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Chemical Engineering, Department of
1-1-1983
A Model of the Bromine/Bromide Electrode
Reaction at a Rotating Disk Electrode
Ralph E. White
University of South Carolina - Columbia, [email protected]
S. E. Lorimer
Texas A & M University - College Station
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Journal of the Electrochemical Society, 1983, pages 1096-1103.
© The Electrochemical Society, Inc. 1983. All rights reserved. Except as provided under U.S. copyright law, this work may not be
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DOI: 10.1149/1.2119890
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1096
J. EIectrochem. Soc.: ELECTROCHEMICAL SCIENCE AND TECHNOLOGY
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This Journal, 110, 190 (1963).
10. V. V. Gorodetskii, I. P. Slutskii, and V. V. Loser,
Elektrokhimiya, 8, 1401 (1972).
11. I. P. Slutskii, V. V. Gorodetskii, and V. V. Loser,
ibid., 13, 205 (1977).
12. S. H. Glarum and J. H. Marshall, This Journal, 128,
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Lestrade, J. Electroanal. Chem. Inter]acial Electrochem., 28, 57 (1970).
15. D. A. Scherson and J. Newman, This Journal, 127,
110 (1980).
16. K. Tokuda and H. Matsuda, J. glectroanal. Chem.
Interracial Electrochem., 82, 157 (1977); 9 0 , 149
(1978) ; 95, 147 (1979).
17. R. D. Armstrong, M. F. Bell, and A. A. Metcalfe,
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19.
20.
21.
22.
23.
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25.
26.
27.
28.
May 1983
"Electrochemistry--Spec. Period, Rpts.," Vol. 6,
The Chemical Society, London (1978).
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297 (1972).
H. P. Agarwal and S. Qureshi, Electrochim. Acta,
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C. A. Wamser, J. Am. Chem. Soc., 70, 1209 (1948).
A. M. Bond and R. J. Taylor, J. Electroanal. Chem.
Interracial EIectrochem., 28, 207 (1970).
M. Pourbaix, "Atlas of Electrochemical Equilibria,"
Pergamon, London (1966).
R. de Levie, Electrochim. Acta, 10, 113 (1965).
M. Fleischmann and J. A. Harrison, ibid., 11, 749
(1966).
H. Gerischer, Z. Phys. Chem., 198, 286 (1951).
J. A. Harrison and D. R. Sandbach, J. ElectroanaI.
Chem. Interracial Electrochem., 85, 125 (1977).
T. D. Hoar, Trans. Faraday Soc., 33, 1152 (1937).
A Model of the Bromine/Bromide Electrode Reaction at a Rotating
Disk Electrode
R. E. White* and S. E. Lorimer*
Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843
ABSTRACT
A mathematical model is presented for the Br._,/Br- electrode reaction at a rotating disk electrode. The model includes
current density-overpotential expressions for thelelectrode reaction according to either the Volmer-Heyrovsky
(V-H) or the
Volmer-Tafel (V-T) mechanism and the transport equations including the effect 0f ionic migration. The model is used to
predict current-overpotential curves for various cases of interest. Qualitative comparison of the model predictions to literature data shows that either the V-H or the V-T mechanism, with V controlling, may be acceptable for the BrjBr- reaction.
Table I presents a summary of the investigations of
the kinetics o f the B r z / B r - electrode reaction. The
three mechanisms that have been proposed frequently
are the Volmer-Heyrovsky (V-H), Volmer-Tafel
(V-T), and the Heyrovsky-Tafel (H-T), which can
be written as follows
V-H mechanism
[1]
B r - ~-Brads -l- e -
(V)
(H)
Brads q- B r - ~
Brs Jc e-
[2]
V-T mechanism
(V)
2 (Br- ~---Brads -{- e- )
[3]
(T)
2Brads ~=~ Br2
[4]
H-T mechanism
(H)
(T)
2(Br2+ e- ~__ B r - + Brads)
2Brads ~---Br2
[5]
[6]
Table I includes the kinetic mechanism(s) proposed
by each group, the type of electrode material, and the
technique used to determine the kinetics. Also included in the table are comments concerning each
group's approach to handling the tribromide ion formation reaction
kf
Brs ~- B r - $=~Br~kb
kf
KeQ -"
[7]
kb
and assumptions made concerning the degree of coverage of the electrode surface with adsorbed bromine
(0) or the rate of adsorption-desorption processes.
* E l e c t r o c h e m i c a l Society A c t i v e Member.
Key words: battery, migration.
Essentially, four different methods have been used
to study the kinetics of the B r J B r - electrode reaction.
The four methods are: galvanostatic or potentiostatic,
impedance, preexponential factor, and coulostatic.
