J. Phys. Chem. 1983, 87,586-591
588
Appendix A. Evaluation of K-’. The quantity K-’given in eq 2.10 is defined as
I?r-’(Q,Q’)
= JdQ g(Q,Q”) K-’(Q’’,Q’)
(Al)
(We suppress the sphere label for convenience). Expanding both g and K-’ in spherical harmonics yields
I?r-’(Q,Q’)
=
C g,K,-’Y,,(Q)Ylm*(Q’)
(A2)
1,m
Using this in eq A7 and expanding K-’ in spherical harmonics leads to
where
where gl[K;’] is the corresponding coefficient in the expansion of g(Q,Q’)[K-’(Q,Q’)]. First we evaluate K-’(Q,Q’). From eq 2.3 specialized to the sphere surface one has, using Fourier representation
In the hydrodynamic approximation Z ( k ) = kf + 6Dk2. Substituting it in eq A3 and using the spherical representation of the exponential factors yields
G(r - r? = @a)-’ C
[ai,(a)k,(a)lYlm(Q)Ylm*(Q’)
(A4)
1,m
with a = (kp2/D)’I2,and il and klare spherical modified Bessel functions.” From the definition of K-l
Now we evaluate g’(Q,V) defined by eq 2.11. Let
B(Q,Q’)= S d Q ” K-’(Q,Q’’)Q(r’? G ( r ” - r? (A7) We can write Q(r’? G(r”- r? as
(16) I. S. Gradshteyn and I. M. Ryzhik,“Table of Integrals Series and Products”, Academic Press, New York, 1965.
Measurement of Ternary Diffuslon Coefficients Using a Position-Scanning Spectrophotometer Masataka Tanlgakl,’ Kazuo Kondo,+Makoto Harada, and Wataru Eguchlt Institute of Atomic Energy, Kyoto Untverslty, Gokasho, Ul, Kyoto, Japan 6 11 (Recetved: July 2, 7982; In Final Form: October 11. 1982)
A new method of measuring the four diffusion coefficients in ternary systems using the position-scanning spectrophotometer (PSS)is presented. Using the concentration profiies of the two solutes from a set of different experiments, one can easily determine the diffusion coefficients without using any elaborate mathematical treatment. Measurements in watel-NiC12-CoC12and water-CoS04-CuS0,systems are reported. The calculated cross phenomenological coefficients satisfied the reciprocal relation of Onsager. Binary diffusion coefficients of NiC12, CoCl2,CoS04, and CuSO., in their aqueous solutions were also measured.
Introduction In a solution composed of more than three components, diffusion of a certain component is known to occur not only due to its own concentration gradient but also due to those Central Research Laboratory, Sumitomo Metal Industries Ltd., Amagasaki, Hyogo, Japan 660. Now at the Department of Chemical Engineering, Kyoto University, Kyoto, Japan 606.
*
of the other components. We then have “cross diffusion coefficients” in addition to “main diffusion Coefficients”. There are a large number of established methods available for the measurement of binary diffusion coefficients and many of them can, in principle, be used for ternary systems and those with more components.’ In (1) Cussler, E. L. “Multicomponent Diffusion”;Elsevier: Amsterdam,
1976.
0022-3654/83/2087-0586$01.50/00 1983 American Chemical Society
Measurement of Ternary Diffusion Coefficients
steady-state methods,2 the steady concentration profiles are established across a channel or a diaphragm between two liquid compartments containing solutions of different constant concentrations. This is the most straightforward method which does not require any elaborate calculation for the determination of diffusion coefficients. The basic disadvantage of this method, however, lies in the difficulty in establishing the steady state, which requires quite a long period of time. Slight disturbances occurring during this long period may become serious sources of error. Another method frequently used in multicomponent systems is the Stokes diaphragm method: which utilizes the pseudo steady state in the membrane between two compartments. The experimental apparatus is simple and the accuracy of the obtained diffusion coefficients can be very good. The disadvantage of this method is the necessity of calibration for the area and the effective width of the membrane. Any change in the state of the porous membrane can cause an error in the determination of the diffusion coefficients. Methods which utilize the refractive index or its gradient, such as Mach-Zehnder, Rayleigh, and Gouy interferometers, are powerful means for the determination of diffusion coefficients. The method which is most reliable and has been used most frequently to date in ternary systems is the Gouy interferometer developed by Gosting and c o - w ~ r k e r s . ~Here ~ ~ the refractive index gradient profile due to the solute concentration gradients developed during the unsteady diffusion process in the cell is measured as optical fringes. The position of these fringes can be determined very precisely, which makes the accuracy of the resulting diffusion coefficients excellent. However, the use of a single kind of information, Le., the refractive index, in a system where more than one component is diffusing at the same time makes the method of data analysis quite elaboratea6 Either the nonlinear least squares or the pseudobinary analysis should be employed to get the multicomponent diffusion coefficients. Many solutes have their characteristic light absorption and this can be applied to the determination of multicomponent diffusion coefficients. A new device named the “position-scanning spectrophotometer” (PSS) has been developed by Eguchi et al.’ This device makes possible the repeated measurements of absorbance profiles at more than one wavelength by the use of an oscillating mirror assembly attached to a commercial spectrophotometer. The present report outlines the method of determining multicomponent diffusion coefficients with the PSS. It is tested in two ternary systems, water-NiC1,-CoCl, and water-CoS04-CuS01, These systems were chosen because of the relatively large cross effect anticipated due to the electrostatic interaction between the constituent ions and the ease of the absorbance measurement.
