Download Electro-Kinetics

Electro-Kinetics
Description of Electrochemical
Techniques
• The technique is named according to the
parameters measured
• E.g.
• Voltammetry – measure current and voltage
• Potentiometry – measure voltage
• Chrono-potentiometry – measure voltage with
time (under an applied current)
• Chrono-amperometry – measure current with
time (under an applied voltage)
Electro-Kinetics
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Movement of Ions
Butler Volmer Equation
Rotating Disc Electrode
Rotating Cylinder Electrode
Voltammetry
Cyclic Voltammetry
Chrono-potentiometry
Chrono-amperometry
Movement of Ions in Solution
• Diffusion – Movement under a concentration gradient. If
an electrochemical reaction occurs the current due to
this reaction is called, id , the diffusion current.
• Migration or Transport – Movement of ions under an
electric field due to coulombic forces. If an
electrochemical reaction occurs the current due to this
reaction is called, im , the migration current.
• Convection – Movement due to changes in density at the
electrode solution interface. This occurs due to depletion
or addition of a species due to the electrochemical
reaction.
The Capacitance Current
• The charging or capacitance current, ic , is due to the
presence of the electrical double layer and it is always
present. This current, of course, is not related to any
movement of ions.
• Ic = Cdl x V
• Where:
• Cdl = the capacitance of the electrical double layer
• V = voltage scan rate
• The capacitance current makes its presence felt when
measuring charge transfer (Faradaic) processes at
concentrations of 10-5 M.
Diffusion
• Molecular diffusion,
often called simply
diffusion, is a net
transport of
molecules from a
region of higher
concentration to one
of lower
concentration by
random molecular
motion.
Migration or Transport
• Is the fraction of current carried by the ions.
• For example in a solution of copper sulphate the transport
number of Cu2+ is 0.4 and that of SO42- = 0.6.
• t+ + t- = 0.4 + 0.6 = 1
• Since the migration current depends on the ionic strength of
the solution it is usually eliminated by addition of excess of an
inert supporting electrolyte (100 – 1000 fold excess in
concentration)
• The current is carried by the inert supporting electrolyte (e.g.
NaCl , KNO3 etc) – because the ions produced do not undergo
any electrochemical reaction the transport current is
effectively removed.
• In excess inert supporting electrolyte, the current measured
due to the electro-active species of interest is due only to
diffusion which can be related to mass transfer.
Voltammetry – the following example shows how the migration
current is eliminated. Pb2+ + 2e → Pb0
• The supporting electrolyte
• Ensures diffusion control of limiting currents by eliminating
migration currents
• Table: Limiting currents observed for 9.5 x 10-4 M PbCl2 as a
function of the concentration of KNO3 supporting electrolyte
Molarity
of KNO3
0
Il
μA
17.6
0.001
12.0
0.005
9.8
0.10
8.45
1.0
8.45
Voltammetry
• The example shown is for the reduction of Pb2+ at an
inert mercury electrode.
• Pb2+ + 2e → Pb(Hg)
• At low inert electrolyte concentration a large fraction
of the total current is due to the migration current,
i.e. the currents due to the electrostatic attraction of
ions to the electrode.
• For solution 1:
• i migration
im 17.6 – 8.45 = 9.2 A
• i diffusion
id = 8.45 A
Fick’s First Law of Diffusion
• Fick's first law relates the diffusive flux to the
concentration field, by postulating that the flux goes
from regions of high concentration to regions of low
concentration, with a magnitude that is proportional
to the concentration gradient (spatial derivative). In
one (spatial) dimension, this is

