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O3 evolving PbO2
Electrochimica Acta, Vol. 30, No. 9. pp. 1213-1217, 1985. Printed in Great Britain.
0013~686/85 $3.00+0.00 © 1985. Pergamon Press Ltd.
ANALYTICAL EXPRESSIONS FOR OVERVOLTAGE DETERMINATION BY THE USE OF SUPERIMPOSED CURRENT PULSES O N CONSTANT FARADAIC CURRENT. A USEFUL EXPRESSION FOR TRANSIENT STUDIES O N 0 2 / 0 3 EVOLVING PbO2 JORDANIS THANOS* Institut ffir organische Chemic und Biochemie der T. U. Miinchen, F.R.G. (Received 28 June 1984; in revisedform 27 February 1985) Abstract--In studies of the kinetics of oxygen and ozone, evolution by the electrolysis of water transient galvanostatic and impedance measurements were made. Under the existing high dc current densities, necessary to obtain substantial amounts of ozone, the treatments of the diffusion problems presented so far in the electrochemical literature are not directly applicable in the evaluation of the kinetic results. In this paper a critical theoretical account is given under which circumstances the existing equations may be used providing a more general solution to the galvanostatic step problem as used in transient measurements.
NOMENCLATURE
Partial pressure of 02, Oa Incremental changes of current, potential Concentration of H + Time Distance from electrode surface Diffusion coefficient of H ÷ Diffusion coefficient of CIO2 Faradaic constant Faradaic current Transformed form of If, Equation (5) Number of transfered electrons Oxidant O Rcductant R Diffusion boundary layers of oxidant, re~luctant ~O, ~R a, b Concentration gradients of oxidant, reductant Standard electrode potential E~ Electrode potential at equilibrium E~ Dc current Icon Capacitive current icap Electrode potential prior to current pulse Econ Co(x, t), CR(x, t) = concentrations of species O, R Cb(t), C~(t) = surface concentrations at x = 0 PO=, Po~ ~i, ~E CH+ t x DI.I+ DCIO~ F If lfe~
The subscripts "f", "cap" and "tot" to the current denote Faradaic, capacitive and total current, whereas low case symbols denote incremental values; eo r/= overvoltage, i = pulsed current. 1. I N T R O D U C T I O N The numerous investigations about the mechanism of the evolution of oxygen may be separated into two categories: (a) determination of rate determining step by stationary electrochemical methods, e9[1-5], and * Dr. Jordanis Thanos, Lehrstuhl III f/Jr organische Chemic und Biochemie der Technischen Universit/it Mfinchen, Lichtenbergstr.4, 8046 Garching bei M/inchen, F.R.G.
(b) estimation of the oversaturation of the gas prior to the formation of gas bubbles by the application of current reversal after pulse electrolysis, taking into account the high irreversibility of the reaction, eg[6]. Most of the existing data have been collected using Pt anodes[4, 5]. PbO2 is more appropriate, however, for the preparation of 0 3 and for this reason systematic data have been collected by us with such anodes. Before the evaluation of the results the effect of two complications arising on ozone evolving PbO2 anodes has to be considered: (a) Moderately high current densities (eg >/100 m A c m -2) are required to produce Oa in quantities higher than 6~o w/w[7]. If in this case we consider the possibility that 0 2 or Oa produce a diffusion overpotential which should be studied by transient techniques it has to be taken into account the kind of existing boundary conditions. For most of the known analytical expressions it has been assumed that at time equal to zero a specially constant concentration exists of the redox species involved in electrochemical reaction. (b) If, on the other hand, it can be shown otherwise that Oa or 0 2 do not cause a diffusion overpotential it still remains the problem of pH variations. With acids more concentrated than 0.5 M, PbO2 electrodes rapidly corrode during anodic polarization. In fact the electrode surface is so unstable that stationary polarization curves with freshly plated electrodes are not reproducible after the first few minutes of polarization. With dilute acids as electrolytes a pronounced diffusion overpotential due to (3pH/c~E)i =~ 0 is expected. Should this polarization be studied by a galvanostatic current pulse method, more general boundary conditions than those introduced by Delahay and Berzins[8] should be considered. Additionally a further suitable transformation of the concentration/current describing equations is needed and is described below.
1213
1214
JORDANISTHANOS
2. F O R M U L A T I O N
OF THE PROBLEM
For the development and solution of the differential equations for the small signal potential/time function under the application of a current pulse it is wished to explicitly define a linear relation of the form:
6i = k ~6E + k 2 6 C 0
+
k 36C R.
