American Journal of Analytical Chemistry, 2011, 2, 93-103
doi:10.4236/ajac.2011.22010 Published Online May 2011 (http://www.SciRP.org/journal/ajac)
Approximate Solution of Non-Linear Reaction Diffusion
Equations in Homogeneous Processes Coupled to Electrode
Reactions for CE Mechanism at a Spherical Electrode
Alagu Eswari, Seetharaman Usha, Lakeshmanan Rajendran*
Department of Mathematics, The Madura College (Autonomous), Madurai, India
Received October 18, 2010; revised January 16, 2011; accepted January 21, 2011
Abstract
A mathematical model of CE reaction schemes under first or pseudo-first order conditions with different
diffusion coefficients at a spherical electrode under non-steady-state conditions is described. The model is
based on non-stationary diffusion equation containing a non-linear reaction term. This paper presents the
complex numerical method (Homotopy perturbation method) to solve the system of non-linear differential
equation that describes the homogeneous processes coupled to electrode reaction. In this paper the approxi-
mate analytical expressions of the non-steady-state concentrations and current at spherical electrodes for
homogeneous reactions mechanisms are derived for all values of the reaction diffusion parameters. These
approximate results are compared with the available analytical results and are found to be in good agreement.
Keywords: Non-Linear Reaction/Diffusion Equation, Homotopy Perturbation Method, CE Mechanism,
Reduction of Order, Spherical Electrodes
1. Introduction
Microelectrodes are of great practical interest for quanti-
tative in vivo measurements, e.g. of oxygen tension in
living tissues [1-3], because electrodes employed in vivo
should be smaller than the unit size of the tissue of inter-
est. Microelectrodes having the geometry of a hemi-
sphere resting on an insulating plane are difficult to fab-
ricate, but their behavior is easily predicted [4]. They
also have advantages in electrochemical measurements
of molten salts with high temperature [5]. Microelec-
trodes of many shapes have been described [6]. Micro-
electrodes of simple shapes are experimentally preferable
because they are more easily fabricated and generally
conformed to simpler voltammetric relationships. Those
shapes with restricted size in all superficial dimensions
are of special interest because many of these reach true
steady-state under diffusion control in a semi infinite
medium [7]. Nevertheless, there is interest in microelec-
trodes of more complicated shapes, only because the
shapes of small experimental electrodes may not always
be quite as simple as their fabricators intended. Moreover,
and ironically, complex shapes may sometimes be more
easily modeled than simpler ones [8]. However, many
applications of microelectrodes of different shapes are
impeded by lack of adequate theoretical description of
their behavior.
As far back as 1984, Fleischmann et al. [9,10] used
microdisc electrodes to determine the rate constant of
coupled homogeneous reactions (CE, EC’, ECE, and
DISPI mechanisms). Fleischmann et al. [9] obtained the
steady-state analytical expression of the concentration of
the species HA and H by assuming the concentration of
the specie A is constant. Also measurement of the cur-
rent at microelectrodes is one of the easiest and yet most
powerful electrochemical methods for quantitative me-
chanistic investigations. The use of microelectrodes for
kinetic studies has recently been reviewed [11] and the
feasibility demonstrated of accessing nano second time
scales through the use of fast scan cyclic voltammetry.
However, these advantages are earned at the expense of
enhanced theoretical difficulties in solving the reaction
diffusion equations at these electrodes. Thus it is essen-
tial to have theoretical expressions for non steady state
currents at such electrodes for all mechanisms.
The spherical EC’ mechanism was firstly solved by
Delmastro and Smith [12]. In electrochemical context
Diao et al. [13] derived the chronoamperometric current

