ISSN 1023-1935, Russian Journal of Electrochemistry, 2018, Vol. 54, No. 1, pp. 62–69. © Pleiades Publishing, Ltd., 2018. Original Russian Text © A.E. Antipov, M.A. Vorotyntsev, 2018, published in Elektrokhimiya, 2018, Vol. 54, No. 1, pp. 73–81.
Bromate Anion Reduction at Rotating Disk Electrode in Steady State under Excess of Protons: Numerical Solution of the Convective Diffusion Equations at Equal Diffusion Coefficients of Components A. E. Antipova, b, * and M. A. Vorotyntseva, b, c, d, ** a
Mendeleev University of Chemical Technology, Miusskaya pl. 9, Moscow, 125047 Russia b Lomonosov State University, Leninskie Gory 1, Moscow, 119992 Russia c Institute of Problems of Chemical Physics, Russian Academy of Sciences, pr. akad. Semenova 1, Chernogolovka, Moscow oblast, 142432 Russia dInstitut de la Chimie Moleculaire, Université de Bourgogne, Dijon, France *e-mail:
[email protected] **e-mail:
[email protected] Received October 14, 2016; in final form, January 31, 2017
Abstract—Process of bromate anion reduction at rotating disk electrode in steady state occurring owing to catalytic cycle is analyzed theoretically. The cycle consists of bromine/bromide reversible redox-couple and comproportionation irreversible reaction. Because of the cycle autocatalytic character (EC''-mechanism: Electrochim. Acta, 2015, 173, 779), the passing current can be enormously large at bromate high bulk concentration (up to the bromate limited diffusion current to electrode surface) even when the bromine concentration in the bulk solution is negligibly small. Unlike the previous theoretical studies of the problem (Electrochim. Acta, 2015, 173, 779; Doklady Chemistry, 2016, 468, 141; Russ. J. Electrochem., 2016, 52, No. 10, 925), in this work the component concentration distributions during the process at a prescribed value of the passing current are calculated on the basis of the convective diffusion equations for bromine and the bromate and bromide anions for the first time. Under the assumption that the component diffusion coefficients are equal, exact interrelations between these concentration profiles are derived, which allows reducing the problem to the solving of nonlinear equation of second order for the bromide concentration with boundary conditions at the electrode surface and in the solution bulk. Thus, obtained concentration profiles of all components for a corresponding set of current densities are used to calculate of steady-state polarization curves, as well, as the maximal current dependence on the rotating disk electrode velocity. Keywords: bromate-anion, bromine/bromide redox-couple, comproportionation, redox-mediator autocatalysis, convective diffusion, kinetic layer DOI: 10.1134/S1023193518010020
INTRODUCTION
occurs, consisting of the bromine/bromide reversible redox-couple:
Recently it was suggested [1] using bromate-anion BrO 3− as a high-energetic multielectron oxidant in flow-through rechargeable redox-cells. The key problem here remains the reactants’ poor activity at positive (with respect to NHE) potentials even at catalytically active electrodes. The solution was found on the basis of the theory suggested in work [2], where it was shown that the situation can be changed drastically when the solution contains, in addition to bromate and proton high concentrations, also bromine even in very low concentration. In particular, a catalytic cycle
(1) Br2 + 2 e ⇄ 2Br – , and irreversible (in highly acidic solutions) comproportionation reaction: (2) BrO 3− + 5Br– + 6H+ → 3Br2 + 3H2O. Because of autocatalytic character of the cycle (EC''-mechanism [2]), even in negligibly small bromine in the solution bulk, yet bromate and proton large bulk concentrations, the passing current may be enormously high, being limited by the bromate limited 62
BROMATE ANION REDUCTION AT ROTATING DISK ELECTRODE
diffusion current to electrode surface where the bromate is rapidly consumed in the comproportionation fast reaction [2]. Analysis for the system was carried out [2] on the basis of approximate analytical calculations. The same set of equations was later analyzed by using numerical integration of this diffusion–kinetic equations [3], which confirmed both principal qualitative conclusions of the analytical approach [2] and its predictions for the dependence of the maximal current on the rotating disk electrode (RDE) angular velocity. Theoretical analysis of interrelations for the system was also carried out [4] for the bromate and proton concentration reverse ratio in the solution bulk, that is, at an excess of bromate, when the current under the conditions of the process very strong acceleration due to the accumulation of the mediator redox-couple (1) is limited by the proton diffusion transfer from the solution bulk to the region of their intense consumption (“the kinetic layer”) in the comproportionation reaction (2). All the studies [2–4] were carried out in the frames of simplified treatment of the transfer equations, which corresponds to the model of the Nernst stagnant layer [5–9]; in other words, the effect of the solution components’ convective transfer, due to the solution hydrodynamic motion near the RDE surface, was replaced by a layer with thickness of zd, through which the components’ transfer occurs solely as a diffusion transfer, whereas at the layer boundary (at z = zd) the components’ concentrations were the same as those in the solution bulk. The thickness of this Nernst diffusion layer zd was assumed to depend on the RDE velocity f by the Levich formula [8], with the using of the bromate-anion diffusion coefficient DA:
z d = 1.61D A ν Ω 13 16
−1 2
, Ω = ( 2π 60 ) f [rpm],
(3)