J  D
x
Fick’s First Law of Diffusion

J  D
x
• where
• J is the diffusion flux in dimensions of [(concentration of
substance) length−2 time-1], example mole (M) m-2 s-1.
• J measures the amount of substance that will flow
through a small area during a small time interval.
• D is the diffusion coefficient or diffusivity in dimensions
of [length2 time−1], example m2 s-1
•  (for ideal mixtures) is the concentration in dimensions
of [(concentration of substance) length−3], example M m-3
• x is the position [length], example m
Fick’s First Law of Diffusion
• D is proportional to the squared velocity of the
diffusing particles, which depends on the
temperature, viscosity of the fluid and the size of the
particles according to the Stokes-Einstein
relationship.
• In dilute aqueous solutions the diffusion coefficients
of most ions are similar and have values that at room
temperature are in the range of 0.6x10-9 to 2x10-9
m2/s.
• For biological molecules the diffusion coefficients
normally range from 10-11 to 10-10 m2/s.
Ficks First Law of Diffusion
• In two or more dimensions we must use, , the del or
gradient operator, which generalises the first derivative,
obtaining
• J = -D  
• The driving force for the one-dimensional diffusion is the
quantity -/x
• which for ideal mixtures is the concentration gradient. In
chemical systems other than ideal solutions or mixtures,
the driving force for diffusion of each species is the
gradient of chemical potential of this species. Then Fick's
first law (one-dimensional case) can be written as:
Dc1 i
Ji  
RT x
Fick’s First Law of Diffusion
Dc1 i
Ji  
RT x
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where the index i denotes the ith species,
c is the concentration (mol/m3),
R is the universal gas constant (J/(K mol)),
T is the absolute temperature (K), and
μ is the chemical potential (J/mol).
Butler-Volmer Equation
• The Butler-Volmer equation is one of the
most fundamental relationships in
electrochemistry. It describes how the
electrical current on an electrode depends on
the electrode potential, considering that both
a cathodic and an anodic reaction occur on
the same electrode:
Butler-Volmer Equation
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where:
I = electrode current, Amps
Io= exchange current density, Amp/m2
E = electrode potential, V
Eeq= equilibrium potential, V
A = electrode active surface area, m2
T = absolute temperature, K
n = number of electrons involved in the electrode reaction
F = Faraday constant
R = universal gas constant
α = so-called symmetry factor or charge transfer coefficient
dimensionless
The equation is named after chemists John Alfred Valentine Butler and
Max Volmer
Butler-Volmer Equation
• The equation describes two regions:
• At high overpotential the Butler-Volmer equation
simplifies to the Tafel equation
• E − Eeq = a − blog(ic) for a cathodic reaction
• E − Eeq = a + blog(ia) for an anodic reaction
• Where:
• a and b are constants (for a given reaction and
temperature) and are called the Tafel equation
constants
• At low overpotential the Stern Geary equation
applies
Current Voltage Curves for Electrode Reactions
Without concentration
and therefore mass
transport effects to
complicate the
electrolysis it is possible
to establish the effects of
voltage on the current
flowing. In this situation
the quantity E - Ee reflects
the activation energy
required to force current i
to flow. Plotted below are
three curves for differing
values of io with α = 0.5.
Voltammetry
• Although the Butler Volmer Equation predicts, that
at high overpotential, the current will increase
exponentially with applied voltage, this is often not
the case as the current will be influenced by mass
transfer control of the reactive species.
• Take the following example of the reduction of ferric
ions at a platinum rotating disc electrode (RDE).
• Fe3+ + e = Fe2+
• The rotation of the electrode establishes a well
defined diffusion layer (Nernst diffusion layer)
• The contribution of the capacitance current will also
be demonstrated in this example.
Effect of the Capacitance Current in Voltammetry. The reduction of Ferric Chloride
is carried out in the presence of 1 M NaCl to eliminate the migration current.
(a)
10-5 M Fe3+ Fe3+ + e → Fe2+
ild
Slope due to ic
Current
Applied Potential → -Ve
10-3
M
Fe3+
Fe3+
+e→
(b)
Fe2+
ild
Current
Applied Potential → -Ve
Note that the iE curve in Fig. (a)
is recorded at a much higher
sensitivity than in Fig. (b).
Charging Current or Capacitance
Current
• Note that due to the presence of the electrical
double layer a charging or capacitance current
is always present in voltammetric
measurements.
Butler-Volmer Equation
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where:
I = electrode current, Amps
Io= exchange current density, Amp/m2
E = electrode potential, V
Eeq= equilibrium potential, V
A = electrode active surface area, m2
T = absolute temperature, K
n = number of electrons involved in the electrode reaction
F = Faraday constant
R = universal gas constant
α = so-called symmetry factor or charge transfer coefficient
dimensionless
The equation is named after chemists John Alfred Valentine Butler and
Max Volmer
Butler Volmer Equation
• While the Butler-Volmer equation is valid over the full
potential range, simpler approximate solutions can be
obtained over more restricted ranges of potential. As
overpotentials, either positive or negative, become larger
than about 0.05 V, the second or the first term of equation
becomes negligible, respectively. Hence, simple exponential
relationships between current (i.e., rate) and overpotential
are obtained, or the overpotential can be considered as
logarithmically dependent on the current density. This
theoretical result is in agreement with the experimental
findings of the German physical chemist Julius Tafel (1905),
and the usual plots of overpotential versus log current density
are known as Tafel lines.
• The slope of a Tafel plot reveals the value of the transfer
coefficient; for the given direction of the electrode reaction.
Butler-Volmer Equation
 1   nF a 
ia `  i0 exp 