Three groups (1, 2, and 3, see Table I) used current
density-potential curves produced either galvanostatically or potentiostatically to study the Br2/Br- reaction. The kinetic mechanism was selected by comparing the experimentally determined slopes of Tafel
curves and reaction orders to those values predicted
for each possible electrode kinetic mechanism. Three
other groups (4, 5, and 6) measured the impedance of
the electrode as a function of the frequency of an applied alternating current and used this to determine
the exchange current density. They then determined
the dependence of the exchange current density on
concentration and overpotential to select the electrode
kinetic mechanism. AUard and Parsons (7) used experimentally determined values of the preexponential
factor to study Br2/Br- electrode kinetics. The order
of magnitude of the experimental value of the preexponential factor, which can be determined from
measurements of the exchange current density at
Table IA. Summaryof Br2/Er- kinetic studies
Investigator(s)
Technique
. Osipov et aL (1)
. Janssen and Hoogland (2)
Potcntiostatic
Galvanostatic
Electrode
Platinum RDE
Graphite
Stationary
Platinized
Ti RDE
3. Faita, Fiori, and Mussini
(3)
Potentiostatic
4. Llopis and Vazquez (4)
Impedance
Platinum
5. Cooper and Parsons (6)
6. Albery et aL (6)
7. Allard and Parsons (7)
8. Rubinstein (8)
Impedance
Platinum RDE
Stationary
Impedance
Platinum RRDE
Preexponential Platinum RDE
Coulostatic
Platinum
Stationary
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VoL 130, No. 5
1097
B R O M I N E / B R O M I D E ELECTRODE REACTION
Table lB. Summary of Br?j'Br- kinetic studies
Investigator(s)
Proposed mechanism
Rate-controlling step
V-H
V-H
V-H
Slightly modified H-T
Both equally
V
Both equally
H
V
6
V-H or V-T
(Low surface c o v e r a g e )
V-T or H-T
(Higja surface coverage)
V-H or V-T
V
7
V-T
V
V-T or V-H
V
I
2
B
4
5
8
(Oxidized electrodes)
Slightly modified
V-T or V-H
(Reduced electrodes)
Comments
Ignored t ri bromi de ion formation a n d adsorption effects
Ignored t ri bromi de ion formation; included surface c o v e r a g e
Low surface coverage assumed; ignored t ri hromi de ion formation
Ignored tribromide ion formation; e x p e r i m e n t a l data indicated adsorption did not effect electrode reaction rate
Ignored t ri bromi de ion formation
T
Reaction t h o u g h t to occur in sparsely filled second l a ye r on top
of an electrode surface completely covered with a layer of unreactive bromine atoms.
Corrections made for t r i b r o m i d e ion formation; e s t i ma t e made of
t h e degree of surface coverage
Used a combined isotherm approach to account for adsorption
V
different temperatures, was then used to identify the
r a t e - d e t e r m i n i n g step of various mechanisms. Rubinstein (8) used the coulostatic method which consists of
charge injection followed by an open-circuit transient
analysis to obtain the kinetic parameters. As before,
these parameters are then compared to those p r e dicted by the proposed mechanisms to d e t e r m i n e acceptable mechanisms.
As can be seen from Table I, no single electrode
kinetic mechanism is supported by all of the investigators. The V-H and V-T mechanisms are, however,
suggested more often than the H-T mechanism. Table I
also shows that most workers neglected the formation
of tribromide ion. This is unfortunate because at the
very least this causes confusion concerning the
bulk concentrations of the reacting species (Br2 and
B r - ) . Finally, Table I entries and other studies reveal
that considerable confusion exists over the role of adsorbed bromine in the electrode reaction. Equilibrium
values for the degree of coverage of platinum surfaces
with adsorbed bromine indicate that as many as 5560% of the available adsorption sites are covered (9,
10). Also, b r o m i n e is known to be strongly chemisorbed on platinum (11-13). On the other hand, the
values of the preexponential factor for the bromine
electrochemical reaction measured by Allard and P a r sons (7) indicate that the degree of surface coverage
is low and that the adsorbed species has a high degree
of mobility across the electrode surface. Also, the
impedance measurements made b y Llopis and Vazquez (4) indicated that the rate of the b r o m i n e / b r o mide reaction is not affected by adsorption. To explain these apparently conflicting results, Allard and
Pro:sons (7) proposed that the electrode reaction involves a loosely adsorbed reactive bromine species
existing in a sparsely filled layer on top of a layer of
unreaetive bromine adsorbed directly on the electrode
surface. A l b e r y et al. (6) and Rubinstein (8) support this proposal.
It is clear that there is still uncertainty concerning
the B r 2 / B r - electrode reaction. The kinetic mechanism and associated issues such as the effect of the
homogeneous tribromide formation reaction and the
role of adsorption-desorption processes in the electrode
reaction are still unresolved.
A model is presented here which may be useful for
analysis of potentiostatic RDE data. The model includes rate expressions for either the V-H or the V-T
mechanisms and is used here to predict c u r r e n t - p o tential curves for both mechanisms for various cases
of interest.
V-T mechanisms, the transport equations, and the
boundary conditions for the RDE system. The model is
presented by first listing the assumptions and then
the equations.
Assumptions
1. The rates of the individual steps in each of the
reaction mechanisms are equal (i.e., iv = i H and iv =
i T ) . A l s o , for the V-H mechanism, i = iv + i H and for
V-T mechanism, i -- iv ----iT (14).
2. The double layer is ignored by assuming that it
is infinitely thin.
3. Equilibrium adsorption of the adsorbed bromine
(Brads) follows the Langmuir isotherm and is very
small (co < < 1).
4. The rates of the adsorption-desorptionprocess are
rapid enough to not interfere with the rates of the
charge transfer steps.
5. The electrolyte is an ideal solution (i.e., concentrations instead of activities are used).
6. Infinitely dilute solution theory (15) applies.
7. The RDE is an equipotential electrode and is uniformly accessible.
8. Steady-state conditions exist.
9. Isothermal conditions exist.
~Srush
l__J
Contact
Shaft
_ Rotating Disk
Working Electrode(+)
Y = Yre
Reference Elect.rode
I
Porous Plug
J
lY sC~
Electrode
I
Model
Figure 1 presents a schematic of the RDE system
in which the position of the reference electrode is
shown. The model presented here consist of currentoverpotential expressions for either the V-H or the
(-)
Fig. I. Schematic ef a rotating disk electrode system
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1098
J. EZectrochem. Soc.:
May 1983
ELECTROCHEMICAL SCIENCE AND TECHNOLOGY
IO. The electrolyte is Newtonian and the physical
and transport properties are constants.