where ν is the solution kinematic viscosity. Unlike the previous theoretical studies of the process [2–4], in this work the convective diffusion equations for the system’s component are analyzed with the accounting for both the electrochemical process (1) at the electrode surface and the comproportionation reaction (2) for the first time. Like in the works [2, 3], the proton concentration variation across the diffusion layer is neglected because of their rather high concentration in solution as compared with that of bromate. To lower the system’s independent parameter number, we assume all diffusion coefficients (of BrO 3− , Br −, and Br2) being equal to each other. EQUATIONS FOR THE REACTANTS’ CONCENTRATIONS AND THE BOUNDARY CONDITIONS The principal scheme of the reactions (1), (2) is described at length in our preceding paper [3], thereRUSSIAN JOURNAL OF ELECTROCHEMISTRY
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fore, it is presented here but briefly. The solution bulk contains the principal oxidant (anion BrO 3− in concentration A°), an excess of a strong acid, as well, as the bromate decomposition product (molecular bromine Br2 in concentration С°), the latter usually at a very small amount. Neither bromate-anion, nor protons react at the electrode in the studied potential range. By contrast, bromine is reduced reversibly at the electrode according to reaction (1). Its reduction product (bromide-anion) in acid medium is involved to the irreversible comproportionation reaction (2), whose product, bromine, reacts at the electrode according to reaction (1). As the cycle (1), (2) is repeated, a redox-mediator couple Br2/Br– is accumulated near the electrode as a result of the excessive bromine atom from the bromate-anion in the product of reaction (2) because of which the reduction of the principal reactant, bromate, occurs. Because the comproportionation reaction (2) occurs as a series of consecutive stages, its rate is proportional to the product of first degrees of the bromate- and bromide-anions’ concentrations [10–13]:
V ≡ –d ⎡⎣BrO 3− ⎤⎦ /dt = k ⎡⎣BrO 3− ⎤⎦ ⎡⎣Br − ⎤⎦ ,
(4)
where the rate constant k is рН-dependent. The rates of other substances consumption or formation are given as products of V and the corresponding stoichiometric coefficients in the reaction (2) with signs “+” or “–”. When the concentrations’ local drops exist in the solution (depending on the spatial coordinate and, possibly, on time), formula (4) relates to local values of all quantities, including k (when рН drops in the solution occur) and V. In the further analysis the concentrations of reactants: BrO 3− , Br −, and Br2 for brevity will be denoted as А, В, and С. Under the conditions of equal access over the entire electrode surface surface (i.e., in the case of RDE) and the process steady state the concentrations depend only on single spatial variable z, normal to the electrode surface. Because of the excess of acid, protons and the acid residue take on a role of a supporting electrolyte; as a result the migration contributions of the rest of components to the fluxes are reduced significantly and their transfer occurs solely by means of diffusion, whereas the proton concentration (hence, the rate constant k) practically is constant over the entire layer bulk. Concentrations of the rest of components vary not only inside the diffusion layer, which holds for the Nernst model [3], but also beyond its boundary, due to combination of the diffusion and convective transfer mechanisms. Divergence of the diffusion–convective flux of the solution each component is equal to the rate of its consumption or formation in the conproportionation reaction (2) subject to its stoichiometric coefficient [8]. With due account for the continuity equation for No. 1
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incompressible liquid, we write the resulting transfer equations for RDE as [8]: DAd2A/dz2 = –αz2dA/dz + V, d2B/dz2
(5a)
+ 5V,
(5b)
DCd2C/dz2 = –αz2dC/dz – 3V,
(5c)
DB
=
–αz2dB/dz
where factor α = 0.51 is determined by the solution viscosity ν and the RDE rotation velocity Ω; the reaction rate V at a constant рН value (due to the proton excess as compared with all bromine compounds) is represented by formula (4), where the coefficient k does not depend on the coordinate z: V(z) = kA(z)B(z). (5d) Unlike the Nernst model analyzed in the preceding paper [3], the concentrations of bromate А° and bromine С° are set not at the diffusion layer outer boundary zd, but at a well remote point (for z zd), where all concentrations are assumed being constant due to the solution convective agitation; the bromide anion is absent here because of reaction (2) in the presence of the bromate excess: (6) A(z) = A°, B(z) = 0, C(z) = C° at z zd. (ν/Ω)1/2
The conservation laws at the electrode/solution interface give three more boundary conditions for the reactants’ concentrations: dA/dz = 0, DBdB/dz = –j/F, DCdC/dz = j/2F at z = 0,(7) where j is the cathodic current density (which is a given parameter under the galvanostatic regime of the process), F is the Faraday constant. The solution of the set (5) of three equations of the second-order with 6 boundary conditions determines all concentration profiles as functions of the system parameters: A°, C°, DA, DB, DC, k, and j [2]. DIMENSIONLESS VARIABLES The number of governing parameters can be diminished significanty by the passing to dimensionless variables and parameters:
y = z z d , x dk ≡ z d z k , z k ≡ [DB 5kA°] , z d is given by formula (3), 12
a ( y ) = A ( z ) A° , b ( y ) = B ( z ) DBF jz k , c ( y ) = C ( z ) DC F jz k ,
(8a)
≡ 5FD A A° z d ,
jClim
≡ 2FDC C ° z d .