Experimental Section Detailed description of the PSS and the slide contact cell used is given elsewhere? The temperature of the liquid in the cell was controlled to be 298 f 0.1 K. All four solute electrolytes employed in the present investigation have their specific light absorbance in the range (2) Graff, R. A.; Drew, T. C. Ind. Eng. Chem. Fundam. 1968, 7,490, for example. (3) Robinson, R. A.; Stokes, R. H. ‘Electrolyte Solutions”; Butterworths: London, 1968; p 254. (4) Gosting, L. J.; Hanson, E. M.; Kegels, G.; Morris, M. S. Reu. Sci. Instrum. 1949,20, 209. (5) Dunlop, P. J.; Gosting, L. J. J. Am. Chem. SOC.1955, 77, 5238. (6) Fujita, H.; Gosting, L. J. J. Am. Chem. SOC.1956, 78, 1099. (7) Eguchi, W.;Harada, M.; Adachi, M.; Tanigaki,M.; Kondo, K. Rev. Sci. Instrum., submitted for publication.
The Journal of Physical Chemistty, Vol. 87, No. 4, 1983 587
DISTANCE (cm)
Flgure 1. Absorbance profiles of NiCI, at 394 nm.
of visible light. The wavelengths of maximum absorbance by NiCl, and CoC12 were 394 and 511 nm, respectively, with small contributions of the other solute. Those by C0S04 and CuS04were 511 and 804 nm, respectively. It was found that the Beer-Lambert law is valid in the concentration range used in the present investigation for all solutes and that the absorbance of a mixture solution is expressed as the linear combination of the contributions of the two solutes encountered at the wavelengths used. All the solutions were prepared by dissolving the weighed reagents of purity higher than 99.9% in deionized water.
Measurement of Mutual Diffusion Coefficients in Binary Systems Before discussing the measurement of diffusion coefficients in ternary systems, we present the result of binary diffusion coefficient measurements to show the applicability of the PSS. An illustration of the original absorbance profile plotted directly by the graphic typewriter is shown in Figure 1. Deionized water in the upper compartment of the cell and the aqueous solution of NiCl, (0.60 kmol/m3) in the lower compartment were made to contact each other at time t = 0. In the water-NiC1, system, the absorbance profile in Figure 1 is directly proportional to the concentration profile of NiC1, in the cell. From the curve at the earliest time (TJ, it can be seen that a nearly stepwise contact was attained by the use of the present slide contact cell. The curves at the subsequent times show the progress of diffusion. The duplicate measurement certified the reproducibility of the absorbance curves, except that a minor difference occurred in some cases at TI.This difference is caused by the slight conditional difference in the contacting procedure but it was smoothed out in the profiles thereafter, causing little difference there. The effect of the failure of the precise stepwise concentration profile at the start of the diffusion process is usually treated as the time correction. This correction was found to be shorter than 1min with the present contact cell and was of negligible importance for the determination of diffusion coefficients because profiles at times later than about 20 min only were used in the analysis. The time necessary for the scanning of the distance of 20 mm from the uppermost to the lowermost positions was approximately 4 s. This nonsimultaneity can safely be neglected except at the earliest time. The vertical line in Figure 1 shows the contact surface. Absorbance profiles at all times crossed each other at this plane as can be seen in the figure. The diffusion in this case is governed by the usual Fick‘s law and the stepwise initial and semiinfinite boundary conditions. The diffusion coefficients can be obtained as a function of the solute concentration by the conventional method. Figure 2 is given to illustrate that the concentration profiles at different times shown in Figure 1 coincide with each other on a plot of the normalized concentration C = ( c - c*)/Ac and Boltzmann’s transformation factor r ) = x / t 1 I 2 : i.e., molecular diffusion took place. Here c* is the mean concentration defined as c* = (cL + cH)/2
588
The Journal of Physical Chemistty, Vol. 87, No. 4, 1983
Tanigaki et ai.
