The results of concentration studies run on several depolarizers are presented in Table 111. Current values at 10-4 and molar concentration were obtained from currentvoltage curves where the value of diffusion-limited current minus extrapolated residual current was used to calculate The values of il;, for 10-6M concentrations were obtained at a constant potential in the diffusion plateau region by determining the difference in current between the supporting electrolyte and 10-6M solution as they were alternately passed through the electrode. The relationship between current and concentration was found to be linear over the 100fold concentration range studied. Assays were run on mixtures of Pb+2and Cu+2 to demonstrate that the electrode could be used to analyze mixtures of ions. Figure 3 shows a typical current-voltage curve obtained for a mixture of PbU2and CuL2. Table IV shows the
analytical results for different ratios of Pb+2to CutZ concentrations. The theoretical value of i l l c for Pb+2 using the diffusion coefficient reported by von Stackelberg et al. (17) was calculated to be 186 which is in agreement with the experimental results demonstrating that the earlier Cu+2 wave had no effect on the Pb+2diffusion current. It is concluded that a tubular platinum electrode can be conveniently coated with a film of mercury, thereby extending the useful cathodic voltage range, and that diffusion limited currents are independent of the quantity of mercury in the coating.
RECEIVED for review June 6, 1967. Accepted August 24, 1967. Thomas 0. Oesterling was a fellow of the American Foundation for Pharmaceutical Education.
Chronoamperometry at Tubular Mercury-Film Electrodes T,0.Oesterling’ and Carter L. Olson CoUege of Pharmacy, The Ohio State Unioersity, Columbus, Ohio The current-time equation for quiet solution electrolysis at a tubular electrode which is infinitely long or has closed ends is derived. At electrolysis times less than one second, currents calculated with the tubular electrode equation agree closely with currents calculated by the Cottrell equation. Currents of experimental chronoamperograms, collected at openended mercury-film tubular platinum electrodes (MPTE) of finite length, agree reasonably well with values calculated using both the tubular electrode and Cottrell equations for electrolysis times less than one second. However, at larger times, experimental currents resulting from the quiet solution electrolysis of Pb(ll), Cd(ll), TI(I), Bi(lll), Cu(ll), Co(NHs)~(lll), and KaFe(CN)6are approximately 10% higher than currents calculated with the tubular electrode equation because of increased mass transfer caused by end diffusion and density gradients.
CURRENT-TIME EQUATIONS have been derived and tested for quiet solution electrolyses at planar ( I , 2), cylindrical (2, 3), and spherical (2, 3) electrodes. In each case experimental data agreed with theoretical equations only during very short time intervals or under precisely controlled experimental conditions. In spite of these limitations many useful applications of current-time measurements at electrodes of various geometries have been utilized (4). The current-time equation for quite solution electrolysis at tubular mercury-film electrodes (MTPE) which are infinitely long or whose contents are physically isolated is presented in this article. Currents of experimental chronoamperograms of ion-ion and ion-amalgam reductions obtained at open1 Present address, Pharmacy Research Unit, The Upjohn Co., Kalamazoo, Mich.
(1) F. G. Cottrell, 2.Physik. Chem., 42, 385 (1902). (2) H. A. Laitinen and I. M. Kolthoff, J . Am. Chem. SOC.,61, 3344 (1939). (3) P. Delahay, “New Instrumental Methods in Electrochemistry,” Chap, 3 , Interscience, New York, 1954. (4) L. B. Anderson and C . M. Reilley, J . Chem. Educ., 44, 9 (1967). 1546
e
ANALYTICAL CHEMISTRY
ended tubular mercury-film electrodes of finite length were compared with the derived expression, and the influence of end diffusion and convection was studied. THEORY Derivation of Current-Time Equation. The differential equation for radial diffusion of an electroactive substance in an infinitely long tubular electrode containing a quiet solution is given by Barrer (5) and Jost (6),
dC(r, t) d2C(r, t ) =Dat ar2
1 X ( r , t)
+;dr
where C(r,f)is the concentration of electroactive substance at a distance r from the axis of the cylinder at time t, and D is the diffusion coefficient of the electroactive substance, To solve Equation 1, the following initial and boundary conditions are imposed :
C(r, f) = C”
C(r,t)=O
t =0 t > O
r
5R
r = R
(2)
(3)
C” is the bulk concentration of electroactive substance and R is the radius of the tube. Equation 2 is obeyed by maintaining a uniform concentration of electroactive substance in the electrode before electrolysis is begun. Condition 3 is attained by applying a potential well out in the limiting current region of the electroactive substance to the electrode so that the charge transfer process is very fast and electrolysis is limited by diffusion of electroactive species to the inside surface of the tube. The solution to Equation 1 for conditions 2 and 3 is given by Barrer (5) and Jost (6),
(5) R. M. Barrer, “Diffusion In and Through Solids,” Macmillan, New York, 1941, p. 34. (6) W. Jost, “Diffusion In Solids, Liquids, Gases,” Academic Press, New York, 1960, p. 52.
