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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• An analysis of the kinetics of electrode processes using activated

complex theory (see Atkins, for details) gives:

• 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

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Butler-Volmer Equation

j = jc - ja = j0[e(1-)nF/RT - e-nF/RT]where– jc and ja are the cathodic and anodic current densities; – j0 is the exchange current density at equilibrium, where the

rates of the forward (cathodic reduction) and reverse (anodic oxidation) reactions are equal (but not zero);

– and is the transfer coefficient (usually ca. 0.5)– The charge transfer coefficient signifies the fraction of the interfacial

potential at an electrode-electrolyte interface that helps in lowering the free energy barrier for the electrochemical reaction. The electroactive ion present in the interfacial region experiences the interfacial potential and electrostatic work in done on the ion by a part of the interfacial electric field. It is charge transfer coefficient that signifies this part that is

utilized in activating the ion to the top of the free energy barrier.

• Remember is negative for reduction.

For a redox active species in solution where electron transfer occurs at an electrode it is possible to establish the effects of voltage on the current flowing. In this situation the quantity reflects the activation energy required to force current i to flow.

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a + b = 1

Does the shape of these curves resemble concepts covered in the previous lecture? (Answer provided in lecture)

Exchange Current• The exchange current can be viewed as an “idle

speed”.

• If we want to draw a current that is only a small fraction of the bidirectional idle current (j/j0 <<1), then a very small overpotential (to unbalance the rates in the two directions slightly) is required.

• If we ask for a net current that exceeds the idle current significantly (j/j0 >1), we need to drive the system more by applying a larger overpotential.

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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

• This theoretical result is in agreement with the experimental findings of the German physical chemist Julius Tafel (1905), and plots of overpotential versus log j 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.

Small

• For x<<1, ex = 1 + x, and the Butler-Volmer equation becomes

j = j0(nF/RT) or = RTj/nFj0• There is a linear relationship between j and at

small overpotentials (< 10 mV).

• This linear region is called "polarization resistance" due to its formal similarity to Ohm’s law

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Problem - Small

• The exchange current density of a Pt|H2(g)|H+

electrode at 298 K is 0.79 mA cm-2. What current flows through a standard electrode of area 5.0 cm2 when the overpotential is 5.0 mV?

Large Oxidation• For large either negative or positive, one of

the exponential terms in the Butler -Volmer equation becomes negligible.

• At large positive potentials (corresponding to oxidation), the first term predominates and

j = j0e(1-nF/RT

or lnj = lnj0 + (1- )nF/RT

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Large - Reduction

• At large negative potentials (for reduction),

j = -j0e-nF/RT

or ln(-j) = lnj0 - nF/RT

• Remember is negative for reduction

• Same form as the Tafel equation

= a + b logi

Large • The Tafel form holds when ja/jc < 0.01 (or vice-versa) –

i.e., for ||>118/n mV.

• If the electrode kinetics are fast (large j0), the current will be limited by mass transfer by the time such a large overpotential is applied, and the Tafel relationship will not be observed.

• When electrode kinetics are slow (small j0) and activation potentials are required, the Tafel equation holds.

• Tafel behaviour is thus an indicator of totally irreversible kinetics.

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Problem - Oxidation

• A solution of 1 M KOH is electrolysed at 250C with a Pt electrode to produce O2 at the anode. At an overpotential of 0.40 V, the current density is 1.0 x 10-3 A cm-2. What does j become when = 0.6 V? Assume = 0.5 and n = 1.

Problem - Reduction• The exchange current density and transfer

coefficient for the reduction of H+ to H2 on Ni are 6.3 A cm-2 and 0.58. What is the current density at an overpotential of 200 mV?

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Tafel Equation

• The overpotential, , increases as the current flowing through the system increases.

• Tafel (1905) found that the overpotential is related to the logarithm of the current:

= a + b logi

where a and b are empirical constants.

Tafel Plots

• lnj = lnj0 + (1- )nF/RT from Butler Volmer

• ln(-j) = lnj0 - nF/RT from Butler Volmer

• A plot of lnj versus has an anodic branch with slope (1-)nF/RT and a cathodic branch with slope - nF/RT. Both linear segments extrapolate to an intercept of lnj0.

• The transfer coefficient, , and the exchange current density, j0, can thus be obtained.

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E

ln|I |

En

ln{In}

slope = −αF/RT

slope = (1−α)F/RT

• Tafel plot: the plot of logarithm of the current density against the over potential.

• Example: The following data are the cathodic current through a platinum electrode of area 2.0 cm2 in contact with an Fe 3+, Fe 2+ aqueous solution at 298K. Calculate the exchange current density and the transfer coefficient for the process. Slope is and intercept is a (=ln i0).

• In general exchange currents are large when the redox process involves no bond breaking or if only weak bonds are broken.

• Exchange currents are generally small when more than one electron needs to be transferred, or multiple or strong bonds are broken.