11. 2'he Nernst,Einstein equation (ui _ D.gRT> is a
valid approximation.
12. The reaction orders are given by the stoichiometry of the individual steps of the overall reaction.
Kinetic Equations
The above assumptions and the rate expressions
given by Jansson and Hoogland (14) for the CI~/CIelectroae reaction can be used to derive curren~ density-potents expressions for the V - H and V-T mechanisms, as dLscussed by Lorimer (16). Essentially the
method consists of using assumption 1 to obtain an
expression for the c u r r e n t density i for each mechanism which depends on, among other things, the surface coverage and the equilibrium surface coverage of
adsorbed bromine atoms. These.equations can then be
simplified by using assumption 3. The resulting expressions for the V - H and V-T mechanisms are as
follows
Table II. Concentration dependence of the
exchange current densities
Reaction
j
Species i
V
V
H
H
T
T
Ur~
BrBr~
BrBr~
DrBrs
I av/Z
1 - a(1 + aH)/2
0
uv
1
RT ~r
-- U~
U~
7~ J
nF
ln(Cl're~)
i 81
~
R,r'
-I- nreF
Po
.--;:--)
( Ci,rs
i
where Ci,rerepresents the concentration of species i in
the reference electrode compartment and both Ci,ref
and el,remust be in tools/literin Eq. [12]. This depen-
V I I mechanism
=
(@me -- @o - V,e,)
exp ( (I +
\
/
RT
(
[, + ,.o,.,.H
(o.._,o).
9
•
- -
CBr2,0
$OreLV \ CBr--,ref
oo-,.,,.,,,>)]
ox,
(1--~V4-
RT
exp
V - T mechanism
~-~
A4- [A2-- (~'oref'v\CBr_,ref)
@o- Uref) )
_,_(.,.Or,,.v( O'r"~) "
CBr2,ref
exp(--(1
-- ~ ) F
~-~
) (@met -- @o _ Uref) )~ ]'/,
[9]
where
A -- ioref,v ( cBr-'O ] e x p (
CBr--,ref /
4-
'a~F (@met - - @o -- Uref) )
"-~
[ iOref.V ~2
f--2(1av)F
- exp
L 2iOref,T J
L
RT
(@met -- @o -- Uref)
'~
/
[10]
The exchange current densities (iorefjt, ioref.v, and
iOref,W) a r e based on the reference concentrations used
in the analysis. These reference exchange current densities are the values that would be obtained from comparison of the predicted and experimental current
density-potential curves. Once values for ioref,j have
been determined they can be adjusted to a standard
electrolyte composition by using the following expression
/ostd.:)----iore,.j '~
Ci,std ~"r'~l
~ . .
)]
(@met -- @o -- Uref)
dence of the open-circuit potential on the reference
concentration instead of the surface concentration
arises naturally in the derivation of Eq. [8] and [9]
due to the concentration dependence of the exchange
current density. Also, it should be pointed out that
Uref is for the overall reaction ( 2 B r - ~--. Br2 4- 2 e - ) .
The above current density-potential expressions can
be simplified by assuming that one of the steps is
rate controlling or that both steps proceed at approximately the same rate. Table III presents the simplified
forms of the rate equations where
( CBr--,O
exp ~"-fiT (r
aH)F
[8]
[11]
where t h e values of "ylj used here are based on the
stoichiometry of the reactions (15, 16) and are presented in Table II. (Even though Br2 does not participate in the Volmer reaction, Eq. [1], a value for ~Br2,V,
as given in Table II, arises because of the use of assumption 3 in the derivation of Eq. [8] and [9]). The
open-circuit potential Uref is based on the reference
concentrations and is given by
"Qref = @met -- @o - - Uref
[13]
Entries IA and IB can be obtained by considering the
denominator of Eq. [8]. For iOref,H > > iOref.V and I~lref[
-- ,~0.1V the term containing the exponential is large
relative to one; however, when l~ref] ~-- ,-,0.1V, the
term containing the exponential is small relative
to one. The reason for this depends on whether 71ref
is negative or positive. When Tlref is negative and is
made to be more and more negative, eventually the
exponential becomes so small that the entire term is
small relative to one. O n the other hand, if "Qref is positive and is made more positive, eventually the surface
concentration of the bromide ion (Csr-,o) becomes so
small that the entire term is small relative to one,
assuming, o f course, that the bromide ion is not present in the electrolyte in abundance. The kinetic p a r a m eters from these simplified forms are presented for
convenience in Table IV and V for cathodic and anodic
values of 11ref,respectively.
These simplified equations and parameters are presented for completeness and to aid in understanding
the system. However, the simplified equations are not
necessary for determination of the kinetic rate constants. Again, the kinetic rate constants should be
determined by comparison of the experimental data
(current density at a set potential difference between
the RDE and a reference electrode) to predicted values
from the complete expressions.
Transport Equations
The steady state, no homogeneous reaction material
balance equation for species i is (15)
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B R O M I N E / B R O M I D E ELECTRODE REACTION
Vol. 130, No. 5
1099
Table III. Simplified forms of the V-H and V-T current density expressions for limiting cases
I. Volmer-Heyrovsky mechanism (Eq. [8])
A. Volmer reaction rate controlling, low ~ret(i0ref,H >>
1Oref.Vj [7}ref[ ~
,~0.1V)
B. Volmer reaction rate controlling, high ~re~(iOr~f,H > > /Or~,v, [~ref] ~ ~ 0 . 1 V )
[ [ car-.o'~"
((I+aH)F
~
(CSr.,O)(--(1-an}F
, = 2,o,o,,~Lk--ycer_,re, exp
RT
~ret/ -- ~
exp '
Ri
)]
,ref
C. Heyrovsky reaction rate c0ntrolling'(iOret,V > > iOref,n)
i
2i0ref,H[(~:~.
