully by the Stokes graphical met hod .lo 3.2. Materials. Our KCl was Mallinckrodt AR Grade Lot 6858 KBCX, while our HCI was Mallinckrodt AR Grade Lot 3560 KCMS. Both were used without further purification. Water, once distilled from a glass system and deaerated by aspiration to a residual conductivity of 1.5 pS/cm, was used as the solvent for all solutions. Standard stock solutions of KC1 were prepared volumetrically from weighed, dried potassium chloride. Stock solutions of HCI were standardized by titration against weighed tris(hydroxymethy1)aminomethane. 3.3. Conductivity Meter. The conductivity meter was a Radiometer Model CDM 83 capable of reading conductivities ranging from 1.3 pS/cm to 1300 mS/cm. The conductivities of standard solutions of KCI and HCl and their serial dilutions were determined with the meter. The molar concentrations, c, were fitted by least squares to polynomial functions of the conductance, G. In the case of KCl, the best fit was obtained using a cubic polynomial function c = -1.308 X + 1.309 X 10-4G + 4.085 X 10-5G2 4.358 X 10”G3 (3.1)
+
which proved satisfactory over the conductivity range 0.149 mS/cm IG I12.89 mS/cm. For HCI, a fit was obtained with the cubic c = 1.398 X 2.441 X 10-3G 2.243 X 104G2 5.463 X IO-’OG3 (3.2)
+
+
which proved satisfactory over the conductivity range 55 pS/cm IG I2.63 mS/cm. Neither eq 3.1 nor eq 3.2 is consistent with the formula for the concentration dependence of the equivalent conductance suggested by Fuoss et al. on theoretical grounds.” Because the latter includes a term involving c log (c), which prevents it from being readily inverted to give c as a function of the conductivity, we have introduced eqs 3.1 and 3.2. These formulas represent our probe calibration data to sufficient accuracy over the conductivity ranges encountered. 3.4. Diaphragm Cells and Procedures. Diaphragm cells were constructed by fusing sintered glass disks into 40-mm4.d. glass tubing. The disks were of medium porosity (10-15-pm glass beads) and were obtained from the Lab Glass Co. During calibration with KCI and all subsequent experiments with HCI, concentration gradients were established as described in the Introduction, and the initial value of the top concentration was always zero. The fixed geometric parameters of cell 1 were frit volume 2.22 cm3, V , = I 1 1.5 cm3, and AI1 = 5.179 cm. Cell 1 was stirred in the standard Stokes method by a motor mounted coaxially above the top ~ompartment.’~ For a given initial HCI concentration on the bottom, the cell was allowed to run for a time t, the contents
Figure 1. Schematic of cell 2: A, diaphragm cell with top compartment showing; B, top magnetic stirring bar; C, bottom magnetic stirring bar; D, glass frit; E, permanent magnets driving the stirring bars; F, cup for positioning the cell; G, U-frame; H, reduction gear; I, magnet drive shaft; J, chain; K, drive gear; L, drive shaft; M, electric motor; N, frame supporting the assembly. The frame and all equipment except for the motor were immersed in a thermostated water bath.
TABLE 11: HCI Data from Cell 2“ t, s
d t ) ,M
cz(t), M
t, s
18000 81900 186300 254700 279000 345900
2.429 2.336 2.198 2.117 2.090 2.0B
0.02209 0.08700 0.1824 0.2390 0.2578 0.3076
426600 451080 513000 554400 601 200
a
M 1.942 1.916 1.863 1.832 1.792
cdt),
M 0.3602 0.3783 0.4148 0.4365 0.4646
CZ(~),
VI = 62.5 cm3, Vz = 90.0cm3, A / / = 1.138 cm, j3 = 0.0308 cm-z, M,cz(0) = 0,?i = 1.009M,temperature = 25.00& 0.01
cl(0) = 2.461 OC.