(1)
H e r e / q , 2.3 are constants to be defined and indicate small variations of the variables following it. When 02 and 03 are evolved on PbO2 we have H 2 0 = 2H ÷ + 1/2 02 + 2 e - , H 2 0 = 2H ÷ + 1/303 + 2e-.
(2)
Using a rotating PbO2 electrode we have shown[7], that
valid in the presence of a constant Faradaic current, and the solution of a simple step galvanostatic problem with more realistic initial conditions is discussed below. Consider the reaction, R"- = O + he.
To a good approximation the concentration profile of the oxidant before the galvanostatic pulse takes the form of Fig. 1 due to the Faradaic leo" current. A similar assumption holds for the reductant. Then at t = 0 we have
Co(x,t)=Cb(t )+x'a
t=0,
0<~x~<6 0 ,
CR(X,t)=CsR(t)--x'b
t =0,
O<~X<~fR,
Co(x,t)=C~)=C°)+~Jo a,
t=0,
CR(X,t)=C~ = C ~ - 6 R b ,
\aPo,],
\aPo,},
= O.
(3)
Hence, for the moment, we will be concerned with potential variations arising from pH changes. According to Equation (2) we can write for the Faradaic current in an, e0 0.5 M HCIO4 electrolyte, the following equations (e919]). 0C H ,
~3t
('32C H .
=
,
D~q (9x2
Dm
-
(3C H,
F
= oo,,
dx
2D N , Dc~o, DH' + Doo,
,
(4)
Dc~o, ,
l~
=
IrD n. + Dc~o~
.
(s)
As shown the Faradaic current is transformed into a variable I f';'4. , which with the parameter D,,q are related . by an equation having a form already used m many well known problems. As a consequence Equations (4) and (5) in combination with a known function of potential vs hydrogen H + concentration can be used in principle in order to correlate the Faradaic current to the other variables in terms of Equation (1) for the special case where irreversible O2 and 03 evolution occurs while the liberated protons cause a shift of the electrode potential because of (?E/OpH)i ¢: 0. 3. S M A L L S I G N A L P O T E N T I A L - T I M E DEPENDENCE FOR A GALVANOSTATIC S I N G L E PULSE, S U P E R I M P O S E D O N STATIONARY H I G H C U R R E N T D E N S I T I E S A straightforward, but algebraically more laborious application of the analytical results of the diffusion galvanostatic single step problem given by Berzins and Delahay[8], provides a more accurate expression for analysing the kinetic results obtained by the use of the double pulse technique[10], according to Matsuda et a l . [ l l ] . Further mathematical treatments by van Leeuwen and Sluyters[12] and more generally by Nagy[13] enabled the overcoming of difficulties leading to misevaluation of kinetic parameters because of wrong measurements arising from inductive effects during the current steps. It has been assumed by all authors, however, that prior to the current pulses a uniform concentration of the reactants existed through the bulk of the electrolyte to the electrode surface. This assumption is no longer
(6)
x > 6 o,
t=O,
(7)
x > 6 R.
Because of the existing steady state it may be written leo n
=
nFO { i~C-°(x" t)
ok,
,~x
L.0' (8)
E,:on = Eov + Ea'
Ea = E~' + R T In C ° ( 0 ' 0) nF C.(O, O)
Here Eov is the overpotential of the reaction (6) before the galvanostatic pulse. When the current pulse itot is applied, a potential change rift) occurs which, for sufficiently small currents, remains much smaller than R T / F and the following small signal equation is valid:
[
if(t)=io-
nFrl(t) ] R~-~-+~bl(O,t)-¢2(O,t).
(9)
Here io is the exchange current density, but for irreversible reaction another mathematical interpretation is given to it[7, 13]. ~kl, ~b2 are the relative concentrations[8]: ¢,~ (x, t) =
qJ2(x, t)
=
Co(x, t) - Co(0, 0)
Co(0, O) Cr(x, t ) - CR(0, 0) CR(0, 0)
(lO)
The solution of Equation (19) is then possible with the initial and boundary conditions (20) and (21). 0t - DO ~ ' Electrode
Ot
Rff~-x2 ,
(11)
Electrolyte
Dis tence
co
.2 ~
co
~
x
Fig. 1. Model of concentration profile of oxidant and reductant in electrolyte.
