LITERATURE CITED (1) (2) (3) (4)
D. W. Underhill, Nucl. Appl., 6, 544 (1969). G. G. Curzio and A. F. Gentili, Anal. Chem., 44, 1544 (1972). L. R. lngersoll and 0. A. Koepp, fhys. Rev., 24, 92 (1924). J. S. Anderson and K. Saddington. J. Chem. SOC.,1949, 5381-6. (5) H. S. Carslaw and J. C. Jaeger, "Conduction of Heat in Solids", 2nd ed., Oxford University Press, 1959. (6) W. E. Bolch, R. E. Seileck, and W. J. Kaufman, "Gas Dispersion in Porous Media", SERL Rep. 67-10 (1967).
(7) S. Kitani. S. Uno, J. Takada, H. Takada, and T. Segawa, Recovery of Krypton by Adsorption Process, JA€R/, 1167 (1968). ( 8 ) G. G. Curzio and A. F. Gentili, Liberation de gaz nobles par les centres nucleaires: quelques remarques sur le functionnement des filtres de charbon de bois, VIe Congres international de la SOC.Francaise de Radioprotection, VI, SFRP/21, 233 (1972).
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RECEIVEDfor review July 30, 1975; Accepted August 29, 1975. This work was supported by CNR Contract No. 73.01279.07.
Determination of Diffusion Coefficients by Diffusion from Tubes N. C. Fawcett' and Roy D. Caton, Jr." Department of Chemistry, University of New Mexico, Albuquerque, N.M. 87 13 1
A new approach for the determination of diffusion coefficients is presented. Since the method employs open tubes containing 0.6-1.4 ml of diffusant solution, ultrasensitive analytical methods need not be used to determine solute concentrations during experiments as in the capillary method. The apparatus consists of stationary glass tubes, filled with a solution of diffusant, in a bath of stirred pure solvent. Results for Pb(ll) in 0.1 M KCV0.005 M HCI and anthracene in acetonitrile are in excellent agreement with those obtained using the capillary method and electrochemical methods. New values for p-nitrobenzene and 4,4'-dinitrozobenzene in acetonitrile are given.
The diffusion coefficient ( D ) for an electroactive species in solution represents an important datum in electroanalytical chemistry. D values may be determined by electrochemical means in many cases. Often, however, the complexity of the system precludes easy evaluation. Even when the diffusion coefficient can be obtained from electrolysis data, confirmstion by a determination independent of electrochemistry is highly desirable. This problem has been addressed by Adams and coworkers ( I , 2) who have had considerable success measuring diffusion rates using the capillary method of Anderson and Saddington ( 3 ) . This appears to be an especially suitable method for determining D values to be applied in electroanalytical chemistry because Bearman ( 4 ) has shown that the identity of the so-called capillary diffusion coefficient corresponds closely to that of the polarographic diffusion coefficient. The capillary method possesses a rather unique set of attributes compared with alternative methods. For example, diffusion rates of both ionic and molecular species may be measured in a variety of solvents, only simple apparatus is required, and the method is absolute in the sense that calibration with a substance of known D is not required. Despite these advantages, application has been somewhat limited by the small total amount of material that can be diffused from a capillary, which is of necessarily small internal volume. This is an especially important restriction to the electroanalytical chemist who desires to measure diffusion rates in very dilute solutions. Because of the small amount of material that can be diffused, past application has been limited to those substances for which very sensitive analysis is available, e.g., radioassay or fluorimetry. Present address, Department of Chemistry, Southwest Texas State University, San Marcos, Texas 78666.
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Another reason for limited use of the capillary procedure is perhaps the possibility of high results arising from partial convective loss of diffusant. Wang et al. (51, and Mills (6) in particular, have thoroughly criticized the capillary method, with special attention being given to convective loss. A differential error analysis of the method has been published (7). In this paper, a new method for determining diffusion coefficients is proposed and tested by the determination of several diffusion coefficients in both aqueous and nonaqueous solutions. The new method is closely related to the capillary procedure, but tubes, several millimeters in diameter, are substituted for capillaries. Although the use of tubes results in much greater convective loss, this loss is accounted for by the method used to calculate D . The greater internal volume of a tube permits more substance to be diffused in the course of an experiment. Since the amount of substance diffused is greater, it is possible to use standard methods of analysis such as titration or spectrophotometry to follow changes in diffusant concentration. This, in turn, should make the new method more widely applicable than the classical capillary procedure, while at the same time retaining the useful attributes of the older method.