A. ESWARI ET AL. 94
at hemispherical electrode for EC’ reaction, whereas
Galceran’s et al. [14] evaluated shifted de facto expres-
sion and shifted asymptotic short-time expression for disc
electrodes using Danckwerts relation. Rajendran et al. [15]
derived an accurate polynomial expression for transient
current at disc electrode for an EC’ reaction. More re-
cently, Molina and coworkers have derived the rigorous
analytical solution for EC’, CE, catalytic processes at
spherical electrodes [16]. Fleischmann et al. [17] dem-
onstrate that Neumann’s integral theorem can be used to
numerically simulate CE mechanism at a disc electrode.
Dayton et al. [18] also derived the spherical response
using Neumann’s integral theorem. In this literature
steady-state limiting current is discussed in [19]
In general, the characterization of subsequent homo-
geneous reactions involves the elucidations of the me-
chanism of reaction, as well as the determination of the
rate constants. Earlier, the steady-state analytical expres-
sions of the concentrations and current at microdisc elec-
trodes in the case of first order EC’ and CE reactions
were calculated [9]. However, to the best of our knowl-
edge, till date there was no rigorous approximate solu-
tions for the kinetic of CE reaction schemes under first or
pseudo-first order conditions with different diffusion
coefficients at spherical electrodes under non-steady-
state conditions for all possible values of reactio-
n/diffusion parameters E
,


S , 1S

, 1E

, 2S

, ,
1

2

and 2E

have been reported. The purpose of this
communication is to derive approximate analytical ex-
pressions for the non-steady-state concentrations and
current at spherical electrodes for all possible values of
parameters using Homotopy perturbation method.
2. Mathematical Formulation of the
Problems
At a range of Pt microelectrodes, the electroreduction of
acetic acid, a weak acid, is strutinized by as in a usual
CE reaction scheme. This reaction is known to proceed
via the following reaction sequence [9]:
1
2
2
HAH A
1
HH
2
k
k
e









(1)
where 1 and 2 are the rate constants for the forward
and back reactiuons respectively and are related to an-
other by the known equilibrium constant for the acid
dissociation [9]. The initial boundary value problems for
different diffusion coefficients () can be
written in the following forms [9]:
k k
HA H A
,,DDD
2
HAHAHA HA
HA1 HA2HA
2
2ccDc
Dkc
trr r
 
 
 
2
HHHH
H1HA
2
2ccDc
Dkc
trr r
 
 
 2HA
kcc
(3)
2
AAAA
A1HA
2
2ccDc
Dkc
trr
r
 
 

2AH
kcc (4)
where HA are the diffusion coefficient of
the species , 1 and 2 are the rate
constant for the forward and back reactions respectively
and HAHA are the concentration of the species
HA, H and A. These equations are solved for the follow-
ing initial and boundary conditions:
H A
, and DD D
HA, H and A
, and cc c
k k
H HHAHAA A
0 ; , , tcccccc



 (5)
HHA A
; 0, 0, 0
S
rrcdc drdcdr


A
c
(6)
HHHAHAA
; , , rccccc



  (7)
where S is the radius of the spherical electrode. We
introduce the following set of dimensionless variables:
r
HA HA
HA HA
HA HA
12
2
HA HA
22
12HA
HA HA HA
22
1HA 2A
1S1
HA HA H
22
1HA 2H
2S2
HA HA A
, , , ,
, , ,
,
, ,
,
S
S
SS
ES
SS
E
SS
E
cccr
uvw r
ccc
Dt DD
DD r
krkc c r
DDc
kcrkc r
D Dc
kcrkcr
D Dc

















(8)
where , u,v,w

and

represent the dimen-
sionless concentrations and dimensionless radial distance
and dimensionless time parameters respectively.
2
2
2
ES
uu u
uv



 
 

w
(9)
2
1
11
2
2
ES
vv v
uv


 
 
 

1
w 
(10)
2
2
22
2
2
ES
www
uv


 
 
 
 2
w  (11)
where
E

, S

, 1
E

, 1S

, 2
E

and 2S

are the di-
mensionless reaction/diffusion parameters and 1

, 2

are dimensionless diffusion coefficients. The initial and
boundary conditions are represented as follows:
0, 1; 1; 1uvw


  (12)


1, 0; 0; 0vu w 

 (13)
kcc (2)
, 1; 1; 1uvw

 (14)
The dimensionless current at the microdisc electrode

A. ESWARI ET AL.95
can be given as follows:


H1
=
SS
InFADrdvd



 (15)
3. Analytical Expression of Concentrations
and Current Using HPM
Recently, many authors have applied the HPM to various
problems and demonstrated the efficiency of the HPM
for handling non-linear structures and solving various
physics and engineering problems [20-25]. This method
is a combination of homotopy in topology and classic
perturbation techniques. The set of expressions presented
in Equations (9)-(14) defines the initial and boundary
value problem. The homotopy perturbation method [26-32]
is used to give the approximate solutions of coupled
non-linear reaction/diffusion Equations (9) to (11). The
dimensionless reaction diffusion parameters E