Analytical expressions for overvoltage determination
xa
a
2~ - C°(O), 0 ~< x ~< ~o"
~ (x, 0) = C~(O) = x2,, 2ifio = KI,
x > 6o
-xb
-b
¢2 x, 0) = C~(0----)= x22, 2~ = ~ ,
22fir = K2,
the equations involving the reduced or the oxidized species. Considering the boundary conditions and Equations (9) and (10) we find 2o~-~---r-~A-~l-~2],
0 ~< x ~< ~R
fir < X,
where the constants K,, K 2 refer to the bulk concentrations. .0¢'2(0,
t)
._
a~,x(0, t) if" Ox ----
_
OX -=C°(0'0)O°
--
_ dr/(t) i to t = lea p -t- if = -- (.J O - - ~
'~O~x01~I=
[
~
nF'
tlA+MYo_NYo,
1/2.+ 2__71=
s
-,
From Equations (18) and (19), and the following substitutions, we eventually get Equation (21) DIO/2
D1/2
= ~1,~. ~a
= a~, 2o21~1 = Bx, '~O 2R AI = --axa2(2R22 +2021), ~q +~t2 = ~t, (20)
1/2 o = io/[nFDoCo(O, 0)],
1/2 R = io/[nF D~ C R(0, 0)], ~l(x,t)=Kx, x > f i o, t~>0, l ~2(x,t)=K2, X>6R, t ~>O.J
F = nFDoCo(O, 0), F (13)
Abbreviating ~k(x,t) to ~ and denoting the Laplace transformed variable of ~k(x, t) as i~ (x, s) or simply i~ we may obtain, with the help of Equations (11) and (12), Equation (14) for 0 ~ x ~< 6 o, after solving the Laplace transformed ordinary differential equations resulting from Equation (11):
/tot -- 21F =- y, F~q A =0,
FB 1
1/2 - ~ - f = - ( S.._~_~ ~ + ~ ' l
\no/
=r[(_
s J
1
•
,t(s)s '/2
Alsi/2-Bls'~
1
~1
,
(21)
r t / S \'/2 l
)
xj
)
x > 6o ,
(14)
-
CDSq (S).
(22)
Noting that itot = itot/s, and using Equations (13), (20), (21) and (22), we find a closed form for ~.
/ s \1,2-]
~b,(f,t)=t~,(6,0),M'=Mexp[-26OtD~) [-
assuming YR ~ Yo = Y, which as further required is correct for large values of s (vide infra). Considering the charging of the double layer capacitance we obviously have 7~p =
with M, M' undetermined constants. Since
J' (15)
=
s(~'+ Yol ~)+ sl/2()'ot - Y-le) y- lOS 2 -- CD0~$5/2 -- C D S 3
(23)
Making further substitutions, Equation (24),
setting Yo = 1 + e x p [ - 26Ot~o ) { s ~tl/2],
(16)
then
~' = M r° exp[ - ( oos--)Y'2x _1 + -s
resp.
i~2=NY, expL-[~-~-Rj
xJ+~-
(17)
resp.
~'+Y-~ =#x,
)'a-Y-le=#2,
- C o ~ = v2,
- C D = v3,
Y-10=#1, (24)
and assuming p~, and P2 are the roots of the algebraic Equation (25):
and similarly
+ Vl = 0, V3 V3 we may express ~ in the following form: y2 _ V 2 y
(25)
, ,l(s) = v~(pl-p2)Ls(s'/2+pz)
where N is a constant. It has been possible to simply consider different 6o, fiR for the algebraic manipulations, which were carried independently for each of Yak 30:9-G
(18)
(19)
1
"
S
Z
0~kl
--q(t)~+01--02
S
-2°MY°
\OR ]
"]" rl l~ U O -~X '
F /e S \1/2 "1
with A = ~ T , l
--2R-~x2 = [ - - q A - - i ~ l - i ~ 2 ] , ]
(12)
CR(0'0)(--DR)
1215
2#2
"t S 3 / 2 ( S 1 / 2 . + p 2 )
S(S1/2+pt) #2
$3/2(S1/2 + p l )
1. _1
(26)
1216
JORDANISTHANOS
It may now be observed that for current pulses ( < 100 #s), ii R, 6 o are sufficiently thick to consider Y as equal to unity. The analytical inversion ofq (s) is thus possible. This renders the solution independent of the absolute size of 3 R and 6 o. The different values, however, of the two diffusion layers affected directly the quantities of a and b and hence the final solution of the equations.
1
L-l
(b) Moreover noticing that when a = b = 0, also /A2/itot = P I + P 2 = v2/Va and so we observe that Equation (29) is equivalent to the less general formula given as a function of time in loc. cit.[8]. This result is obtained only when the first bracket in Equation (3 l) is predominant. A simple evaluation shows that this is approximately equivalent to itot ~> Icon
= 01/2 e ~` erfc[(at)l/2],
,+(;)
L - ' ~ / 2
=\nt/
which however is not the rule for kinetics studied at higher constant current densities.