OBSERVATION OF BOUNDARY FORMATION IN TUBES A glass tube, several centimeters long and a few millimeters in diameter, may be sealed a t one end, filled with dye solution, and immersed vertically in a large container of pure solvent. If the solvent is stirred in a uniform, nonturbulent manner and if the solution in the tube is not less dense than the solvent, the following phenomenon is observed. Convection caused within the mouth of the tube by the stirring motion of the solvent causes dye solution to be swept from the tube. Upon continued observation, the loss of dye subsides and a sharp boundary forms between a column of stationary dye solution in the lower part of the tube and a moving column of solvent in the upper part. Figure 1 illustrates this phenomenon. Also included in Figure 1 is an illustration of what is observed if the outer fluid is more dense than the fluid in the tube. In that case, the boundary is not stable, and the tube's entire content is swept into the surrounding fluid. How much of the bore will be swept by convection depends in part on tube diameter, as shown in Figure 2. For a very small diameter tube, Le., a capillary, only a very small portion of the bore is swept out. For example, about 1 mm of the bore is swept from a 1-mm i.d. capillary under conditions given for Figure 2.
A
0
Figure 1. Appearance of boundaries
1
A = forming, B = stabilized, and C = broken. Arrow indicates direction of outer fluid flow
Provided the stirring speed remains constant and the system is not subjected to disturbing temperature gradients, the position of the boundary remains stable. Once the boundary has stabilized, the only way dye molecules may exit from the tube is by diffusion to the boundary from where they are swept by convection into the moving fluid that surrounds the tube. The result is the formation of a diffusion gradient between the boundary and the lower end of the tube. APPLICATION OF BOUNDARY FORMATION TO DETERMINATION OF D The boundary formation phenomenon described above may be used to obtain diffusion coefficients in a manner similar to the classical capillary procedure. For diffusion from a capillary, the ratio of final average concentration of diffusant, C, to initial average concentration, C,, is given by ( 3 )
where t is the diffusion time and 1 is the bore length of the capillary. Application of Equation 1 to tubes requires that 1 and t be redefined as explained below. The time scale for diffusion from a tube may be defined as in Figure 3, where t , is the time a t which the tube is immersed, t o is the starting time for diffusion, Le., the time a t which the boundary may be said to have stabilized, t is some time after t o and t’ is some time after t . Both t and t’ in Figure 3 may be referenced to t o provided t o is known. In practice, it is easy to determine t o by visual observation of the boundary formation process with the aid of dye solutions as described in the experimental section of this paper. By repeated observation, t o can be determined with a standard deviation of about f 3 min. The systematic error in t o ,however, is likely to be larger than the random error. For a tube, the length of the bore, L , is not the same as the length of the diffusion column, I . These quantities are defined in Figure 4.Although it may be possible to measure 1 directly, perhaps with the aid of a dye solution, in practice the boundary is difficult to locate exactly because of its slightly parabolic surface. Therefore we chose to solve for the effective value of 1 as explained below. Equation 1 may be rewritten in functional form as
(3)
The average concentration of diffusant a t t o is C., If Cf is defined as the analytical concentration of diffusant in the solution used to fill the tube, it may be seen by referenci to Figure 4 that 1 = L(C,/Cf)
3
4
5
6
SWEPT
7
8
9
OUT, mm
Figure 2. Effect of tube diameter upon length of bore swept by convection for M crystal violet in water in an apparatus similar to that shown in Figure 7 Stirring speed = 200 rpm, solvent = water, and tube length = 60 mm
3
C
6s
t
t o
t‘
Flgure 3. Time scale for diffusion from tubes t, = time of immersion, to = time of boundary stabilization. t = time at which first concentration is determined, and t‘ = time at which final concentration is determined
Figure 4. Definition of bore length, L, and length of the diffusion column, I C = region of convective transport, D = region of diffusive transport
I
IO
zc TIME
3c ~
4c
5c
hours
Figure 5. Time rate of change of average diffusant concentration for c m 2 s-’, C, = 1 X the case where D = 1 X M, and I =
Likewise, for some other time, t’, C‘ = f(C,,t’,l,D)
2
DISTANCE
(4)
Equations 2, 3, and 4 represent a set of nonlinear equations which may be solved simultaneously for D, l , and C,
60 m m
if L , Cf, C, C’, t and t‘ are known. This is the basis for determination of D using tubes rather than capillaries. The approach outlined above requires that two separate experiments, each beginning at t , (see Figure 3), be carried out for each value of D obtained. In practice, this involves little additional time and effort. If t is too close to to, the effect of an error in t o will be magnified, because near tothe concentration is changing ANALYTICAL CHEMISTRY, VOL. 48, NO. 3, MARCH 1976
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I 100
I
I
10 TIME,
Figure 6. Maximum relative tration is determined
I
1000
5000
MINUTES
error in D vs.
