fundamental & applications of electrochemistry
TRANSCRIPT
MSE 673 IIT Kanpur
Instructor: Dr. Shobit Omar
Lecture 21
Venue: Online Lecture
Week: 7
Materials Science & Engineering Department
Fundamental & Applications of
Electrochemistry
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Linearized i-t Curve
* 2 1/2( ) exp( ) ( )f Oi t FAk C H t erfc Ht=
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Linearized i-t Curve
For small values of H2t, the factor exp(H2t)erfc(Ht1/2) can be linearized.
2
1/2
2exp( ) ( ) 1
xx erfc x
−For small values of x
1/2*
1/2
2( ) 1f O
Hti t FAk C
= −
R is initially absent
The forward rate constant can be determined from the intercept of the
plot between i versus t1/2.
* 2 1/2( ) exp( ) ( )f Oi t FAk C H t erfc Ht=
1/2 1/2
f b
O R
k kH
D D= +Please remember
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Linearized i-t Curve
(1 )
2 1/2( ) [exp exp ]exp( ) ( )F F
RT RToi t i H t erfc Ht
−−
= −
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Linearized i-t Curve
For small values of H2t, the factor exp(H2t)erfc(Ht1/2) can be linearized.
2/1
2 21)()exp(
xxerfcx −For small values of x
)()exp(]exp[exp)( 2/12
)1(
HterfctHiti RT
F
RT
F
o
−−
−=
−−=
−−
2/1
2/1)1(2
1]exp[exp)(
Ht
iti RT
F
RT
F
o
Both O and R are
initially present
• A plot of i at t =0 versus η can be used to obtain io• The intercept of the plot between i versus t1/2 can provide kinetically-
controlled current free of mass transfer effects.
For small values of Ht1/2 and η
−−=
2/1
2/121)(
Ht
F
RTiti o
Complete linearized
form
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Sampled Current Voltammetry for
Quasireversible Electrode Reaction
Earlier we have derived
1/2 1/2
f b
O R
k kH
D D= +
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Sampled Current Voltammetry for
Quasireversible Electrode Reaction
( )1/2
1/2 1/2 1/21 1
f fO
O R O
k kDH
D D D
= + = +
( )
* 2 1/2
* 1/21/2 1/2 2 1/2
1/2 1/2
( ) exp( ) ( )
( ) exp( ) ( )1
f O
O O
i t FAk C H t erfc Ht
FAC Di t H t H t erfc Ht
t
=
=+
Assuming, 1/2Ht = ( )1( ) ( )
1
dii t F
=+
)()exp()( 22/1
1 erfcF =
Inputting kf
from here
Earlier we have derived
1/2 1/2
f b
O R
k kH
D D= +
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Simplified Expression for i(t) Response
( ))(
1)( 1
F
iti d
+=
A compact representation of
the way in which the current in
potential step experiment
depends on time and potential
This equation hold for all
the kinetic regimes, i.e.
reversible, quasireversible
and irreversible.
2/1Ht=
)()exp()( 22/1
1 erfcF =
Small value of λ implies a
strong kinetic influence on the
current, and a large value of λ
corresponds to the situation
where the kinetics are facile
and the response is controlled
by diffusion
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Reversible Reaction
( ))(
1)( 1
F
iti d
+=
For a reversible reaction, λ is very large & therefore F1(λ) is always close to unity
( )+=
1)( diti
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Sampled Current Voltammetry
( )1/2
1/21
f
O
k
D
= +At a fixed sampling time ( ), λ
becomes
• At a very positive E relative to Eo/, θ is very large. As a result, i is
nearly zero since the denominator becomes far greater than
numerator in the simplified expression.
Vice-versa
occurs in
oxidation
reactions
• At a very negative E relative to Eo/, kf is very large which makes
the value of F1(λ) close to unity and the value of i approximately
equal to id.
( )1( )
1
dii F
=+
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Sampled Current Voltammetry
( )1/2
1/21
f
O
k
D
= +At a fixed sampling time ( ), λ
becomes
( )
1( )1
dii F
=+
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Sampled-Current Voltammograms
for Various Kinetics
The sampled current voltammogram to have a sigmoidal shape as observed
in the reversible case.
For a very facile reaction, where
DO = DR, E1/2 is equal to Eo/.
For a smaller values of ko, the
kinetics must be driven. The
observed shift in the E1/2 relative
to Eo is the extra potential which
is proportional to the required
kinetic activation.
