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Journal of Electroanalytical Chemistry

journal homepage: www.elsevier.com/locate/jelechem

Analytical solutions for the study of homogeneous first-order chemicalkinetics via UV–vis spectroelectrochemistry

A. Molinaa,⁎, E. Labordaa, J.M. Gómez-Gila, F. Martínez-Ortiza, R.G. Comptonb

a Departamento de Química Física, Facultad de Química, Regional Campus of International Excellence “Campus Mare Nostrum”, Universidad de Murcia, 30100 Murcia,Spainb Department of Chemistry, Physical & Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, UK.

A R T I C L E I N F O

Keywords:Analytical theoryUV–vis spectroelectrochemistryReaction mechanismConcentration profilesConstant potential chronoamperometry

A B S T R A C T

Analytical solutions are reported for the identification and study of homogeneous chemical reactions via UV–visspectroelectrochemistry. Expressions are presented for the concentration profiles and the absorbance response ofany the species of the CEC mechanism when applying a potential-pulse perturbation under semi-infinite lineardiffusion conditions. With this theory it is possible to relate easily the absorbance transient to the physico-chemical dynamics of the redox system. In particular, the influence of the chemical reactions involving theoxidized and/or the reduced species will be investigated both in the normal and parallel beam configurations.Expressions for the reaction schemes CE, EC and E can be deduced as particular cases. The EC solutions areapplied to the study of facilitated ion transfers across liquid|liquid interfaces where the likely different diffu-sivity of the ion in the two phases has been considered.

1. Introduction

In electrochemical measurements it is frequently found that theelectroactive species and their products are involved in chemical re-actions in solution. Hence, for the comprehensive analysis of the chargetransfer process, the reaction mechanism must be elucidated and thechemical kinetics and equilibrium constants must be determined. Forthis, monitoring the species involved or affected by the electrode re-action via spectroscopic methods (UV–vis, IR, Raman, EPR [1,2], etc.[3–6]) can be a very convenient, complementary approach. Indeed, thealteration of the species concentration profiles upon driving the elec-trode process is the ultimate response of the system. Thus, spectro-electrochemical (SEC) methods have already been successfully appliedto the study of isomerisations, dissociation/association reactions andelectrocatalytic processes [3], as well as to the investigation of chargetransfer reactions at liquid|liquid interfaces [7].

The SEC signal is evidently a function of the concentration profilesof the absorbing species. Thus, theoretical expressions for such profilesare needed in order to derive analytical relationships between the ab-sorbance transient and the physicochemical processes that define thedynamic response of the system (namely, mass transport, chargetransfer reaction and homogeneous chemical processes in the absenceof adsorption). Assuming that a beam of light is incident normal to anoptically transparent macroelectrode placed in the yz plane (Scheme

1a), the UV–vis absorbance response of species i, Ai, N(λ, t), is propor-tional to the integral of its concentration profile along x [8]:

∫=λ t ε λ c x t dxA ( , ) ( ) ( , )i i

l

i,N0 (1)

where l is the path length of the beam through the electrolyte and εi(λ)is the wavelength dependent extinction coefficient of species i.

When the light beam samples the solution parallel to the electrode,then [9]:

∫=

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟−

λ t w

dxA ( , ) log

10i w

ε λ l c x t,P

0

( ) ( , )i i

(2)

where w is the height of the light beam in the parallel configuration andl the path length over the electrode surface (Scheme 1b).

So far, analytical solutions for the SEC signal have been reported forthe E and catalytic mechanisms. Thus, when the electrogenerated spe-cies R is the photoactive one:

+ ⇄− ∗eO R (3)

the normal beam configuration (N) absorbance of the E mechanism isgiven by Eq. (1) (and also by integration of the current with respect to

http://dx.doi.org/10.1016/j.jelechem.2017.10.031Received 23 June 2017; Received in revised form 11 October 2017; Accepted 14 October 2017

⁎ Corresponding author.E-mail address: [email protected] (A. Molina).

Journal of Electroanalytical Chemistry 819 (2018) 202–213

Available online 16 October 20171572-6657/ © 2017 Elsevier B.V. All rights reserved.

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time, that is, by the converted charge) that leads to [8,10]:

∫= =+

∗λt ε λ

F AI t dt Dt

πε λ

ce

A ( ) ( ) ( ) 2 ( )1

tR ηR,N

E R0

O

(4)

with = − ′η E E( )FRT

0 . In the parallel configuration (P), the SEC re-sponse of the E mechanism under limiting current conditions and withthe restrictions of large w-values (compared to the diffusion layer) andlow absorbance [9] is given by [11]:

= ∗λt lw

ε λ Dtπ

cA ( ) 2 ( )R,PE,lim

R O (5)

In the case of the first-order catalytic mechanism where thehomogeneous reaction is irreversible:

+ ⇄

+ ⎯ →⎯⎯ + ′

− ∗

∗

e

Z

O R

R Z Okcat (6)

the following expressions are obtained for normal configuration (N)SEC [12]:

= ∗∗

∗λ t ε λ c Dk c

k c tA ( , ) ( ) erf( )Z

ZR,Ncat,lim

R Ocat

cat(7)

and for the parallel configuration with large w-values and low absor-bance [13]:

= ∗∗

∗λ t ε λ c lw

Dk c

k c tA ( , ) ( ) erf( )R,Pcat,lim

R Ocat Z

cat Z(8)

For other charge transfer mechanisms, the theoretical treatment ofthe A(λ, t) response has been carried out by means of numerical simu-lations [14–17] (except for the EC mechanism under galvanostaticconditions [18,19]) to the best of our knowledge. Here, analytical so-lutions for the concentration profiles of the CEC mechanism are re-ported for the SEC response upon the application of a constant potentialpulse:

⇄ = =

+ ⇄

⇄ ′ ′ = = ′ ′

− ′

′

′

′

Z O K c c k k

O e R E

R Z K c c k k

; / /

;

; / /

k

kZeq

Oeq

k

k

Req

Zeq

2 1

0

2 1

2

1

2

1

(9)

Note that by studying the CEC mechanism we can evaluate the ef-fects of chemical reactions affecting any of the electroactive species, aswell as analyse the E (K→0, K′≫1), CE (K′≫1) and EC′ (K→0) me-chanisms as particular cases. As will be shown, having at our disposalanalytical expressions for the concentration profiles will enable simpleand rapid interpretation of the experimental spectro-electrochemicalresponse.