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Transport in Electrochemistry

• The rate of redox reactions is influenced by the cell potential difference.

• However, the rate of transport to the surface can also effect or even dominate the overall reaction rate and in this class we look at the different forms of mass transport that can influence electrolysis reactions.

• There are three forms of mass transport which can influence an electrolysis reaction:– Diffusion

– Convection

– Migration

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Diffusion• In essence, any electrode reaction is a

heterogeneous redox reaction. If its rate depends exclusively on the rate of mass transfer, then we have a mass-transfer controlled electrode reaction. If the only mechanism of mass transfer is diffusion (i.e. the spontaneous transfer of the electroactive species from regions of higher concentrations to regions of lower concentrations), then we have a diffusion controlled electrode reaction.

• Diffusion occurs in all solutions and arises from local uneven concentrations of reagents.

Diffusion• The rate of movement of material by diffusion can be predicted

mathematically and Fick proposed two laws to quantify the processes. The first law:

this relates the diffusional flux Jo (i.e. the rate of movement of material by diffusion) to the concentration gradient and the diffusion coefficient Do. The negative sign simply signifies that material moves down a concentration gradient i.e. from regions of high to low concentration. However, in many measurements we need to know how the concentration of material varies as a function of time and this can be predicted from the first law.

• The result is Fick's second law:

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in this case we consider diffusion normal to an electrode surface (x direction). The rate of change of the concentration ([O]) as a function of time (t) can be seen to be related to the change in the concentration gradient.

• Fick's second law is an important relationship since it permits the prediction of the variation of concentration of different species as a function of time within the electrochemical cell. In order to solve these expressions analytical or computational models are usually employed.

• How do we apply this to electrochemistry and the oxidation or reduction of a redox active species at an electrode surface?

When an electrode is polarised, the surface concentration of the species that is either being oxidized or reduced falls to zero. Additional material will then diffuse to the electrode surface towards this region of lower concentration. If the experiment is carried out in an stirred solution or the electrode is rotated then the resulting concentration-distance profile at the electrode surface is shown below.

Schematic covered in the lecture

This concept of the diffusion layer was introduced by Nernst in 1904.δ is the thickness of the Nernst diffusion layer

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Diffusion• The thickness of the Nernst diffusion layer varies within the range

0.1-0.001 mm depending on the intensity of convection caused by agitation of the electrodes or electrolyte.

• According to the definition of the Nernst diffusion layer the concentration gradient may be determined as follows:

Where: C0 - bulk concentration, Cc - concentration of the ions at the cathode surface; δc - thickness of the Nernst diffusion layer.

• Therefore the flux of ions toward the cathode surface is:

• Each ion possesses an electric charge. The density of the electric current formed by the moving ions:

Where: F - Faraday’s constant, F = 96485 Coulombs; z - number of elementary charges transferred by each ion.

• The maximum flux of the ions may be achieved when Cc= 0 therefore the electric current density is limited by the value:

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Convection• Convection results from the action of a force on the solution.

This can be a pump, a flow of gas or even gravity. There are two forms of convection the first is termed natural convection and is present in any solution.

• This natural convection is generated by small thermal or density differences and acts to mix the solution in a random and therefore unpredictable manner.

• In the case of electrochemical measurements these effects tend to cause problems if the measurement time for the experiment exceeds 20 seconds.

• It is possible to drown out the natural convection effects from an electrochemical experiment by deliberately introducing convection into the cell. This form of convection is termed forced convection.

• It is typically several orders of magnitude greater than any natural convection effects and therefore effectively removes the random aspect from the experimental measurements. This of course is only true if the convection is introduced in a well defined and quantitative manner.

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• If the flow is controlled, after a smalltime period, the profile will become stable with no mixing in the lateral direction, this is termed laminar flow.

• For laminar flow conditions the mass transport equation for (1 dimensional) convection is predicted by:

where vx is the velocity of the solution which can be calculated in many situations by solving the appropriate form of the Navier-Stokes equations. An analogous form exists for the three dimensional convective transport.

Migration• The final form of mass transport we need to consider is

migration. This is essentially an electrostatic effect which arises due the application of a voltage on the electrodes. This effectively creates a charged interface (the electrodes). Any charged species near that interface will either be attracted or repelled from it by electrostatic forces. The migratory flux induced can be described mathematically (in 1 dimension) as:

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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) – why?

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

Molarityof KNO3

Il

μA0 17.60.001 12.00.005 9.80.10 8.451.0 8.45

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• 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 what is the migration current contribution?

Mass Transport in Electrochemical Cells

To gain a quantitative model of the current flowing at the electrode we must account for the electrode kinetics, the 3 dimensional diffusion, convection and migration, of all the species involved. This is currently beyond the capacity of even the fastest computers - and will be for some time. However, as we will discover electrochemical cells and experimental conditions can be employed to cheat the

mass transport equations. We can effectively remove much (but not all) of the mass transport complexity by carefully designing and controlling the electrochemical experiment.