A large displacement is observed
in the case of the totally
irreversible reaction.
Please remember the negative shift in the potential will
accelerate the reduction kinetics or forward reaction or
cathodic current.
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Totally Irreversible Reactions
The very displacement in potential that activates kf simultaneously
suppresses the kb. As a result the backward reaction becomes significantly
less important than the forward reaction.
For a smaller value of ko, a large activation of kf is required to achieve
appreciable current flow without much contribution from kb
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Totally Irreversible Reactions
The very displacement in potential that activates kf simultaneously
suppresses the kb. As a result the backward reaction becomes significantly
less important than the forward reaction.
For a smaller value of ko, a large activation of kf is required to achieve
appreciable current flow without much contribution from kb
As a result, 0b
f
k
k= = over the whole range of the voltammetric wave
( )1 1( ) ( ) ( ) ( )
1
dd
ii t F i t i F
= =
+
1/2 1/2 /
1/2 1/2
exp[ / ( )]o of
O O
k t k t F kT E E
D D
− −= =
1/2 2
1( ) exp( ) ( )F erfc =Where,
1
( )0.5 ( )
d
i tF
i= =
At this condition
o == 433.0 =t
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Calculation of ko in Totally
Irreversible ReactionsNow,
1/2 /
1/2
1/2
exp[ / ( )]0.433
o oo
O
k F kT E E
D
− −= =
1/2/
1/2 1/2
2.31ln
oo
O
kT kE E
F D
= +
One can determine the ko value if α is known
1/2/
1/2 1/2
2.31exp[ / ( )]
oo
O
kF kT E E
D
− =
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Kinetic Regimes
Reversible Regime Quasireversible Regime Irreversible Regime
By focusing on the particular value of λ at Eo/, it is possible to distinguish
the three kinetic regimes
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Kinetic Regimes
Reversible Regime Quasireversible Regime Irreversible Regime
By focusing on the particular value of λ at Eo/, it is possible to distinguish
the three kinetic regimes
Let us call the λ at Eo/ as λo
1/2 / 1/2
1/2 1/2
(1 ) exp[ / ( )](1 )
o o oo
O O
k t F kT E E k
D D
+ − −= = +
Let us assume DO = DR
2/1
2/1
2/1
2/1 2)1(
O
o
O
oo
D
k
D
k
=+=
For the reversible reaction, F1(λ) is close to unity
29.0)(1 oF 12/1
2/1
O
o
D
k Value of λo should be greater
than 2 for the reversible reaction.
θ = 1 at E = Eo/
Now,
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Irreversible Regime
For the irreversible reaction, θ is close to zero when E = E1/2
/exp[ / ( )]oF kT E E = −
Let us take the value of θ = 0.01 and estimate the E1/2 value
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Irreversible Regime
For the irreversible reaction, θ is close to zero when E = E1/2
This is the condition for
the total irreversibility
/exp[ / ( )]oF kT E E = −
Let us take the value of θ = 0.01 and estimate the E1/2 value
/
1/2
4.61( )o kTE E
F− −
But we know for the irreversible reactions,
Thus,
1/2
1/2
1 2.31ln 4.61
o
O
k
D
−
1/2
1/2
2 oo
O
k
D
=
But,
1/2/
1/2 1/2
2.31ln
oo
O
kT kE E
F D
= +
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
22
31.2log −
o
Neglecting this term
log 2o −
−−
2
31.2log2log o
Value of λo should be less than
10-2α for the irreversible reaction.
Irreversible Regime
1/2
1/2
1 2.31ln 4.61
o
O
k
D
−
1/2
1/2
2 oo
O
k
D
=
MSE 673 IIT Kanpur
Fundamental & Applications of Electrochemistry Week 7 Lecture 21
Value of λo Various Kinetic Regimes
210−o210 2 −
o 2o
Reversible RegimeQuasireversible RegimeIrreversible Regime
1/2
1/2
2 oo
O
k
D
=
•Please remember that beside the intrinsic electrode kinetics, λo value is
also dependent on sampling time.
• If the sampling time is large, than the reaction tends to become reversible.
MSE 673 IIT Kanpur
Instructor: Dr. Shobit Omar
Lecture 21
Venue: Online Lecture
Week: 7
Materials Science & Engineering Department
Fundamental & Applications of
Electrochemistry