Acknowledoements--I thank Prof. Dr. H. P. Fritz, Dr. D. W. Wabner and Dr. W. D. Fleischmann for useful discussions on the problem of O2/O 3 evolution and the "Fonds der Chemischen Industrie" as well as the "Foundation Alexander Onasis" and the "Freunde der Technischen Universit~it Mfinchen" for financial support.
-ae"'erfc[(at)l/2], (27)
L-to(sl=o(t)-v3(p _p2,(-t,,[p21~(ea2 erfc(p2t:)-l)+~t':21 1
l
2t
t 2
-pt[~(e p:terfc(ptt'/z)-I + ~ t LPl \ 2 t,2_ [ P~1 (ep,, +p~nt/~ \ erfc(p~t~/2) - 1
JJ+/~2 )
~2
It,:2 nt2# ]]}
+p~
ep~terfc(p2t'/2)-I (28)
.
Equation (28} may be rewritten as n(t) -
[[ _ l+ ep~,erfclp2t,/2)l[#~2 +
1 --CD(Pt --P2)
+[eP~terfc(ptt'12)--ll[--P2O _ _2 + # ~l ] ] '
(29)
noting that #, -- it,,,, ioi,o,[ 1 m = ~F b~o;~co(0.0) +
1
]
io[
v~
.F
1
I
P"P~ - " ~ L OU(o(O, 0)
F, i 2 o [
+
'
1
2
o~c-~(o, o)) +
nFio 11/2
n = 6E, we obtain after
oulF I i f = - - RT .
D[/2CR(O, O)
1
For small variations it=b1, differentiation
]
1]
1
I = io exp [ct(E - E~JF/[RT)].
(31)
DUCo(O, O) + o~:~c-.(0, o)
~i0 [
For an irreversible reaction away from equilibrium we have
o~:'c.lo, o) -io
1_C ~ b _ 0 ) + C,(0, 0 ) J ' v2
APPENDIX
(30)
(32)
In Equation (9) we have to interpret - nFtl (t)/RTas aFI/RT so that i0 stands for - Ict/n instead of an exchange current density. Furthermore, for the case of the 02/03 evolution where only H + produces diffusion overpotential, we interpret R as H20 and O as H + having a diffusion coefficient Deq. Since then in the applied equations a "'transformed" Faradaic current is used and the other two quantities Icon,/cap have to be normalized by the same factor. This is easy since (e0 for small signals) / c a p = -- CD d ~t
4. D I S C U S S I O N or
From the derivation in Section 3 the following observations are made: (a) Since at short times t, the predominant terms of the linearized form of Equation (29) is of the order of t 1/2, it may be expected that the Delahay extrapolation formula (for the double pulse problem) vs t t/2 is also valid for the present case with the more complicated initial conditions.
,
oc,o eaPDn+ +DcIo; / =
oc,o )7, ~, DDrt. +Dcto;
and hence the quantities in parentheses can be used as the new variables in the place of iono and C D. A final modification should be also made in the latter case. Since CH2o is in high concentration it is expected that hardly affects the kinetics. So instead of the term (RT/nF)
Analytical expressions for overvoltage determination In(Co/CR) in Equation(8) the corresponding term - [(dE/dpH)i In C H+] is required. With these formal substitutions it is shown, at least formally, that the results derived here can be applied for the irreversible electrochemical reactions discussed here.
REFERENCES 1. J. O'M. Bockris, J. chem. Phys. 24, 817 (1956). 2. A. Damjanovic, A. Day and J. O'M Bockris, Electrochim. Acta 11, 791 (1966). 3. A. I. Krasirshchikov, Russian J. phys. Chem. 37, 273 (1963). 4. A. T. Kuhn, The Electrochemistry of Lead, p. 217. Academic Press, New York (1979).
1217
5. J. P. Hoare, The Electrochemistry of Oxyoen. Wiley, New York (1968). 6. S. Shibata, Electrachim. Acta 23, 619 (1978). 7. I. C. Thanos, Dissertation, T. U. Mfinchen (1981). 8. T. Berzins and P. Delahay, J. Am. chem. Sac. 77, 6449 (1955). 9. Veniamin G. Levich, Physicochemical Hydrodynamics, pp. 286-287. Prentice Hall, N. J. (1962). 10. H. Gerischer and M. Krause, Z. phys. Chem. NF 10, 264 (1957); 14, 184 (1958). 11. H. Matsuda, S. Oka and P. Delahay, J. Am. chem. Sac. 81, 5077 (1959). 12. H. P. van Leeuwen and J. H. Sluyters, J. electroanal. Chem. 42, 313 (1973). 13. Z. Nagy, J. electrochem. Soc. 126(7), 1148 (1979). 14. L. A. Pipes and L. R. Harvill, Applied Mathematics for Engineers and Physicists, 3rd Edition, McGraw-Hill, New York (1970).
