Handout

CHAPTER 7
CHRONOPOTENTIOMETRY
In this technique the current flowing in the cell is instantaneously stepped from
zero to some finite value. The solution is not stirred and a large excess of supporting
electrolyte is present in the solution; diffusion is the only mass transfer process to be
considered. Electrolysis at constant current is conducted with the apparatus schematically
presented in Figure (1). where P is a power supply whose output current remains constant
regardless of the processes occurring in the cell. The potential of the working electrode
E1 against the reference electrode E2 is recorded by means of instrument V.
For a simple reaction as described by Equation (1), a chronopotentiogram will
typically look like the plot in Figure (2).
O + ne − = R
(1)
As the electrolysis proceeds, there is a progressive depletion of the electrolyzed species at
the surface of the working electrode. As the current pulse is applied there is an initial
sharp decrease in the potential as the double layer capacitance is charged, until a potential
at which O is reduced to R is reached. There is then a slow decrease in the potential
determined by the Nernst Equation, until the surface concentration of O reaches
essentially zero. The flux of O to the surface is then no longer sufficient to maintained the
1
applied current, and the electrode potential again falls more sharply, until a further
electrode process occurs.
7.1 Initial and Boundary Conditions
Unless otherwise stated, the following conditions are assumed to be achieved: (1)
the solution is not stirred; (2) a large excess of supporting electrolyte is present in
solution, and the effect of migration can be neglected; (3) conditions of semi-infinite
linear diffusion are achieved.
Substance O is reduced at a plane electrode and the product of electrolysis R is
soluble in solution. Since the current density is maintained constant during electrolysis,
the following equation
 ∂C ( x, t ) 
io = nFDo  o

 ∂x  x =0
(2)
can be written from the definition of the flux. Equation (2), which is the first boundary
condition, can be written in the following form
2
 ∂C o ( x, t ) 

 =λ
 ∂x  x =0
(3)
with
λ=
io
nFDo
(4)
The second boundary condition is obtained by expressing that the sum of the
fluxes for substance O and R at the electrode surface is equal to zero. Thus
 ∂C ( x, t ) 
 ∂C ( x, t ) 
Do  o
 + DR  R
 =0
 ∂x  x =0
 ∂x  x =0
(5)
The initial conditions can be selected a priory, and generally one can assume that the
concentration of substance R is equal to zero before electrolysis and that the
concentration of substance O is constant. Thus:
(1) CR(x, 0) = 0, and Co(x, 0) = Co.
(2) The functions Co(x, t) and CR(x, t) are bounded for large values of x. Thus,
Co(x, t) → Co and CR (x, t) → 0 for x → ∞
Variation of The Concentrations Co(x, t) and CR(x, t).
The solution of the above boundary value problem was reported by Karaoglanof,
who calculated the concentrations of both species. The concentrations are
C o ( x, t ) = C o −
C R ( x, t ) =
2λDo1 2 t 1 2
π12


 x2 
x

 + λxerfc
exp −
12 12 
 4 Do t 
 2 Do t 



2λDo t 1 2
x 2  λxDo
x

 −

erfc
exp
−
12 12
12 12 

DR π
 4 DR t  DR
 2 DR t 
where the notations "erf" and "erfc" represent the error integral defined by the formula:
3
(6)
(7)
erfλ =
2
π
12
∫
λ
0
(
)
exp − z 2 dz
(8)
The function erf{ λ } is defined under the form of a finite integral, having zero as lower
limit and as upper limit of integration. Therefore, values of erf{ λ } are determined only
by the variable λ , "z" being simply an auxiliary variable. The variations of erf { λ } with
λ are shown in Figure (3) for values of λ comprised between 0 and 2. The error
function is zero when its argument is equal to zero, and the function approaches unity
when λ becomes sufficiently large. "erfc" is defined by the equation:
erfc(λ ) = 1 − erf (λ )
(9)
An example values of Co(x, t) are plotted against x in Figure (4) for various times
of electrolysis and for the following data: io =10-2 A cm-2, n=1, D = 10-5 cm2 s-1, Co =
5x10-5 M /cm3. All the curves of Figure (4) have the same slope at x=0, because the flux
and consequently the derivative ∂C o ( x, t ) ∂t is constant at x=0.
4
7.2 Potential - Time Curves
The potential is calculated from the Nernst equation, the concentrations Co(0, t)
and CR(0, t) being written from (6) and (7). Thus
E = Eo +
foDR1 2 RT C o − Pt 1 2
RT
ln
+
ln
nF
f R Do1 2 nF
Pt 1 2
P=
π
12
(10)
2io
nFDo1 2
(11)
The sum of the first two terms on the right-hand of equation (10) is precisely the potential
E1/2 defined by equation:
E = E1 2 +
E1 2
RT id − i
ln
nF
i
RT  f R  Do
ln 
=E +
nF  f o  DR
o
(12)



