Determination of diffusion coefficients of the electrode reaction products by the double potential step chronoamperometry at small disk electrodes

www.elsevier.nl/locate/jelechem Journal of Electroanalytical Chemistry 493 (2000) 93 – 99

Determination of diffusion coefficients of the electrode reaction products by the double potential step chronoamperometry at small disk electrodes Haruko Ikeuchi *, Mitsuhiro Kanakubo 1 Department of Chemistry, Faculty of Science and Technology, Sophia Uni6ersity, 7 -1, Kioicho, Chiyoda-ku, Tokyo 102 -8554, Japan Received 20 June 2000; received in revised form 27 July 2000; accepted 31 July 2000

Abstract We propose an easy method for determining the diffusion coefficients of electrode reaction products by double potential step chronoamperometry at small disk electrodes. The necessary theoretical relationship between current and time for this method was obtained by digital simulation of a hopscotch algorithm. We quantified the digital results and showed the routine of the measurement. The method was verified successfully by an experiment in which the diffusion coefficients of both the elements of the [Fe(CN)6]4 − /[Fe(CN)6]3 − redox couple were determined by the usual potential step chronoamperometry and by this method. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Diffusion coefficient; Digital simulation; Double potential step chronoamperometry; Hexacyanoferrate(II) ion; Hexacyanoferrate(III) ion

1. Introduction Diffusion coefficients (Ds) of species that have the same chemical compositions and similar sizes but different charges will give valuable information about the transport mechanisms in solutions. These species often appear as redox couples of electrode reactions. Therefor we commonly measure D of the reactant of the electrode reaction by using an electrochemical method such as chronoamperometry [1 – 3], chronopotentiometry [4], normal pulse polarography [5,6], or steady-state amperometry at ultra-micro electrodes [7]. The D of the electrode reaction product can also be measured by these electrochemical methods after the bulk of the solution has been electrolyzed [8]. But only one and not both species of this kind can be stable for a long time in the usual ambience, except in some special cases, for example the [Fe(CN)6]4 − /[Fe(CN)6]3 − redox couple. So * Corresponding author. Tel.: + 81-3-32383370; fax: +81-332383361. E-mail address: [email protected] (H. Ikeuchi). 1 Present address: Tohoku National Industrial Research Institute, 4-2-1 Nigatake, Miyagino-ku, Sendai 983-8551, Japan.

that, if we could measure the Ds of the electrode reaction products in a short time of several seconds, that would be very helpful. The data of this kind are important not only in solution chemistry but also are necessary for rigorous analysis of the electrode kinetics. For this purpose, a method that uses the scanning electrochemical microscope has been proposed [9,10], but this method demands expensive instruments and sophisticated techniques. Michael and Wightman showed by digital simulation how the shape of the cyclic voltammogram at the micro-disk electrode changes when the D of the electrode reaction product differs as much as several times from the D of the reactant [11]. This fact implies that cyclic voltammetry by the micro-disk electrode could be used for the same purpose. But the difference in the diffusion coefficients of the usual redox couple is not so large that it can cause a significant difference on the cyclic voltammogram. It might be difficult to precisely measure the D of the electrode reaction product, even if the phenomena could be quantified. The technique of double potential step (DPS) chronoamperometry has been developed and used for the determination of the rate constants of homogeneous

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chemical reactions that couple with electrode reactions [12 – 17]. If we apply this method to an electrode reaction not coupled with a homogeneous chemical reaction, by using a small disk electrode of appropriate size, we can determine the D of the electrode reaction products. Actually, Park et al. determined the Ds of the metal atoms in mercury by DPS chronoamperometry at hanging mercury drop electrodes (HMDEs) [18]. They had mathematically derived an equation for the current –time relationship for DPS chronoamperograms at a spherical electrode. Though they used an HMDE of as large a diameter as 1 mm, a smaller electrode will work better for more precise measurements, because radial diffusion becomes more effective the smaller the electrode. From this point of view, a disk electrode is preferable because preparation of a disk electrode of desirable size is rather easy, while that of a small ideal sphere HMDE is almost impossible. However, we have not been able to use disk electrodes for this measurement, since no theoretical equation that represents the current –time relationship has yet been derived. The Shoup –Szabo equation [19] has been commonly regarded as the most precise theoretical equation of the current –time curve at a disk electrode for potential step (SPS) chronoamperometry. Shoup and Szabo devised this equation by combining the long time and the short