~B'-'~
-..o.. .)~
- cxp
, )((I-
-- ()__C~r.,ret
Car.,O
+RT
-~]e
(.--(I--RTeX~)F~re,)]
exp
.
iOref,H)
D. Both reactions approximately equally rate controlling (iOref,V ~
1. Cathodic overpotential (anodic back reaction neglected)
[ Car'.O,
~ (I +aH)F
i = -2~o..,.~k~
1., e ~ p
~
)
~..,
2. Anodic overpotential (cathodic back reaction neglected)
(avF
exp - - ~
c~r-,o
~= 2 / o , , , , v ( ~ )
r., )
CBr~reg
II. Volmer-Tafel mechanism (Eq. [9] and [10])
A. Volmer reaction rate controlling (iore~,~ ~
r ( cs._,o ) [ a v F
ior~,v)
"~ ( _
CBr-~ref --
CSr,,O ~'/'
( (1--av)F--
CBr~,ref
)]
R-T-
~ref
B. Tafel reaction rate controlling (iOret,v ~ > i0ret,z)
[(C~r-,O]~ex p
(
2F
careD )
C. Both reactions approximately equally rate controlling (lO~e~,V ~ lores,T)
1. Cathodic overpotentials (anodie back reaction neglected)
No simplified form of the V-T current density expressi on is available for this case except near limiting current where
(
t
c~r,,o
)
--~Oref,T
CBr~pret
2. Anodic overpotentials (cathodic back reaction neglected)
i = toref,v
dNt
--
- -
=
dy
(o..o)exp
~ ' - " ~ - - _ 7]ref
eBr',ref
dc~
DiFci dr
0
[14]
Nt -- --zi
R--'-T dy
Di ----s
dy ~- 'OyCi
[15]
w h e r e Ni is the flux of species i. The flux of species i
occurs by three mass transfer processes: ionic migration, diffusion, and convection and is given by
Newman (15), for example, shows that at high
Schmidt numbers, the component of fluid velocity in
Table IV. Electrode kinetic parameters from simplified forms
of current density expressionsfor cathodic ~ref
Table V. Electrode kinetic parameters from simplified forms
of current density expressionsfor anodic Tlref
2.303
Kinetic
mechanism
V-H
Rate-determining step
Volmer, -~re~
~-- ~0.1V
Volmer, -~ret
Tafel slope
(in volts)
(2 - av)F
RT
RT
(1
2.303
Reaction
orders
qBrs
qs~"
2iOref,V
1
-1
2iOre t,H
1
0
(i - ~H)F
---~,,~0.1V
Heyrovsky
Tafel
line intercept
Kinetic
mechanism
Rate determining step
V-H
Volmer, ~re~ --~0.1V
Volmer, ~e~ ~-~0.1V
Heyrovsky
aH)F
-
RT
(I cz~)F
21Oret,H
1
0
21oret,s
1
0
equal
V-T
equal
Volmer
Tafel
Approximately
equal
RT
(1 - ~v)F
RT
0
0
~Or~,V
~
0
io,e~,T
fOre~.T
1
1
0
O
V-T
Volmer
Tafel
Approximately
equal
Tafel
line intercept
Reaction
orders
pBrs
PBr-
a~F
RT
(1 + aH)F
RT
(i + aH)F
RT
Approximately
-
Approximately
Tafel slope
(in volts)
2/Oref,v
0
I
2~orer,~
0
2
2ioret.n
0
2
2iOref,V
0
1
tOre~,V
0
1
$Oref,T
0
2
toret,v
0
1
avF
RT
~vF
RT
2F
~
RT
avF
RT
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II00
May 1983
J. Electrochem. Sac.: E L E C T R O C H E M I C A L S C I E N C E A N D T E C H N O L O G Y
the d i r e c t i o n a l n o r m a l to the electrode surface is given
infinitelythindiffusedoublelayerat the electrodesurface
by
vy
"
--al2 (42/v) ~l=y2
-
[16]
r
w h e r e y is the distance from the electrode into the
electrolyte in a direction n o r m a l to the electrode s u r face. This coordinate can be r e w r i t t e n in dimensionless
f o r m as
F(oF
=
I ~
/
r o
T
[17]
td
[18]
"1"
0
E
Z
Potential of
the workingelectrode
Potentialin the solutionat
~
y=O
or
"-- Y/~D
$
where ~D is the diffusion layer thickness and is given
by
=
(3DR)l/3
~ )l/s
~
Potentialdistributionwithinthe solution
[19]
S u b s t i t u t i o n of Eq. [I5], [16], and [18] into Eq. [14]
yields
Bulk
o
=
~o
#2re
solution
I
d2ci
dei
Di--~- -I- 3DR~2
Diffusion
Layer
d'-'~--
dSr + de,d__T d~
+ z,DiP [
0
= o
[20]
0
-
'
I
I
[
Y = ~D
I
Y = Yre
~._
Fig. 2. Schematic of the solution potential profile near an electrode
which is the m a t e r i a l balance equation used h e r e for
species i. The e l e c t r o n e u t r a l i t y condition
ZZiCi -- 0
t
[21]
completes the set of i -t- 1 equations to be solved subject to the b o u n d a r y conditions to obtain ci and r
Boundary Conditions in the Bulk Solution
Bulk solution conditions are assumed to exist at
Y ~ Yre w h e r e Yre is the n o r m a l distance f r o m the RDE
to the reference electrode (see Fig. 1). The c o n c e n t r a tion o f species i in the b u l k solution is designated as
Ci,r~f and the p o t e n t i a l in the b u l k solution at y ---- Yre
is d e s i g n a t e d as ere.