TABLE I : Binary Diffusion Coefficients at 298 Ka waterNiCl,, run 1
-
D
C
0 0.06 0.12 0.18 0.24 0.30 0.36 0.42 0.48 0.54 0.60
1.00) 1.02 1.02 1.02 1.03 1.01 1.01 1.02 0.99)
concn range
run 1
waterCoSO,, run 1
run 2
C
D
C
D
C
D
C
D
0 0.06 0.12 0.18 0.24 0.30 0.36 0.42 0.48 0.54 0.60
(1.03) 1.05 1.01 1.01 1.01 1.02 1.02 1.03 (0.99)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
(0.69) 0.68 0.63 0.60 0.59 0.59 0.59 0.56 (0.51)
0 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80
(0.59) 0.52 0.49 0.46 0.46 0.44 0.44 0.42 (0.40)
0 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80
(0.52) 0.47 0.46 0.44 0.42 0.42 0.40 0.40 (0.38)
1.02
1.01
DCOnSt
w ater-CuSO
waterCoCl,, run 1
water-CuSO,
waterNiCl,, run 2
waterCoCl,, run 2
run 3
run 4
run 5
0.10-0.50 0.99
0.10-0.40 1.01
0.10-0.20 0.53
0.20-0.40 0.48
0.40-0.80 0.42
mz/s, respectively. Values in parentheses a Units of concentrations and diffusion coefficients are in kmol/m3 and may involve errors because of the poor accuracy in graphical differentiation and/or integration. Dconstrefers to the diffu. sion coefficient determined by the method of constant diffusion coefficient.
1
J
7
i
0.5
-G5
P.
-10
'
-5
I
0
x/JI x 1 6
I
I
5
1C
I
0.1
0
I
Flgure 2. Normalized Concentration profile of NiCI, plotted against Xlt",.
and Ac = cH - cL, cL and cH being the lower and higher end concentrations, respectively. The obtained binary diffusion coefficients for the above water-NiC1, system are given in Table I and are plotted in Figure 3, together with the data by Stokes et a1.8 The agreement of the present values with those by Stokes et al. is quite satisfactory. From the above result, the dependence of the diffusion coefficients of NiClz upon the solute concentration is relatively small in the concentration range used in the present work. This fact makes the analysis of constant diffusion coefficients possible. The value was found to be 1.01 X m2/s by this procedure. A different experimental run with the narrower concentration range of 0.10-0.50 kmol/m3 was carried out and the diffusion coefficient in this case was determined to be 0.99 X m2/s by the same procedure. Similar measurements of binary diffusion coefficients were also carried out for water-CoC12, water-CuS04, and water-CoS04 systems and the results are shown in Table I. Data in the water-CoC1, system are quite similar to those in the water-NiCl, system in their absolute values and in the concentration dependence, as is easily anticipated from the similarity in the two solutes. The binary diffusion coefficients in the water-CuSO, and water(8) Stokes,R. H.; Phang, s.; Mills, R. J. Solution Chem. 1979,8,489.
0.3
I
I
0.4
0.5
0.6
c (kmoi/m3)
1
(m/s:'2:
I
0.2
Flgure 3. Binary diffusion coefficients in the water-NiCi, system at 298 K. 0 represents the diffusion coefficients in run 1, determined by eq 4. V is the value in run 2, determined by the method of constant diffusion coefflclent. I t is plotted as the value at the mean concentration of the two end concentrations. A shows the values of Stokes et ai.' X is the value at infinite dilution calculated from the ionic mobility.