, S

,
1
E

, 1S

, 2
E

and 2S

are related to one another,
since the bulk solution is at equilibrium in the non-steady
state. Using HPM (see Appendix A and B), we can obtain
the following solutions to the Equations (9) to (11).



 

2
1
1
1
1
2
1
1
1
(,) 1exp
14
π
1
exp1 exp2
1
+exp1 exp
12
1
1
exp
4
π
 
  




 









 






 
 












 




E
E
u
erfc
erfc 


(16)


1
2
1
1
1
11
,1 2
1
11
1exp4
2π

 






 








 
 

 

E
verfc



(17)
 





2
2
12
12 2
2
1
1
2
2
12
2121
2
1
1
,1 exp4
π
1
1
exp4
π
+exp1exp
1
2


 
 





 

 




 


















  




E
E
w
erfc 2
(18)
The Equations (16)-(18) satisfies the boundary condi-
tions (12) to (14). These equations represent the new
approximate dimensionless solution for the concentration
profiles for all possible values of parameters
E

, S

,
1
E

, 1S

, 2
E

, 2S

, 1

and 2

.
e
From Eq
dimnsionless curr
is as follows:
uations
ent, w
(15)
and (17), we can obtain the hich
HH
1
11
0.28217 0.56419
1
SS
E
Ir nFDAC 

 


  (19)
Equation (19) represents the new approximate expres-
sio
Fleischmn et al. hri
s:
n for the current for small and medium of parameters.
4. Comparison with Fleischmann Work [9]
an [9]ave deved the analytical ex-
pressions of dimensionless steady-state concentrations u
and v as follow


11
1
11
11ES

11
1
1exp1
E
E
ES
u

 



 



 (20)




 







 
11
11
11
1e xp 1
E
ES
ES
v  
 
1
 
E










(21)
not arrived upon in the third specie A.
The normalized current is given by
Fleischmann assumed that the concentration profiles
of w is constant. So the definite solution for concentra-
tion profiles of w is



HH
11 11
11
1
SS
EE ES
ES
Ir nFDAC 

 
 




(22)
When 11
E
S



the above equation becomes
 
11
1
1exp1
EE
u1







 





 (23)
 
11
1
1exp 1
EE
v







 





 (24)
The normalized current is given by
HH 1
1
SSE E
Ir nFDAC

1


 (25)
Previously, mathematical expressions pertaining to
steady-state analytical exprestrations
annt at micated by
Fleischmann et al. [9]. In addition, we have also pre-
sions of the concen
d currerodisc electrodes were calcul

A. ESWARI ET AL. 96
sented an approximate solution for the non-steady state
concentrations and current.
5. Discussions
Equations (16)-(18) are the new and simple approximate
soisomers calculated
using Homotopy perturbation method for th
boundary conditions Equations (12)-(14).
approxim
lution of the concentrations of the
e initial and
The closed
ate solution of current is represented by the
Equation (19). The dimensionless concentration profiles
of u versus dimensionless distance

are expressed in
Figures 1(a)-(d). From these figures, we can infer that
the value of the concentration decreases when

and
distance

increases when 1
E

. Moreover when
1
E

 and 1

, the conceains the steady-
res 2(a)-(dalized cn-
isomers v us values pa-
ntr
), the
fo
ation att
norm
r vario
state value. In Figu
tration profiles of
once
of
rameters are plotted. From these figures, it is inferred
that the concentration v increases abruptly and reaches
the steady-state value when 5

. In Figures 1(a)-(d)
a)-(d),values of dimensionless concentrations
u and v for various values of
and 2( the
E

,
E

and

and for
11


are reported and a satisfactory agreement with the
available [9] estimates of Fleischmann et al. is noticed
when

is large. Figures 3(a)-(d) show the normalized
dimensional concentration prof w in file o

space cal-
culated using Equation (18). The plot was constructed for
various values 20.1, 1
E