d2a/dy2 = –α0 y2 da/dy + Jxdk a(y) b(y),
(10a)
2 a(y) b(y), d2b/dy2 = –α0 y2 db/dy + x dk
(10b)
2 d2c/dy2 = –α0 y2 dc/dy – 0.6 x dk a(y) b(y),
(10c)
where α0 = 2.13, b(y) = 0 at y 1,
(11b)
a(y) = 1, c(y) = c° ≡ 0.5xdkJCA/J at y 1,
(11ac)
db/dy = –xdk at y = 0,
(12b)
da/dy= 0, dc/dy = 0.5 xdk at y = 0.
(12ac)
Dimensionless concentration profiles a(y), b(y), and с(y) depend on three dimensionless parameters: xdk (the kinetic layer and diffusion layer thickness ratio), JCA (the ratio of the limiting diffusion currents of ions С and А), and J (the ratio of the passing current to the limiting diffusion current of ion А, that is, bromate). By combining equations (10a) and (10b), we obtained homogeneous differential equation for a combination of a(y) and b(y), by the excluding of nonlinear member: d 2 {a ( y ) − J x dk b ( y )} d y 2
(13)
+ a0 y 2d {a ( y ) − J x dk b ( y )} d y = 0. By separating the variables we obtained:
(8b)
lim J = j j Alim j Alim (C ) , J CA ≡ jC (C ) = 2DC C ° 5D A A° ,
j Alim (C )
Notice that the definition of с(y) in formula (8b) differs from that of с(х) in formula (9а) in the preceding papers [2, 3] not only because of the difference in the normalization of the dimensionless coordinates y and x, but also because the member with C° is absent in the definition. For the sake of simplification of the subsequent formulas, let us assume the diffusion coefficients of all system’s parameters being equal to each other: DA = DB = DC ≡ D (the distinction between them will not alter the below-given conclusions qualitatively). Then equations (5) and the boundary conditions (6), (7) can be written as:
(9)
Here J is the dimensionless current density, zk is the lim kinetic layer thickness, j Alim are the formally (C ) and jC defined limiting diffusion currents for А or С ions, respectively (for the case when they react at the electrode) in compliance with their bulk concentrations.
d{a(y) – J/xdk b(y)}/dy = C1exp{–α0 y3/3}.