RT


at high anodic overpotent ial
 nF c 
ic `  i0 exp 

RT


at high cathodic overpotent ial
ia and ic are
the exhange
current
densities for
the anodic
and cathodic
reactions
These equations can be rearranged to give
the Tafel equation which was obtained
experimentally
Butler Volmer Equation - Tafel Equation
c 
RT
RT
ln i0 
ln ic
 c nF
 c nF
c 
0.059
0.059
log i0 
log ic at 250 C for the cathodic process
cn
cn
0.059
0.059
a 
log i0 
log ia at 250 C for the anodic process
an
an
The equation is the well known Tafel equation
  a  b log i
and
0.059
a
ln io
n
0.059
b
n
Tafel Equation
• The Tafel slope is an intensive parameter and does not
depend on the electrode surface area.
• i0 is and extensive parameter and is influenced by the
electrode surface area and the kinetics or speed of the
reaction.
• Notice that the Tafel slope is restricted to the number of
electrons, n, involved in the charge transfer controlled
reaction and the so called symmetry factor, .
• n is often = 1 and although the symmetry factor can vary
between 0 and 1 it is normally close to 0.5.
• This means that the Tafel slope should be close to 120
mV if n = 1 and 60 mV if n = 2.
Tafel Equation
• We can write:
RT

ln i i0  or   b ln i i0 
nF
where
2.303RT
b
 the Tafel slope
nF
ln i  2.303 log i
Current Voltage Curves for Electrode Reactions
Without concentration
and therefore mass
transport effects to
complicate the
electrolysis it is possible
to establish the effects of
voltage on the current
flowing. In this situation
the quantity E - Ee reflects
the activation energy
required to force current i
to flow. Plotted below are
three curves for differing
values of io with α = 0.5.
Tafel Equation
• The Tafel equation can be also written as:
• where
• the plus sign under the exponent refers to an anodic
reaction, and a minus sign to a cathodic reaction, n is
the number of electrons involved in the electrode
reaction k is the rate constant for the electrode
reaction, R is the universal gas constant, F is the
Faraday constant. k is Boltzmann's constant, T is the
absolute temperature, e is the electron charge, and α
is the so called "charge transfer coefficient", the
value of which must be between 0 and 1.
Tafel Equation
• The following equation was obtained
experimentally
  a  b log i
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Where:
 = the over-potential
i = the current density
a and b = Tafel constants
Tafel Equation
• Applicability
• Where an electrochemical reaction occurs in two half reactions on
separate electrodes, the Tafel equation is applied to each electrode
separately.
• The Tafel equation assumes that the reverse reaction rate is
negligible compared to the forward reaction rate.
• The Tafel equation is applicable to the region where the values of
polarization are high. At low values of polarization, the dependence
of current on polarization is usually linear (not logarithmic):
• This linear region is called "polarization resistance" due to its formal
similarity to Ohm’s law
Stern Geary Equation
• Applicable in the linear region of the Butler Volmer
Equation at low over-potentials
icorr
B