where a, is the nth root of the equation Jo(a,r) = 0. The first ten roots of Jo(a,r) = 0 are given in Table I. The functions JO ( Y ) and J1 ( Y ) are Bessel functions of the zero and first order, respectively and their values for any values of Y are given in tables in the literature (7). The current flowing at any time after electrolysis has begun is given by
Table I. Roots of J&,r) n 1
anr
2.405
2 3 4 5 6
5.520
8.654 11.792 14.932 18.071
21.211 24.352 27.493 30.634
7
where n is the number of electrons involved in the charge transfer process, F is the Faraday constant, and A is the area w r ,0 of the electrode in sq cm. The quantity D[
7 1
7-R
is the flux of the electroactive species at the tubular electrode surface and may be obtained from Equation 4, =
-D
T=R
since ___
dY
=
2C”
c3
~
R
i
exp (- Dan2t)
(6)
-aJ1 ( a y ) (8).
Combining Equations 5 and 6 gives the expression for instantaneous current at any time after the electrolysis has begun, FJ
exp ( - D a n 2 t )
i = - 4 m F X COD 1
= 0
8 9 10
Table 11. Experimental Conditions Employed to Obtain Chronoamperograms in 0.1M KNOa Diffusion Applied coefficient, Concentration, potential, cm Metal cation mole/liter volt cs. SCE sq Pb(I1) 0,00010 -0.800 0.86 0.00010 -0.900 1.82 TKI) 0.00010 -0,820 0.69 Cd(I1) 0,000088 -0.800 0.82 CoWHah (111) -0.100 0.70 K8Fe(CN),p 0.00010 0,00010 -0,400 0.69 Cu(I1) 0.00010 -0.250 0.52 Bi(II1) a Background electrolyte was 0.1M KCl.
sec
(7)
where X i s the length of the electrode in cm. In Equation 7, i is pA when concentration is expressed in mole/cc. Short Electrolysis Times. At very short electrolysis times,
the thickness of the diffusion layer is very small and current values calculated through the use of Equation 7 should be equal to currents calculated through the use of the Cottrell equation ( I ) for the same values of t. Using the same values of n, electrode area, diffusion coefficient, and depolarizer concentration, currents calculated by the Cottrell equation are identical to currents calculated by Equation 7 at small values of t and gradually diverge as t increases. For example, at 0.1 second, ilube = i C o t t r e l l ; at 1 second, = 0.97 i C o t t r e l l ; at 3 seconds, itube = 0.93 iCottrell; and at 6 seconds, itube = 0.90 i c o t t r e l l . EXPERIMENTAL Apparatus and Materials. The apparatus, reagents, and electrodes were identical to those described previously (9). Current-time curves of some studies at very short electrolysis times of a 7.5 X 10-5M TI(1) solution in 1M KN08 and in 1M K N 0 3 alone were obtained using a Model 564 Textronix Oscilloscope. Procedure. Initially current-voltage curves were obtained for each depolarizer to determine the voltage region where the current was diffusion limited. This was done by pumping a solution containing the depolarizer plus background electrolyte through the electrode and running a current-potentia1 curve. Well defined diffusion limited current regions were obtained with all of the depolarizers, The potential chosen to be applied to the electrode was at least 200 mV more cathodic than the half-wave potential where the current-time curves were potential independent. The experimental con-
ditions employed for each depolarizer and the value used for their diffusion coefficients in the calculations are given in Table 11. The procedure used to obtain chronoamperograms for all depolarizers except K3Fe(CN)e is given below. A deaerated solution containing depolarizer and background electrolyte was pumped through the elcctrode at 3 mlimin for 5 minutes at an applied potential of 0.0 volt (cs. SCE). This serves to flush any traces of oxygen from the flow system, and also to strip any previously deposited metals from the mercury-film electrode. The cell circuit was then opened and the pumping was stopped. After 30 seconds had elapsed, a predetermined potential in the limiting current region (Table 11) was set, the electrode circuit closed, and the resulting current-time curve was recorded. Current-time curves were obtained for solutions of background electrolyte treated ucder identical solutions. Corrected current values were obtained by subtracting the current for the background electrolyte from the values obtained for the solution containing depolarizer plus background electrolye. These corrected values were then used in the calculations. The procedure used to obtain current-time curves for K3Fe(CN)ewas identical to the above procedure except that the pre-electrolysis potential was 0.5 volt us. SCE, and ferricyanide was reduced in 0.1M KCl at a tubular platinum electrode. Each experimental value reported represents the average of at least two measurements. Current values of replicate runs measured on the same film did not vary by more than 0.5 in one day or by more than 5 % from day to day.