(14)
Similarly, from equations (10b) and (10c) for b(y) and c(y) we obtained homogeneous differential equation for the combination of b(y) and c(y): d2{c(y) + 0.6b(y)}/dy2 + α0 y2 d{c(y) + b(y)}/dy = 0.(15) Similarly, by separating the variables we obtained d{c(y) + 0.6b(y)}/dy = C2exp{–α0 y3/3},
RUSSIAN JOURNAL OF ELECTROCHEMISTRY
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BROMATE ANION REDUCTION AT ROTATING DISK ELECTRODE
Boundary conditions for equations (13) and (15) can be obtained from equations (11), (12): b(y) = 0 at y 1, (17b) a(y) – J/xdkb(y) = 1, c(y) + 0.6b(y) = c° at y 1,(17ac) db/dy = –xdk at y = 0,
(18b)
d {a ( y ) − J x dk b ( y )} d y = J , (18ac) d {c ( y ) + 0.6b ( y )} d y = − 0.1x dk at y = 0. Constants C1 and C2 can be found from the boundary conditions (17), (18): C1 = J and C2 = –0.1xdk; correspondingly: d{a(y) – J/xdk b(y)}/dy = Jexp{–α0 y3/3},
(19)
3
d{c(y) + 0.6b(y)}/dy = –0.1xdkexp{–α0 y /3}. (20) By integrating equations (19) and (20) we obtained expressions for a(y) or с(y) through b(y): (21) a(y) = 1 – JI(y) + b(y)J/xdk,
c ( y ) = c° + 0.1x dk I ( y ) − 0.6b ( y ) , ∞
⎛ α y3 ⎞ I ( y ) = exp ⎜ − 0 ⎟ d y. ⎝ 3 ⎠
∫
(22)
y
Thus, to find all dimensionless concentration profiles it is reasonable integrating equation (10b), upon the substituting of expression (21) for a(y) therein, with the boundary conditions (11b) and (12b). Thereafter, the profile с(y) can be found from expression (22). In what follows we substitute, to facilitate the interpreting of the obtained results, the corresponding dimensional concentration profiles A(y), B(y), C(y), assigned to the bromate concentration in the solution bulk A0, for the dimensionless concentrations a(y), b(y), c(y). In view of the definitions (8), the quantities are unequivocally expressed through a(y), b(y), c(y) and dimensionless parameters xdk and J, with regard for the equality of the diffusion coefficients of the components: (y)/A0 = a(y), (23) B(y)/A0 = 5b(y)/ x dk J,
(24)
C(y)/A0 = 5c(y)/ x dk J.
(25)
RESULTS AND DISCUSSION By using numerical methods (described in Appendix), in this work dimensionless concentration profiles of all components were calculated as function of the system’s characteristic dimensionless parameters: xdk, J, and JCA (Fig. 1). Polarization curves were also obtained (Fig. 2); they give the value of dimensionless maximal current Jmax that can be passed through the system at given values of xdk and JCA. Parameter xdk that is the ratio of the diffusion layer and kinetic layer thicknesses was varied from 1 to 10; the dimensionless RUSSIAN JOURNAL OF ELECTROCHEMISTRY
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current density, from the “weak current” mode (J = JCA) to the maximal current (J = Jmax). These dependences were derived for small values of the parameter JCA (JCA = 10–3), because, bearing applications in mind, of interest is the case when the bromine bulk concentration (C0) is small as compared with that of bromate (A0). These calculations allow comparing the concentration profiles for the system’s each component, calculated numerically with regard for the convective transfer, with the numerical calculations carried out in the frames of the Nernst immovable layer model ignoring the convective transfer [3]. We recall that in accord with the model the system’s component concentrations vary only inside the diffusion layer (at z < zd, that is, y < 1), whereas beyond the limits of the diffusion layer (at z > zd, that is, y > 1) they remain constant, equal to the corresponding bulk concentrations. In Fig. 1а we demonstrate the comparison for the bromide concentration profiles B(y) for the case when the kinetic layer thickness zk is comparable with that of the