time mathematical solutions derived by Aoki et al. [20] while taking into account the results of digital simulation obtained by themselves and others [2,21]. It would be much more difficult to mathematically solve the diffusion equation of the double potential step technique since the Shoup –Szabo equation is not a pure mathematical solution. In this situation, the technique of digital simulation is the most suitable for this problem. Here, we perform digital simulation to obtain a relationship between the current and time functions for DPS chronoamperometry, and on the basis of the results we quantify the dependence of current function on the ratio of the Ds of the redox couple. We propose an easy method for determining the D of the electrode reaction product by showing the routine of the measurement. Computer-processed potentiostats that are now quite common in laboratories will help in the experiments and in data analysis. We verify the method by comparing the D values measured by SPS and DPS chronoamperometry for the [Fe(CN)6]4 − /[Fe(CN)6]3 − redox couple.

2. Digital simulation When a diffusion controlled electrode reaction of A such as Eq. (1) occurs at the first potential step, E1, the species B generated at the electrode diffuses into the solution around the electrode, while we observe current I1. A= B+ ne −

(1)

where n is the charge number of the electrode reaction: positive for oxidation and negative for reduction. At the second potential step, E2, the species B diffuses towards the electrode and the reverse electrode reaction occurs. If E1 and E2 are properly set, the observed currents at these potentials, I1 and I2, become diffusioncontrolled currents. The scheme is sketched in Fig. 1. The profile of I1 at a microdisk electrode is well reproduced by the equation: I1/ynFDAacA = 1+y − 1/2(a 2/DAt)1/2 + 0.2732 exp{− 0.3911(a 2/DAt)1/2} (2)

Fig. 1. (a) A cyclic voltammogram of [Fe(CN)6]4 − in 1 mol dm − 3 KCl aqueous solution at a Pt Disk electrode of 0.241 mm radius and scan speed of 50 mV s − 1 at 25°C. (b) Schematics of the potentials and the currents of double potential step chronoamperometry.

which was presented by Shoup and Szabo [19]. Here F is the Faraday constant, DA and cA are the diffusion coefficient and the bulk concentration of the species A, respectively, a is the radius of the disk electrode and t is the electrolysis time. This equation is plotted in Fig. 2 along with the plot of each term. When (a 2/DAt)1/2 is much larger than 1, then t is short compared to a 2/DA, the second term, the linear diffusion term, is predominant in the right side of Eq. (2). As is well known, if the

H. Ikeuchi, M. Kanakubo / Journal of Electroanalytical Chemistry 493 (2000) 93–99





#ci #2ci 1 #ci #2ci = Di + + #t #r 2 r #r #z 2

Fig. 2. Plot of Eq. (2). The thick solid line represents the profile of Eq. (2). The thin solid lines, 1, 2 and 3 represent the first, second and third terms of the equation, respectively.

i= A, B

95

(3)

in the cylindrical coordinates (r, z) of which the origin lies on the center of the electrode disk. This equation is solved for the electrolysis of the solution for which the bulk concentration of A is cA,0 and that of B is zero. The digital simulation was performed in two parts; they are for the first and the second potential steps, respectively. The initial and boundary conditions for both the potential steps are as follows. Here, ~ is the duration time of the first potential step (see Fig. 1). For the first potential step, 05t5~, cA(r, z, 0)=cA,0 cB(r, z, 0)= 0 cA(r, 0, t)= 0 for 05 r5 a For the second potential step, t] ~. cA(r, z, ~)=cA,~ (r, z) cB(r, z, ~)= cB,~ (r, z) The concentrations, cA,~ (r, z) and cB,~ (r, z), are given as the results of the calculation of the first potential step at time ~. cB(r, 0, t)= 0 for 05 r5 a

Fig. 3. Discretization of the space adjacent to the electrode, for digital simulation.

diffusion is completely linear, the diffusion current at the second potential step is solely controlled by DA, and not by D of B, DB [22,23]. Therefore if the pulse width of the first potential step, ~, is so small that I1 can be regarded as almost a linear diffusion current, I2 is controlled mainly by DA, and a little by DB. If ~ is long enough, I2 is controlled by not only DA but also DB. If the equation that represents the I2 – t curve is known, we can then extract DB from it. We applied the digital simulation on DPS chronoamperometry at a disk electrode for deriving a theoretical relationship between the currents and electrolysis time functions. The digital simulation was performed by the simple but efficient hopscotch algorithm that was originally proposed by Gouray [24] and has been applied to chronoamperometry at a disk electrode and some other electrochemical problems by Shoup and Szabo [19,25,26].