time at which first concen-
The value above each curve is the absolute value of c t used to compute the curve. tC = 0.01 C,e,, = 0.01 C.C,= C, = 1 X M, / = 70 mm. ‘t = om2 s-’ 120 h. and D = 1 X
rather rapidly. This is illustrated by Figure 5 in which the rate of change in concentration given by - =at
“$J) 5= O exp[
-(2n
n
+
412
n2Dt]
(5)
is plotted vs. time elapsed from t o , assuming typical values for D , C , and 1. Provided that D > 0 and 1 < a,for t = 0,
and for t =
aClat = - m
(6)
aClat = 0
(7)
a,
I t would not be a good idea to obtain C , directly by stopping the experiment a t the estimated t o point and analyzing the tube contents because, as Figure 5 shows, the concentration is changing so rapidly near t o that a small error in time becomes quite significant. To minimize the relative effect of an error in time, et, the value of t should be large, and to minimize the effect of experimental errors, ec and ec,, in C and C’, the difference between C and C’ should be large. With t’ fixed a t some practical value a priori, the requirement that t be large is in opposition to the requirement that C minus C’ be large, because as t increases, the quantity C minus C‘ must decrease. This situation results in the error in D being a minimum for some value o f t between t oand t’. It is possible to demonstrate the value of t needed to minimize the error in D by computing for various values of t , the maximum error in D due to et, eC, and tC, with t’ constant. The results of such a computation are plotted in Figure 6. The computation was based on a hypothetical system in which cc and eC’ were set a t fl% of C and C’, respectively; the true value of D was set a t 1 X 10-5 cm2 s-l; L and 1 were set equal to 70 mm, and Cf and C , were both 1 X lop3 M. Setting Cf equal to C , and L equal to 1 is equivalent to assuming no convective loss. This simplified the computation without affecting the validity of the results. The actual computation of the error curves shown in Figure 6 was carried out as follows. Using the assigned true value of D , concentrations C and C‘ a t times t and t‘ were computed according to Equation 1. Erroneous concentrations, C ec, and C’ + eC,, and erroneous times, t et and t’ + et’, were then used to compute D by simulated, simultaneous solution of Equations 2 and 3. The signs of the errors were selected to maximize the error in D . The relative error in D was obtained by comparison of the true value of D with the simulated, erroneous value. The above process was
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Figure 7. Apparatus for determining diffusion coefficients by diffusion from tubes A = synchronous motor, B = Plexiglas motor support, C = stainless steel ball bearing, D = 45 mm Teflon disk, E = Epoxy base, F = Tygon tubing coupling, G = metal coupling, H = Teflon lid, I = 18 mm i.d. Pyrex tube, J = 95 mm i.d. Pyrex cylinder, K = 6.35 mm diam. stainless steel shaft, L = Teflon support for diffusion tubes, and M = collars
repeated for incremental values o f t starting a t zero and increasing until the minimum in the error curve was located. Notice that an error in time causes the curves (Figure 6) to rise on the left, and an error in concentration causes a rise on the right. The bottom curve is for zero time error and, therefore, does not rise on the left. As an example of how the curves in Figure 6 could be applied in an experimental situation, suppose e t was estimated a t f 1 5 min. From the 15-min curve, it can be seen that C should be measured about 100 min after t o to minimize the error in D . Figure 6 also shows that, for the assumed conditions, a 10%error in t contributes several times less to the maximum relative error in D than does a 1%error in C and C’. As part of our experimental procedure, we attempted to optimize the value of t by calculating an approximate error curve like those in Figure 6 prior to each experiment. In cases where a published D value was available for the substance and solvent of interest, the published value was used as an estimate for D in the error calculation. In other cases, only a very rough estimate of D was made by comparison with data published for structurally similar substances in the same solvent. Fortunately, assignment of values to the various parameters used in the error computation is not critical because the region of minimum error is quite broad, allowing considerable flexibility in the value chosen for t . Estimates for the other parameters were chosen to be consistent with the experiment to be performed.