diffusion layer zd (xdk ≡ zd/zk = 1). Near the electrode surface (at y 1) the curves are in satisfactory agreements with each other. The concordance between the results obtained for the two models is by no means unexpected because the convective member –α0 y2 db/dy in equation (10b) cannot make any significant contribution to the solution at y 1 because of the quadratic dependence from y; in other words, in both cases the concentration profile B(y) is determined only by the substance diffusion transfer. The convective transfer manifests itself only well away from the electrode surface in the diffusion layer outer part (y ~ 1), which may result in a divergence between the numerical results in this region. Indeed, such a divergence in this region was obtained at the diffusion and kinetic layer comparable thicknesses, xdk ~ 1 (Fig. 1а, curves 1–3(B) and 1'–3'(B)). In this case the profile B(y) within the frame of the Nernst model varies nearly in linear fashion, the slope being determined by the current density J in the system, up to the point y = 1, where B(y) becomes equal to the concentration of the component B in the solution bulk (B0), that is, it zeroes (Fig. 1а, curves 1 '–3 '(B)). This “purely diffusion” kind of the dependence B(y) is due to the weakness of the effect of the comproportionation reaction (2) because of relatively thin diffusion layer, so that Br– ions produced at the electrode cross the layer without entering the reaction, that is, they react beyond the diffusion layer. The accounting for the convection results in nonlinear shape of the concentration profile B(y) with soft passing to the zero value at y → ∞ (Fig. 1а, curves 1–3(B)). When the diffusion layer thickness increased with respect to that of the kinetic layer (the xdk value grows) the above-described difference at y ~ 1 vanished, so both models give nearly the same results. At xdk @ 1 the profile B(y) both without regard for the convective No. 1
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(b) 1 ', 2 ', 3 '(A)
(a) 6
3
1, 2, 3(A)
1 ', 2 ', 3 '(C) 1, 2, 3(C)
5 2, 2 '(B) 4 1, 1 '(B)
0.9999 2
3
A(y)/A0
103C(y)/A0
103B(y)/A0
3, 3 '(B)
1.0000
0.9998
1
2
0.9997 1 2, 2 '(C) 3, 3 '(C) 0
0.5
0.9996
0 1.5
1.0
0
0.5
y = z/zd (c) 5
1.000
12 3, 3 '(B) 4
1 ', 2 ', 3 '(C)
2
4 2, 2 '(B) 2
0.997 A(y)/A0
3
0.998 103C(y)/A0
103B(y)/A0
1, 2, 3(C)
8 2, 2 '(B) 6 1, 1 '(B)
0.996 2, 2 '(A) 0.995 0.994
1 0.993
1, 1 '(B) 0 3, 3 '(C) 0.5
1.0 y = z/zd (e)
0
0.25
3, 3 '(B) 3, 3 '(C)
1, 2, 3(A)
0.9
0.10
0.2 2, 2 '(B)
1.0 y = z/zd
2, 2 '(A) 0.6
0.05 0.4
1, 2, 3(C)
3, 3 '(A) 0
0.5
0.7
0.5
1 ', 2 ', 3 '(C)
1, 1 '(B, C)
A(y)/A
0.3
C(y)/A
0.15
2, 2 '(C)
0
0.8 0
B(y)/A0
1.0 1.5 y = z/zd (f) 1 ', 2 ', 3 '(A)
1, 1 '(A)
0.5
0
0.5
1.0
0.20
0.1
3, 3 '(A)
0.992
0 1.5
0.6
0.4
1, 2, 3(A)
0.999 1, 1 '(A)
10
0.7
1.0 1.5 y = z/zd (d) 1 ', 2 ', 3 '(A)
1.5
0.3 0
0.5
1.0 y = z/zd
1.5
Fig. 1. Concentration profiles A(y) (b, d, f), B(y) and C(y) (а, c, e) divided by the bromate concentration in solution bulk (A0). Solid curves 1–3 are calculated from equations of convective diffusion; dashed curves 1'–3', in terms of the Nernst layer model [8]. The symbols А, В or С after curve numbers indicate to which one of the concentrations the curve relates. Parameter JCA equals 10–3 (low bromine concentration in the solution bulk). Parameter xdk equals 1 (а, b), 4 (c, d) or 10 (e, f), which corresponds to the maximal current Jmax values of 1.4 × 10–3, 1.0 × 10–2 or 0.8, respectively. Dimensionless current J equals JCA (curves 1, 1'), 0.5 (Jmax – JCA) (curves 2, 2') or Jmax (curves 3, 3'). RUSSIAN JOURNAL OF ELECTROCHEMISTRY
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Δ A ≡ { ANL ( 0) − Aconv ( 0)} = Δ BJ x dk , Δ B ≡ {BNL ( 0) − Bconv ( 0)} .