3.81
4
I
I
I
I
I
I
I
I
I
I
-
1322-
i
32
04
06
08
13
c :kmo /m3) Figure 4. Binary diffusion coefficients in water-CuSO, system at 298 K. 0 and show the result of runs 1 and 2, determined with eq 4. shows 'the result of runs 3-5, determined with the method of constant diffusion coefficients. They are plotted at the mean concentrations. Literature data: (A)Eversole et ai.? (0) Emanuei and Olander,'o (V)Woolf and Hoveling," (0)Awakura et ai.," ( 0 ) Miller et ai.13
+
+
C0S04 systems are much more dependent upon the solute concentrations. Data of the water-CuSO, system are plotted in Figure 4 together with those reported in the (9)Eversole, W. G.;Kindsvater, H. M.; Peterson, J. D. J. Phys. Chem. 1942, 46,370. (10) Emanuel, A.; Olander, D. R. J . Chem. Eng. Data 1963, 8, 31. (11) Woolf, L. A.; Hoveling, A. W. J . Phys. Chem. 1970, 74, 2406. (12) Awakura, Y.; Ebata, A,; Morita, M.; Kondo, Y. Denki Kagaku 1975, 43, 569.
The Journal of Physical Chemistry, Vol. 87,No. 4, 1983 589
Measurement of Ternary Diffusion Coefficients
TABLE 11: Four Cases of Diffusion Experiments" case IIc
case ~b
IC for
x < 0 x>o
case I I I ~
case IVe
BC a t
x = -- c , = C,L, c , = c,L x=c , = C,H, c , = C,H
c , = C , L , c , = C,H c , = C,H, c , = c,L
c , = c , * , c , = c,L
c , = clL, c , = c 2 *
c , = c , * , c , = C,H
c , = C,H, c, = c,*
*
cp = (112) erf { ~ / [ 2 ( ~ + t ) " ~ =] (} 1, / 2 ) erf { ~ / [ 2 ( 0 _ t ) ~ ' ~ A Z] l} P ,,8 P= = A,,Q, A , , = 1 / A 2 , = A c , / A c , , A c j = cjH - ciL. Solutes diffuse i n same direction. Solutes diffuse in opposite directions. [ 11 constant. [ 2 ] constant.
literature. Agreement is again quite satisfactory. The uncertainty of the determined diffusion coefficients in all binary systems is estimated to be f0.02 X lo* m2/s. Values listed in Table I for the water-CuS04 system are the averages of several duplicate runs. They also agreed with each other within the above range.
Method of Determining Diffusion Coefficients in Ternary Systems In the previous section, the applicability of the PSS to the measurement of binary diffusion coefficients was verified. The method of obtaining four diffusion coefficients in ternary systems is presented in this section. Diffusion in a ternary system is governed by the generalized Fick's law and we have two main diffusion coefficients, Dll and D22,and two cross terms, D12and D21.14 The diffusion equations together with the stepwise initial and semiinfinite boundary conditions can be solved to give eq l.15 Here q again is Boltzmann's factor, ci* is the mean ~1 - c1* (1 + A21P)CP - A21P(1 + A12Q)q cl=-= ACl (1 - PQ) ~2 - c2* (1 + A12Q)Q - A12Q(1 + A21PN c2=-= Ac2 (1 - PQ) CP = (1/2) erf (q/[2D+1/2]) 9 = (1/2) erf (q/[2D-1/2]) = 1/A21 = A c ~ / A c ~ (1) concentration of component i in the two compartments, and Aci is the difference between two end concentrations, Aci = ciH- c;L. The four parameters P, Q, D+, and D- are related to the four diffusion coefficients as shown in eq 2. D* = (1/2)[(Dii + 0 2 2 ) Fl P = (D+ - D11)/D21 = (1/2021)[(D22 - D l J + Fl A12
Q = (D- - D22)/D12
= (1/2D12)[(D11 - D22) - Fl
F = [(Dll - D2J2 + 4D12D21]1/2 (2) Now the four parameters and consequently the four diffusion coefficients can, in principle, be determined from a single experiment. This effort, unfortunately, had to be abandoned because the preliminary test using the computer program to determine the four parameters by the least-squares method revealed that four or more significant digits with respect to the normalized concentrations are required for this method to be applicable. This was far (13) Miller, D. G.; Rard, J. A.; Eppstein, L. B.; Robinson, R. A. J. Solution Chem. 1980, 9, 467. (14) Fitta, D. D. "Nonequilibrium Thermodynamics"; McGraw-Hill: New York, 1962. (15) Equivalent expressions are given in ref 1, p 44.