2. Theoretical treatment: the CEC mechanism

The differential equation system that describes the evolution withtime of the concentration profiles of the CEC mechanism under semi-infinite, linear diffusion when applying a potential pulse E is:

∂∂

= ∂∂

− +

∂∂

= ∂∂

+ −

∂∂

= ′∂∂

− ′ + ′

∂∂

= ′∂∂

+ ′ − ′′ ′

′

′

ct

D cx

k c k c

ct

D cx

k c k c

ct

D cx

k c k c

ct

Dcx

k c k c

Z ZZ O

O OZ O

R RR Z

Z ZR Z

2

2 1 2

2

2 1 2

2

2 1 2

2

2 1 2 (10)

= ∀> → ∞

⎫⎬⎭

∞ = ∞ =∞ = ∞ =′

t xt x

c c c cc c

0,0,

( ) , ( )( ) 0 , ( ) 0

eq eqO O Z Z

R Z (11)

t > 0,x=0:

=c e c(0) (0)ηO R (12)

⎛⎝

∂∂

⎞⎠

= − ′⎛⎝

∂∂

⎞⎠

D cx

D cx

O

0

R

0 (13)

⎛⎝

∂∂

⎞⎠

= ⎛⎝

∂∂

⎞⎠

=′cx

cx

0Z

0

Z

0 (14)

where D and D′ are the diffusion coefficients of the oxidized (O and Z)and reduced (R and Z′) species, respectively. Note that considering inthe theory that the values of D and D′ can be different is important forits application to the study of ion transfer reactions between water andorganic phases [7], where species may show very different diffusivity ineach phase [20].

Note that it has been assumed that only the reagent species Z and Oare initially present and also that the spectro-electrochemical cell isconsiderably larger than the depletion layer and semi-infinite diffusionholds (Eq. (11)). Further, the presence of sufficient supporting elec-trolyte to suppress migration is presumed [21] as well as that theduration of the SEC experiment is controlled to avoid the onset ofnatural convection [22].

The boundary value problem Eqs. (10)–(14) will be solved analy-tically in this work by extending the procedure given in ref. [23], ob-taining expressions for the concentration profiles of all the chemicalspecies. First, it is convenient to define the functions:

= += −

ς c cϕ c cK

O Z

Z O (15)

and:

Scheme 1. Normal-beam (N) and parallel-beam (P) configurations.OTE: optically transparent electrode; WE: working electrode.

A. Molina et al. Journal of Electroanalytical Chemistry 819 (2018) 202–213

203

′ = +′ = − ′

′

′

ς c cϕ c K c

R Z

R Z (16)

that fulfil the following differential equations:

∂∂

=∂∂

∂ ′∂

= ′∂ ′∂

ςt

Dς

xςt

Dς

x;

2

2

2

2 (17)

∂∂

=∂∂

−∂ ′∂

= ′ ∂ ′∂

− ′ ′ϕt

Dϕ

xκ ϕ

ϕt

Dϕx

κ ϕ;2

2

2

2 (18)

with:

= +′ = ′ + ′

κ k kκ k k

1 2

1 2 (19)

and they are subject to the following boundary conditions:

= ∀> → ∞

⎫⎬⎭

→

⎧

⎨⎪

⎩⎪

∞ = = +′ ∞ =

∞ =′ ∞ =

∗

t xt x

ς ς c cςϕϕ

0,0,

( )( ) 0( ) 0( ) 0

eq eqO Z

(20)

t > 0,x=0:

⎜ ⎟⎛⎝

∂∂

⎞⎠

= −⎛⎝

∂ ′∂

⎞⎠

γς x t

xς x t

x( , ) ( , )2

0 0 (21)

⎛⎝

∂∂

⎞⎠

= − ⎛⎝

∂∂

⎞⎠

ς x tx K

ϕ x tx

( , ) 1 ( , )

0 0 (22)

⎜ ⎟ ⎜ ⎟⎛⎝

∂ ′∂

⎞⎠

= ⎛⎝

∂ ′∂

⎞⎠

ς x tx

ϕ x tx

( , ) ( , )

0 0 (23)

− = ⎛⎝

++ ′

⎞⎠

′ ′ + ′ς t ϕ e KK

K ς t ϕ(0, ) (0) 11

{ (0, ) (0)}η(24)

with

=′

γ DD

2(25)

In the present mathematical treatment, functions ϕ and ϕ′, whichaccount for the magnitude of the perturbation of the chemical equili-bria, are assumed to be independent of time such that Eq. (18) becomeinto [23,24]:

∂∂

= ′∂ ′∂

= ′ ′Dϕ

xκ ϕ D

ϕx

κ ϕ;2

2

2

2 (26)

Then, functions ϕ and ϕ′ are given by:

= −

′ = ′ −

⎫

⎬

⎪⎪

⎭⎪⎪

′′( )

( )ϕ ϕ exp x

ϕ ϕ exp x

(0)

(0)

κD

κD (27)

The above assumption has been reported to be very appropriate fordifferent reaction mechanisms [25] since it enables us to obtain simplerexpressions for the concentration profiles and the current response.

Taking into account the above, the surface boundary conditions forEq. (17) simplify to:

> =t x0, 0:

⎜ ⎟⎛⎝

∂∂

⎞⎠

= −⎛⎝

∂ ′∂

⎞⎠

γς x t

xς x t

x( , ) ( , )2

0 0 (28)

⎛⎝

∂∂

⎞⎠

=ς x t

x Kϕ

δ( , ) 1 (0)

r0 (29)

⎜ ⎟⎛⎝

∂ ′∂

⎞⎠

= −′

′ς x t

xϕ

δ( , ) (0)

r0 (30)

− = ⎛⎝

++ ′

⎞⎠

′ ′ + ′ς ϕ e KK

K ς ϕ(0) (0) 11

{ (0) (0)}η(31)

where δr and δr′ are the thickness of the so-called linear reaction layers[25], which reflect the extent of the region of the solution next to theelectrode|solution (or liquid|liquid) interface where chemical equili-brium conditions are perturbed by the heterogeneous charge transferprocess:

=

′ = ′′

δ

δ

rDκ

rDκ (32)

Note that δr(′)≤δ(′) [25].