(26)
In other words, their difference is caused by the difference by ΔB in the concentration profiles at the elecRUSSIAN JOURNAL OF ELECTROCHEMISTRY
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1 (Jmax)3
3, 3 '
10–1 10–2
(Jmax)2
J
transfer (the Nernst layer model) and with the allowance thereof drops approximately at an exponential rate down to its zero value deep inside the diffusion layer, so that the convective member in equation (10b) is practically zero. This is the reason for the practical coincidence of the corresponding curves 1–3(B) and 1'–3'(B) for xdk = 4 (Fig. 1c) and xdk = 10 (Fig. 1e). We now proceed to the comparing of the bromine concentration profiles C(y) (Figs. 1а, 1c, 1e). As for B(y), because of smallness of the convective member in equation (11c) at small y 1 near the electrode surface the results of numerical calculations both with the allowance for the convective transfer and without regard thereof (the Nernst layer model) well agree with each other. When moving off the electrode surface and approaching the diffusion layer outer surface (y ~ 1) the contribution from the convective transfer member increased, which results in change of the С(y) profile and divergence of the results of numerical calculations for all values of the xdk parameter in this region (see curves 1–3(C) and 1'–3'(C) in Figs. 1а, 1c, and 1e). Recent conclusions derived for B(y) on the convergence of results for the case of thin kinetic layer (xdk @ 1) are inapplicable here because the condition of smallness of dC(y)/dy in a region well remote from the electrode surface is not fulfilled, whereas the bromine concentration in the solution bulk still is finite (C0 > 0), even if very small. Finally, we compare the bromate concentration profiles а A(y) (Figs. 1b, 1d, and 1f). In the case when the kinetic and diffusion layer thicknesses are comparable, xdk = 1 (Fig. 1b), even the density of the maximal current that can pass under these conditions corresponds to the “weak current mode”: J 1, when the bromate concentration A(y) does hardly change: A(y) ≈ 1 for all values of the coordinate y. When the diffusion layer thickness increased with respect to that of kinetic layer, that is, with the increasing of the parameter xdk (for example, xdk = 4, Fig. 1d, or xdk = 10, Fig. 1f), the currents emerging in the system can be rather large (and comparable with the limiting diffusion current of the bromate j Alim ). As a result, because of the occurring of intense reaction (2) the bromate consumption inside the thin kinetic layer increased drastically, that is, the value А(0) drops down with the increasing of the parameter xdk during the passing of the maximal current Jmax. At that, because of smallness of convective member in equation (10a) we may give an estimate for y = 0, that is, near the electrode surface: results of numerical calculations both in respect to the convective transfer (the profile Aconv(y)) and not taking into account (the common Nernst layer model, the profile ANL(y)) are in good agreement with each other, differing by a constant value of
67
2, 2 '
(Jmax)1 10–3
1, 1 '
10–4 –5
0 5 10 2F(E – E° – ΔE)/(2.3RT)
Fig. 2. Polarization curves for different values of the thickness ratio for diffusion and kinetic layers: xdk equals 1 (curves 1, 1'), 4 (curves 2, 2') or 10 (curves 3, 3'). Solid curves 1, 2 and 3 are calculated from equations of convective diffusion; dashed curves 1', 2' and 3', in terms of the Nernst layer model [8].
trode surface B(0). The latter value does not exceed 2%. Because of different scales of the ordinate axis, the quantity ΔA is represented at best in Fig. 1b, where the scale of the ordinate axis is the smallest. When moving off the electrode surface and approaching the diffusion layer outer surface (y ~ 1) the contribute from the convective-transfer member increased, which leads to a change in the shape of the profile A(y) and divergence of results of numerical calculations for the two models in this region over entire range of xdk values(see curves 1–3(A) and 1'–3'(A) in Figs. 1b, 1d, 1f). Recent conclusions derived for B(y) on the closeness of results for the case of thin kinetic layer (xdk @ 1) are inapplicable here by the same reasoning as for C(y). In Fig. 2 we show polarization curves for the two models at different values of the parameter xdk: xdk = 1 (curves 1, 1'), 2 (curves 2, 2') or 10 (curves 3, 3'). Results of numerical calculations demonstrate excellent coincidence of the corresponding curves for all values of the parameter xdk over wide range of currents emerging in the system, both in the “weak” mode currents (J ! 1) and at current densities comparable with the bromate limiting diffusion current j Alim (C ) .
CONCLUSIONS Basing on the above-given comparing of results of numerical calculations both with the allowance for the convective transfer of substance and without explicit regard thereof (the Nernst layer model [8]) at coincident values of the diffusion coefficients for all solution components, we may conclude on the effect of convection in the system on regularities of changes concentrations in the concentrations of the system’s principal components inside diffusion layer near the electrode surface. The effect of convection on the No. 1
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components’ concentration profiles is well pronounced at a distance from electrode surface, by order of magnitude, of diffusion layer thickness, whereas near the electrode surface the effect becomes negligibly small under properly chosed parameter zd in the nernst layer model, in particular, when calculated by the Levich formula (3). Unexpected prediction [2–4, 14] of analytical theory on the possible achieving of enormously large current densities in the solution bulk at BrO 3− high concentration in the presence of trace amount of molecular bromine Br2 (up to the bromate limiting diffusion current j Alim (C ) ) still holds provided consistent accounting for the substance convective transfer in the system is carried out. Numerical algorithm developed in this work allows its using hereafter in the analysis of more complicated situations in other types of systems. APPENDIX Formulation of the Problem The problem (10b) for any function a(y) and given set of parameters xdk and j can be written as a set of differential equations of first order ⎧d b ' ( y ) = −α y 2b ' ( y ) + x 2 a ( y ) b ( y ) , 0 dk ⎪⎪ d y (A.1) ⎨ ( ) ⎪d b y = b ' ( y ) , ⎪⎩ d y with two boundary conditions (A.2) b ' ( y ) y =0 = − x dk , and (A.3) b ( y ) y →∞ = 0. The discussed problem relates to “boundary value problems” because the boundary conditions (А.2) and (А.3) are set in two different points of segment: y = 0 and y → ∞. We note that the functions a(y) and b(y) represent the solution components’ concentrations, so that addition of their nonnegativity constraint is required: a(y) ≤ 0, b(y) ≥ 0 for all values of y(0 ≤ y ≤ ∞). (A.4) Now we pass from the “boundary value problem” (A.1–A.3) to the next Cauchy problem:
⎧d u ' ( y ) = −α y 2u ' ( y ) + x 2 α ( y ) u ( y ) , 0 dk ⎪⎪ d y (A.1') ⎨ ( ) ⎪d u y = u ' ( y ) ⎪⎩ d y with two boundary conditions on the same extremity of interval: u ' ( y ) y →∞ = − 1
(A.2')
u ( y ) y →∞ = 0.
