### Butler–Volmer equation & Current density

At 298 K the exchange current density (i_{o})
of an electrode having a surface area of 5 cm^{2}, used in an
electrodeposition process is 0.79 mA cm^{–2}.

Estimate the applied current through the electrode if the overpotential (η) is 5 mV.

Required formula derived from the Butler–Volmer equation is:

i = (i_{o} η F) / (R T)

where, ‘i’ is the applied current density (mA cm^{–2})

i_{o} is exchange current density = 0.79 mA cm^{–2}

η is overpotential = 5 mV

F is Faraday = 96485 C mol^{–1}

R is gas constant = 8.314 J K^{–1} mol^{–1}

T is temperature (in K) = 298 K

Current density = Current (mA) / Surface area (cm^{2})

Surface area = 5 cm^{2}

Substituting these values in the equation, value of the applied current density is:

i = (0.79 mA cm^{–2 }× 5
mV × 96485 C mol^{–1}) / (8.314 J K^{–1}
mol^{–1 }× 298 K).

Since, Joule = Volt × Coulomb or J = V × C.

So, i = (0.79 mA cm^{–2 }× 5 mV × 96485 C mol^{–1}) / (8.314 V C K^{–1}
mol^{–1 }× 298 K)

And, 1 V = 1000 mV

i = (0.79 mA cm^{–2 }× 5
mV × 96485 C mol^{–1}) / (8.314 × 1000 mV C K^{–1} mol^{–1 }× 298 K)

Hence, i = 0.15 mA cm^{–2}

And the applied current (mA) = Current density × Surface area.

So, the applied current = 0.15 mA cm^{–2 }× 5 cm^{2 }= 0.75 mA.

Dr. M Kanagasabapathy

Associate Professor of Chemistry

Rajapalayam Rajus' College

Madurai Kamaraj University

Rajapalayam (TN) India