Then, substituting Eqs. (29) and (30) into Eq. (31), one obtains:

⎜ ⎟− ⎛⎝

∂∂

⎞⎠

= ⎛⎝

++ ′

⎞⎠

⎧⎨⎩

′ ′ − ′ ⎛⎝

∂ ′∂

⎞⎠

⎫⎬⎭

ς Kδςx

e KK

K ς δςx

(0) 11

(0)rη

r0 0 (33)

The problem given by Eqs. (17), (20), (28) and (33) can be solved asindicated in the Appendix A introducing the variable changes:

=

′ =

=

′

s

s

χ κ t

xDt

xD t

2

2

(34)

and the following expressions are obtained for the concentration pro-files of the pseudo-species ς and ς′:

∑ ∑

∑ ∑

= ++ ′

⎧⎨⎩

−

∏− ⎫

⎬⎭

′ = −+ ′

⎧

⎨⎪

⎩⎪

−

∏− ′ ′

⎫

⎬⎪

⎭⎪

′

′

∗∗

++

=

∞

= =

∞

∗

++

=

∞

=

=

∞

( )

( )

ς ςς

K γe

χ

pd p e s

ς γς

K γe

χ

pd p e s

1

( 1)( ) s

1

( 1)( ) s

η KK

jlj

l mj m j j m

m

η KK

j

l

j

l

m j m j j mm

11

1

jCECj

1 0, ,

11

1

jCECj

1

0 , ,

(35)

such that:

+ ′ ′ = ∗D ς D ς D ς(0) (0) (36)

with:

=⎛

⎝⎜⎜

+ ′

+

⎞

⎠⎟⎟

++ ′

′++ ′

( )( )

χ κ tK γ e

K γe2

1 η KK

κκ

η KK

CEC

11

11 (37)

and:

= =

= =

+−

+ +

+− −

+ +

d d d

e e e

1;

1;

j j m j mj m

m m

j j m j mj m

m m

,0 , 2 ,2( )

( 1)( 2)

,0 , 2 ,2( 1)

( 2)( 3) (38)

=++

( )( )

pΓ

Γ

2 1j

j

j2

12 (39)

with =p π2/0 and pj pj+1=2(j+1).The surface gradients of pseudo-species ς and ς′ can be easily ob-

tained from Eq. (35)


204

= =

= = −

′

′ ′ ′′ ′ ′

∂∂

∂∂

∂∂ +

∂∂

∂∂

∂∂ +

∗

++ ′

∗

++ ′

( ) ( )( )

( ) ( )( ) ( )

( )

F χ

F χ

( )

( )

ςx

sx

ςs πDt

ς

K γe

ςx

sx

ςs πD t

γς

K γe

0 0

1

1CEC

0 0

1

1CEC

η KK

η KK

11

11 (40)

such that the solutions for ϕ and ϕ′ are (see Eqs. (27), (29) and (30)):

= −

′ = −

′

′′ ′ ′

+

+

∗

++ ′

∗

++ ′

( ) ( )

( )( )

( )

ϕ K F χ exp

ϕ γ F χ exp

( )

( )

δδ

ς

K γeCEC

xδ

δδ

ς

K γeCEC

xδ

1

1

rη K

K

rη K

K

11

r

11

r

(41)

with:

∑= −

∏= ⎛

⎝⎞⎠

⎛⎝

⎞⎠=

∞ +

=

F x x

pπ x x x( ) ( 1)

2exp

2erfc

2j

j j

l

j

l0

1

0

2

(42)

and =δ π D t and ′ = ′δ π D t correspond to the thickness of thelinear diffusion layer. From the above expressions, the concentrationprofiles of the species can be finally calculated considering the fol-lowing relationships:

=

=

−+

++

c

c

ς ϕK

ς ϕK

O 1

ZK1 (43)

=

=

′ ′ ′′

′ ′′

++

′−+

c

c

K ς ϕK

ς ϕK

R 1

Z 1 (44)

Regarding the current-potential response of the CEC mechanism,this is immediately obtained from the surface gradient of species ς (Eq.(40)):

=+ ′

∗

++ ′( )

I FADςδ

F χ

K γ e

( )

1 η KK

CECCEC

11 (45)

that under limiting current conditions (eη→0) becomes into:

=∗

I FADςδ

F χ( )CEClim

CEClim

(46)

with =χ κ tKCEC

lim 2 .Considering Eq. (1), the following explicit expressions for the ab-

sorbance response when the light beam is incident normal to the elec-trode surface are obtained by introducing Eqs. (35), (41), (43) and (44)in Eq. (1):

∑ ∑ ⎜

⎟⎜ ⎟

=+

⎧

⎨⎪

⎩⎪ + ′

⎛

⎝

⎜⎜⎜

−

∏

⎛⎝ +

−+

⎞⎠

⎛⎝

⎞⎠

⎞

⎠

⎟⎟⎟

+

+ − ⎛⎝

⎞⎠ + ′

⎫

⎬⎭

′

′

∗

++

=

∞

=

=

∞

+

++

( )

( )

A λε λ ς δ

π K γe

χ

pm

m Dt Dt

δδδ K γe

F χ

( , t)( )

11 K

2 1

1

( 1) d1

e2

p l2

l2

l K 1

1( )

O N

O

η KK

j

l

j

l

mj m

j mj

m

r

η KK

CEC

,

11

1

jCECj

1

0,

,1

2

11

(47)

∑ ∑ ⎜

⎟⎜ ⎟

=+

⎧

⎨⎪

⎩⎪ + ′

⎛

⎝

⎜⎜⎜

−

∏

⎛⎝ +

−+

⎞⎠

⎛⎝

⎞⎠

⎞

⎠

⎟⎟⎟

+

+ + ⎛⎝

⎞⎠ + ′

⎫

⎬⎭

′

′

∗

++

=

∞

=

=

∞

+

++

( )

( )

A λε λ ς δ

π K γe

χ

pm

m Dt Dt

δδδ K γe

F χ

( , t)( )

K1 K

2 1

1

( 1) d1

e2

p l2

l2

l 1

1( )

Z N

Z

η KK

j

l

j

l

mj m

j mj

m

r

η KK

CEC

,

11

1

jCECj

1

0,

,1

2

11

(48)