EXPERIMENTAL Apparatus. A cross-section view of the apparatus is shown in Figure 7 . The volume of the outer cylinder was slightly greater than 2 1. The lid and tube support were machined from solid Teflon stock. The diffusion tubes (usually four) were supported vertically by insertion into holes of appropriate size drilled symmetrically around the perimeter of the support. Twelve holes of three different sizes were used, holes of the same size being located a t 90° intervals. The central Pyrex tube surrounding the stirring shaft was press-fit to the Teflon lid and Teflon tube support. The stirring shaft, hearings, coupling, and collars were purchased as precision ground, stainless steel parts from PIC Design Corporation, Van Nuys, Calif. The central shaft was 270 mm long and 6.35 mm in diameter. The upper shaft hearing was press-fit to the lid. The lower bearing was held in place with epoxy cement. The stir-
Table I. Summary of Diffusion Data Obtained by t h e T u b e M e t h o d a n d Comparison with Similar D a t a Obtained b y Comparable M e t h o d s Diffusant (solvent)
na
D X io5b
D X 105c
D X 105e
D X 105d
2 . 4 1 i ?f 8 2.84 i 0 . 1 2 2 . 9 1 i. 0.10.f 2 . 8 8 i ?f Anthracene (MeCN) Pd2+ 4 0.954 * 0.005 ... 0 . 9 7 0 i 0.12h 0 . 8 6 7 0.03g ( 0 . 0 1 % gelatin) (0.1 M KCl/O.OOS M HCl) PNA 11 2 . 2 5 i 0.07 ... ... ... (MeCN) DNAB 4 1 . 9 4 i 0.01 ... ... (MeCN) Unless otherwise s t a t e d t h e i figures are s t a n d a r d deviations. a T h e n u m b e r o f observations used to c o m p u t e t h e first colu m n of D values o n t h e right. b T h e m e a n value of D f o r n observations b y t h e t u b e m e t h o d . CValue obtained using t h e capillary m e t h o d with fluorescence analysis. T h e q u o t e d precision is a n average deviation. d Values o b t a i n e d using t h e capillary m e t h o d with radiotracer analysis. e Values o b t a i n e d by electrochemical m e t h o d s . fValues o b t a i n e d by Bacon a n d Adams ( 2 ) . g V a l u e o b t a i n e d by V o n Stackelberg e t al. ( 1 2 ) .hVa1ue o b t a i n e d b y Wang ( 1 3 ) . .
ring blade was a flat disk of Teflon, 1.5 mm thick and 45 mm in diameter. The blade was held on the shaft between two set screw collars. Stirring was accomplished with a 200-rpm synchronous motor. In operation, the stirring motor and shaft assembly were essentially free from vibration. The lower two thirds of the assembly was supported in a constant temperature bath by means of a bottle clamp. The clamp was fixed to a rigid support, permanently attached to the bench top and not in direct contact with any part of the temperature bath. This allowed for minimal transfer of vibration from the bath-stirring motor to the cell assembly. Temperature control of the bath was slightly better than 10.1 "C and was estimated to be accurate within 0.1 "C as determined by a 0.1 "C Beckmann thermometer which had been calibrated against an N.B.S. standardized thermometer. Diffusion tubes were made from conventional Pyrex glass tubing. For quantitative work, two different sets of four tubes each were used. One set had a nominal inside diameter of 5 mm and a nominal length of 76 mm. The other set had a nominal inside diameter of 3.5 mm and a nominal length of 70 mm. Each tube was sealed a t one end. The bore length of each tube was determined with a vernier caliper to the nearest 0.05 mm. This measurement was estimated to be accurate within f 0 . 3 mm. Tube volumes were calculated from the weight of mercury held by each tube. Volumes were determined to four significant figures. Each volume determination was repeated two or more times. In no case was the standard deviation of the mean volume greater than f0.0004 ml, a typical value being f0.0002 ml. The 3.5-mm tubes had volumes of about 0.71 ml, and the 5-mm tubes had volumes of about 1.45 ml. Tube volumes were used in calculating concentrations from analysis data. Reagents. With the exceptions noted below, all solutions were prepared from reagent grade chemicals without further purification. p-Nitroaniline (PNA) was purified by column chromatography of technical grade PNA on anhydrous alumina. The column was loaded with a benzene solution of PNA. 0 - and rn-nitroaniline were eluted with 20% ethanol in benzene followed by elution of PNA with 40% ethanol in benzene. PNA was recrystallized from benzene. 4,4'-Dinitroazobenzene (DNAB) was prepared by electrooxidative coupling of PNA using the procedure of Wawzonek and McIntyre (8). Acetonitrile was purified according to the method of O'Donnell e t al. (9). Water was ion-exchanged and distilled. Solvents were degassed before use, either by boiling or evacuation. Chemical Analyses. Anthracene, PNA, and DNAB concentrations were determined spectrophotometrically in the ultraviolet region using a Beckman Model DU spectrophotometer. Lead was determined by thermometric titration with 0.00943 M K2Cr04 using a Tronac Solution Calorimeter. Procedure. The apparatus was filled with 2 1. of solvent and then placed into the constant temperature bath for equilibration. After several hours of equilibration with stirring, the cell lid was removed, and tubes filled with diffusant solution were inserted into the tube support. The tubes were immersed in the solvent by reassembly of the cell, and the stirring motor was turned on. After a predetermined (see below) time ( t )had elapsed, the stirring was stopped and the tubes were removed for analysis of diffusant con-
I
.