2.1. Procedure of the digital simulation The partial differential equation is formulated as:

For both the potential steps,

    #cA #z #ci #z

=−

  #cB #z

z=0

=0

for 05r5 a z=0

i= A, B for r\ a

z=0

cA(r“ , z “ , t)= cA,0 cB(r“ , z“ , t)=0 The quantity we observe is the diffusion current at each potential step; that is given by

&  a

Ij = 2ynFD

0

#ci #z

r dr

i =A, B for j= 1, 2

z=0

The technique of the digital simulation is conventional and is the same as that used by Shoup and Szabo [19]. The space adjacent to the disk electrode surface is divided into concentric rings and cylinders as shown in Fig. 3, where L is the number of rings and a cylinder on the disk, and is related to a by: L=

a +0.5 Dr

(4)

H. Ikeuchi, M. Kanakubo / Journal of Electroanalytical Chemistry 493 (2000) 93–99

96

The electrolysis time, t, is also divided by finite time Dt. t is represented by: (5)

t = kDt

where k is the number of iterations. We chose the iteration numbers of the first potential step, k1, so that the first term of Eq. (2) becomes comparable with the second term. Thus the duration time of the first potential step, ~ is given by: ~ = k1Dt

(6)

From Eqs. (4) and (6) with the aid of the equation: lA =

DADt (Dr)2

(7)

we can obtain Table 1 Sets of parameters for the digital simulation

  a2 DA~

1/2

=

(L− 0.5) (lAk1)1/2

(8)

from which we can calculate the variable (a 2/DA~)1/2. We chose three parameter sets, as shown in Table 1, and performed the digital simulation for eight different ratios of DB:DA from 0.4 to 2.0 for each parameter set. The values of the variable, (a 2/DA~)1/2, are also listed in Table 1, and are indicated in Fig. 2. The output for the first potential step is the current function at each iteration, I1(k), and the concentration function of each ring and cylinder at the end of the first potential step, c(i, j, k1). One can calculate the current function at the second potential step, I2(k), by using the values of the concentration functions c(i, j, k1) as the initial condition. In order to relate the results of the digital simulation to the experimental DPS chronoamperogram, we introduce the ratios, x and y:

Parameter set no.

L

k1

lA

(a 2/DA~)1/2

x=

1 2 3

50 75 100

4000 4000 5000

0.2 0.2 0.2

1.75 2.48 3.15

y=

   t− ~ t

1/2

=

k− k1 k



1/2

(9)

I2(t) I2(k) = I1(t−~) I1(k−k1)

(10)

where, t\~ and k\ k1. Programs for digital simulation were compiled by Lahey Fortran, and the calculation was performed by personal computer, AV1/55CD-95 (Sharp Co., CPU: Pentium 75 MHz, 16 MB memory).

2.2. Results of the digital simulation

Fig. 4. The results of the digital simulation for parameter set number 1 represented by y versus x curves.

The results of the digital simulation for the DPS chronoamperogram cannot be unified into one equation, because I2 is a function not only of DB, t, cA.0 and a but also of DA and ~. Therefore, the results are represented by the group of curves for three different digital simulation parameter sets. For example the results for parameter set no. 1 are shown in Fig. 4. The curves are characterized by the ratios, DB/DA, and the wider they are separated, the smaller are the values of (a 2/DA~)1/2. We can determine DB/DA then DB when DA is known, by superimposing the experimental I2(t)/ I1(t−~) versus {(t − ~)/t}1/2 curves on the correspondTable 2 The values of ~, for the different digital simulation parameter sets when a =0.101 mm

Fig. 5. Relationship between y and DB/DA at different x values. Filled circles are the values of y on the curves at the definite x values read from the curves of Fig. 4(a), and the solid lines are fitted quadratic curves.

Species

1010DA/m2 s−1

Parameter set no.