11and

. Frm these
s it is confirmed that the value of the concentration
profile of w increases when
o
figure

and 2
E

increases. Al-
so from the Figures 1(a)-(d) and 2(a)-(d), it is evident
that the concentration of species HA and H increases
when the radius of the electrode (S
r) decreases. There-
fore, the use of thef the smdius is clearly
advangeous for the study of CE reaction mechanism. The
concentration of specie A deeases wn the radius of
the electrode decreases. It reaches the steady state value
when 1
electrode o
c
all
h
r
e
a
r


. The dimensionless current log

versus

for various values of 1
E

is givenin Figure 4. From
these figure, it is evident that the value of the current

decreases abruptly and reaches the steady-state value.
Dimensionless distance ρ
Dimensionless concentration u
1.005
1
0.995
0.99
0.98
τ = 1
τ = 0.5
τ = 0.1
γ
E
= 0.1, ε
1
= 0.01
Dimensionless distance ρ
Dimensionless concentration u
1
0.96
0.94
τ = 1
τ = 0.5
τ = 0.1
98
1 1.5 2 2.5 3 3.5 4 4.5 5
5
0.98
0.975
0.97
τ = 10, 100
1 1.5 2 2.5 3 3.5 4 4.5 5
τ = 10, 100 γ
E
= 0.1, ε
1
= 0.5
0.
0.92
0.9
0.88
0.86
(a) (b)
Dimensionless distance ρ
Dimensionless concentration u
1 1.5 2 2.5 3 3.5 4 4.5 5
1
0.9
0.95
τ = 1
τ = 10, 50
τ = 0.5
τ = 0.1
γ
E
= 1, ε
1
= 0.01
0.85
0.8
0.75
0.7
Dimensionless distance ρ
Dimensionle
τ = 10, 100
0.7
ss concentration u
1 1.5 2 2.5 3 3.5 4 4.5 5
1
0.9
0.95
τ = 1
τ = 0.5
τ = 0.1
γ
E
= 1, ε
1
= 0.5
0.85
0.8
0.75
0.65
0.6
0.55
(c) (d)
Figure 1. Normalized concentration u at microelectrode. The concentrations were computed using Equation (16) for various
values of

and for some fixed small value of 1

E
when the reaction/diffusion parameter and dimensionless diffusion coef-
ficient (a) , 1
0.1 0.01


E (b) , 1
0.1 0.5


E (c)1 , 1
10.0


E

 (d) 5, 1 10.


E

. The key to the graph: ( __ )
represen+ts Equation (16) and () represents Equation (23) [9].

A. ESWARI ET AL.97
Dimensionless concentration v
1 1.1 1.2 1.3 1.4
1
τ = 1
τ = 10, 50
τ = 0.5
τ = 0.1
γ
E1
= 0.1, ε
1
= 0.01
0.8
0.6
0.4
0.2
0
Dimensionless distance ρ
1.5 1.6 1.7 1.8 1.9 2
Dimensionless distance ρ
(a) (b)
Dimensionless concentration v
3 3.5 4 4.5 5
1
τ = 1
τ= 10, 50
τ = 0.5
τ = 0.1
γE1 = 0.1, ε1 = 0.5
0.8
0.6
0.4
0.2
1 1.5 2 2.5
0
Dimensionless distance ρ
Dimensionless concentration v
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
1
τ = 1
τ = 10, 100
τ = 0.5
τ = 0.1
γ
E1
= 1, ε
1
= 0.01
0.8
0.6
0.4
0.2
0
Dimensionless distance ρ
Dimensionless concentration v
1 1.5 2 2.5 3 3.5 4
1
τ = 1
τ = 10, 100
τ = 0.5
τ = 0.1
γ
E1
= 1, ε
1
= 0.5
0.8
0.6
0.4
0.2
0
(c) (d)
Figure 2. Normalized concenr various
values of
tration v at microelectrode. The concentrations were computed using Equation (17) fo

and for some fixed small value of

E
when the reaction/diffusion parameter and dimensionless diffusion coef-
ficient (a) ,
11
0.1 0.01