x
RESULTS AND DISCUSSION
Comparison of Experimental Data with Equation 7. It (7) A. Gray, G. Matthews, and T. MacRobert, “Treatise on Bessel
Functions,” Macmillan, New York, 1922. (8) F. Bowman, “Introduction to Bessel Functions,” Dover Publications, New York, 1958, p. 10. (9) T. 0. Oesterling and C. L. Olson, AKAL.C H E M39, . , 1543 (1967).
i/c exp (-Da,Zt m
can be seen in Equation 7 that the term
1
should be constant for all values of t , and the value of the constant can be calculated with a knowledge of n, the length of the electrode, bulk concentration, and diffusion coefficients VOL. 39, NO. 13, NOVEMBER 1967
1547
of the different depolarizers. Results of the experimental current-time studies are given in Table I11 for the ion-amalgam reductions and in Table IV for the ion-ion reductions, m
Values for the term i/cexp (- D a n 2 f were ) obtained by divid1
ing corrected experimental currents by the sum of the ex-
ponential terms at a given time. Theoretical values for the m
i/cexp (- Danzt)terms given in Tables I11 and IV were ob1
tained by inserting the appropriate values into the presummation term of Equation 7 and solving the resulting equation. Values of diffusion coefficients, listed in Table 11, were calculated from the limiting current equation for reduction of electroactive substances in solutions flowing through tubular electrodes (9).
-
Ideally, experimental values of
1
for Ion-Amalgam Reductions Pb(I1) Cd(I1) Tl(1) Bi(II1) Cu(I1) Time, seconds (2.64)~ (2.12)a (2.80)a (2.40)a (2.12)4 2.51 3.59 2.70 2.57 3 3.32 6 2.97 2.30 3.24 2.51 2.40 2.32 3.19 2.55 2.42 9 2.80 2.33 3.10 2.53 2.43 12 2.70 2.30 3.08 2.53 2.38 15 2.71 2.26 3.05 2.54 21 2.69 2.36 2.27 3.06 2.54 2 26 30 3.20 2.31 3.05 2.53 2.37 36 3.19 2.28 3.09 45 3.23 2.74 2.35 3 04 2.31 2.66 2.38 51 3.27 2.33 3.06 2.26 2.39 60 3.28 3.16 1.44 90 2.37 2.42 120 2.46 3.40 2.56 2.43 150 2.55 3.87 2.57 2.49 180 2 65 5.23 2.57 3.43 210 2.81 6.23 2.59 3.84 240 2.47 8.62 4.13 270 3.24 12.14 5.69 300 3.50 17.25 6.93 Average Value for Times between 3 and 60 Seconds 2.30 3.09 2.54 2.22 3.03 zk0.06 irO.10 -10.05 -10.08 10.02 GTheoretical values.
,..
I
Table IV.
Experimental Values of i/cexp (- Dan2t) 1
for Ion-Ion Reductions KaFe(CN)e (1.07)~ 2.57 1.47 1.36 1.40 1.43 1.40 1.43 1,36 1.49 1.46 51 1.43 60 1.39 90 120 1.23 1.35 150 1.25 1.35 165 1.35 190 1.27 1.35 195 1.26 1.36 210 1.25 1.39 225 1.26 1.37 240 1.30 1.38 255 1.43 270 1.47 300 1.54 Average Value for Times between 3 and 60 seconds 1.27 J .42 ir0.03 10.04 a Theoretical values. Co("a)6
Time, seconds 3 6 9 12 15 21 30 36 45
(111)
(1.12)a 1.67 1.33 1.32 1.30 1.30 1.29 1.25 1.26 1.24 1.25 1.24
1
constant and agree with the theoretical value for all values of time. Experimentally it was found that the determined values of the constant were larger than the theoretical value for times greater than three seconds and were reasonably constant only during limited time intervals. The ratios of the average value
ANALYTICAL CHEMISTRY
i/cexp (- Dcun2t)in the time interval 3 to 60 m
of the constant
1
seconds to the theoretical value for ion-amalgam reductions were 1.14, 1.08, 1.10, 1.06, and 1.05 for Pb(II), Cd(II), T1 (I), Bi(III), and Cu(II), respectively. The average of these values is 1.09 0.03. Similarly, the ratio of the experimental m
value of
i/cexp (- Dan2t)to the theoreticalvalue for the ion1
ion reductions in the range 3 to 60 seconds were 1.13 and 1.31 for hexamminocobaltic and ferricyanide ions, respectively, m
Values of i/cexp (- D a n 2 f )for ion-ion reductions remained 1
constant for considerably longer periods of time than in the case of ion-amalgam reductions. The major reason for measured currents being higher than theoretically predicted probably stems from the fact that Equation 7 holds only for electrodes that are infinitely long or have closed ends. Since the volume element within the MTPE is not physically isolated, additional quantities of depolarizer may enter from the ends after concentration gradients have been established. This increased flux of electroactive substance occurs at least as early as three seconds after electrolysis has begun as shown in Tables I11 and IV. Close agreement between experimental and theoretical currents could obviously be obtained by devising an electrode with closed ends or solving the differential equation for electrolysis