of the compartments were emptied, and the concentrations were determined from conductivity measurements. To obtain the next t vs Ac p i n t , as required to evaluate eqs 2.2 and 2.7, the cell had to be refilled with the initial concentrations exactly as before and restarted. The values o f t and c l ( t ) and c 2 ( t ) measured with cell 1 are given in Table I, where it is also demonstrated that they satisfy eq 2.1. The fixed geometric parameters of cell 2 were frit volume 0.43 cm3, V, = 62.5 cm3, and AI1 = 1.138 cm. As shown in Figure 1 , cell 2 was designed so that the shaft of the overhead stirring motor was not concentric with the axis of the cell. Instead, the motor shaft was connected through gears and a chain to a shaft below the cell which drove the stirring magnets. This arrangement provided free access for the probe of the conductivity meter to be put in continuous contact with the top solution. The concentration in the top compartment, cz(t), was determined with the conductivity meter, and the concentration in the bottom compartment, c , ( t ) , was calculated from eq 2.1 and the initial conditions. Measurements using cell 2 were less time-consuming than those with cell 1 . This is because, for a given set of initial concentrations, a series of t vs Ac determinations could be made without sacrificing the contents of the cell and restarting the experiment. Both cells were immersed in thermostated water baths which maintained the temperature at 25.00 f 0.01 OC. The calibration of the conductivity probe and all concentration determinations using it were carried out in one of these baths. 4. Results
(31) Lind, J. E.; Zwolenik; Fuoss, R. M. J . Am. Chem. Soe. 1959, 81. 1557.
4.1. Test o f E q 2.2. The data obtained for solutions of HCI with cells 1 and 2 are summarized in Tables I and 11. From these
6102 The Journal of Physical Chemistry, Vol. 94, No. 15. 1990 TABLE III: Linear Remession for HCI Solutions" cell 1 B,,, 104 s 8 .00 BL, 10' s -1.97 B , , 103 S / M -6.78 Bz,IO3 s/M2
Clunie et al.
a
6ool '0°
cell 2 86.8 -10.8 17.9 9.89
500
'These results were obtained by fitting data in Tables I and 11 to eq. 2.2 with Bo, BL.B , , and Bz taken as least-squares parameters.
t-
200
100
1
0 1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
( C I - ~ 2(M ) 1
Figure 3. Diffusion time as a function of concentration difference across the frit for the cell 2 experiment. The experimental data are denoted by the symbol (+). The curve is a plot of eq 2.2 with the B coefficients evaluated from the second column of Table Ill.
TABLE I V Differential Diffusion Coefficients and Derivatives for
0 0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
HCI Solutions'
1.3
p,
(c, -c,)(M)
Figure 2. Diffusion time as a function of concentration difference across the frit for the cell 1 experiment. The experimental data are denoted by the symbol (+). The curve is a plot of eq 2.2 with the B coefficients evaluated from the first column of Table 111.
data, values of Ac were computed from eq 2.3, and the pairs (Ac,t) were formed. These pairs were fitted by linear regression to eq 2.2 using the coefficients Bo, BL, B , , B2, etc., as least-squares parameters. Due to the fact that eq 2.2 is an infinite series, it needs to be truncated at some finite order to accommodate the least-squares method. Reflecting this, our numerical procedure consisted of the following iteration on the coefficients in eq 2.2: The experimental (Ac,t) pairs were first fitted to eq 2.2 including only the terms involving Bo and B,. The linear regression correlation coefficient was noted. Since both 0 and D(E) are positive definite, the sign of BL was checked to make sure that it was negative as required by eq 2.4. If so, in the second iteration, the term involving Bl(Ac) was included, the correlation coefficient noted, and the sign of BLchecked. This procedure was repeated in subsequent iterations, adding each time the next higher term from eq 2.2. Due presumably to the random error in our data, we always experienced a point in our routine where BL turned positive. We then terminated our procedure at this number of terms and searched the previous iterations of lower order for the largest correlation coefficient. In no case did this search justify including terms beyond E2(Ac)? The values of Bo, EL,B,, and B2 from these best iterations of lower order are reported in Table 111. In the case of the data obtained from cell 1, we found it impossible to proceed beyond B , . The best least-squares truncated forms of eq 2.2 have been plotted in Figures 2 and 3 for comparison with the experimental (Ac,t) data points. Equations 2.4-2.6 may be solved as a linear system to obtain formulas for D(?) and its derivatives. The results are 1
D(F) = -BBL
(4.1) (4.2)
We have combined eqs 4.1-4.3 with the least-squares coefficients
M
D(4 D(')(Z) D'2'(T)
cell 1 0.989 3.38 6.09
cell 2 1.009 3.00 5.50 8.13
"The results reported in this table were obtained by substituting the least-squares parameters in Table I11 into eqs 4.1-4.3. D(Z), D(')(Z), and D(2)(ii)have units of IO-' cm2/s, IO" cm2/(s M), and IO" cm2/(s M2), respectively.