beyond the limit of the present experimental technique using the light absorbance. This difficulty can be avoided if the results of two different experiments are combined. Let us consider the four cases shown in Table 11. In case I, the solutes 1 and 2 diffuse in the same direction. Two solutes diffuse in opposite directions in case 11. Cases I11 and IV are those where the initial concentrations of one solute in the two compartments are set to be equal to ci* = ( c t ciH)/2.16 The normalized concentrations for the four cases defined in the table can be expressed neatly as in the table using the same four parameters as before. Cases I11 and IV are especially interesting because the effect of cross diffusion Coefficients appears directly. The combination of the expressions in the two cases gives the following interesting relation valid for each value of q:
+
P=
+
A12[f{(C1(2)- C2(1))2 4C1(1)C2(2)]1~2(C1(2) - C2('))]/2C2(2)
Q = -P(C2'2'/Ci(1))(A2i/Ai2)
(3) The two parameters P and Q are expressed by the concentrations only in eq 3 and they can be determined easily from the observed concentrations at each point of 17.' Once P and Q are known, D+ and D- can be determined by the same method as in binary systems, because the concentrations can be arranged to give Cl(2)+ A21PC2(2) = (C1(l) + A21PC2(1))/A21P = (1/2) erf (q/2D+lI2)
C2(l)+ A12QC1(l)= (C2(2)+ A12QC1(2))/A12Q = (1/2) erf ( s / ~ D J / ~(4) ) This method of obtaining the four diffusion coefficients in ternary systems is straightforward and has the advantage that the reliability of the determined values is known directly. Examination of the expressions in Table TI reveals that the concentrations in cases I and I1 can be combined to give C1(+) - C1(-)= 2C1(1) C1(+) + C1(-) = 2C1'2' (5) C,(+) - C2(-) = 2C2(2) C2(+)+ C2(-) = 2C2(l) Thus, the combination of cases I and I1 has exactly the same information as the combination of cases I11 and IV. The four diffusion coefficients can be obtained by the same (16) The same end concentrations, :c and c,", were used in the four cases of Table I1 for neatness of the resulting expressions, though it is not at all necessary. (17) A word of caution may be valuable at this point. Equation 11 gives rise to two values for P, i.e., P and P', and the corresponding Q values of Q and Q'. However, the following relations hold, resulting in single independent values for P and Q, respectively: P'= l / Q Q'= 1 / P
590
The Journal of Physical Chemistry, Vol. 87, No. 4, 1983
Tanigaki et ai.
TABLE 111: Results of Ternary Diffusion Experiments NiCl, (1)CoSO, (1)COCl, ( 2 ) cuso, (2) C,La 0.10 0.15 C,H a 0.50 0.85 C,*a 0.30 0.50 C,La 0.10 0.20 C,H a 0.42 0.80 c,*a 0.26 0.50 c o s o , (1)NiCI, (l)-CoCl, ( 2 ) CUSO" (21. . ,, cases I and I1 cases I11 and IV cases I and I1
y-_.;z*:./.
-"l;e.:a L 1 1
.
P
0.93 i 0.07 -0.89 0.20
109D+b 109D-b lO9DlIb 109D,,b 109D,,b 109D,,b
1 , O O i 0.03 0.57 f 0.03 0.81 c 0.03 0 . 2 2 c 0.03 0.21 c 0.03 0.76 i 0.03 0.53 i 0.03 -0.16 f 0.02 -0.14 i 0.03 0.46 i 0.03
Q
10'oL,,c 10'oLl,c 10'oL,,c 10'oL,,c
a Units: kmol/m3. m3). I
z
1.00 i 0.17 -0.91 0.13 0.99 c 0.04 0.57 i 0.03 0.79 i 0.03 0 . 2 2 c 0.04 0.20 f 0.03 0.77 i. 0.03 0.52 i 0.03 - 0 . 1 6 0.02 - 0 . 1 5 i 0.03 0.46 c 0.03
Units: mz/s.