12
(13)
When a mercury electrode is used, the potential E1/2 is the polarographic halfwave potential. Hence, equation (10) can be written as follows
5
E = E1 2
RT C o − Pt 1 2
ln
+
nF
Pt 1 2
(14)
The potential calculated from equation (14) is infinite when the numerator in the
logarithmic term is equal to zero, i.e. when the time t has the value τ defined by the
following relationship
τ1 2 = Co P
(15)
Actually the potential at time τ increases toward more cathodic values until a new
reaction occurs at the electrode: such a process can be the reduction of water or the
supporting electrolyte.
By introducing in Equation (14) the value of Co expressed in terms of the time τ
defined by equation (15), one obtains the following potential-time relationship
E = E1 2 +
RT τ 1 2 − t 1 2
ln
nF
t1 2
(16)
The above equation has the same form as the equation of a reversible polarographic
wave, the diffusion current and the current being replaced by τ 1 2 and by t1/2,
respectively. Thus, the properties of potential-time curves can be deduced by simple
transposition of the theory of reversible polarographic waves. The potential E1/2
corresponds to a value of t equal to τ 4 as can be seen from Figure (5). Equation (16)
[(
)
]
also shows that a plot of the decimal logarithm of the quantity τ 1 2 − t 1 2 t 1 2 versus
potential should yield a straight line whose reciprocal slope is 2.3
RT/nF. Logarithmic plots are linear as predicted by equation (16), and the potentials E1/2
as shown in Figure (6) are in good agreement with the polarographic half-wave
potentials.
6
Butler and Armstrong coined the term "transition time" to designate the time
defined by equation (15). According to equations (11) and (15) the transition time is
τ12 =
π 1 2 nFC o D 1 2
2io
(17)
The square root of the transition time is proportional to the bulk concentration of
substance reacting at the electrode and inversely proportional to the current density io.
Thus, transition times can be greatly changed by variation of the current density. The
limits between which the transition time can be adjusted are determined by the
experimental conditions:
(1) convection should not interfere with diffusion.
(2) the fraction of current corresponding to the charging of the double layer
should remain negligible in comparison with the total current through the cell. In practice
the transition time should not exceed a few minutes. Because of the charging of the
7
double layer, transition times shorter than one millisecond cannot be measured with
reasonable precision.
According to equation (17), [called Sand equation (1-3)], the product ioτ 1 2
should be independent of the current density.
Potential-Time Curves for Totally Irreversible Processes:
The rate of totally irreversible process is correlated to the current density by the equation:
io
 αn FE 
= k of ,h C o (0, t ) exp − α

RT 
nF

(18)
The combination of (16) and (18) yields
io
 αnFE 
= k of ,h C o − Pt 1 2 exp −