∑ ∑ ⎜

⎟⎜ ⎟

⎜ ⎟

′

=+ ′

⎧

⎨⎪

⎩⎪

−′

+ ′

⎛

⎝

⎜⎜⎜

−

∏

⎛⎝ +

−+ ′

⎞⎠

⎛⎝ ′

⎞⎠

⎞

⎠

⎟⎟⎟

+

+ ⎛⎝

′′⎞⎠ + ′

⎫

⎬⎭

′

′

∗

++

=

∞

=

=

∞

+

++

( )

( )

A λε λ ς δ

γπ

K

K γe

χ

pm

m D t D t

δδ K γe

F χ

( , t)( )

1 K2

1

( 1) d1

e2

p l2

l2

1

1( )

R N

R

η KK

jCEC

l

j

l

mj m

j mj

m

rη K

K

CEC

,

11

1

j j

1

0,

,1

2

11

(49)

∑ ∑ ⎜

⎟⎜ ⎟

⎜ ⎟

′

=+ ′

⎧

⎨⎪

⎩⎪

−+ ′

⎛

⎝

⎜⎜⎜

−

∏

⎛⎝ +

−+ ′

⎞⎠

⎛⎝ ′

⎞⎠

⎞

⎠

⎟⎟⎟

−

− ⎛⎝

′′⎞⎠ + ′

⎫

⎬⎭

′

′

′

′

∗

++

=

∞

=

=

∞

+

++

( )

( )

A λε λ ς δ

γπ K γe

χ

p m

m D t D t

δδ K γe

F χ

( , t)( )

1 K2 1

1

( 1) d1

e2

p l2

l2

11

( )

Z N

Z

η KK

jCEC

l

j

l

mj m

j mj

m

rη K

K

CEC

,

11

1

j j

1

0,

,1

2

11

(50)

Note that for values >′l/2 D t 2( ) (as it is the case here) it is foundthat the value of the sum in m does not change significantly; hence, inorder to avoid convergence issues of this sum, the term ′l D t/2 ( ) in Eqs.(47)–(50) can be replaced by 2 without compromising the accuracy ofthe results.

The SEC response in the parallel beam configuration, Ai,Pkss(λ, t),

can be calculated by numerical integration of Eq. (2) with ci(x, t) beinggiven by Eqs. (43)–(44). By comparison with rigorous solutions, it hasbeen found that the theoretical treatment developed above leads toaccurate values for (k1+k2)t > 5 and/or (k1′+k2′)t > 5 [26,27].

2.1. Diffusive-kinetic steady state solution

For relatively fast chemical kinetics ( >κ K2 t / 10, κ′t > 6), thediffusive-kinetic steady state (dkss) theoretical treatment can be used to


205

obtain simpler, closed-form expressions for the concentration profilesand the absorbance response [23]. Following the dkss approach, theform of the solutions for ϕ and ϕ′ are assumed to be given by Eq. (27)and, additionally, the mathematical form of ςdkss and ς′dkss are supposedto be equivalent to that of a species only subject to diffusion:

⎜ ⎟

⎜ ⎟= + − ⎛⎝

⎞⎠

′ = ′ ⎛⎝ ′

⎞⎠

∗ ∗ς ς ς ς xDt

ς ς xD t

( (0) ) erfc2

(0) erfc2

dkss dkss

dkss dkss

(51)

so that:

=

⎛⎝

⎞⎠

=′′

∂∂

−

∂∂

−

∗( )ςx

ς ςδ

ςx

ςδ

0

(0)

0

(0)

dkss dkss

dkss dkss

(52)

Taking the above into account and applying the surface boundarycondition (33), the following expressions are obtained for the surfacevalues ςdkss(0) and ς′dkss(0):

=

′ =

′

′

′

∗+ ⎛

⎝+ ⎞

⎠

+ + ⎛⎝

+ ⎞⎠

∗

+ + ⎛⎝

+ ⎞⎠

++ ′

′′

++ ′

′′

++ ′

′′

( )( )

( )

ς ς

ς ς

(0)

(0)

dkssK γ K

K γ K

dkss γ

K γ K

e

1 e

1 e

δδ

KK

η δδ

δδ

KK

η δδ

δδ

KK

η δδ

r 11

r

r 11

r

r 11

r(53)

and also for ϕdkss(0) and ϕ′dkss(0) (see Eqs. (29), (30) and (53)):

=⎛

⎝⎜

⎞

⎠⎟

′ =⎛

⎝⎜

⎞

⎠⎟

′

′′ ′

+ + ⎛⎝

+ ⎞⎠

+ + ⎛⎝

+ ⎞⎠

∗

++ ′

′′

∗

++ ′

′′

( )

( )

( )

ϕ K

ϕ γ

(0)

(0)

dkss δδ

ς

K γ K

dkss δδ

ς

K γ K

1 e

1 e

rδδ

KK

η δδ

rδδ

KK

η δδ

r 11

r

r 11

r(54)

Finally, the concentration profiles of the species O, R, Z, and Z′ arederived by introducing Eqs. (27), (51), (53) and (54) in the relation-ships (43) and (44):

=⎧

⎨⎩

−⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎫

⎬⎭

=⎧

⎨⎩

−⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎫

⎬⎭

′

′

+

⎛⎝

⎞⎠

+ ⎛⎝

− ⎞⎠

+ + ⎛⎝

+ ⎞⎠

+

⎛⎝

⎞⎠

− ⎛⎝

− ⎞⎠

+ + ⎛⎝

+ ⎞⎠

∗

++ ′

′′

∗

++ ′

′′

( )

( )

c 1

c 1

dkss ςerfc exp

γ K

dkss ςerfc exp

γ K

O 1 K

K

1 K e

ZK

1 K 1 K e

Dtδrδ

xδ

δδ

KK

η δδ

Dtδrδ

xδ

δδ

KK

η δδ

x2 r

r 11

r

x2 r

r 11

r

(55)

=⎧⎨⎩

⎫⎬⎭

=⎧⎨⎩

⎫⎬⎭

′

′

′

′ ′

+

⎛⎝

⎞⎠

+ ⎛⎝

− ⎞⎠

+ + ⎛⎝

+ ⎞⎠

′ +

⎛⎝

⎞⎠

− ⎛⎝

− ⎞⎠

+ + ⎛⎝

+ ⎞⎠

∗ ′′′ ′

++ ′

′′

∗ ′′′ ′

++ ′

′′

( )