0013~686/85 $3.00+0.00 © 1985. Pergamon Press Ltd.
ANALYTICAL EXPRESSIONS FOR OVERVOLTAGE DETERMINATION BY THE USE OF SUPERIMPOSED CURRENT PULSES O N CONSTANT FARADAIC CURRENT. A USEFUL EXPRESSION FOR TRANSIENT STUDIES O N 0 2 / 0 3 EVOLVING PbO2 JORDANIS THANOS* Institut ffir organische Chemic und Biochemie der T. U. Miinchen, F.R.G. (Received 28 June 1984; in revisedform 27 February 1985) Abstract--In studies of the kinetics of oxygen and ozone, evolution by the electrolysis of water transient galvanostatic and impedance measurements were made. Under the existing high dc current densities, necessary to obtain substantial amounts of ozone, the treatments of the diffusion problems presented so far in the electrochemical literature are not directly applicable in the evaluation of the kinetic results. In this paper a critical theoretical account is given under which circumstances the existing equations may be used providing a more general solution to the galvanostatic step problem as used in transient measurements.
NOMENCLATURE
Partial pressure of 02, Oa Incremental changes of current, potential Concentration of H + Time Distance from electrode surface Diffusion coefficient of H ÷ Diffusion coefficient of CIO2 Faradaic constant Faradaic current Transformed form of If, Equation (5) Number of transfered electrons Oxidant O Rcductant R Diffusion boundary layers of oxidant, re~luctant ~O, ~R a, b Concentration gradients of oxidant, reductant Standard electrode potential E~ Electrode potential at equilibrium E~ Dc current Icon Capacitive current icap Electrode potential prior to current pulse Econ Co(x, t), CR(x, t) = concentrations of species O, R Cb(t), C~(t) = surface concentrations at x = 0 PO=, Po~ ~i, ~E CH+ t x DI.I+ DCIO~ F If lfe~
The subscripts "f", "cap" and "tot" to the current denote Faradaic, capacitive and total current, whereas low case symbols denote incremental values; eo r/= overvoltage, i = pulsed current. 1. I N T R O D U C T I O N The numerous investigations about the mechanism of the evolution of oxygen may be separated into two categories: (a) determination of rate determining step by stationary electrochemical methods, e9[1-5], and * Dr. Jordanis Thanos, Lehrstuhl III f/Jr organische Chemic und Biochemie der Technischen Universit/it Mfinchen, Lichtenbergstr.4, 8046 Garching bei M/inchen, F.R.G.
(b) estimation of the oversaturation of the gas prior to the formation of gas bubbles by the application of current reversal after pulse electrolysis, taking into account the high irreversibility of the reaction, eg[6]. Most of the existing data have been collected using Pt anodes[4, 5]. PbO2 is more appropriate, however, for the preparation of 0 3 and for this reason systematic data have been collected by us with such anodes. Before the evaluation of the results the effect of two complications arising on ozone evolving PbO2 anodes has to be considered: (a) Moderately high current densities (eg >/100 m A c m -2) are required to produce Oa in quantities higher than 6~o w/w[7]. If in this case we consider the possibility that 0 2 or Oa produce a diffusion overpotential which should be studied by transient techniques it has to be taken into account the kind of existing boundary conditions. For most of the known analytical expressions it has been assumed that at time equal to zero a specially constant concentration exists of the redox species involved in electrochemical reaction. (b) If, on the other hand, it can be shown otherwise that Oa or 0 2 do not cause a diffusion overpotential it still remains the problem of pH variations. With acids more concentrated than 0.5 M, PbO2 electrodes rapidly corrode during anodic polarization. In fact the electrode surface is so unstable that stationary polarization curves with freshly plated electrodes are not reproducible after the first few minutes of polarization. With dilute acids as electrolytes a pronounced diffusion overpotential due to (3pH/c~E)i =~ 0 is expected. Should this polarization be studied by a galvanostatic current pulse method, more general boundary conditions than those introduced by Delahay and Berzins[8] should be considered. Additionally a further suitable transformation of the concentration/current describing equations is needed and is described below.
1213
1214
JORDANISTHANOS
2. F O R M U L A T I O N
OF THE PROBLEM
For the development and solution of the differential equations for the small signal potential/time function under the application of a current pulse it is wished to explicitly define a linear relation of the form:
6i = k ~6E + k 2 6 C 0
+
k 36C R.
(1)
H e r e / q , 2.3 are constants to be defined and indicate small variations of the variables following it. When 02 and 03 are evolved on PbO2 we have H 2 0 = 2H ÷ + 1/2 02 + 2 e - , H 2 0 = 2H ÷ + 1/303 + 2e-.