(A.3')
and
Then it can be shown by the substituting of expression in formula (А.5) into equations (А.1)–(А.3) that for any type of the function a(y) functions b ( y ) and u ( y ) are interrelated as follows:
b ( y ) ≡ − x dku ( y ) u ' ( 0) .
(A.5)
For the function u ( y ) the Cauchy problem occurs because both boundary conditions (А.2', А.3') are set in the same point. On solving the problem we can unambiguously determine the function b ( y ) and its derivative b ' ( y ) from the known function u ( y ) and its derivative u ' ( y ) . In what follows we shall write for brevity that we solve the problem directly for b ( y ), without stipulating the fact that actually we always find the function b ( y ) through u ( y ). For the numerical calculations the condition y → ∞ should be defined explicitly. It was shown empirically that in the numerical procedure we used the function b(y) can be thought of as zeroed at y ≥ 2 , which allows numerically integrating equations (A.1') only over the interval [0; 2], by carrying the boundary conditions (А.2') and (A.3') to the point у = 2. Actually, the function a(y) is not known; it should be found by the simultaneous solution of the set of equations (10a), (10b). At that, the set of equations is 2 nonlinear because of the member x dk a ( y ) b ( y ) . Generally, the problem can have both one solution or a few (even infinitely many) solutions and no solutions at all. To solve the problem, we shall look for the profile b(y) and its derivative b'(y) by using iterative technique in combination with the explicit Runge–Kutt method of fourth-order accuracy for the procedure of numerical integration (see the following section). Iterative Technique of Calculating the Concentration Profiles and the Maximal Current Density Jmax Now we discuss the iterative procedure we used. It must be started from some preset initial profile b0(y). It is but natural using the analytical expression for concentration profile of component b, calculated analytically [2] for the Nernst model, as this initial profile b0(y). According to work [2], the profile shape depends on both the current j passing in the system and the parameter xdk. When the current density in the system is small ( j 1) , the shape of the initial profile b0(y) for any value of xdk is: b0 ( y ) = sinh ( x dk (1 − y )) cosh ( x dk ) .
(A.6)
For a pretty thin kinetic layer ( x dk 1) the expression (А.5) can be simplified: (A.7) b0 ( y ) = exp ( − yx dk ) , The current passing in the system may be rather high only when the reaction (2) rate is sufficiently large,
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BROMATE ANION REDUCTION AT ROTATING DISK ELECTRODE
that is, when the condition of thin kinetic layer holds ( x dk 1). Here, according to [2], the initial profile shape b0(y) for the value xdk is as follows:
yx (A.8) b0 ( y ) = x k j exp ⎛⎜ − dk ⎞⎟ , x k j > 1, ⎝ x kj ⎠ where xkj by definition is (A.9) b0 ( 0) = x k j . To find the value xkj, one should solve the transcendental equation derived in [2], which links the current density J to the parameter xkj: −1
x ⎞ ⎛ (A.10) j ≈ (1 − x k−j2 ) ⎜1 − k j ⎟ . x dk ⎠ ⎝ Thus, for any given set of parameters xdk and j, using formulas (А.6), (А.7) or (А.8)–(А.10), the initial approximation b0(y) can be found for the calculating of b(y) by the iteration method. The initial approximation corresponding to that of b0(y) can be found by using expression (21) connecting the profiles b(y) and a(y). By substituting a0(y) to the set of equations (А.1) with the boundary conditions (А.2) and (А.3) we do away with the nonlinear character of the first equation in the set (A.1), which allows integrating numerically this set of equations by using the explicit Runge–Kutt method of fourth-order accuracy. the procedure of numerical integration of a set of differential equations is described at length in our preceding paper [14]. Eventually we obtained the functions b1 ( y ) and b1' ( y ) corresponding to the initial approximation a0(y), for which the procedure of search for the i-th approximation bi(y) for