~/s

[Fe(CN)6]4−

6.609

1 2 3

5.039 2.225 1.559

[Fe(CN)6]3−

7.744

1 2 3

4.301 1.899 1.331

H. Ikeuchi, M. Kanakubo / Journal of Electroanalytical Chemistry 493 (2000) 93–99 Table 3 Parameters of the quadratic Eq. (11) fitted to the y versus (DB/DA) curve for different values of x

97

a

Parameter set no.

x

p

q

s

|2

1

0.1 0.2 0.3 0.4

1.101(10) 0.858(6) 0.673(4) 0.518(3)

−0.378(19) −0.275(11) −0.214(8) −0.166(6)

0.0857(76) 0.0602(44) 0.0462(33) 0.0357(26)

0.0007 0.0002 0.0001 B0.0001

2

0.1 0.2 0.3 0.4

1.049(7) 0.845(3) 0.684(3) 0.543(2)

−0.265(13) −0.189(7) −0.149(5) −0.118(4)

0.0561(54) 0.0381(28) 0.0293(21) 0.0228(16)

0.0003 0.0001 B0.0001 B0.0001

3

0.1 0.2 0.3 0.4

1.022(6) 0.837(3) 0.686(2) 0.551(2)

−0.217(11) −0.157(6) −0.125(4) −0.100(3)

0.0440(43) 0.0303(24) 0.0236(18) 0.0185(14)

0.0002 0.0001 B0.0001 B0.0001

a

The standard error in the least significant digits is given in parentheses.

ing group of y versus x curves. For convenience, the values of y on the curves of different values of DB/DA at a definite value of x, for example 0.1, 0.2, 0.3 and 0.4 are plotted against DB/DA, and the plots are fitted by a quadratic curve. One example is shown in Fig. 5, and the values of the parameters of the quadratic equation (Eq. (11)) are given in Table 3 along with the variance of the fitting for the three cases of the digital simulation. y= p+q

    DB D +s B DA DA

2

(11)

3. Experimental Potassium hexacyanoferrate(III), potassium hexacyanoferrate(II) trihydrate (Tokyo Chemical Industry Co. Ltd., reagent grade, B99% up) and potassium chloride (Wako Pure Chemical Industries, reagent grade, B 99.5%) were used as purchased. A potentiostat, digital universal signal processing unit HECS 326 with head box 326-2 Fuso Electro Chemical System Co. was remodeled to an auto-rangechange system of 16 ranges from 32.25 nA to 1.024 mA. Test electrodes were platinum disk electrodes embedded in glass. Their radii were measured with a measuring optical microscope. Two electrodes of 0.101 and 0.241 mm radii were used in SPS chronoamperometry and the former one in DPS chronoamperometry. The electrode was polished with aluminum oxide of average grain size of 0.03 mm on a turntable for several minutes before every series of measurements. A reference electrode, Ag AgCl 3 M NaCl (BAS Co.), and platinum wire counter electrode were used. The procedure for the determination of DB is as follows. 1. Determine DA by the usual SPS chronoamperometry at the potential of the first step.

2. Choose a disk electrode of suitable size. The radius a is known. 3. Choose a parameter set of the digital simulation. The values of L, k1 and lA are known. 4. Calculate ~ from ~=

a 2lAk1 DA(L−0.5)2

(12)

5. Acquire the DPS chronoamperogram for which the ~ value is calculated in procedure (4), by using the disk electrode chosen in procedure (2). 6. Calculate the value of I2(t)/I1(t−~) at the time when the value of {(t − ~/t)}1/2 is for example 0.1, 0.2, 0.3 and so on. 7. Insert the value of I2(t)/I1(t−~) obtained in procedure (6) to y, and the corresponding parameter values (Table 3) to p, q and s in Eq. (11) and then calculate DB/DA. The value of DB can be obtained since DA is known by procedure (1). In procedure (1), we used the SPS technique that had been developed for the determination of reliable and precise diffusion coefficients [27,28]. In this technique, the product of the concentration of the diffusing species (c) and the charge number of the electrode reaction (n) is determined together with the diffusion coefficient from the chronoamperogram, and the value is used for the assessment of the reliability of the diffusion coefficient obtained. If the determined value (ncobs) agrees with the known value (nccalc), the diffusion coefficient obtained with it should be correct. The chronoamperograms were recorded over a time period of about 0.1 –10 s. The applied potentials are as follows: for the measurement of [Fe(CN)6]4 − , E0 =80 mV, E1 = 400, 450 or 500 mV, and for [Fe(CN)6]3 − , E0 = 450 or 500 mV, E1 = 50, 75, 100 or 125 mV. As we chose a platinum electrode of radius 0.101 mm for DPS chronoamperometry, ~ calculated by means of

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98

Table 4 Diffusion coefficients in 1 mol dm−3 KCl aqueous solution at 25°C determined by SPS chronoamperometry a Species

E1/mV

1010D/m2 s−1

ncobs/nccalc Average

[Fe(CN)6]4−

[Fe(CN)6]3−

a

400 450 500

1.004(5) 1.012(9) 1.007(7)

125 100 75 50

0.996(12) 0.997(8) 1.011(7) 1.016(11)

Reference Average

6.62(5) 6.61(10) 6.57(10)

1.006(4)

7.67(19) 7.68(12) 7.83(12) 7.82(18)

1.004(4)

6.61(4)

6.32(3) [1]

7.74(6)

7.84(2) [2] 7.63(2) [1]

The 95% confidence limit to the least significant digit is given in parentheses.