nF
 RT 
(
)
8
(19)
The transition time τ is determined by the condition Co(0, ) = 0 as for reversible
processes. Hence, Co = P τ 1 2 , and the corresponding value of τ can be introduced in
equation (19). The equation for the potential-time curve is thus
E=
RT
RT π 1 2 Do1 2
ln(τ 1 2 − t )1 2 −
ln
αnF
αnF
2k of ,h
(20)
or in view of equation ()
12
RT
RT   t  
12
12
E=
ln(τ − t ) −
ln 1 −   
αnF
αnF   τ  
An example of potential time curve is shown in Figure (7) for the reduction of iodite.
The potential should rise at time t=0 according to equation (21). It is seen from
equation (21) that the shape of the potential-time curve depends on the product α n, and
that the transition time is independent of the kinetic of the electrochemical reaction.
9
(21)
The potential at time zero depends on the parameters α n, k of ,h and on the current
density. Potential-time curves for the totally irreversible processes may thus be shifted to
a certain extent in the potential scale by variation of the current density.
A plot of decimal logarithm of {1-(t/ τ )1/2} versus potential is a straight line
[Figure (8)] whose reciprocal value is 2.3RT/ α nF. Thus, α n is readily calculated. The
rate constant k of ,h is calculated from the potential at time zero by application of equation
(21).
7.3 Two Consecutive Electrochemical Reactions Involving Different Substances
When two substances O1 and O2 are reduced at different potentials, the potentialtime curve exhibits two distinct steps. The first transition time τ 1 corresponding to the
reduction of substance O1 can be calculated from the treatment explained above. The
second step cannot be determined from this simple treatment. As the electrolysis
10
proceeds after the transition time τ 1 , the potential of the polarizable electrode adjusts
itself to a value at which substance O2 is reduced. Substance O1 continues to diffuse
toward the electrode at which it is immediately reduced. As a result the constant current
through the cell is the sum of two contributions corresponding to the reduction of
substances O1 and O2, respectively. The transition time τ 2 for the second step in the
potential-time curve is reached when the concentration of substance O2 becomes equal to
zero at the electrode surface.
Initial and boundary conditions have to be described to derive the transition time
τ 2 . In the writing of these conditions it is convenient to take as origin of the transition
time τ 1 . The time in the new scale will be represented by the symbol t', and the
relationship between t and
t' is
t' = t −τ 1
(22)
The transition time for the second step of the potential-time curve τ 2 is determined by the
condition C O2 (0,τ 2 ) = 0 by the equation
(τ 1 + τ 2 )
12
−τ
12
1
=
π 1 2 n2 FDO C 2o
2
2io
(23)
The quantity on the right-hand is proportional to
{(τ
+τ 2 )
12
1
− τ 11 2
}
As a result, the transition time τ 2 depends on the concentration of substance O1 which is
reduced at less cathodic potentials. The order of magnitude of the increase in τ 2 which
results from the contribution of O2 can be judged from the particular case in which
11
(24)
C1o = C 2o , n1 = n 2 , DO1 = DO2 ;
(25)
equation (23) yields for such conditions τ 2 = 3τ 1 .
Stepwise Electrode Processes-The Boundary Value Problem:
A substance O is reduced in two steps involving n1 and n2 electrons, the reduction
product being R1 and R2. Potential-time curves for such processes exhibit two steps,
when R1 is reduced at markedly more cathodic potentials than substance O. After the
transition time τ 1 for the first step substance O1 continues to diffuse toward the electrode
at which it is directly reduced to substance R2 in a process involving n1 + n2 electrons.
Furthermore, substance R1 produced during the first step diffuses toward the electrode at
which is reduced in a process involving n2 electrons.
The distribution of substance R1 at the transition time τ 1 is given by equation (7)
in which t is made equal to τ 1 . The resulting expression is the initial condition for the
present problem. The boundary condition is obtained by expressing that the current is the
sum of two contributions corresponding to the reduction of substances R1 and O,
respectively. Thus
 ∂C R1 ( x, t ') 
 + (n1 + n 2 )Do  ∂C o ( x, t ')  = io
n2 DR 1 


∂x
 ∂x  x =0 F
 x =0

Functions C o ( x, t ) and C R 1 ( x, t ) are bounded for large values of x. The following
equation for
the concentration of substance R1 at the electrode surface is reported in the literature (4)
12
(26)
C R 1 (0, t ') =
π
12
 n1 + n2 1 2

2io
τ 1 − (τ 1 + t ')1 2 
12 
n2 FDR1  n1

(27)
Transition Time for the Second Step of the Potential-Time Curve:
The transition time τ 2 is obtained by equating the right-hand member in equation (7) to
zero. The resulting expression can be written in the form:
C R 1 (0, t ') =
π
12
 n1 + n2 1 2

2io
τ 1 − (τ 1 + t ')1 2 
12 
n2 FDR1  n1

which shows that the relationship between the transition times τ 1 and τ 2 is remarkably
simple. When n1=n2, the transition time τ 2 is equal to 3 τ 1 .
Experimental data for the reduction of oxygen confirm the correctness of the
foredoing analysis. Potential-time curves for these substances are given in Figure (9).
Oxygen is reduced in two steps involving two electrons each, and consequently τ 2 = 3 τ 1 .
Cathodic Process Followed by Anodic Oxidation:
A substance O is reduced to R and the direction of the current through the electrolytic cell
is reversed at the transition time τ corresponding to the reduction of O. Substance R is
now oxidized and a potential-time curve is observed for this process.
The concentration of substance R at the transition time τ is expressed by
equation (7) in which the time t is made equal to τ . The resulting expression is the initial
condition for the present boundary value problem, since it is now the concentration of
substance R which is to be calculated. Initial and boundary conditions are the same as for
single electrochemical reaction (see equation 4).
13
(28)
i '
 ∂C R ( x, t ') 

 = λ' λ' = o
∂x
nFDR

 x =0
(29)
the intensity io ' in the reoxidation process may not necessarily be adjusted at the same
value as in the reduction of O and R.
Hence the current density io ' is introduced in equation (29). The function
C R ( x, t ') 6 0 for x 6 ∞
The concentration of substance R during the re-oxidation is:
 DR(τ + t ) 
∂C R ( x, t ') = 2θ 