in Table 111. The results are summarized in Table IV. Note that the experiments with cells 1 and 2 give comparable values for both D ( E )and D(I)(E). The two experiments were designed with a common value of Z, so that, within experimental error, D(E) and its derivatives would be the same for both. By reference to Tables I and 11, we see that VI was greater than Vz for cell 1, whereas for cell 2 it was the reverse. According to eq 2.5, since o")(?) was positive, we should expect for an experiment carried out with cell 1 that B I would be negative, while for an experiment with cell 2 it would be positive. Table 111 demonstrates that this effect was indeed found. 4.2. Test of Eq 2.7. Having determined the least-squares values of the coefficients in eq 2.2, we were in a position to test eq 2.7. For the cell 1 experiment, we used the Taylor series coefficients listed in Table 111 and
D(c) = D(P)
+ D(')(P)(C- Z) + j/zD(Z)(E)(c- E ) 2
(4.4)
to evaluate D ( c ) at c1 and cz, respectively, as is required by the first two terms on the right-hand side of eq 2.7. Because D(2)(P) was unknown for the cell 1 data, we took Dm(E) = 0 when evaluating eq 4.4. The ratio of time derivatives required to evaluate the third term on the left-hand side of eq 2.7 was computed by implicit differentiation of eq 2.2 with respect to t . The result was d2Ac/dt2 -dAc/dt BL - ~ & ( A c )-~... - n(n - I)B,,(Ac)" - ... [BL + B,(Ac)
+ 2B2(Ac)*+ ... + nB,(Ac)n + ...I2
(4.5)
In evaluating eq 4.5, eq 2.3 was combined with the values of cI and c2 listed in Table I to compute Ac, while the least-squares coefficients BL and B, (higher coefficients set to zero) were taken from the first column of Table 111. Table V shows the results of evaluating eqs 4.4 and 4.5 at the six values of cI and cz encountered in the cell 1 experiment.
Differential Diffusion Coefficient
The Journal of Physical Chemistry, Vol. 94, No. 15, 1990 6103
Izl < 1. We may use eq A. 18, which gives z correct to second order, to test whether or not our experimental data lie within a unit radius of convergence. Since the experiments with cells 1 and 2 were both started with c2(0) = 0, and since the maximum value of Ac encountered in any run occurs when t = 0, we take Ac = r l ( 0 )so as to be on the conservative side in evaluating eq A. 18. Other data required to determine z appear in the captions to Tables I and I1 and in the body of Table IV. When these data are substituted into eq A. 18, we find z < 0.05 for the cell 1 data and z < 0.08 for the cell 2 data. Assuming that the higher order VI = 11 1.5 cm3, V2 = 50.0 cm3, AI1 = 5.179 cm, j3 = 0.150 terms in eq A.17, which were ignored in eq A.18, do not contribute D ( c l ) and D(c2) were computed by using eq 4.4 and the Taylor series substantially to z , it is apparent that both of our calculated values coefficients listed in Table IV. The derivatives of Ac with respect to 1 for z lie well within the radius of convergence of eq A.19. were computed by using eq 4.5 and the coefficients listed in Table Ill. According to the Debye-Huckel-Onsager theory, D(c) for The third and fourth columns have units IO-' cm-I s-l. electrolytes at low concentration depends linearly upon c1/2.3s Derivatives with respect to c of linear functions of c1/2 are unTABLE VI: Test of Ea 2.7 for HCI h t a Obtained with Cell 2" defined as c 0, implying that D(c) approaches D(0) with infinite D ( c l ) / V l+ ( I d2Ac/dt2)/ slope. Square-root c behavior can never be represented by the c,. M c,. M D ~ c ,/) V, ( A dAcldt) Taylor series in eq 4.4, since it is restricted to integral powers of 2.429 0.0221 14.9 -13.9 c. Because eq 2.2 relies upon the convergence of this series (in 2.336 0.0870 13.9 -1 2.7 the form of eq A.12), it would thus seem to apply to binary 2.198 0.1824 12.5 -1 1.4 solutions of electrolytes not too close to infinite dilution and to 2.1 17 0.2390 11.8 -10.7 nonelectrolytes at any concentration. 0.2578 11.6 -10.6 2.090 We note from eq 2.5 that B , = 0 at a mean concentration ?, 0.3076 11.0 -10.1 2.018 where D(')(?) = 0 [an extremum of D ( c ) ] . Unless ? happens also 1.942 0.3602 10.5 -9.71 to be a point of inflection, the first nonzero polynomial term in 1.916 0.3783 10.3 -9.58 1.864 0.4149 10.0 -9.35 eq 2.2 is then B2(Ac).ZA diaphragm cell experiment operating 1.832 0.4365 9.85 -9.23 at a mean concentration ?where D(F) has an extremum would 1.792 0.4646 9.64 -9.09 presumably provide a good opportunity for the direct determination of B2. VI = 62.5 cm3, V2 = 90.0 cm3, AI1 = 1.138 cm, j3 = 0.0308 We suggest that the condition VI = V,, which causes B I to D ( c l ) and D(c2) were computed by