( )

γ

c γ

cdkss ςK

K erfc exp

K γ K

dkss ςK

erfc exp

K γ K

R 1 1 e

Z 1 1 e

D tδrδ

xδr

δδ

KK

η δδ

D tδrδ

xδr

δδ

KK

η δδ

x2

r 11

r

x2 r

r 11

r

(56)

and, from Eq. (1) with l≫δ,δ′ and l≫δr,δr′, the contribution of eachspecies to the absorbance response in the normal beam configuration isdeduced:

=⎧

⎨⎩

−⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎫

⎬⎭

=⎧

⎨⎩

−⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎫

⎬⎭

′

′

+

+

+ + ⎛⎝

+ ⎞⎠

+

−

+ + ⎛⎝

+ ⎞⎠

∗ ++ ′

′′

∗ ++ ′

′′

( )

( )

( )

( )

A λ

ε λ ς δlδ γ K

A λ

ε λ ς δlδ γ K

( t)

( )1

1 K

K

1 K e

( t)

( )K

1 K 1 K e

dkss

O

πδδ

δδ

KK

η δδ

dkss

Z

πδδ

δδ

KK

η δδ

O,N2 r 2

r 11

r

Z,N2 r 2

r 11

r

(57)

=⎛

⎝⎜⎜

⎞

⎠⎟⎟

=⎛

⎝⎜⎜

⎞

⎠⎟⎟

′ ′′

′

′ ′ ′

+

+ ⎛⎝

⎞⎠

+ + ⎛⎝

+ ⎞⎠

+

− ⎛⎝

⎞⎠

+ + ⎛⎝

+ ⎞⎠

∗

′′

++ ′

′′

′

′ ∗

′′

++ ′

′′

( )

( )

A λ

ε λ ς δγ

K

K

γ K

A λ

ε λ ς δγ

K γ K

( t)

( ) 1 1 K e

( t)

( ) 1 1 K e

dkss

R

πδδ

δδ K

η δδ

dkss

Z

πδδ

δδ K

η δδ

R,N2 r

2

r 1 K1

r

Z ,N2 r

2

r 1 K1

r

(58)

With regard to the response in the parallel beam configuration,Ai,P

dkss(λ, t), this can be calculated numerically from Eqs. (2) and(55)–(56).

3. Results and discussion

With the expressions given in Section 2, the concentration profilesof all the species in the CEC mechanism when applying a potential pulseperturbation at a macroelectrode can be calculated as well as the SECresponse in both normal and parallel beam configurations. Given thegenerality of the reaction scheme considered, the analytical solutionscover situations where none, one or both electroactive species take partin homogeneous chemical reactions (CEC, EC, CE and E mechanisms)and where one or more species are photoactive. In the following sec-tions, the influence of the chemical rate and equilibrium constants willbe examined for some specific situations of interest. Limiting currentconditions will be considered in all cases (i.e., eη→0) since this enablesus to avoid the influence of uncertainties in the electrode kinetics andformal potential, the SEC response being only a function of the char-acteristics of the homogeneous chemical reactions.

In Fig. 1, the concentration profiles of all the species are plottedfrom which the reaction layer can be immediately identified as theregion close to the electrode surface where equilibrium conditions arebroken (that is, where cZ≠cO and cZ′≠cR in Fig. 1a since K=K′=1).Note that such a region is larger for reactants than for products inFig. 1a due to the slower kinetics of the preceding chemical reaction:κ′=10κ. In Fig. 1b, the constant K′ is assumed to be very large (=100)such that the formation of species Z′ is negligible and the system be-haves as a CE-like mechanism. On the other hand, the EC mechanismcorresponds to the limit case where the constant K is very small (=0.01in Fig. 1c).

3.1. E mechanism

The case of simple charge transfer processes (i.e., the E mechanism)where the species O and R do not undergo any chemical process insolution can be derived from the solutions obtained for the CEC me-chanism (Eqs. (43), (44), (47)–(50) and Eqs. (55)–(58)) by making K→0 and K′≫1. Thus, the following expressions are obtained for theconcentration profiles [8,25]:

⎜ ⎟

⎜ ⎟= ⎧⎨⎩

−+

⎛⎝

⎞⎠

⎫⎬⎭

=+

⎛⎝ ′

⎞⎠

∗

∗

γerfc

Dt

γγ

erfcD t

c c 1 11 e

x2

cc

1 ex

2

Eη

Eη

O O

RO

(59)

and for the normal-beam SEC signal for any applied potential (see Eq.(4)):


206

= −

=′

+

+

∗

∗

A λ

ε λ ς δlδ π γ

A λ

ε λ ς δ πγγ

( t)

( )2 1

1 e

( t)

( )2

1 e

E

O η

E

R η

O,N

R,N

(60)

3.2. CE-like mechanisms

⇄ = =

+ ⇄− ′

Z O K c c k k

O e R E

; / /

;

k

kZeq

Oeq

2 1

02

1

(61)

The CE mechanism is another particular case of the CEC schemewhere K′≫1; hence, the analytical expressions (47)–(50) for the ab-sorbance response in the normal beam configuration become into:

∑ ∑ ⎜

⎟⎜ ⎟

=+

⎧⎨⎩

+ +⎛

⎝⎜

−

∏⎛⎝ +

−+

⎞⎠

⎛⎝

⎞⎠

⎞

⎠⎟ +

+ − ⎛⎝

⎞⎠ + +

⎫⎬⎭

∗

=

∞

==

∞

+

A λε λ ς δ

K γe K πχ

p m

m Dt Dt

δδδ γe K

F χ

( , t)( )

11

11 (1 )

2 ( 1) d1

e2

p l2

l2

l K 11 (1 )

( )

O NCE

O

η jlj

lm

j m

j mj

m

rη CE

,

1

jCEj

10

,

,1

2

(62)

∑ ∑ ⎜

⎟⎜ ⎟

=+

⎧⎨⎩

+ +⎛

⎝⎜

−

∏⎛⎝ +

−+

⎞⎠

⎛⎝

⎞⎠

⎞

⎠⎟ +

+ + ⎛⎝

⎞⎠ + +

⎫⎬⎭

∗

=

∞

==

∞

+

A λε λ ς δ

π γeχ

p m

m Dt Dt

δδδ γe K

F χ

( , t)( )