(2)
Using a rotating PbO2 electrode we have shown[7], that
valid in the presence of a constant Faradaic current, and the solution of a simple step galvanostatic problem with more realistic initial conditions is discussed below. Consider the reaction, R"- = O + he.
To a good approximation the concentration profile of the oxidant before the galvanostatic pulse takes the form of Fig. 1 due to the Faradaic leo" current. A similar assumption holds for the reductant. Then at t = 0 we have
Co(x,t)=Cb(t )+x'a
t=0,
0<~x~<6 0 ,
CR(X,t)=CsR(t)--x'b
t =0,
O<~X<~fR,
Co(x,t)=C~)=C°)+~Jo a,
t=0,
CR(X,t)=C~ = C ~ - 6 R b ,
\aPo,],
\aPo,},
= O.
(3)
Hence, for the moment, we will be concerned with potential variations arising from pH changes. According to Equation (2) we can write for the Faradaic current in an, e0 0.5 M HCIO4 electrolyte, the following equations (e919]). 0C H ,
~3t
('32C H .
=
,
D~q (9x2
Dm
-
(3C H,
F
= oo,,
dx
2D N , Dc~o, DH' + Doo,
,
(4)
Dc~o, ,
l~
=
IrD n. + Dc~o~
.
(s)
As shown the Faradaic current is transformed into a variable I f';'4. , which with the parameter D,,q are related . by an equation having a form already used m many well known problems. As a consequence Equations (4) and (5) in combination with a known function of potential vs hydrogen H + concentration can be used in principle in order to correlate the Faradaic current to the other variables in terms of Equation (1) for the special case where irreversible O2 and 03 evolution occurs while the liberated protons cause a shift of the electrode potential because of (?E/OpH)i ¢: 0. 3. S M A L L S I G N A L P O T E N T I A L - T I M E DEPENDENCE FOR A GALVANOSTATIC S I N G L E PULSE, S U P E R I M P O S E D O N STATIONARY H I G H C U R R E N T D E N S I T I E S A straightforward, but algebraically more laborious application of the analytical results of the diffusion galvanostatic single step problem given by Berzins and Delahay[8], provides a more accurate expression for analysing the kinetic results obtained by the use of the double pulse technique[10], according to Matsuda et a l . [ l l ] . Further mathematical treatments by van Leeuwen and Sluyters[12] and more generally by Nagy[13] enabled the overcoming of difficulties leading to misevaluation of kinetic parameters because of wrong measurements arising from inductive effects during the current steps. It has been assumed by all authors, however, that prior to the current pulses a uniform concentration of the reactants existed through the bulk of the electrolyte to the electrode surface. This assumption is no longer
(6)
x > 6 o,
t=O,
(7)
x > 6 R.
Because of the existing steady state it may be written leo n
=
nFO { i~C-°(x" t)
ok,
,~x
L.0' (8)
E,:on = Eov + Ea'
Ea = E~' + R T In C ° ( 0 ' 0) nF C.(O, O)
Here Eov is the overpotential of the reaction (6) before the galvanostatic pulse. When the current pulse itot is applied, a potential change rift) occurs which, for sufficiently small currents, remains much smaller than R T / F and the following small signal equation is valid:
[
if(t)=io-
nFrl(t) ] R~-~-+~bl(O,t)-¢2(O,t).
(9)
Here io is the exchange current density, but for irreversible reaction another mathematical interpretation is given to it[7, 13]. ~kl, ~b2 are the relative concentrations[8]: ¢,~ (x, t) =
qJ2(x, t)
=
Co(x, t) - Co(0, 0)
Co(0, O) Cr(x, t ) - CR(0, 0) CR(0, 0)
(lO)
The solution of Equation (19) is then possible with the initial and boundary conditions (20) and (21). 0t - DO ~ ' Electrode
Ot
Rff~-x2 ,
(11)
Electrolyte
Dis tence
co
.2 ~
co
~
x
Fig. 1. Model of concentration profile of oxidant and reductant in electrolyte.