Thus, the b o u n d a r y conditions in the b u l k solution
are:
at y = Yre
Ci -- CLref
[22]
"- ere -- aset value (0.01V, e.g.)
[23]
where CLref must satisfy the electroneutrality condition
(Eq. [21]).
Boundary Conditions at the Electrode Surface
The b o u n d a r y conditions at the e l e c t r o d e surface are
as follows:
aty = 0
nF ----Nt
[24]
Zzici : 0
[25]
!
~met : a set value (0.1V, e.g.)
J
[26]
w h e r e N i i s given b y Eq. [15] with vy = 0 and i is
given b y either Eq. [8] o r [9].
The potentials in the b o u n d a r y conditions given by
Eq. [23] and [26] a r e shown s c h e m a t i c a l l y in Fig. 2
for a r o t a t i n g disk electrode being o p e r a t e d anodically.
Note that the p o t e n t i a l difference b e t w e e n the w o r k ing e l e c t r o d e a n d the r e f e r e n c e electrode ( ~ e t -- r
is t h e q u a n t i t y set b y a potentiostat. In the model,
however, both Omet and ~re a r e set i n d e p e n d e n t l y such
t h a t t h e i r difference is equal to t h a t set b y t h e p o t e n tiostat.
Solution Technique
The i -F 1 set of g o v e r n i n g equations for ci and
(Eq. [20] and [21]) can be solved n u m e r i c a l l y subject
to the b o u n d a r y conditions (Eq. [22]-[26]) once
values have been specified for the following p a r a m eters: Ci,ref, Di, #o, v, Y, U ~ si, n, t y p e of r e f e r e n c e electrode, Si,re, el,re, Yre, iOreLj, aV, all, (if necessary) ~2, and
Cmet, and ~re (again, only ~met -- d2re is i m p o r t a n t ) .
N e w m a n ' s (15) n u m e r i c a l technique [see also White
(17)] can be used to solve this set of equations which
yields values for the v a r i a b l e quantities r CBr-,O, and
CBr2,O. These values t o g e t h e r with e i t h e r Eq. [8] or [9]
and the values for ~rnet, ~re, and the calculated value
Ure f yield values for the c u r r e n t d e n s i t y i at a specified
value of ~met - - ~ r e or Ilref calculated according to Eq.
[13].
Results and Discussion
The m o d e l p r e s e n t e d above can be used to p r e d i c t
the c u r r e n t d e n s i t y for a set p o t e n t i a l difference b e tween the w o r k i n g electrode and a reference electrode
( ~ m e t - - ~Pre) once the p a r a m e t e r values have been set.
A l t e r n a t i v e l y , the m o d e l could be used to d e t e r m i n e
the kinetic p a r a m e t e r s b y c o m p a r i n g the p r e d i c t e d
c u r r e n t d e n s i t y values to observed values and finding
the p a r a m e t e r values that minimize the s u m of the
squares of the difference b e t w e e n the p r e d i c t e d and
m e a s u r e d c u r r e n t densities. However, the model is
used h e r e to p r e s e n t p r e d i c t e d c u r r e n t d e n s i t y - p o t e n tial curves which illustrate some of the capabilities of
the model. Also, to s i m p l i f y the presentation, the trib r o m i d e ion f o r m a t i o n reaction is neglected c o m p l e t e l y ( a d d i t i o n a l w o r k which includes the f o r m a tion of B r 3 - m a y be p r e s e n t e d in the f u t u r e ) .
Table VI presents the fixed p a r a m e t e r values used
in the m o d e l to produce the p r e d i c t e d c u r r e n t - p o t e n tial curves shown in Fig. 3-7. F i g u r e 3 presents the
effect of v a r y i n g the m a g n i t u d e of the exchange c u r r e n t densities in the V - H m e c h a n i s m w i t h iOref,H/iOref,V
= 1.0. As shown in Fig. 3, an o r d e r of m a g n i t u d e
change in the exchange c u r r e n t densities yields noticea b l y different cathodic c u r r e n t - p o t e n t i a l curves. F i g u r e 4 shows the effect of the m a g n i t u d e of the e x change c u r r e n t densities for the V - T m e c h a n i s m with
the V o l m e r step controlling (iOref,V/iOref.W = 0.01).
Notice that the p r e d i c t e d limiting c u r r e n t densities
depend on the m a g n i t u d e of the exchange c u r r e n t d e n sities e v e n though t h e V o l m e r step is rate controlling.
Note also t h a t the limiting c u r r e n t d e n s i t y p r e d i c t e d
b y Levich equation (15)
iLD = --0.62FCBr2,ref (,a~')
1/=( DBr2 ) 2/3
.
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Vo/. 130, No. 5
BROMINE/BROMIDE ELECTRODE REACTION
1101
Table VI. Fixed parameter values
-20
Parameter
9
0
9
O
(~
-18
cK+,ref = 2.01 x
Reference concentrations
10~
mol/cma
CBr-,ref = 1.0 X 10 -5 m o t / c m
cso~2-,ref
Diffusion and
stotchiometric
=
1.0
a
-16
X 1 0 -8 m o l / c m a
D~+ = 1.957 x 10 -~ c m ~ / s e c ~
coefficients
S~ + =
Dso~--'=
1;065
D~,- = 2.084
x
DB,~ = 1.2 x 10-~ em~/seo~
=
(r
Kinematic viscosity (~)
(T)
Equilibrium electrode potential ( U ~ )
Reference electrode
t i o n ~ (/#re).