E (b) , 11
0.1 .50


E (c)1 , 11
10.0


E

 (d) .5,
11
10


E

. The key to the graph: ( __ )
represen+) rep ts Equation (17) and (resents Equation (24) [9].
Dimensionless distance ρ
1 1.5 2 2.5 3 3.5
1
τ = 1
τ = 0.01
τ = 0.5
τ = 0.1
γ
E2
= 0.1, ε
1
= 0.01
1.035
1.03
1.025
1.02
1.015
1.01
1.005
0.995 4 4.5 5
Dimensionless concentration w
mensionless concentration w Di
Dimensionless distance ρ
1 1.5 2 2.5 3 3.5
1
τ = 1
τ= 0.01
τ = 0.5
τ = 0.1
γE2 = 0.1, ε1 = 0.5
1.003
1.0025
1.002
1.0015
1.001
1.0005
(a) (b)
0.9995
4 4.5 5
0.999

A. ESWARI ET AL. 98
Dimensionless distance ρ
Dimensionless concentration w
1.08
1.07
1 1.5 2 2.5 3 3.5 4 4.5 5
1
τ = 1
τ = 0.01
τ = 0.1
γ
E2
= 1, ε
1
= 0.01
0.99
1.06
1.05
1.04
1.03
1.02
1.01
Dimensionless distance ρ
Dimensionless concentration w
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
1
τ = 1
τ = 0.01
τ = 0.1
γ
E2
= 1, ε
1
= 0. 5
1.045
1.04
1.005
0.995
1.035
1.03
1.025
1.02
1.015
1.01
(c) (d)
Figure 3. Normalized concentration w at microelectrode. The concentrations were computed using Equation (18) for various
values of

,
21
and for some fixed value of the reaction/diffusion parameter and dimensionless diffusion coefficient (a)
0.1 0.01

 
E (b) , 1
0.1 0.5
2

  (c) ,
21
10.01


E

 (d) ,
21
10.5


E

.
E
Dimensionless time τ
Dimensionless Current log ψ
0 1 2 3 4 5 6 7 8 9 10
ε
1
= 0.01
10
4
γ
E1
= 10
10
3
10
2
10
1
10
0
γ
E1
= 1
γ
E1
= 0.5
γ
E1
= 0.1
Dimensionless time τ
Dimensionless Current log ψ
0 1 2 3 4 5 6 7 8 9 10
ε
1
= 0.5
γ
E1
= 10
10
3
10
2
10
1
10
0
γ
E1
= 1
γ
E1
= 0.5
γ
E1
= 0.1
a
(a) (b)
Figure 4. Vriation of normalized non-steady-state current response log

as a function of the dimensionless time

for
various values 1) 10.5


10.01


(b
of

E
and foa) r the fixed values of (. The curvwere
he key to the represents Equation (19) and (+) resention (25) [9].
when the values of
es computed using Equation (19).
ts EqT
graph: ( __ )repua
10.1 E

. Also, the value of the
current

2S

, 1

, 2

and

e
and c
r va
dgements
ppor
DS
r. M
o
hiaga
tics, The Madura
based on the Homotopy perturba-
his mthod can be easily extended to find
the concetrationsurrent for all mechanism for all
s forious complex boundary condi-
e
rks suted by the Department of Science
gy (T) Government of India. The au-
ank M. S. Meenakshisundaram, Secre-
ura Cllege Board, T. V. Krishnamoorthy,
S. Trajan Head of the Department of
College, Madurai, for their
tion m
mi
tions.
This
and
thors als
tary
e
croelectro
7. Acknow
wo
Tech
o
, Th
pal an
a
thod. T
n
de
l
wa
nolo
th
e Mad
d
increae reaction diffusion pa-
rameter
ses when th
1
E

increa
6. Conclusions
The time dependent non-linear reaction/diffusion equa-
tions for spherical microelectrodes for CE mechanism
has been formulated and solved using HPM. The primary
result of this work is simple approximate calculation of
concentration profiles and current for all values of fun-
damental parameters. We have presented approximate
solutions corresponding to the species HA, H and A i
terms of the parameters of
ses
n Princi
Mathem
, S

