1. The document discusses electrode kinetics and the Butler-Volmer equation, which describes the relationship between current and overpotential during a redox reaction. 2. It explains how applying a potential above the equilibrium potential Eo lowers the activation barrier for oxidation and raises it for reduction. 3. The Butler-Volmer equation contains terms for the forward and backward rate constants that are dependent on the applied potential relative to Eo, as well as the exchange current density io which is proportional to the surface concentrations.

1. kf kb
v = k C(0,t) = i/nFA
mole/cm2-s = cm/s × mole/cm3 = (coul/s)(mole/coul)(1/cm2)
 is a transmission coefficient
Electrode Kinetics: Charge transfer control (Activation Polarization)

2. 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.

3. O+e- R
Apply and E > Eo so that (E-Eo ) > 0. The activation barrier for the anodic reaction
has been reduced by and that for the cathodic reduction reaction
Has been increased by .
Reaction coordinate Reaction coordinate
The effect of applied potential on activation free energy

4. f = F/RT
Application of a potential greater
than Eo reduces the Act. Barrier for
the oxidation (anodic) process and
the activation barrier for the reduction
(cathodic) process.
and these Act. Free energies we
write the forward and backward
rate constants as,
From our general expression for the
rate constant,
where

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

6. 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 equal
In magnitude. We can express this so-called exchange current (or exchange current density)
In terms of either ia or ic,
or

7. Raising both sides of to the –α power yields
And substitution into the expression for the exchange current yields an expression
For io in terms of the bulk concentrations of O and R,
dividing
by the expression for io

8. and after a bit of algebraic manipulation
If the surface and bulk concentrations of O and R are equal then there are NO mass
transfer effects. The system is under activation (charge transfer)control.
Butler-Volmer Equation

9. 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.

10. 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