K1 K

2 11 (1 K)

( 1) d1

e2

p l2

l2

l 11 (1 )

( )

Z NCE

Z

η jlj

lm

j m

j mj

m

rη CE

,

1

jCEj

10

,

,1

2

(63)

∑ ∑ ⎜

⎟⎜ ⎟

′= −

+ +

⎛

⎝

⎜⎜⎜

−

∏

⎛⎝ +

−+ ′

⎞⎠

⎛⎝ ′

⎞⎠

⎞

⎠

⎟⎟⎟

∗=

∞

=

=

∞

+

A λε λ ς δ π

γγe K

χ

pm

m D t D t

( , t)( )

21 (1 )

( 1) d1

e2

p l2

l2

R NCE

Rη

j

CE

l

j

lm

j m

j mj

m

,

1

j j

1

0

,

,1

(64)

with:

= + +χ κ tK

γe K2 (1 (1 ))ηCE (65)

and the dkss solutions (57) and (58) into:

=⎧

⎨⎩

−⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎫

⎬⎭

=⎧

⎨⎩

−⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎫

⎬⎭

+

+

+ + +

+

−

+ + +

∗

∗

( )

( )

A λ

ε λ ς δlδ K γ

A λ

ε λ ς δlδ K γ

( t)

( )1

1 K

K

1 K (1 ) e

( t)

( )K

1 K 1 K (1 ) e

dkss CE

O

πδδ

δδ

η

dkss CE

Z

πδδ

δδ

η

O,N, 2 r 2

r

Z,N, 2 r 2

r

(66)

Fig. 1. Concentration profiles (Eqs. (35) and (41), (43) and (44)) of the CEC mechanismunder different conditions of chemical kinetics and thermodynamics; note that cases b)and c) correspond to CE-like and EC-like mechanisms, respectively. D = D′, limitingcurrent conditions (eη→0).


207

′=

+ + +∗

A λε λ ς δ π

γγ

( , t)( )

21 K (1 K) e

dkss CE

Rδδ

ηR,N

,

r(67)

Fig. 2 shows the SEC response of a CE mechanism where species R isphotoactive. As the K-value increases, AR(λ, t) decreases since there islower ‘availability’ of the species O for the electrode reaction andsmaller amounts of the species R are generated. Eventually, when theinterconversion between Z and O is instantaneous (χ≫1, δr→0 andeη→0 in Eq. (67)), the following expression is obtained forAR,N

CeqE, lim(λ, t)

= ∗λ t Dtπ

ε λ ςA ( , ) 2 ( )R,NC E,lim

Req

(68)

which is equivalent to the expression for the E mechanism with the bulkconcentration of the electroactive reagent being ς∗.

It is worth mentioning that for the case considered in Fig. 2, theabsorbance response can be calculated, apart from the concentrationprofile (as in Eq. (58)), also from the current response since the onlyabsorbing species is the electrogenerated one that is stable in solution.Hence, as in the case of the E mechanism, the absorbance is propor-tional to the total charge passed by the electrode reaction:

∫= =λ t ε λ Q tFA

ε λFA

I t dtA ( , ) ( ) ( ) ( ) ( )R Rt

R,NCE

0CE,kss

(69)

where the expression for the current-potential-time response of the CEmechanism is given by (see Eq. (45) with K′≫1) [23]:

⎜ ⎟= ⎛⎝ + +

⎞⎠

∗I t FAς DK γe π t

F χ( ) 11 (1 )

1 ( )ηCE CE(70)

with:

= + +χ K γe κtK

2(1 (1 ) )ηCE (71)

Note that in the limit κ≫1, it holds that F(χCE)→1 and Eq. (69)with eη→0 leads to (68).

Another situation of interest is that where the photoactive species isone of the reactants initially present (i.e., O or Z) [28]. In such situa-tion, the normal beam configuration may not be appropriate to inspectthe chemical kinetics since the length of the ‘bulk solution’ is muchlarger than the reaction layer so that the absorbance response is scar-cely affected by the electrode process (see inset in Fig. 4). As shown inFigs. 3 and 4, the parallel configuration would be more suitable since by

decreasing the w -value one can sample a few tens of microns of thesolution adjacent to the electrode surface and so the SEC signal is morerevealing of the physicochemical dynamics. Thus, the SEC signal doesdepend significantly on the value of κ (see Fig. 4) and so it enables thedetermination of the chemical rate constants.

3.3. EC-like mechanisms

+ ⇄

⇄ ′ ′ = = ′ ′

− ′

′

′

′

O e R E

R Z K c c k k

;

; / /k

k

Req

Zeq

0

2 12

1

(72)

The EC mechanism can be viewed as a particular case of the CECmechanism where K→0; thus, for example, the dkss expressions for thenormal beam absorbance transients turn into:

= −⎛

⎝

⎜⎜ + + ′

⎞

⎠

⎟⎟∗

+ ′′′( )

A λε λ δ

lδ π γ K

( t)( )c

2 1

1 e

dkss EC

O OK

η δδ

O,N,

11

r(73)

=⎛

⎝⎜⎜

⎞

⎠⎟⎟

=⎛

⎝⎜⎜

⎞

⎠⎟⎟

′ ′′

′

′ ′ ′

+

+ ⎛⎝

⎞⎠

+ ⎛⎝

+ ⎞⎠

+

− ⎛⎝

⎞⎠

+ ⎛⎝

+ ⎞⎠

∗

′′

+ ′′′

′

′ ∗

′′

+ ′′′

A λ

ε λ δγ

K

K

γ K

A λ

ε λ δγ

K γ K

( t)

( ) c 1 1 e

( t)

( ) c 1 1 e

dkss EC

R O

πδδ

Kη δ

δ

dkss EC

Z O

πδδ

Kη δ

δ

R,N, 2 r

2

11

r

Z ,N, 2 r

2

11

r

(74)

This situation is particularly interesting here since the limitingcurrent chronoamperometry of EC-like mechanisms does not informabout the chemical process [8,25,29] and the use of SEC can be veryvaluable to detect and characterize the latter.