π

12




x2
x
exp −

 − θxerfc  1 2
 4 DR (τ + t ') 
 2 DR (+ t ') 
(30)



x 
x
 D t' 
 + (θ + λ ')xerfc  1 2 1 2 
− 2(θ + λ ') R  exp −
 π 
 4 DR t ' 
 2 DR t ' 
2
with
θ=
io
nFDR
(31)
Variations of the concentration CR(x, t) with distance from the electrode are
shown in Figure (10) for the same data as those used in the construction of Figure (4) and
14
for the following numerical values: io ' = io = 10 −2 A cm 2 , DR=Do=10-5cm2s-1. The CR(x,
t') versus x curves of electrolysis larger than t' = 0 exhibit a maximum. The concentration
of R at a sufficient distance from the electrode becomes slightly larger than the
corresponding initial concentration at time t'=0; this results from diffusion of substance R
toward a region of the solution in which the concentration of R is lower than at the
maximum of the CR(x, t') vs x curve.
The concentration of substance O during reoxidation when the diffusion
coefficients Do and DR are equal is
C o ( x, t ') = C o − C R ( x, t ')
(32)
7.4 Transition Time for the Re-oxidation Process
The transition time is determined by the condition CR(0, τ ')=0. By writing (32)
for x=0 and solving for the transition time τ ' for the re-oxidation process one obtains
τ '=
θ2
(θ + λ ')2 − θ 2
When θ = λ ', when the current densities io and io ' are equal, equation () takes very
simple form
15
(33)
τ ' = 1 3τ
which shows that the transition time for the re-oxidation process is equal one third of the
transition time for the initial cathodic process, the current density being the same in both
processes. An example of potential-time curve is given in Figure (11).
16
(34)
EXPERIMENTAL
Required equipment and supplies
Pt working electrode, A=0.5 cm2.
1x10-3 M
[Fe(CN ) ] = 1x10
−3
6
-3
M.
[KCl] = 1M
PAR Model 352
Three compartment electrochemical cell.
Objectives:
The objectives of this experiment are by using chronopotentiometry and
chronopotentiometry with current reversal to determine: (1) the dependence of the
transition time on the bulk concentration of electroactive species reacting at the electrode
and (2) the dependence
of the transition time on applied current density.
The electrochemical cell employed for these studies should be conventional threecompartment design with contact between the working
electrode compartment and
the reference electrode via a Luggin probe. The chronopotentiometric experiments should
[
be carried out using standard calomel electrode (SCE) in (1) 1x10-3 M Fe(CN )6
−3
] and
1M KCl.
Data Analysis:
Values of (1) the transition times as a function of the applied current densities for
[
constant concentrations of Fe(CN )6
−3
]
and (2) for transition times as a function of
17
[
different concentrations of Fe(CN )6
−3
] at constant current densities and (3) potential vs.
log τ 1 / 2 − t 1 2 t 1 2 were obtained by Popov and Laitinen (unpublished results) and are given
in Table (1), Table (2) and Table (3).
Table 1. Values of Transition Time for Different Cathodic Currents Obtained in
1x10-3M Fe(CN)6-3 + 1M KCl
Concentration
Current Density
Transition Time
Relative Standard
10-3M
µ A/cm3
Measured, (s)
Deviation
1.0
60
5.2
1.2%
1.0
55
6.2
0.6%
1.0
50
7.5
1.4%
1.0
45
10.7
0.8%
1.0
40
12.2
1.2%
1.0
35
15.5
0.3%
Table 2. Values of Transition Time Obtained for Different Concentrations of
Fe(CN)6-3 at Constant Cathodic Current Density of 50 A/cm2.
[Fe(CN)6-3 ]
Applied Current
Transition Time
Relative Standard
10-3M
µ A/cm3
Measured, (s)
Deviation
1.5
50
15.5
0.5%
2.0
50
31.0
1.16%
2.5
50
51.8
1.1%
3.0
50
72.5
1.8%
18
τ 1 2 − t1 2
Table 3. Potential vs log
τ12
Potential (mv) [SCE]
τ 1 2 − t1 2
, (s)
log
τ12
180
-0.78
190
-0.6
210
-0.3
227
0.0
255
0.4
265
0.6
275
0.8
Plot iτ 1 2 vs i; iτ vs 1/i; iτ 1 2 /C vs C; τ 1 2 vs C
Compute (a) the diffusion coefficient of the electroactive species in 1M KCl. (b)
the number of electrons involved in the process (c) Discuss the logarithmic plot for
reversible electrode processes (d) compare the "n" value obtained from the slope of the
logarithmic plot for reversible electrode processes and "n" value obtained from Sand
Equation.
19
REFERENCES
1. B. N. Popov and H. A. Laitinen, J. Electrochem. Soc., 117, 4, 482, (1970).
2. B. N. Popov and H. A. Laitinen, J. Electrochem. Soc., 120, 10, 1346, (1973).
3. R. Cvetkovic, B. N. Popov and H. A. Laitinen, J. Electrochem. Soc., 122, 12, 1616,
(1975).
4. Boris B. Damaskin, "The Principles of Current Methods for the Study of
Electrochemical Reactions”, Editor, Gleb Maamntov, McGraw-Hill Book Co., New
York, (1968)
20