Eq. (12) for the parameter set nos. 1, 2 or 3 became the values shown in Table 2. The D values of [Fe(CN)6]4 − and [Fe(CN)6]3 − in 1 mol dm − 3 KCl aqueous solutions were determined with DPS chronoamperometry by using solutions that contained 1 mol m − 3 [Fe(CN)6]3 − and [Fe(CN)6]4 − , respectively. The applied potentials are as follows: for the measurement of: [Fe(CN)6]4 − , E0 = 500, E1 = 75, 100 or 125, E2 =500 mV; and for [Fe(CN)6]3 − , E0 = 80, E1 =450, E2 =80 mV.

tion of the diffusion coefficient of electrode reaction products works successfully. In this work, we used a disk electrode of 0.101 mm radius taking into account the precision of the size measurement and the capacity of our potentiostat. A smaller electrode would work more efficiently, because

4. Results and discussion The values of D determined by SPS chronoamperometry are shown in Table 4. The deviations of the ratio ncobs/nccalc and D by E1 were all within the limits of experimental error as shown in Table 4. Also, the radius of the electrode did not affect these values significantly. The ratios for both the species are close to 1, and thus the obtained D values are reliable. One example of the DPS chronoamperograms is shown in Fig. 6. The I2(t)/I1(t − ~) versus {(t −~)/t}1/2 curves calculated from the observed DPS chronoamperograms for [Fe(CN)6]3 − and [Fe(CN)6]4 − are superimposed on the theoretical y versus x curves in Fig. 7. The curve for [Fe(CN)6]4 − lies between the curves for which DB/DA values are 0.8 and 1.0, and that for [Fe(CN)6]3 − lies between those whose values are 1.0 and 1.25. We calculated DB/DA values by Eq. (11) from the values of I2(t)/I1(t−~) at the values of {(t −~)/t}1/2 of 0.2 and 0.3. The values of DB/DA were independent of the applied potentials as in the case of SPS chronoamperometry. The results are shown in Table 5. We calculated DBs by using DB/DA and the corresponding DA values listed in Table 4. The errors in DB shown in Table 5 were estimated from the errors of DB/DA and of DA. The diffusion coefficients determined by DPS chronoamperometry agree within 3% for [Fe(CN)6]4 − and within 2% for [Fe(CN)6]3 − with those determined by SPS chronoamperometry. We can conclude that the method that has been developed here for the determina-

Fig. 6. (a) An example of a DPS chronoamperogram of [Fe(CN)6]3 − in 1 mol dm − 3 KCl aqueous solution at a Pt-disk electrode of 0.101 mm radius with ~ of 4.302 s, at 25°C. (b) The residual current of (a).

Fig. 7. Fitting of the observed I2(t)/I1(t −~) versus {(t−~)/t}1/2 curves on the y versus x curves. Measured from a solution of [Fe(CN)6]3 − (open circle), and [Fe(CN)6]4 − (filled circle) at a Pt-disk electrode of 0.101 mm radius, ~ of 4.302 s, at 25°C.

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99

Table 5 Diffusion coefficients determined by DPS chronoamperometry by three digital simulation parameter sets a Species B

Parameter set no.

Number of runs

1010DB/m2 s−1

DB/DA

Average [Fe(CN)6]4+

[Fe(CN)6]3+

a

1 2 3

7 6 6

0.869(8) 0.889(6) 0.887(10)

0.882(5)

6.82(9)

1 2 3

9 9 9

1.15(2) 1.16(3) 1.15(4)

1.15(2)

7.61(15)

The 95% confidence limit in the least significant digit is given in parentheses.

a shorter ~ can make the first term of Eq. (1) smaller compared with the second term. For this purpose we have to overcome the problem of precise measurement of the electrode size and the roughness of its surface that must be small enough compared with the thickness of the diffusion layer formed in the time ~. A smaller electrode would make this technique usable for measurement of the diffusion coefficient of the electrode reaction products coupled with a homogeneous chemical reaction.

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