Analytical expressions for overvoltage determination
xa
a
2~ - C°(O), 0 ~< x ~< ~o"
~ (x, 0) = C~(O) = x2,, 2ifio = KI,
x > 6o
-xb
-b
¢2 x, 0) = C~(0----)= x22, 2~ = ~ ,
22fir = K2,
the equations involving the reduced or the oxidized species. Considering the boundary conditions and Equations (9) and (10) we find 2o~-~---r-~A-~l-~2],
0 ~< x ~< ~R
fir < X,
where the constants K,, K 2 refer to the bulk concentrations. .0¢'2(0,
t)
._
a~,x(0, t) if" Ox ----
_
OX -=C°(0'0)O°
--
_ dr/(t) i to t = lea p -t- if = -- (.J O - - ~
'~O~x01~I=
[
~
nF'
tlA+MYo_NYo,
1/2.+ 2__71=
s
-,
From Equations (18) and (19), and the following substitutions, we eventually get Equation (21) DIO/2
D1/2
= ~1,~. ~a
= a~, 2o21~1 = Bx, '~O 2R AI = --axa2(2R22 +2021), ~q +~t2 = ~t, (20)
1/2 o = io/[nFDoCo(O, 0)],
1/2 R = io/[nF D~ C R(0, 0)], ~l(x,t)=Kx, x > f i o, t~>0, l ~2(x,t)=K2, X>6R, t ~>O.J
F = nFDoCo(O, 0), F (13)
Abbreviating ~k(x,t) to ~ and denoting the Laplace transformed variable of ~k(x, t) as i~ (x, s) or simply i~ we may obtain, with the help of Equations (11) and (12), Equation (14) for 0 ~ x ~< 6 o, after solving the Laplace transformed ordinary differential equations resulting from Equation (11):
/tot -- 21F =- y, F~q A =0,
FB 1
1/2 - ~ - f = - ( S.._~_~ ~ + ~ ' l
\no/
=r[(_
s J
1
•
,t(s)s '/2
Alsi/2-Bls'~
1
~1
,
(21)
r t / S \'/2 l
)
xj
)
x > 6o ,
(14)
-
CDSq (S).
(22)
Noting that itot = itot/s, and using Equations (13), (20), (21) and (22), we find a closed form for ~.
/ s \1,2-]
~b,(f,t)=t~,(6,0),M'=Mexp[-26OtD~) [-
assuming YR ~ Yo = Y, which as further required is correct for large values of s (vide infra). Considering the charging of the double layer capacitance we obviously have 7~p =
with M, M' undetermined constants. Since
J' (15)
=
s(~'+ Yol ~)+ sl/2()'ot - Y-le) y- lOS 2 -- CD0~$5/2 -- C D S 3
(23)
Making further substitutions, Equation (24),
setting Yo = 1 + e x p [ - 26Ot~o ) { s ~tl/2],
(16)
then
~' = M r° exp[ - ( oos--)Y'2x _1 + -s
resp.
i~2=NY, expL-[~-~-Rj
xJ+~-
(17)
resp.
~'+Y-~ =#x,
)'a-Y-le=#2,
- C o ~ = v2,
- C D = v3,
Y-10=#1, (24)
and assuming p~, and P2 are the roots of the algebraic Equation (25):
and similarly
+ Vl = 0, V3 V3 we may express ~ in the following form: y2 _ V 2 y
(25)
, ,l(s) = v~(pl-p2)Ls(s'/2+pz)
where N is a constant. It has been possible to simply consider different 6o, fiR for the algebraic manipulations, which were carried independently for each of Yak 30:9-G
(18)
(19)
1
"
S
Z
0~kl
--q(t)~+01--02
S
-2°MY°
\OR ]
"]" rl l~ U O -~X '
F /e S \1/2 "1
with A = ~ T , l
--2R-~x2 = [ - - q A - - i ~ l - i ~ 2 ] , ]
(12)
CR(0'0)(--DR)
1215
2#2
"t S 3 / 2 ( S 1 / 2 . + p 2 )
S(S1/2+pt) #2
$3/2(S1/2 + p l )
1. _1
(26)
1216
JORDANISTHANOS
It may now be observed that for current pulses ( < 100 #s), ii R, 6 o are sufficiently thick to consider Y as equal to unity. The analytical inversion ofq (s) is thus possible. This renders the solution independent of the absolute size of 3 R and 6 o. The different values, however, of the two diffusion layers affected directly the quantities of a and b and hence the final solution of the equations.
1
L-l
(b) Moreover noticing that when a = b = 0, also /A2/itot = P I + P 2 = v2/Va and so we observe that Equation (29) is equivalent to the less general formula given as a function of time in loc. cit.[8]. This result is obtained only when the first bracket in Equation (3 l) is predominant. A simple evaluation shows that this is approximately equivalent to itot ~> Icon
= 01/2 e ~` erfc[(at)l/2],
,+(;)
L - ' ~ / 2
=\nt/
which however is not the rule for kinetics studied at higher constant current densities.
Acknowledoements--I thank Prof. Dr. H. P. Fritz, Dr. D. W. Wabner and Dr. W. D. Fleischmann for useful discussions on the problem of O2/O 3 evolution and the "Fonds der Chemischen Industrie" as well as the "Foundation Alexander Onasis" and the "Freunde der Technischen Universit~it Mfinchen" for financial support.
