Question

derive butler-volmer equation from Arrhenius equation

please show all steps

Answer 1

Arrehnius equation :

k = A exp (-E_a/ (RT) )

Where

• k is the rate constant
• T is the absolute temperature (in kelvin)
• A is the pre-exponential factor, a constant for each chemical reaction that defines the rate due to frequency of collisions in the correct orientation
• E_a is the activation energy for the reaction (in Joules)
• R is the universal gas constant

At the electrode surface, the redox reaction occurs: Ox + e^-=Red. The reaction rates of the forward direction k_f and backward direction k_b depend on the applied potential E at the electrode:

,

Where E_a= transfer coefficient*no of electron transferred*Faradays constant*(E-E₀)

where (in V) is the standard potential of the redox reaction,

(in V) is the applied potential,

is the gas constant ( J/K mol),

is the Faraday constant ( C/mol),

(in K) is the temperature,

is the number of electrons transferred,

A=is the standard heterogeneous rate constant (in m/s),

is the transfer coefficient.

The net current at the electrode is the sum of the currents in the forward (cathodic current) and backward directions (anodic current).

Then , where (m​​) is the surface area of the electrode, [Ox] and [Red] are the concentrations of the oxidant and reductor, and, replacing and , we have

.

This is called the Butler–Volmer equation