Electro kinetics

Electro-KineticsMEHRAN UNIVERSITY OF MEHRAN UNIVERSITY OF

ENGINEERING AND TECHNOLOGY, ENGINEERING AND TECHNOLOGY, JAMSHOROJAMSHORO

DEPARTMENT OF METALLURGY AND DEPARTMENT OF METALLURGY AND MATERIALS ENGINEERINGMATERIALS ENGINEERING

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

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 currentsTable: Limiting currents observed for 9.5 x 10-4 M

PbCl2 as a function of the concentration of KNO3 supporting electrolyteMolarity

of KNO3

I lμA

0 17.60.001 12.00.005 9.80.10 8.451.0 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

xDJ


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

xDJ


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: xRT

DcJ ii

1


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).

xRTDcJ i

i

1

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:


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.

Slope due to ic

Applied Potential → -Ve

Current

10-5 M Fe3+ Fe3+ + e → Fe2+

Current

10-3 M Fe3+ Fe3+ + e → Fe2+

Applied Potential → -Ve

(a)

(b)

Note that the iE curve in Fig. (a) is recorded at a much higher sensitivity than in Fig. (b).

ild

ild

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.


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.


ialoverpotent cathodichigh at

exp

ialoverpotent anodichigh at

1exp

0`

0`

RTnFii

RTnFii

cc

aa

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

nb

in

a

iba

in

in

in

in

inFRTi

nFRT

o

aaa

a

ccc

c

ccc

c

059.0

ln059.0and

logequation Tafelknown well theisequation The

process anodic for the C25at log059.0log059.0

process cathodic for the C25at log059.0log059.0

lnln

00

00

0

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

iinFRTb

iibiinFRT

log303.2ln

slope Tafel the303.2 where

lnor ln 00

We can write:

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

iba log

The following equation was obtained experimentally

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

resistanceon polarisati measured the 3.2

constant Tafel theWhere

a

a

iE

R

B

RBi

p

c

c

pcorr

Applicable in the linear region of the Butler Volmer Equation at low over-potentials

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

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.

FRTb

303.2

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:

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 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 214463.0 RTnFvDnFACip

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

ACDvnip212123510687.2

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 – Stationary Electrode• 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 0.028/n• Sign is + for a reduction

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/levich http://www.calctool.org/CALC/chem/electrochem/

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:

iinFRTb

iibiinFRT

log303.2ln

slope Tafel the303.2 where

lnor ln 00

Evans Diagrams

Electro kinetics

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