-ae"'erfc[(at)l/2], (27)
L-to(sl=o(t)-v3(p _p2,(-t,,[p21~(ea2 erfc(p2t:)-l)+~t':21 1
l
2t
t 2
-pt[~(e p:terfc(ptt'/z)-I + ~ t LPl \ 2 t,2_ [ P~1 (ep,, +p~nt/~ \ erfc(p~t~/2) - 1
JJ+/~2 )
~2
It,:2 nt2# ]]}
+p~
ep~terfc(p2t'/2)-I (28)
.
Equation (28} may be rewritten as n(t) -
[[ _ l+ ep~,erfclp2t,/2)l[#~2 +
1 --CD(Pt --P2)
+[eP~terfc(ptt'12)--ll[--P2O _ _2 + # ~l ] ] '
(29)
noting that #, -- it,,,, ioi,o,[ 1 m = ~F b~o;~co(0.0) +
1
]
io[
v~
.F
1
I
P"P~ - " ~ L OU(o(O, 0)
F, i 2 o [
+
'
1
2
o~c-~(o, o)) +
nFio 11/2
n = 6E, we obtain after
oulF I i f = - - RT .
D[/2CR(O, O)
1
For small variations it=b1, differentiation
]
1]
1
I = io exp [ct(E - E~JF/[RT)].
(31)
DUCo(O, O) + o~:~c-.(0, o)
~i0 [
For an irreversible reaction away from equilibrium we have
o~:'c.lo, o) -io
1_C ~ b _ 0 ) + C,(0, 0 ) J ' v2
APPENDIX
(30)
(32)
In Equation (9) we have to interpret - nFtl (t)/RTas aFI/RT so that i0 stands for - Ict/n instead of an exchange current density. Furthermore, for the case of the 02/03 evolution where only H + produces diffusion overpotential, we interpret R as H20 and O as H + having a diffusion coefficient Deq. Since then in the applied equations a "'transformed" Faradaic current is used and the other two quantities Icon,/cap have to be normalized by the same factor. This is easy since (e0 for small signals) / c a p = -- CD d ~t
4. D I S C U S S I O N or
From the derivation in Section 3 the following observations are made: (a) Since at short times t, the predominant terms of the linearized form of Equation (29) is of the order of t 1/2, it may be expected that the Delahay extrapolation formula (for the double pulse problem) vs t t/2 is also valid for the present case with the more complicated initial conditions.
,
oc,o eaPDn+ +DcIo; / =
oc,o )7, ~, DDrt. +Dcto;
and hence the quantities in parentheses can be used as the new variables in the place of iono and C D. A final modification should be also made in the latter case. Since CH2o is in high concentration it is expected that hardly affects the kinetics. So instead of the term (RT/nF)
Analytical expressions for overvoltage determination In(Co/CR) in Equation(8) the corresponding term - [(dE/dpH)i In C H+] is required. With these formal substitutions it is shown, at least formally, that the results derived here can be applied for the irreversible electrochemical reactions discussed here.
REFERENCES 1. J. O'M. Bockris, J. chem. Phys. 24, 817 (1956). 2. A. Damjanovic, A. Day and J. O'M Bockris, Electrochim. Acta 11, 791 (1966). 3. A. I. Krasirshchikov, Russian J. phys. Chem. 37, 273 (1963). 4. A. T. Kuhn, The Electrochemistry of Lead, p. 217. Academic Press, New York (1979).
1217
5. J. P. Hoare, The Electrochemistry of Oxyoen. Wiley, New York (1968). 6. S. Shibata, Electrachim. Acta 23, 619 (1978). 7. I. C. Thanos, Dissertation, T. U. Mfinchen (1981). 8. T. Berzins and P. Delahay, J. Am. chem. Sac. 77, 6449 (1955). 9. Veniamin G. Levich, Physicochemical Hydrodynamics, pp. 286-287. Prentice Hall, N. J. (1962). 10. H. Gerischer and M. Krause, Z. phys. Chem. NF 10, 264 (1957); 14, 184 (1958). 11. H. Matsuda, S. Oka and P. Delahay, J. Am. chem. Sac. 81, 5077 (1959). 12. H. P. van Leeuwen and J. H. Sluyters, J. electroanal. Chem. 42, 313 (1973). 13. Z. Nagy, J. electrochem. Soc. 126(7), 1148 (1979). 14. L. A. Pipes and L. R. Harvill, Applied Mathematics for Engineers and Physicists, 3rd Edition, McGraw-Hill, New York (1970).