, 1
E

, 1S

, 2
E

,
E


A. ESWARI ET AL.99
constant encouragement. It
hn
alytical Chemistry, Marcel Dekker,
New York, Vol. 11, 1972, p. 85.
I. A. Silver, “Microelectrodes and
trodes used in Biology,” In: D. J. G. Ives and G. J. Jane
is our pleasure to thank the
referees for their valuable comments.
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A. ESWARI ET AL.101
ppendix A: Solution of the Equations (9) to
1) Using Homotopy Perturbation Method
this Appendix, we indicate how Equations (16) to (18)
this paper are derived. To find the solution of Equa-
ons (9) to (11) we first construct a Homotopy as fol-
ws:
A
(1
In
in
ti
lo

2
2
2
2
2
1
20
ES
d ududu
pdd
d
dudu du
pu
dd d
 


 









vw 
(A1)

2
1
12
2
1
11
2
2
1
20
ES
dvdv dv
pdd
d
dvdv dv
pu
dd d

 



 









1
vw  
(A2)

2
2
22
2
1dwdw dw pdd
d

 





(A3
2)
2
22 2
2
20
ES
dw dw dw
puvw
dd
d

 
 


 


and the initial approximations are as follows:
00 0
0; 1; 0; 1uv w

 (A4)

00 0
1; 0; 0, 0vduddwd     (A5)
00 0
; 1; 0; 1uv w

 (A6)
0; 0; 0; 0
ii i
uvw

 (A7)

1; 0; 0, 0
ii i
vdud dwd     (A8)
; 0; 0; 0 1,2,
ii i
uvw i

  (A9)
and
(A10)
Substituting Equation (A10) into Equations (A1) and
(A2) and (A3) and arranging the coefficients of powers
uations.
23
01 2 3
23
01 2 3
23
01 2 3
uupu pupu
vv pvpv pv
wwpw pwpw
 

 

 





p, we can obtain the following differential eq
2
0000
2
2
:
d ududu
pdd
d
 

0 (A11)
2
1111
000S
vw    (A12)
2
0000
1
12
2
:0
d vdvdv
pdd
d

 


 (A13)
2
11111
110
2
2
:0
ES
dvdv dv
pu
dd d100
vw


 

  (A14)
and
2
0000 2
22
2
:0
d wdwdw
pdd
d

 


 (A15)
2
11211
220
2
2
:0
ES
d wdwdw
pu
ddd


 
200
vw 

  (A16)
Subjecting Equations (A11) to (A16) to Laplace
transformation with respect to results in

2
00
0
2
210
dudu su
d
d



 (A17)
2
00
0
2
11
21
0
dvdvsv
d
d   


(A18)
2
00
0
2
22
21
0
dwdw sw
d
2
2
:0
E
d ududu
pu
dd d

  

and
d  

(A19)



11 2
11
1
2
21
0
s
E
S
du due
su
dsssd


 
(A20)










11 2
1 11 1
1
2
111
21
s
S E
dvdv se
v
dsss d











(A21)
0
 
11
2
2 11 2
1
2
222
21
0
s
S E
dwdw se
w
dsss d

 
 








(A22)
Now the initial and boundary conditions become
00 0
0; 1; 0; 1uv w


  (A23)


000
1; 0; 0, 0vduddwd 

 (A24)
000
; 1; 1; 1usvsws

 (A25)
0; 0; 0; 0
ii i
uvw


  (A26)


0
i
dwd  1; 0; 0,
ii
vdud   (A27)

; 0; 0; 0 1,2,
ii i
uvw i

  (A28)
where s is the Laplace variable and an overbar ind
a Laplace-t
to
icates
ransformed quantity. Solving equations (A17)
(A22) using reduction of order (see Appendix-B) for

A. ESWARI ET AL. 102
initial and
boundary conditions (A26) to (A28), we can find the
following results
solving the Equation (A20), and using the

01 us
  (A29)

 









1
1
1
122
1
11
11
11
11
11 1
s
SS
E
ss
SS
e
uss
s
ee
sss

 
 


 
 

 




(A30)
and



11
0
1s e
vss



 (A31)

 