~
-14
g
-12
.~
-10
2
SBr~ =
Miscellaneous:
Pure solvent density
1.0 g/eraa
1.0 x 10~ em-~
298.15 K
0.904V, for reference concentrations and a saturated calomel
X
X
X
X
X
~0-~A/crn 2, io~T~
10-:ZA/cm 2, io,~f.T=
10-3A/cm 2, i0,eLT=
10-4A/cm 2, i0ref.T =
10-SA/crn2~ ioreLT=
1.
"i.
1.
1.
1
•
X
X
X
X
10~A/cm ~
10~
~
10-~A/cm 2
10-~A/cm 2
10-3A/crn 2
,o_
o
_o,
2--0._
E
0.01
Newman (15).
Osipov et at. (1).
Picked
1.
1.
1.
1.
1.
-- 1
reference electrode
posi-
io,~tv =
Joref.v=
io,em =
imet.v=
Jaref.v=
d"
0
X 10 "~ e m ~ / s e c a
sso, '~- = 0
10-~ c m ~ / s e c a
SBr-
Temperature
i
Value
-4
for c o n v e n i e n c e ,
is n o t o b t a i n e d u n t i l the values of the exchange c u r r e n t
densities are l a r g e r t h a n expected. This means that
the exchange c u r r e n t density for the Tafel reaction
is p r o b a b l y l a r g e r t h a n the exchange c u r r e n t density
for Volmer reaction b y more t h a n two orders of m a g nitude.
F i g u r e 5 shows the effect of the Heyrovsky transfer
coefficient (~H) in the V - H m e c h a n i s m for the case
w h e r e the Heyrovsky step is r a t e controlling and Fig. 6
shows the effect of ~v w h e n the Volmer step is controlling. Both figures show that transfer coefficients
change the shape of the entire curve.
F i g u r e 7 presents the effect of av on the predicted
c u r r e n t d e n s i t y - p o t e n t i a l curves for the V - T mechanism. Notice that the s m a l l exchange c u r r e n t density
for the Tafel step indicating that it is the r a t e - c o n trolling step p r e v e n t s a noticeable change in the
curve w h e n ~v is changed from 0.I to 0.9.
Qualitative comparison of predicted c u r r e n t d e n s i t y - p o t e n t i a l curves based on the model presented
here to those given b y Osipov et al. (1) reveals that
either the V - H or the V - T m e c h a n i s m with the Volmer
step rate controlling m a y be a n acceptable mechanism
for the B r 2 / B r - electrode reaction. Additional r a w
data are necessary for a complete comparison which
m a y p e r m i t a more definitive statement about the
B r J B r - mechanism,
-2
0
0
-0.1
-0,2
Overpotential,
-0.3
-0.4
r/fef (V)
Fig. 4. Effect of a change in the exchange current density magnitude on cathodic current density curves predicted with the model
for V-T kinetics. Volmer reaction rate controlling. ~v ----- 0.5, rotation speed ~ 1000 rpm.
-16
-14
-12
<
E
v
-10
~6 -6
-4
-16--
i
i
i
7
-2
-14
0
E -12
I
0
-0.1
I
|
-0,2
-0,3
-0.4
Overpotential, r/,ef (V)
E,E -10
Fig. 5. Effect of a change in Heyrovsky transfer coefficient value
on cathodic current density curves predicted with the model for
V-H kinetics. Heyrovsky reaction rate controlling, ioref,v = 1 X
10-1 A/cm 2, ioref,ff = I X 10 - 2 , rotation speed = 1000 rpm.
-S
c~ -6
_41
Conclusion
o
-0.1
-0.2
-0.3
Overpotential, r/tel (V)
-0.4
-0.5
Fig. 3. Effect of the magnitude of the exchange current densities
for the V-H mechanism with iOreLff/iOreLV ~ 1.0, ~V ~ ~n
0.5, rotation speed ~ 1000 rpm.
The c u r r e n t d e n s i t y expressions presented here for
the V - H a n d V - T mechanisms can be used to predict
c u r r e n t d e n s i t y - o v e r p o t e n t i a l curves which are similar
to data. Qualitative comparison of the predictions of
the model presented here to data shows that either
the V - H mechanism or the V - T m e c h a n i s m with the
Volmer reaction controlling is a n acceptable m e c h a nism for the B r 2 / B r - reaction.
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J. EIectrochem. Sac.: E L E C T R O C H E M I C A L S C I E N C E A N D T E C H N O L O G Y
1102
E
-16
-16
-14
-14
E
-12
Transfer Coefficients
Aav = 0.1, aH= 0.5
Oar= 0.5, all= 0.5
13av= 0.9, CON=0.5
-4
~
...6.
0
-4
-2
-2
0
0
i
-0.1
i
-0.2
Overpotential,
i
-0.3
0
0
-0.4
-0.2
Acknowledgments
-0.3
-0,4
r/Fef (V)
Fig. 7. Effect of a Change in Volmer transfer coefficient value on
cathodic current density curves predicted with the model for V-T
kinetics using two sets of exchange current density values, ioref,v
- - I • 10 - 1 A/cm 2, iOref,T = I • 10 - 3 A/cm 2 (- - - ) ; ioref,v
=
| X 10-3 A/cm 2, iOref,T = 1 X 10-1 A/cm 2 (
,).
Vy
This w o r k was s u p p o r t e d b y N S F G r a n t N u m b e r
ENG-7908071 a n d b y the Center for E n e r g y and M i n eral Resources of Texas A&M University.