in tubular electrodes of finite length, where axial as well as radial diffusion must be considered. Another factor contributing to higher experimental currents than expected could be the existence of convection currents. In solving Equation l , diffusion was assumed to be the only mode of mass transfer at the electrode surface. However, for the case of reduction at the MTPE, density gradients may be established at the electrode surface causing stirring and mass transport by convective diffusion. These effects may become significant a short time after electrolysis has begun, and may explain the rise in the calculated constant in Table I11 after 60 seconds for the listed ion-amalgam reductions. Since the reduced form of ion-ion reductions diffuses back into the bulk solution from the electrode surface, it would seem reasonable that smaller density gradients would be formed at the electrode surface, and the fact that i/Z exp (- Dan2t)remains constant for longer periods of time for ion-ion reductions (Table IV) supports this hypothesis. Long Electrolysis Times. It can be seen in Equation 7 that at sufficiently large values of f the summation term reduces to one term and the equation becomes i
1548
i/cexp (- Dan2t)should be m
m
Table 111. Experimental Values of i/cexp (- Dan2t)
=
- 4 m F X D C " exp (- Da12t)
(8)
0.31-
Table V.
Current-Time Relationship at Short Electrolysis Times" i/g exp 1
Time, seconds 0.3 0.4 0.5 0.6
0.7 0.8
0.9 a
b
I
0.7
I
I
1
1
170
150
I
I
I
I
1
190 210 SECONDS
I
1
230
Figure 1. Log i-time plot for ion-ion reductions at long electrolysis times 0 0
10d4MCo(NH& (111) in 0.lM KNOs 10d4:MKaFe(CN)oin O M 4 KCl
where alz = 992 (Table I), Thus a plot of In i 6s. t should give a straight line with slope of -Da12. An additional criteria that must be applied for Equation 8 to hold is that the
i/cexp m
value of
(- Daa2t)must remain constant in the time
1
interval during which the equation is being tested. At values of time greater than 180 seconds for Co (NH& (111) and 150 m
seconds for K3Fe(CN)e, the term
i/cexp ( - D a n 2 t ) reduces 1
to one term and the value of this term remains constant for 90 seconds thereafter as shown in Table IV. Since the data from these two depolarizers met the necessary criteria to apply Equation 8, plots of log i cs. t were made for each depolarizer (Figure 1) in the appropriate time interval and the values of diffusion coefficient, calculated from the slopes, were com-
it"2,
pa sec1/2 9.80b 9.30 9.35
( - - D a n 2 z ) , pu.A
1.86b 1.80 1.81
1.76 1.74 1.74 1.74
9.10 9.00
8.95 8.95 9.30
1.81
7.5 X Tl(1) in liM KNOI. Theoretical values.
pared with literature values. The value of the diffusion coefficient of hexamminocobaltic ion was calculated from the slope of the line in Figure 1 to be 8.07 x 10-6 sq cmjsec, which is in excellent agreement with the value of 8.2 x sq cmisec calculated from the equation for limiting current at tubular electrodes (9). The value of the diffusion coefficient of ferricyanide was calculated to be 6.87 x 10-6 sq cm/sec which is in good agreement with 7.10 X 10-6 sq cmisec, calculated from the tubular electrode limiting current equation (Y), and with Von Stackelberg's value of 7.6 X sq cmisec (10). Short Electrolysis Times. At short electrolysis times when density gradients and end diffusion have little effect on flux of electroactive substance, experimental currents should not be higher than predicted. In addition, as pointed out, experimental currents flowing at electrolysis times of less than 1 second should agree closely with current calculated by the Cottrell equation. Some preliminary data collected at electrolysis times less than 1 second are shown in Table V
-,"
where it can be seen that i t 1 / 2as well as i/cexp (- Dan2t)re1
main constant, Even though the experimental constants (and currents) differ by slightly greater than 5 from the predicted values, the fact that the experimental values are smaller than theoretical indicates that end diffusion and density gradients are not strongly influencing flux of electroactive substance. RECEIVED for review June 6,1967. Accepted August 24,1967. Thomas 0. Oesterling was a Fellow of the American Foundation for Pharmaceutical Education. (10) M. von Stackelberg, M. Pilgram, and V. Toome, 2. Electrochem., 57, 342 (1953).
VOL. 39, NO, 13, NOVEMBER 1967
0
1549