In Fig. 5 the SEC response of the EC mechanism is studied when thephotoactive species is the electrogenerated one (i.e., species R). Asexpected, the SEC signal is smaller than in the absence of homogeneouschemical reaction (E mechanism, grey line) and the decrease is moreapparent as the chemical kinetics is faster (larger κ′–values) and theK′–value decreases. In the limit of very fast chemical kinetics (κ′≫1),chemical equilibrium conditions hold and the normal configurationabsorbance is easily obtained by making δr′→0 in Eqs. (73) and (74);for example, the response under limiting current conditions when R isthe photoactive species is given by (see Eq. (74) with eη→0 and δr′→0):

Fig. 2. Dimensionless chronoamperometric SEC response of the CE mechanism with R being the absorbing species as a function of the chemical kinetics (κ) and thermodynamics (K)plotted using Eq. (64) for the normal configuration and Eqs. (2) and (44) for the parallel one. Other conditions as in Fig. 1.


208

=′ ′

+ ′∗λ t c ε λ D t

πK

KA ( , ) ( )2

1R,NEC ,lim

O Req

(75)

The AR,NECeq, lim-value obviously becomes null when the chemical

equilibrium is displaced towards the non-absorbing species Z (K′→0),whereas it is coincident with the response of the E mechanism (Eq. (4))in the opposite limit (K′≫1).

3.3.1. Ion transfer at ITIES

⇄

⇄

+ +∗

+∗

+

′ ′w o

o o

X ( ) X ( )

X ( ) XL( )L

κ K,

(76)

In the study of the transfer of an ion X+ across the interface betweentwo immiscible electrolyte solutions (ITIES), it is frequent to use a verylipophilic ligand L that facilitates the transfer (so-called facilitated iontransfer, scheme (76)). Also, the diffusivity of the ion in the water andorganic media can be very different (i.e., D≠D′), especially when li-quid membranes are employed. Both aspects are included in the theoryhere developed such that a facilitated transfer mechanism with

different diffusion coefficients can be studied with the expressions givenin Eqs. (73) and (74) with γ≠1.

In Fig. 6, the influence of the diffusivity of the ion in the organicphase (D′) on the spectro-electrochemical response of the facilitatedtransfer of a cation X+ is studied. As expected, Fig. 6a shows how X+

accumulates closer to the liquid|liquid interface (s=0) as the value ofD′ decreases. While this does not affect the absorbance in the normalconfiguration (see inset in Fig. 6), the SEC signal in the parallel con-figuration is quite sensitive and it shows a complex dependence withthe shape of the concentration profile (see Eq. (2)).

The cases above-considered point out how spectroelectrochemistrycan greatly assist in the elucidation and characterization of the kinetics(rate laws and rate constants), reaction mechanism (identification ofintermediates and products of charge transfer process) and masstransport conditions of electrochemical systems. First, SEC adds a newsource of information for the study of complex situations and for givingfurther consistency to the results of the electrochemical analysis.Specifically, the use of SEC is of particular value in those caseswhere the current response is not informative about some of the on-going physicochemical processes, such as comproportionation/

Fig. 3. Influence of the w-value on the chronoamperometric SEC response of the CE mechanism when (a) Z or (b) R are the absorbing species. Eqs. (2) and (43) and (44). Other conditionsas in Fig. 1.

Fig. 4. Dimensionless chronoamperometric SEC response of the CEmechanism with Z being the absorbing species as a function of thechemical kinetics (χ) and thermodynamics (K) plotted using Eq.(63) for the normal configuration (inset) and Eqs. (2) and (43) forthe parallel one. Other conditions as in Fig. 1.


209

disproportionation reactions in multi-electron transfers or the chemicalreactivity and diffusivity of the product of irreversible electrode pro-cesses. Finally, it is worth noting that SEC enables the determination ofthe absorbing properties of (highly) unstable species that are electro-generated in situ.

4. Conclusions

Analytical explicit expressions for the concentration profiles of thereaction mechanisms CEC, CE and EC have been deduced and applied tothe analysis of the spectroelectrochemical (SEC) UV–vis signal. With theexpressions here reported, it is possible to relate the absorbance tran-sient of any of the species to the physicochemical dynamics of thesystem (specifically, the mass transport and homogeneous chemicalkinetics) upon the application of constant potential pulse perturbationof any value. Two possibilities have been considered namely that thebeam light can be incident either in a normal or in a parallel way to the

electrode.The equations presented provide accurate results for (k1+k2)

t > 5. For faster chemical reactions, the diffusive steady state treat-ment (dkss) has also been employed to obtain very simple closed-formexpressions for the concentration profiles and for the absorbancetransients. The influence of the chemical kinetics on the SEC signalunder limiting current conditions (eη→0) has been analysed for the CEand EC mechanisms when only the electrogenerated species is photo-active. The latter case is particularly interesting since in such situationsthe electrochemical signal is not sensitive to the coupled chemical re-action. The value of the SEC parallel configuration, especially when thephotoactive species is initially present in solution, has also been dis-cussed. Finally, the study of ion transfer processes at ITIES via SEC hasbeen considered pointing out that large differences between the iondiffusivity in each phase has an apparent effect on the parallel-beamsignal in contrast to that in the normal configuration.

Fig. 5. Dimensionless chronoamperometric SEC response of the EC mechanism with R being the absorbing species as a function of the chemical kinetics (κ′) and thermodynamics (K′)plotted using Eq. (74) for the normal beam configuration and Eqs. (2) and (56) for the parallel one. Other conditions as in Fig. 1.

Fig. 6. (a) Dimensionless concentration profile and normal-beam SEC response (inset) and (b) parallel-beam SEC response associated with the free cation X+ in the organic phase for threedifferent values of D′: D′= D (black line); D′ = 0.1D (red line) and D′ = 0.01D (blue line). Eq. (74) for the normal beam configuration and Eqs. (2) and (56) for the parallel one and theconcentration profiles. Limiting current conditions (eη→0). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)


210

Acknowledgments

The authors greatly appreciate the financial support provided by theFundación Séneca de la Región de Murcia (Project 19887/GERM/15) aswell as by the Ministerio de Economía y Competitividad (Project CTQ-

2015-65243-P). EL thanks the Ministerio de Economía yCompetitividad for the fellowship “Juan de la Cierva-Incorporación2015”. JMGG thanks the Ministerio de Educación, Cultura y Deporte forthe fellowship ‘Ayuda de Formación de Profesorado Universitario2015’.