- 0 . 5 1 i 0.34 0.51 0.27 0 . 3 8 f 0.02 0.35 i 0.02 0.37 % 0.02 -0.01 c 0.01 -0.01 c 0.01 0.36 c 0.02 0.74 c 0.11 -0.03 f 0.02 -0.03 0.02 0.72 c 0.11
Units: moll s/(kg I
I
T3: 1242
lower 4
D I S T A N C E tcm) F w e 5. Absorbance proRles of case I V in water-NiCI&oCi, system at 298 K. The absorbance of the reference solution (0.10 (NICI,) and 0.25 (CoCi,) kmoi/m3) has been subtracted. Vertical lines show the contact surface. Times shown in the figure are the mean values of those at 394 and 511 nm, which differ about by 100 s.
method as in the combination of cases I11 and IV with the help of eq 5.
Measurement of Ternary Diffusion Coefficients Experiments which correspond to the four cases mentioned above were carried out in the water-NiC12 (1)-CoC12 (2) system18at 298 K. The initial concentrations used are listed in Table 111. These concentrations were chosen because the relatively small concentration dependence of the binary diffusion coefficients in this range gave the hope of expecting constant diffusion coefficients even in the ternary system. Care was taken to avoid flow due to the density inversion. The original absorbance profiles for cases I and I1 have been shown earlier.7 That for case IV (with constant concentration of CoC1,) is shown in Figure
c
-3
1
'A.>3Sm]
5
1c
x / J i x lo5 ( f r / s " 2 ) Figure 6. Normalized concentration profiles of NiCI, and CoCI, in the four cases. Superscripts (+), (-), (l), and (2) refer to cases I-IV, respectively. C /+),C /-),C (), and CJ2)are plotted in the symmetrical position with respect to the axis 9 = 0.
5. In the lower curve at 511 nm, data at three times only were reproduced for visual simplicity. The effect of cross diffusion coefficients is clearly seen although the curves show the combined contribution of absorbances by both NiClz and CoC12. Duplicate measurements certified the reproducibility of the absorbance curves. The normalized concentrations of both solutes in the four cases are shown in Figure 6 as a function of q. Note that the curves for C1(+), Cl(-), C1@), and C2(2)are plotted in the symmetrical position with respect to the axis q = 0, Le., against -?, for ease of visual observation. The concentrations at different times again agreed with each other, showing the occurrence of molecular diffusion. The accelerated diffusion in case I and the hindered diffusion in case I1 compared with case I11 or IV show the nonnegligible magnitude of the cross diffusion coefficients. Curves C1")and C2@)show the so-called uphill diffusion. From either the combination of cases I and I1 or that of cases I11 and IV, values of P and Q can be determined at each value of I] using eq 3. The obtained values were averaged in the range of 0.1 X lo* I171 I0.5 X lo* m/s1i2, because those outside this range differed significantly due to the small concentration differences appearing in the calculation. Once these mean values of P and Q are determined, the two characteristic diffusion coefficients D+ and D- can be determined in a similar fashion as in the case of binary diffusion. The determined values of the four diffusion coefficients are shown in Table 111, together with the parameter values. The uncertainties in the four parameter values and the probable errors in the four diffusion coefficients due to these uncertainties in the parameters are also listed in the table. The solid curves in Figure 6 are the concentration profiles recalculated by using the diffusion coefficients listed in Table 111. Both main terms are nearly equal to each other as has been anticipated from the similarity in the two solutes. Their values, however, are considerably smaller than those in binary systems. The two cross terms are very similar to each other and are about 25% of the main terms. These large cross terms are due to the electrostatic interactions between the constituent ions present in the system. Multicomponent diffusion coefficients are related to the phenomenological coefficients relative to the solvent, L: , and the latter in a ternary system can be expressed asi4 L'll
L'12
= =
L'22
(P22D'll
- P21D112)/S
(PllD112- Pl2Dt1l)/S t~22Dhi- ~21D'22)/S
L'zi
(18)A word is necessary for the reason that this system is a ternary one. Both solutes in this system are considered to be completely ionized and the constituting species are the three ions of NiZ+,Co2+, C1- and water. The condition of electrical neutrality at any position in the liquid phase, however, reduces the number of independent species to three.