Rp
Where
B  the Tafel constant
a  c

2.3 a   c 
R p  the measured polarisati on resistance
 E i
Tafel Equation
• Overview of the terms
• The exchange current is the current at equilibrium,
i.e. the rate at which oxidized and reduced species
transfer electrons with the electrode. In other words,
the exchange current density is the rate of reaction
at the reversible potential (when the overpotential is
zero by definition). At the reversible potential, the
reaction is in equilibrium meaning that the forward
and reverse reactions progress at the same rates.
This rate is the exchange current density.
Tafel Equation
• The Tafel slope is measured experimentally; however,
it can be shown theoretically when the dominant
reaction mechanism involves the transfer of a single
electron that
2.303RT
b
F
• T is the absolute temperature,
• R is the gas constant
• α is the so called "charge transfer coefficient", the
value of which must be between 0 and 1.
Levich Equation
• The Levich Equation models the diffusion and solution
flow conditions around a rotating disc electrode (RDE). It
is named after Veniamin Grigorievich Levich who first
developed an RDE as a tool for electrochemical research.
It can be used to predict the current observed at an RDE,
in particular, the Levich equation gives the height of the
sigmoidal wave observed in rotating disk voltammetry.
The sigmoidal wave height is often called the Levich
current.
• In work at a RDE the electrode is usually rotated quite
fast (1000 rpm) in order to establish a well defined
diffusion layer.
• The scan rate is relatively slow – typically 2-5 mV s-1
Current Voltage Curve at a RDE
It is important to remember that in order to determine the
diffusion current and the mass transfer coefficient using
volatmmetry, excess inert supporting electrolyte must be
present to eliminate the migration current
Levich Equation
• The Levich Equation is written as:
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where
iL is the Levich current
n is the number of electrons transferred in the half reaction
F is the Faraday constant
A is the electrode area
D is the diffusion coefficient (see Fick's law of diffusion)
w is the angular rotation rate of the electrode
v is the kinematic viscosity
C is the analyte concentration
While the Levich equation suffices for many purposes,
improved forms based on derivations utilising more terms in
the velocity expression are available.[1][2]
Rotating Disc Electrode
Levich Equation
Levich Equation
Levich Equation
• It is important to note that the layer of solution immediately
adjacent to the surface of the electrode behaves as if it were
stuck to the electrode. While the bulk of the solution is being
stirred vigorously by the rotating electrode, this thin layer of
solution manages to cling to the surface of the electrode and
appears (from the perspective of the rotating electrode) to be
motionless.
• This layer is called the stagnant layer in order to distinguish it
from the remaining bulk of the solution. The act of rotation
drags material to the electrode surface where it can react.
Providing the rotation speed is kept within the limits that
laminar flow is maintained then the mass transport equation is
given by the Levich equation.
Levich Equation RDE
• The Levich equation takes into account both the rate of
diffusion across the stagnant layer and the complex solution
flow pattern. In particular, the Levich equation gives the
height of the sigmoidal wave observed in rotated disk
voltammetry. The sigmoid wave height is often called the
Levich current, iL, and it is directly proportional to the analyte
concentration, C. The Levich equation is written as:
• iL = (0.620) n F A D2/3 w1/2 v–1/6 C
• where w is the angular rotation rate of the electrode
(radians/sec) and v is the kinematic viscosity of the solution
(cm2/sec). The kinematic viscosity is the ratio of the solution's
viscosity to its density.
Current Voltage Curve at a RDE
Ilimiting vs ½ (electrode rotational velocity)
Levich Equation - RDE
• The linear relationship between Levich current and the square
root of the rotation rate is obvious from the Levich plot. A
linear least squares fit of the data produces an equation for
the best straight line passing through the data. The specific
experiment shown, the electrode area, A, was 0.1963 cm2, the
analyte concentration, C, was 2.55x10–6 mol/cm3, and the
solution had a kinematic viscosity, v, equal to 0.00916
cm2/sec. After careful substitution and unit analysis, you can
solve for the diffusion coefficient, D, and obtain a value equal
to 4.75x10–6 cm2/s. This result is a little low, probably due to
the poor shape of the sigmoidal signal observed in this
particular experiment.
• The kinematic viscosity is the ratio of the absolute viscosity of
a solution to its density. Absolute viscosity is measured in
poises (1 poise = gram cm–1 sec–1). Kinematic viscosity is
measured in stokes (1 stoke = cm2 sec–1). Extensive tables of
solution viscosity and more information about viscosity units
can be found in the CRC Handbook of Chemistry and Physics.
Cyclic Voltammetry
• Cyclic Voltammetry is carried out at a stationary
electrode.
• This normally involves the use of an inert disc
electrode made from platinum, gold or glassy carbon.
Nickel has also been used.
• The potential is continuously changed as a linear
function of time. The rate of change of potential with
time is referred to as the scan rate (v). Compared to a
RDE the scan rates in cyclic voltammetry are usually
much higher, typically 50 mV s-1
Cyclic Voltammetry
• Cyclic voltammetry, in which the direction of the
potential is reversed at the end of the first scan. Thus,
the waveform is usually of the form of an isosceles
triangle.
• The advantage using a stationary electrode is that the
product of the electron transfer reaction that occurred in
the forward scan can be probed again in the reverse
scan.
• CV is a powerful tool for the determination of formal
redox potentials, detection of chemical reactions that
precede or follow the electrochemical reaction and
evaluation of electron transfer kinetics.
Cyclic Voltammetry
Cyclic Voltammetry
For a reversible
process
Epc – Epa = 0.059V/n
The Randles-Sevcik equation Reversible systems
The Randles-Sevcik equation Reversible systems
i p  0.4463nFACnFvD RT 
12