centration. This was accomplished by quantitative transfer of each tube's contents into separate volumetric flasks. Five-, 25-, or 50-ml flasks were used, depending on the sensitivity of the analysis to be used, The resulting solutions were analyzed separately to obtain a value of diffusant concentration, C, for each tube. The solvent in the apparatus was renewed, and, after temperature equilibration, the tubes were again filled with diffusant solution and placed in the cell as before. This second diffusion was continued for 3 to 9 days ( t ' ) depending on the estimated rate of diffusion. The tubes were then removed and their contents analyzed as before to obtain a value of C' for each tube. The diffusion coefficient was computed by simultaneous solution of Equations 2, 3 and 4. A FORTRAN IV grid-search, functionminimization routine (IO)was used in this computation. The input data consisted of values for Cf,C, C', t , t', and L . The solution consisted of values for C,, 1 , and D . The optimum time, t , at which to take the first concentration point, C, was determined in advance of each series of experiments. This was done by computing an error curve like those shown in Figure 6. The principle and method of this computation is explained in the preceding section. The time required for stabilization of the diffusion boundary ( t o )was determined separately for each combination of tube diameter and solvent. This was done by substituting a small amount of M) for diffusant, which made it possible dye (crystal violet to observe formation of the diffusion boundary. The time a t which the tubes were immersed was then noted. At first, observations were made every 10 min, then every five, and finally every minute until the dye plume rising from each tube could no longer be seen. The elapsed time required for disappearance of the dye plume was taken as to, the starting time of diffusion. Other times ( t and t') were referenced to to. The value of t o did not have to be redetermined so long as the stirring rate, solvent type and volume, and mounting position of the tubes were not changed.
RESULTS AND DISCUSSION Diffusion coefficients of several species were d e t e r m i n e d as a test of the proposed t u b e method. The results of t h e s e d e t e r m i n a t i o n s a r e s u m m a r i z e d i n T a b l e I. C o m p a r a b l e lite r a t u r e values, where available, a r e also shown. There is no significant difference b e t w e e n values obt a i n e d i n t h i s work and c o m p a r a b l e values o b t a i n e d b y othe r s using the capillary procedure. The electrochemically o b t a i n e d D value for Pb(I1) is not strictly comparable, as i t was m e a s u r e d i n t h e presence of 0.01% gelatin. U n f o r t u nately, the D value for a n t h r a c e n e was d e t e r m i n e d before the i m p o r t a n c e of optimizing t w a s realized. T h e r e f o r e , the precision of t h i s result m a y not be representative. The relative standard deviation for t u b e D values r a n g e d f r o m a b o u t 0.5% for 4,4'-dinitroazobenzene (DNAB) to 6% for
Pb(I1). In general, t h e precision r e p o r t e d for D values d e t e r m i n e d using the capillary method has ranged f r o m 2 to 596, a l t h o u g h Mills (11) was able to o b t a i n a precision of better than 1%for the self diffusion coefficient of 22Na ion. Obviously, e n o u g h data has not y e t been o b t a i n e d to give a reANALYTICAL CHEMISTRY, VOL. 48, NO. 3, MARCH 1976
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liable estimate of what precision on the average may be expected from the tube method; however, the values in Table I suggest that the precision will be a t least as good as that obtained from capillaries. The tubes used in determining the coefficient for Pb(I1) had nominal bore dimensions of 5 X 76 mm. Appr'oximately 6 mm of the bore was swept by convection, making the diffusion column 70 mm long and the effective volume about 1.41 ml. The tubes used for anthracene, p-nitroaniline (PNA), and 4,4'-dinitroazobenzene (DNAB) had bore dimensions of 3.5 X 70 mm with an effective volume of about 0.6 ml. About the same length of bore (6 mm) was swept from these tubes as for the larger 5-mm tubes. This is because the solvent used with the larger tubes was water while that used with the smaller tubes was acetonitrile. This suggests that solvent viscosity may be an important factor in determining how much of the bore will be swept by convection. The value of t used for anthracene, Pb(II), PNA, and DNAB was