M a n u s c r i p t s u b m i t t e d Oct. 14, 1982; r e v i s e d m a n u script r e c e i v e d Jan. 3, 1983.
A n y discussion of this p a p e r will a p p e a r in a
cussion Section to be p u b l i s h e d in the D e c e m b e r
JOUaNAL. All discussions for the D e c e m b e r 1983
cussion Section should be s u b m i t t e d b y Aug. 1,
-0.1
Overpotential,
r/F~f (V)
Fig. 6. Effect of a change in Volmer transfer coefficient value on
cathodic current density curves predicted with the model for V-H
kinetics. Volmer reaction rate controlling, ioref,V = 1 • 10 - 3 A /
cm2, iOref,H - - i • 10 - 1 A/cm 2, rotation speed - - 1000 rpm.
Dis1983
Dis1983.
LIST OF SYMBOLS
0.51023
s t a n d a r d concentration of species i ( m o l / c m 3)
concentration of species i at distance y f r o m
electrode ( m o l / c m 3)
Di
diffusion coefficient of species i (cm2/sec)
DR
diffusion coefficient of l i m i t i n g r e a c t a n t (cmS/
sec), chosen to be Br~ h e r e
F
F a r a d a y ' s constant, 96,487 C / m o l
i
c u r r e n t d e n s i t y ( A / c m 2)
iLD
l i m i t i n g c u r r e n t d e n s i t y d e t e r m i n e d b y convection and diffusion, value given b y Levich e q u a tion ( A / c m 2)
iOstd.J exchange c u r r e n t d e n s i t y for reaction j at s t a n d a r d concentrations ( A / c m 2)
ioref, j exchange c u r r e n t d e n s i t y for reaction j at r e f erence concentrations ( A / c m e)
Keq
e q u i l i b r i u m constant for homogeneous t r i b r o m i d e f o r m a t i o n r e a c t i o n (cm3/mol)
kb
b a c k w a r d reaction r a t e constant for t r i b r o m i d e
homogeneous r e a c t i o n (sec - 1 )
kf
f o r w a r d reaction rate constant for t r i b r o m i d e
homogeneous reaction (cm~/mol-sec)
Ni
flux of species i (mol/cm2-sec)
n
n u m b e r of .electrons t r a n s f e r r e d in the o v e r a l l
e l e c t r o d e r e a c t i o n (n -- 2 h e r e )
Pi
anodic reaction o r d e r of species i
qi
cathodic reaction o r d e r of species i
R
u n i v e r s a l gas constant, 8.3143 J / r e a l K
si
stoichiometric coefficient for species i in o v e r a l l
electrochemical reaction
T
t e m p e r a t u r e (K)
U
s t a n d a r d potential of an electrode reaction on
h y d r o g e n e l e c t r o d e scale (V)
U%e
s t a n d a r d p o t e n t i a l of r e f e r e n c e e l e c t r o d e r e a c tion on h y d r o g e n electrode scale (V)
U%ef e q u i l i b r i u m p o t e n t i a l difference b e t w e e n w o r k ing electrode and reference electrode e v a l u a t e d
w i t h chosen reference concentrations (V)
ul
m o b i l i t y of species i ( c m 2 - m o l / J - s e c )
a
Ci,std
ci(y)
// / ~ c /
-12
I
-8
"~
(~
I
-lO
~,~ -10
9~
I
May 1983
component of velocity of e l e c t r o l y t e flow in a
direction n o r m a l to the electrode surface ( c m /
se C)
y
distance from electrode in a direction n o r m a l to
the electrode surface (cm)
zi
charge on species i
aj, (I -- aj) anodic and cathodic t r a n s f e r coefficients of
reaction j
71j
p o w e r on concentration t e r m for species i in r e action j in expressions for exchange c u r r e n t
densities
6D
diffusion l a y e r thickness (cm)
~ref
o v e r p o t e n t i a l expressing d e p a r t u r e of electrode
r e a c t i o n d r i v i n g force (Cruet ~ ~o) f r o m e q u i l i b r i u m value corresponding to chosen reference
concentrations (V)
~o
e q u i l i b r i u m surface coverage of a d s o r b e d b r o mine
k i n e m a t i c viscosity of the electrolyte (cm2/sec)
dimensionless distance f r o m electrode surface
(Y/6D)
~o
p u r e solvent d e n s i t y ( g / c m 3)
r
p o t e n t i a l in the solution as sensed b y a r e f e r ence e l e c t r o d e (V)
ere
potential in the b u l k solution at the reference
electrode as sensed b y a reference e l e c t r o d e at
Y = Yre (V)
~o
potential in the e l e c t r o l y t e i m m e d i a t e l y a d j a cent to w o r k i n g electrode surface as sensed b y a
reference eIectrode (V)
~met
p o t e n t i a l of the w o r k i n g electrode (RDE) w i t h
respect to the same t h e r m o d y n a m i c potential
scale as the reference electrode in the solution
(v)
disk r o t a t i o n a l speed ( r a d / s e c )
Subscripts
H
R
re
ref
std
T
V
o
identifies the value of a q u a n t i t y associated with
the H e y r o v s k y reaction
denotes q u a n t i t y associated w i t h the l i m i t i n g
reactant
refers to the reference electrode
identifies a v a l u e associated with a chosen set
of reference concentrations (e.g., CLref, iOref,.i)
identifies a value associated w i t h a chosen set of
s t a n d a r d concentrations
identifies the value of a q u a n t i t y associated with
the Tafel reaction
identifies the value of a q u a n t i t y associated with
the V o l m e r reaction
subscript, denoting a q u a n t i t y w h i c h is e v a l u a t e d w i t h the c o n c e n t r a t i o n a d j a c e n t to the electrode surface (except for 0o and po)
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VoZ. I30, No. 5
BROMINE/BROMIDE ELECTRODE REACTION
REFERENCES
1. O. R. Osipov, M. A. Novitskii, Yu. M. Povarov, and
P. D. Lukovtsev, Elektrokhimiya, 8, 327 (1972).
2. L. J. J. Janssen and J. G. Hoogland, Electrochim.
Acta, 15, 1677 (1972).