1
1
1
11 11
122 22
1
1
1
11
2
s
SS EE
s
S
e
v
s
sss
e
s



 











(A32)
and

01ws
  (A33)












1
2
2
1
221 2
122
2
1
212
221
1
21 2
s
SS E
s
S
s
S
e
wss s
e
s
e
ss


 



 



 


 (A36)
After putting Equations (A29) and (A30) into Equa-
tion (A35) and Equations (A31) and (A32) into Equation
(A36) and Equations (A33) and (A34) into Eq
(A37). Using inverse Laplace transform [33], the final
results can be described in Equations (16) to (18) in the
text. The remaining components of and
1
22 1
According to the HPM, we can conclude that
 
01
1
lim
p
uuuu 

 (A35)
 
v  
 

(A34)
01
1
lim
pvvv


 
01
1
lim
p
www 

w (A37)
uation

n
ux


n
vx
is deter-be completely determined such thterm
mined by the previous term.
Appendix B
In this Appendix, we derive the solution of Equation
(A20) by using reduction of order. To illustrate the basic
concepts of reduction of order, we consider the equation
at each
2
2
dd
d d
cc
PQc


R


where P, Q, R are function of r. Equation (A20) can be
lified to
(B1)
simp

 11
1
0
s
Ee





2
11
1
2
dd 2
d
dS
uu
su sss
 




Using reduction of order, we have
 (B2)
2; PQ

s


and


11
1s
E
S
e
Rsss





 


(B3)
Let ucv

(B4)
Substitute (B4) in (B1), if u is so chosen that
d
20
d
cPc


 (B5)
Substituting the value of P in the above Equation (A7)
become
1c


(B6)
The given Equation (B3) reduces to
11
vQvR



 (B7)
where
2
11
0,
42
PP R
QQ R

c

 (B8)
Substituting (B8) in (B7) we obtain,
 
11s
S E
S
e
vsv
 
ss s

  



 

(B9)
Integrating Equation (B9) twice, we obtain
2
S
SE
vAe Be
s





 



11
 

1
s
Se
 




 (B10)
g (B6) and (B10) in (B4) we have,
2
11
S
ss




Substitutin

A. ESWARI ET AL. 103
 

1
1
2
s
S
S
e





2
1
1
1
S
S
E
Ae Be
us
ss








 



(B11)
ing the boundary conditions Equations (A27) and
), we can obtain the value of the constants A and B.
Substituting the value of the constants A
Equation (B11) we obtain the Equation (A
we can solve the other differential Equations (A17),
(A and
or
Appendix C
N

Us
(A28
and B in the
30). Similarly
18), (A19), (A21)(A22) using the reduction of
der method.
omenclature
Symbols
HA
H
c Concentration of the species H (mole
c Concentration of the species HA
cm–3)
Concentration of the spec
Bulk concentration of the species HA
(molecm–3)
Bulk concentration of the species H
–3
(molecm–3)
(molecm–3)
A
c ies A (molecm–3)
HA
c
H
c (molecm )
Bulk concentration of the species A
A
c
HA
D
Diffusion coefficient of the species HA (cm2sec–1)
H
D
Diffusion coefficient of the species H (cm2sec–1)
A
D
Diffusion coefficient of the species A (cm2sec–1)
D Diffusion coefficient (cm2sec–1)
R Radial distance(cm)
e (s)
Rate constant for the forward reactions (cm3/molesec)
r
u, v, w s concentrations (dimensionless)
T Tim
1
k
2
k Rate constant for the backward reactions
3
(cm/molesec)
S
r Radius of spherical electrode (cm)
Distance in the radial direction (cm)
Dimensionles

Dimensionless radial distance (dimensionless)

Dimensionless time (dimensionless)
S
I
Current density at a sphere (ampere/cm2)
A
F Faraday constant (Cmole –1)
n
Greek sb
Area of the spherical electrode (cm2)
Number of the electron (dimensionless)
ymols
1

Dimensionless diffusion coefficient
(dimensionless)
2

Dimensionless diffusion coefficient
(dimensionless)
11
22
, ,
, ,
,
ES
ES
ES






Dimensionless reaction/diffusion parameters
(dimensionless)