.\ %-*=
'2,
=
tPllDh2
(6)
- PlZDhl)/S
s = PllP22 - 1112P21 Here,
pt,
= apL/ac,,pLIbeing the chemical potential of
Measurement of Ternary Diffusion Coefficients
The Journal of Physical Chemistry, Vol. 87, No. 4, 1983
591
species i in the ternary system. Diffusion coefficients relative to the solvent, D:,, are related to those relative to the volume-averaged reference frame, Dij, as
7-
where Vi is the partial molar volume of component i and r refers to the solvent. The phenomenological coefficients were calculated from the observed diffusion coefficients by using eq 6 and 7. The partial molar volumes were calculated by assuming that the density in the ternary system is expressed as the linear combination of those in the two corresponding binary systems. Data on the binary systems were taken from a standard reference b00k.l~ Activity coefficients in the ternary system were estimated by using the LietzkeStoughton equationmwith the data in binary systems from a standard reference book.21 Refer to the article by Millerz2for the details of the methods of calculating L The calculated L{j values are shown in Table 111, together with the probable errors calculated by the method used by Miller.22 In the estimation of the errors, uncertainties in ci, Vi were neglected. Activity values in the present mixture are missing in the literature and were estimated as stated earlier. The error in the activity factors necessary for the calculation of p i j were arbitrarily chosen to be IO%, which, we believe, would be more than the maximum possible error. Fortunately, this error in the activity factors causes a large error in neither p i j nor the calculated L :j values. Note that Onsager's reciprocal relation L'12 = L'21 (8) is very closely satisfied within the probable error by the present result. This strengthens the reliability of the observed diffusion coefficients in the present investigation. Methods of estimating the phenomenological coefficients in a multicomponent system from those in the corresponding binary systems are available.% The comparison of the observed values with these estimations will be discussed in a forthcoming separate report. Experiments in the water-CoSO, (1)-CuS04(2) system were also canied out as another example. Preliminary runs indicated much smaller cross diffusion coefficients compared with the water-NiC1,-CoC12 system. Large differences of the initial concentrations, therefore, were chosen as are shown in Table I11 to exaggerate the differences in the concentration profiles. The constancy of the diffusion (19) Nihon Kagaku-kai (the Chemical Society of Japan). 'Kagaku Binran"; Maruzen: Tokyo, Japan, 1975. (20) Lietzke, M. H.; Stoughton, R. W. J.Solution Chem. 1972,1,299. In the present ternary system, the use of this method is very close to assuming that the activity coefficient of component i in the ternary system is the same as that in the corresponding binary system at the same ionic strength, because the two solutes are similar to each other. (21) See ref 3, pp 499, 502. (22) Miller, D. G. J.Phys. Chem. 1959, 63, 570. (23) Wendt, R. P. J.Phya. Chem. 1965,69,1227. Miller, D. G. Ibid. 1967, 71, 616.
ot
Figure 7. Normalized concentration profiles of cases I and I1 in water-CoS0,-CuSO, system at 298 K.
coefficients in this case is doubtful as can be suggested from the data in the binary systems. The normalized concentration curves for cases I and I1 are shown in Figure 7. Again, C1(+) and Cl(-) are plotted in the opposite position of 7. It is interesting to notice that the curves for the opposite diffusion of case I1 show that accelerated diffusion occurred in this case compared with the diffusion in the same direction of case I. This shows that both cross diffusion coefficients are negative. The diffusion coefficients and the phenomenological coefficients determined for the data shown in Figure 7 are shown in Table 111, together with the probable errors estimated by using the same method as before. The 10% error used again in the activity factors gives much larger errors in the main phenomenological coefficients, but not in the cross terms. When the errors in the activity factors are neglected, the probable errors in L;, and L i 2 are f0.04,those for L'12 and L i l being unchanged. The main diffusion coefficients again are a little smaller than those in the corresponding binary cases. Cross terms are negative and are small in magnitude. The two cross phenomenological coefficients are equal to each other, again verifying Onsager's reciprocal relation.
Conclusion A new method of measuring the four diffusion coefficients in ternary systems using the position-scanning spectrophotometer is presented. Using the concentration profiles of the two solutes from a set of different experiments, we could easily determine the diffusion coefficients without using any elaborate mathematical treatment. The main diffusion coefficients thus determined in the system of water-NiC12-CoC12 were found to be considerably smaller than those in the corresponding binary systems. Cross diffusion coefficients were about 25% of the main terms. The calculated cross phenomenological coefficients were almost the same in magnitude, showing the validity of the reciprocal relation of Onsager. Measurements in the water-CoS04-CuS04system revealed small negative cross diffusion coefficients. Registry No. CoC12, 7646-79-9; NiCl,, 7718-54-9; CuS04, 7758-98-7; COSO~, 10124-43-3.