3. G. Faita, G. Fiori, and T. Mussini, ibid., 13, 1765
(1968).
4. J. Llopis and M. Vasquez, ibid., 6, 167, 177 (1962).
5. W. D. Cooper and R. Parsons, Trans. Faraday Soc.,
66, 1968 (1970).
6. W. J. Albery, A. H. Davis, and A. J. Mason, Faraday
Discuss. Chem. Soc., 56, 317 (1973).
7. D. Allard and R. Parsons, Chem.-Ing.-Tech., 44,
201 (1972).
8. I. Rubinstein, J. Phys. Chem., 85, 13 (1981).
9. V. S. Vilinskaya and M. R. Tarasevich, Elektro-
1103
khimiya, 7, 1515 (1971).
10. V. S. Bagotzky, Yu. B. Yassilyev, J. Weber, and
J. N. Pirtskhalava, J. Electroanal. Chem. Interfacial Electrochem., 27, 31 (1970).
11. M. W. Breiter, Electrochim. Acta, 8, 925 (1963).
12. N. A. Balashova and K. E. Kazarinov, Electroanal.
Chem., 3, 135 (1969).
13. R. F. Lane and A. T. Hubbard, J. Phys. Chem., 79,
808 (1975).
14. L. J. J. Janssen and J. G. Hoogland, ELectrochim.
Acta, 15, 941 (1972).
15. J. S. Newman, "Electrochemical Systems," Prentice-Hall, Englewood Cliffs, NJ (1973).
16. S. E. Lorimer, M.S. Thesis, Texas A&M University,
College S,tation, TX (1982).
17. R, E. White, Ind. Eng. Chem. Fundam., 17, 367
(1978).
Technical Notes
I
I
I
A Technique for Calculating Shunt Leakage and Cell Currents in
Bipolar Stacks Having Divided or Undivided Cells
E. A. Kaminski and R. F. Savinell*
Department of Chemical Engineering, The University of Akron, Akron, Ohio 44325
The design of high-output voltage electrochemical
batteries calls for series electrical connection of separate cells. Likewise, in scaling up electrosynthesis
processes, simplicity of construction may warrant the
building of reactors containing several cells arranged
in series. A convenient design for supplying reactants
to a cell assembly is to parallel feed the cells with
electrolyte distributed by an inlet manifold and collected by an outlet manifold. The electrolyte-filled
piping furnishes secondary series electrical connections
among the cells, Consequently, the ionic currents generated by electrode processes are driven by the intercell potential gradients of the assembly through conductive paths in the piping. These ionic shunt or bypass currents short-circuit each cell in the assembly
causing power loss, corrosion, current inefficiency, and
nonuniform cell-to-cell current distribution. The magnitude of the bypass current in an assembly is a function of the number of cells in the assembly, cell voltage
(which includes current dependent polarizations),
conductivity of the electrolyte, and the geometry of
the electrolyte feed system. Thus, bypass current is a
parameter in the design of assemblies.
Analysis of bypass current proceeds by applying
Kirchhoff's laws to electrical circuit analogs of assemblies. The analog circuits are devised to represent
current flow paths by standard electrical components.
Most investigators (!) model assemblies by linear,
passive, d-c networks, the work of Katz (2) and
Onishchuk (3) being exceptions. Katz incorporates into
the analog imaginary Zener diodes with different polarization characteristics in the forward and backward
current direction. The diodes account for the polarizations of the anodic and cathodic reactions which occur
at the ends of the shunt paths. Onishchuk (3) models
* Electrochemical ~ociety Active Member.
Key words: shunt currents, bipolar cell, separated cells, current
distribution.
a cell by a voltage source in series with a n o n l i n e a r
current dependent resistive element.
Analysis of the analog results in a large set of
simultaneous linear or nonlinear equations. Two
methods have usually been employed in obtaining solutions. One is simultaneous solution of the governing
equations (1, 2, 4, 5). The other is approximating the
system of equations by a single differential equation
(3, 6). The continuous approximations apply to assemblies with a sufficiently large number of cells.
This study presents a treatment of bypass current
based on the solution of finite difference equations.
The formalism is applied to assemblies of divided and
undivided cells. The analysis yields a set of compact
equations for determining branch currents in the circuit analog. Implementation of this computational
procedure requires a minimum of programming effort
and computer storage as simultaneous solution of
equations is avoided. The same equations can be used
to simulate an assembly of cells operating as energy
or substance producers.
Theory
Circuit analogs are constructed making the usual
assumptions (2, 4, 5, 6). Surfaces of the electrolyte
distribution system are nonconducting. The electrolyte
flow paths are conducting and are represented by resistor elements. All resistance paths between the cells
and a manifold are identical, and each manifold segment between a pair of branches is represented by an
identical resistor. Cells of assemblies are assumed to
have a linear polarization characteristic, while each
cell is represented as an ideal voltage source in series
with this resistor. Electrodes offer no resistance to
current flow. The electrolyte in a cell compartment is
assumed to be at a uniform potential throughout.
An analog circuit of an N cell assembly of two-compartment cells which incorporates these assumptions
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