Appendix A. Theoretical treatment

We will make use of Koutecký's dimensionless parameter method [24] to solve the boundary value problem corresponding to the CEC mechanismunder linear, semi-infinite diffusion conditions (see Section 2). Thus, by introducing the dimensionless variables s, s′ and χ (Eq. (34)), the differentialequation problem becomes into:

+ − =

+ ′ − =′′

′′

′

∂∂

∂∂

∂∂

∂

∂

∂∂

∂∂

s χ

s χ

2 2 0

2 2 0

ςs

ςs

ςχ

ς

s

ςs

ςχ

2

2

2

2 (A1)

→ ∞′ → ∞

∞ =′ ∞ =

∗}ss

ς ςς

( )( ) 0 (A2)

=′ = }s

s00

⎜ ⎟⎛⎝

∂∂

⎞⎠

= −⎛⎝

∂ ′∂ ′

⎞⎠

γςs

ςs0 0 (A3)

⎜ ⎟⎜ ⎟− ⎛⎝

∂∂

⎞⎠

= ⎛⎝

++ ′

⎞⎠

⎛⎝

′ ′ −′

⎛⎝

∂ ′∂ ′

⎞⎠

⎞⎠

ς Kχ

ςs

e KK

K ςχ

κκ

ςs

(0)2

11

(0) 12

η

0 0 (A4)

Following Koutecký's dimensionless parameters method [23,24], the solution for ς(s,χ) is supposed to be a functional series of the variable χ asfollows:

= ∑

′ ′ = ∑ ′

=∞

=∞

ς sχ σ s χ

ς s χ ρ s χ

( ) ( )

( ) ( )

j jj

j jj

0

0 (A5)

with:

= +

′ = ′ +′′

∞

∞

→∞

′→∞

σ s a Ψ s

ρ s c Ψ s

( ) ( )

( ) ( )

j j jσ L s

lim L s

j j jρ L s

L s

( ) ( )( )

( ) ( )

lim ( )

j j

sj

j j

sj (A6)

where functions Ψj are given by:

∑ ∑= − ⎫⎬⎭

′

=

∞′

=

∞′ +Ψ s d s p e s( ) ( ) ( )j

mj m

mj

mj m

m( )

0,

( )

0,

( ) 1

(A7)

having the following properties:

=∞ =

= −

=

⎫

⎬

⎪⎪

⎭⎪⎪

′−

′

′ ′

ΨΨ

Ψ s p Ψ s

Ψ s s

(0) 1( ) 0

( ) ( )

( ) erfc ( )

j

j

j j j( )

1( )

0( ) ( )

(A8)

with coefficients dj,m and ej,m being given by Eq. (38) and pj by Eq. (39). Also, Lj are given by:

∑=′

=

∞′ +L e(s ) (s )j

mj m

m( )

0,

( ) 1

(A9)

Considering the application of the bulk conditions (Eq. (A2)) to Eqs. (A5)–(A6) leads to:

∞ =∞ =

⎫⎬⎭

∞ =

∗

≥

σ ςσ

ρ

( )( ) 0

( ) 0j

j

0

1

(A10)

such that:


211

⎧⎨⎩

= += >

′ = ′ ∀

∗σ s a Ψ s ς sσ s a Ψ s j

ρ s c Ψ s j

( ) ( ) erf ( )( ) ( ) ( 0)

( ) ( )

j j j

j j j

0 0 0

(A11)

Then, taking into account the surface conditions (see Eqs. (A3) and (A4)), Eqs. (A5) and (A11), the following expressions for coefficients aj areobtained:

==

⎫⎬⎭

= −

=⎛

⎝⎜

⎞

⎠⎟

⎫

⎬⎪

⎭⎪

≥′

′

∗

+

−∏

+

+

∗

++ ′ =

++ ′

′++ ′( )

( )( )

ca ςc a γ

aj

0

1

j j

jς

K e γ p

K e γ

K e γ

j

0

0

1

( 1) 2 1

η KK

j j

lj

l

η KK

κκ

η KK

11 1

11

11 (A12)

From Eqs. (A5), (A7)–(A8), (A11) and (A12), the concentration profiles of the oxidized species O and Z can be calculated from Eq. (43) and of thereduced species R and Z′ from Eq. (44).

Appendix B. Major symbols

A(λ, t) UV–visible absorbance response.Ai, N(λ, t) UV–visible absorbance response of species i (i≡O,Z,RorR′) considering normal incidence of the light beam.Ai,N

dkss(λ, t) UV–visible absorbance response of species i (i≡O,Z,RorR′), considering the normal incidence of the light beam for the CEC me-chanism, obtained with the dkss treatment.

Ai, P(λ, t) UV–visible absorbance response of species i (i≡O,Z,RorR′), considering incidence of the light beam parallel to the electrode surface.Ai,P

dkss(λ, t) UV–visible absorbance response of species i (i≡O,Z,RorR′), considering the incidence of the light beam parallel to the electrodesurface for the CEC mechanism, obtained with the dkss approach.

ci(x, t) Concentration profile of species i (i≡O,Z,R,R′).ci∗ Bulk concentration of species i (i≡O,Z,R,R′).D Diffusion coefficient.D′ Diffusion coefficient of the reduced species (R and Z′).εi(λ) Wavelength dependent extinction coefficient of species i (i≡O,Z,R,R′).K (Conditional) chemical equilibrium constant of the reaction involving the oxidized species (O and Z).K′ (Conditional) chemical equilibrium constant of the reaction involving the reduced species (R and Z′).κ Sum of the chemical rate constants of the reaction involving the oxidized species.κ′ Sum of the chemical rate constants of the reaction involving the reduced species.l Path length of the beam through the material sample in the normal mode. Path length of the beam over the electrode surface in the parallel

configuration.δ Thickness of the linear diffusion layer for the oxidized species (O and Z).δ′ Thickness of the linear diffusion layer for the reduced species (R and Z′).δr Thickness of the linear reaction layer related to the perturbation of the chemical equilibrium involving the oxidized species (O and Z).δ'r Thickness of the linear reaction layer related to the perturbation of the chemical equilibrium involving the reduced species (R and Z′).w Height of the light beam in the parallel configuration.ς∗ Sum of the bulk concentration of the oxidized species (O and Z).

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