the profile b(y) through the corresponding (i – 1)th approximation of ai – 1(y)for the profile a(y) can be repeated the right amount of times. The condition of completing of the suggested iteration procedure is the smallness of changes in the profile bm ( y ) found in the m-th iteration of the described procedure as compared with the preceding profile bm −1 ( y ), which corresponds to the preceding (m – 1)th iteration: bm ( y ) − bm − 1 ( y ) ≤ ε b for any y ∈ [0;2], where εb is determined in terms of required accuracy of the calculations, e.g., ε b ~ 10 −9 . Finally, upon the calculating of the final profile b(y) for the given set of parameters xdk and j by the iteration procedure, one can obtain expressions for the corresponding profiles a(y) and c(y), in particular, the value of c(0), by using formulas (21), (22). It is known from the analytical examination of the problem [2] that the current density in the system reached its maximal value jmax when the dimensionless concentration at the electrode surface c(0) goes to zero. Thus, the value jmax for a fixed value of xdk can be calculated from the condition of the zeroing of c(0). The corresponding procedure for jmax calculation is described at length in our preceding paper [14]. RUSSIAN JOURNAL OF ELECTROCHEMISTRY
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ACKNOWLEDGMENTS This work was financially supported by the Counsil of grants of the President RF (project no. 14.W01.16.6741-MK). REFERENCES 1. Tolmachev, Y.V., Piatkivskyi, A., Ryzhov, V.V., Konev, D.V., and Vorotyntsev, M.A., Energy cycle based on a high specific energy aqueous flow battery and its potential use for fully electric vehicles and for direct solar-to-chemical energy conversion, J. Solid State Electrochem., 2015, vol. 19, p. 2711. 2. Vorotyntsev, M.A., Konev, D.V., and Tolmachev, Y.V., Electroreduction of halogen oxoanions via autocatalytic redox mediation by halide anions: novel EC" mechanism theory for stationary 1D regime, Electrochim. Acta, 2015, vol. 173, p. 779. 3. Antipov, A.E. and Vorotyntsev, M.A., Bromate anion electroreduction on inactive RDE under steady-state conditions. Numerical study of ion transport processes and comproportionation reaction, Russ. J. Electrochem., 2016, vol. 52, № 10, 925. 4. Antipov, A. E., Vorotyntsev, M. A., Tolmachev, Y. V., Antipov, E.M., and Aldoshin, S.M., Electroreduction of bromate anion in acidic solutions at the inactive rotating disc electrode under steady-state conditions: Numerical modeling of the process with bromate anions being in excess compared to protons, Doklady Chemistry, 2016, vol. 468, p. 141. 5. Nernst, W., Theorie der reaktionsgeschwindigkeit in heterogenen systemen, Z. Phys. Chem., 1904, vol. 47, p. 52. 6. Nernst, W. and Merriam, E.S., Zur theorie des reststroms. (nach versuchen von Herrn Merriam), Z. Phys. Chem., 1905, vol. 53, p. 235. 7. Bard, A.J. and Faulkner, L.R., Electrochemical methods, 2nd ed., New York: Wiley, 2001. 8. Levich, V.G., Physicochemical hydrodynamics, New York: Prentice Hall, Englewood Cliffs, 1962. 9. Damaskin, B.B., Petrii, O.A., and Tsirlina, G.A., Elektrokhimiya (Electrochemistry), Moscow: Khimiya, Koloss, 2006. 10. Cortes, C.E.S. and Faria, R.B., Revisiting the kinetics and mechanism of bromate-bromide reaction, J. Braz. Chem. Soc., 2001, vol. 12, p. 775. 11. Cortes, C.E.S. and Faria, R.B., Kinetics and mechanism of bromate-bromide reaction catalyzed by acetate, Inorg. Chem., 2004, vol. 43, p. 1395. 12. Schmitz, G., Kinetics of the bromate-bromide reaction at high bromide concentrations, Int. J. Chem. Kinet., 2007, vol. 39, p. 17. 13. Pugh, W., The stability of bromic acid and its use for the determination of bromide in bromates and in chlorides, Trans. Roy. Soc. South Afr., 1932, vol. 20, p. 327. 14. Antipov, A.E. and Vorotyntsev, M.A., Generalized Nernst layer model for convective-diffusional transport. Numerical solution for bromate anion electroreduction on inactive RDE under steady state conditions, Russ. J. Electrochem., 2017, vol. 53 (in press).
Translated by Yu. Pleskov No. 1
2018
