kf kb

v = k C(0,t) = i/nFA

mole/cm2-s = cm/s × mole/cm3 = (coul/s)(mole/coul)(1/cm2)

κ is a transmission coefficient

Electrode Kinetics: Charge transfer control (Activation Polarization)

v = kC(0,t) = i/nFA i = nFAkC(0,t)

inet = nFA[kf CO(0,t) - kbCR(0,t)]

vf = kf CO(0,t)

O + e- = R

vb= kbCR(0,t)

vnet=vf-vb = kf CO(0,t) - kbCR(0,t)

Since v is measured in moles/cm2-s if we multiply by the number of coulombs per mole(nFA) in the rate equation we get a current.

Now consider a 1e- redox reaction

We can write a rate expression for each and the net rate is just the difference.

O+e- R

Apply and E > Eo so that (E-Eo ) > 0. The activation barrier for the anodic reactionhas been reduced by and that for the cathodic reduction reactionHas been increased by .

Reaction coordinate Reaction coordinate

The effect of applied potential on activation free energy

f = F/RT

Application of a potential greaterthan Eo reduces the Act. Barrier forthe oxidation (anodic) process and the activation barrier for the reduction(cathodic) process.

and these Act. Free energies wewrite the forward and backwardrate constants as,

From our general expression for therate constant,

where

Inserting these relations in to inet = nFA[kf CO(0,t) - kbCR(0,t)] yields,

Note that if we set i = 0

Then E = Eequil and we recover the Nernst Equation.

Even though the net current is zero at the equilibrium potential, the oxidation and reduction currents are not – they just add together to yield zero since they are equalIn magnitude. We can express this so-called exchange current (or exchange current density) In terms of either ia or ic,

or

Raising both sides of to the –α power yields

And substitution into the expression for the exchange current yields an expressionFor io in terms of the bulk concentrations of O and R,

dividing

by the expression for io

and after a bit of algebraic manipulation

If the surface and bulk concentrations of O and R are equal then there are NO masstransfer effects. The system is under activation (charge transfer)control.

Butler-Volmer Equation

The symmetry factor, α

α is dependent on the energy landscape in transitioning from O to R and vice versa.This term together with the io term in the B-V equation contains lots of buried physics,much if it poorly understood.

Butler-Volmer Equation

There are two regimes of behavior: High η: At 298 K this occurs when

Tafel Equation

Low η:

Tafel Plot

charge transfer resistance Rct