using eq 4.4 and the Taylor series vanish, should be adopted for future experimental work. With coefficients listed in Table IV. The derivatives of Ac with respect to t were computed by using eq 4.5 and the coefficients listed in Table 111. B , = 0, we can argue that the higher order terms beginning with The third and fourth columns have units IO-' cm-I s-I. & ( A C ) ~ ,which distinguish eqs 1.2 and 2.2, are probably small enough in many cases to be ignored in comparison with BL In (Ac). Table VI shows the analogous results for the cell 2 experiment. With the higher terms omitted, the initial conditions imply that In constructing Table VI, eq 4.4 was evaluated as shown (B3)(?) Bo = (l/pD(Z) In ( ~ ' ( 0 )- cz(0)), and eqs 1.2 and 2.2 are funcand higher set to zero), and eq 4.5 was evaluated with B3 and tionally identical. Assuming the conditions above, repeated runs higher set to zero. The Ac values came from Table 11, while the using different values of E, as computed from eq 2.1, may be used least-squares coefficients were taken from the second column of to map out D(c) simply by analyzing the data from each run Table 111. according to eq 1.2 using the identification D = D(T). In Tables V and VI, some small systematic discrepancies be5.4. Comparison between Experiment and Eq 2.7. Equation tween columns are evident. We believe these differences to arise 2.7 is satisfied identically by eqs 4.4 and 4.5 only when both of because of the truncations applied to eqs 4.4 and 4.5. In any case, the latter contain an infinite number of terms. In evaluating D(cl) it is apparent from Tables V and VI that eq 2.7 is nicely satisfied and D(c2) for the cell 1 data, using eq 4.4 we have ignored terms by the data. involving D(2)(c)and higher. By consulting eqs 2.2 and 2.6 in addition, it is apparent that through truncation of eq 4.5 at Bz, 5. Discussion and Conclusions we have neglected not only the effect of the higher derivatives 5.1. Cell Design. Our cell 2 permits the continuous monitoring of D(c) everywhere but also the effect of D(')(F) in B2 and in each of electrolyte concentration in the top compartment by conductivity coefficient beyond it. As evidenced by Table V, these various measurement and, hence, the easy collection of long sequences truncations, nonetheless, fail to spoil the consistency of our data of ( A c J ) data for a given set of initial conditions. It is similar with eq 2.7. Although the truncations were of higher order but in this sense to the cell of Collings et aLs2 and to the modified the differences more pronounced, a similar conclusion can be Lewis introduced by Tanaka et al.34 In our previous work, drawn from the cell 2 data displayed in Table VI. This consistency we have given the theory for the determination of the concentration between the data taken with the two cells and eq 2.7 serves to dependence of the interdiffusion coefficient from continuous Lewis establish the rapid convergence of the series given by eq 4.4 over cell measurement^.^^ the concentration range to which we have applied it. Evidence 5.2. The Stokes-Gordon Method. The Gordon definition of in support of convergence for HCI is gratifying because, as we D,expressed by eq 1.3, has no general significance within the have pointed out, this Taylor series in the guise of eq A.12 lies formalism of our theory. Only in the special case where DcZ)(?) at the heart of the method by which we have integrated eq A.16 and higher derivatives are identically zero and the volumes V, and to obtain eq A.21. V2 are equal can we readily make a comparison. Within the The theoretical status of eq 2.7 is quite separate from that of limitations of this special case, we find D = D(F) by combining eq 2.2, however. Having been derived by direct differentiation eqs 1.3, 1.4, and 4.4. of eq A.11 without any resort to Taylor series, eq 2.7 should suffer 5.3. Comparison between Experiment and Eq 2.2. An essential no limitation when applied to electrolytes. To avoid reference to step in the integration of eq A.16 to obtain eq 2.2 is the expansion eq 2.2, the third term on the left of eq 2.7 can be computed of (1 + z)-I in a geometric series (eq A.19) which converges for numerically by second differences directly from the data. Finally, for an arbitrary system of two components, little is usually known a priori about the relative sizes of the coefficients (32) Collings, A. F.; Hall, D. C.; Mills, R.; Woolf, L.A. J. Phys. E 1971, 4, 425. in eq 2.2, except for the special case where VI = V2and B I 0. (33) Lewis, J. B. J . Appl. Chem. 1955, 5, 228. (34) Tanaka, K.; Hashitani, Y.;Tamamushi, R. Trans. Furaduy SOC.