i p  2.687 105 n3 2v1 2 D1 2 AC
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n = the number of electrons in the redox reaction
v = the scan rate in V s-1
F = the Faraday’s constant 96,485 coulombs mole-1
A = the electrode area cm2
R = the gas constant 8.314 J mole-1 K-1
T = the temperature K
D = the analyte diffusion coefficient cm2 s-1
The Randles-Sevcik equation Reversible systems
As expected a plot of peak height vs the square root of the scan rate
produces a linear plot, in which the diffusion coefficient can be obtained
from the slope of the plot.
Cyclic Voltammetry
Cyclic Voltammetry
Cyclic Voltammetry
Cyclic Voltammetry – Stationary Electrode
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Peak positions are related to formal potential of redox
process
• E0 = (Epa + Epc ) /2
• Separation of peaks for a reversible couple is 0.059/n volts
• A one electron fast electron transfer reaction thus gives
59mV separation
• Peak potentials are then independent of scan rate
• Half-peak potential Ep/2 = E1/2
• Sign is + for a reduction
0.028/n
Cyclic Voltammetry – Stationary Electrode
• The shape of the voltammogram depends on the transfer
coefficient 
• When  deviates from 0.5 the voltammograms become
asymmetric -cathodic peak sharper as expected from Butler
Volmer eqn.
Web Sites
• http://calctool.org/CALC/chem/electrochem/l
evich
• http://www.calctool.org/CALC/chem/electroc
hem/cv1
Tafel Equation
• The Tafel slope is an intensive parameter and does not
depend on the electrode surface area.
• i0 is and extensive parameter and is influenced by the
electrode surface area and the kinetics or speed of the
reaction.
• Notice that the Tafel slope is restricted to the number of
electrons, n, involved in the charge transfer controlled
reaction and the so called symmetry factor, .
• n is often = 1 and although the symmetry factor can vary
between 0 and 1 it is normally close to 0.5.
• This means that the Tafel slope should be close to 120
mV if n = 1 and 60 mV if n = 2.
Tafel Equation
• We can write:
RT

ln i i0  or   b ln i i0 
nF
where
2.303RT
b
 the Tafel slope
nF
ln i  2.303 log i
Evans Diagrams
Evans Diagrams
Evans Diagrams