TABLE V Test of Ea 2.7 for HCI Data ObtaiRed with Cell 1" D(cl)/Vl + ( I d2Ac/dr2)/ CI. M CZ. M D ( c J / Vi ( A dAc/dr) 8.93 -8.98 1.386 0.1157 0.2027 9.02 -9.06 1.346 9.14 -9.17 1.281 0.3290 0.3934 9.20 -9.22 1.249 0.5157 9.32 -9.34 1.202 0.6136 9.42 -9.42 1.154 ~~
~
-
1910, 66, 14.
(35) Reference 15, p 114.
6104 The Journal of Physical Chemistry, Vol. 94, No. 15, 1990
If our experience with HCI is at all indicative, however, we are led to the hopeful conclusion that the convergence properties assumed by our theory occur generally for binary systems. The essential role played by experiment in determining this, nevertheless, should not be ignored. Acknowledgment. Support from the National Aeronautics and Space Administration through Grant NAGW-81 with the Consortium for Materials Development in Space at the University of Alabama in Huntsville is gratefully acknowledged. Appendix: Derivation of Theoretical Results We wish to derive and solve the equations of motion for the diaphragm cell with unequal volumes and arbitrary D ( c ) . Let the normal to the plane of the frit point from VI into V2, and let x be the distance coordinate in that direction. The bottom and top bounding surfaces of the frit are located at x = 0 and x = I , respectively. A.I. Initial Value Problem. Fick's first law states
where c is the solute concentration and J is the flux. Because of the steady-state concentration distribution within the frit established by the "prediffusion" operation, the continuity of the flux is expressed by dJ/ax = 0. Noting that J is independent of x , we multiply both sides of eq A. 1 by dx and integrate the left with respect to x from 0 to 1 and the right with respect to c from cl(t) to c,(t) to obtain
Conservation of solute mass in the volumes VI and V2 is expressed by the equations VI
2
= -JA
Clunie et al. ('4.9) which is the standard equation of motion for the cell. If we let c = + y, where i; is given by eq A.8, then dc = dy, since c? is independent of time. Using Ac(t) = cl(t) - c2(t) to introduce the variable, w, defined by Ac w=(A.lO) 1 1 I
-+i VI
v2
the limits of integration in eq A.9 transform from c = cl and c = c2 t o y = w/Vl and y = -w/V2, respectively, and the equation of motion becomes dAc (A.11) dt = -fllWiy1D(-w/ vz c + Y) dy We expand the integrand on the right-hand side of eq A.ll in a Taylor series about i; D(")(e) D(F+y) = D(T) + -y" (A.12) ,,=I n! where D(")(E) = (d"D(c)/dc"),,,. This series may be integrated term by term to obtain
-
-
D(")(i;) ,,=l(n E-[ + l)!
(;)I
-
(
(A.13)
which when substituted into eq A.11 gives dAc _ -- -
(A.3)
and (A.4) respectively. Substitution of eq A.2 into eqs A.3 and A.4 leads to the coupled equations of motion for the cell:
Combined with the initial conditions c,(O) and c2(0), eqs A S and A.6 specify an initial value problem satisfied by the two functions, ci(t) and cz(t). As a pair of coupled differential equations, eqs A.5 and A.6 are solved by two integrals. A.I.I. First Integral. To find the first integral, we add eqs A.5 and A.6 to obtain dcl d~2 V1-+V,-=O (A.7) dt dt This equation has the solution Vlcl(t) V2c2(t)= constant, which can be written in terms of the volume average concentration, i;, as
+
VICl(0) + V2C2(0) (A.8) VI + v2 The volume average concentration defined by eq A.8 is a constant independent of time and is completely determined by the initial conditions. A.1.2. Second Integral. To calculate the second integral, we divide eq A S by VI and eq A.6 by V2. subtract the results, and introduce eq 1.1 to obtain T = V,c,(t) + V2CAt) VI + v2
-
(A.14) where we have used eq A.lO. The reciprocal of dAc/dt satisfies dt - = -1 dAc/dt dAc
(A. 15)
which permits us to writ eq A.14 in the form 1 dt _ -dAc PD(?)Ac(l ~ ( A c ) )
(A.16)
+
where
The first few terms of the sum defining z(Ac) give us
z(Ac) =
I'
Ac)
+ ... (A. 18)
The Journal of Physical Chemistry. Vol. 94, No. IS, 1990 6105
Differential Diffusion Coefficient
+
The factor of (1 z)-I on the right-hand side of eq A.16 may be expanded in the geometric series
- -1 - I - z + 2 2 1 +z
23
+ ...
(A.19)
which when combined with eq A.18 and substituted into eq A.16 gives
dt = -
d Ac
-A
D(l)(z)
1
($- (;)
eq A.21 constitutes a series solution to the initial value problem for the cell. A.1.3. Eq A.21 in the Case VI = V2 V. When VI = V2 = V, eq 1 . 1 takes the form = 2A/IV (A.25) and eqs A.23 and A.24 become BI = 0
(A.26) (A.27)
A
BD(z)Ac ’ 2 ! 8 ( D ( ~ ) ) ~
while eq A.22 remains the same except that 8 is given by eq A.25. We see that the coefficients of all odd powers of Ac in eq A.21 vanish, a result which we obtained p r e v i o ~ s l y . ~ ~ A.2. Concentration Dependence of the Differential Diffusion Coefficient at Different Times. Employing Leibnitz’s rule,36we differentiate both sides of eq A.11 with respect to t to obtain
ImlAc );(:)( &)!$ I$!)$(): - I
U I
[
D(I)(S)
2[
2! ( D ( C ) ) 2 ]
-+-
+ *’*
(A’20)
- 8[ D( E
+
- D(
-
(A.28)
The combination of eqs 1 . 1 , A.8, and A.10 with eq A.28 then leads to
where terms of like order in Ac have been collected together. Equation A.20 is a differential equation with variables separable, which can be integrated to the form t = Bo
+ BL In Ac + Bl(Ac) + B , ( A C )+~ B3(Ac)3+ ... (A.21)
which contains both even and odd powers of Ac. Here, we list explicit expressions for a few of the coefficients:
Equation A.29 links the differential diffusion coefficient evaluated at the bottom and top concentrations present at some time t to the first and second derivatives of Ac evaluated at the same time. In the limit t 0 and c2(0) = 0, eq A.29 becomes
-
(A.22)
Bl = -
D(’)(?)
(
V I - Vz)
2!/3(D(Z))2 VI
+ v2
(A.23)
DC2)(C) VI*- VIV2 + Vz2 (A.24)
etc. After evaluation of Bo by setting t = 0 and Ac = cl(0) - c2(0),
which is a result connecting the differential diffusion coefficient at infinite dilution to its value at the concentration ~ ~ ( 0 In ) . the case of nonelectrolytes, D(0) is linked to the mobility, p, through the Nernst-Einstein relation, D(0) = pkT, where k is Boltzmann’s constant and T is the absolute temperature. Where there is ambipolar diffusion, as in the case of strong electrolytes, the equivalent relation is the Nernst-Hartley formula.37 (36) Arfken, G. Mathematical Methods for Physicists, 3rd ed.; Academic Press: New York, 1985; p 478. (37) Reference 13, pp 286-288.
