Electrochemistry Calculations Using the Nernst Equation

The Nernst equation is used to calculate the voltage of an electrochemical cell or to find the concentration of one of the components of the cell.

The Nernst Equation

The Nernst equation relates the equilibrium cell potential (also called the Nernst potential) to its concentration gradient across a membrane. An electric potential will form if there is a concentration gradient for the ion across the membrane and if selective ions channels exist so that the ion can cross the membrane. The relation is affected by temperature and whether the membrane is more permeable to one ion over others.

The equation may be written:

E_cell = E⁰_cell - (RT/nF)lnQ

E_cell = cell potential under nonstandard conditions (V)
E⁰_cell = cell potential under standard conditions
R = gas constant, which is 8.31 (volt-coulomb)/(mol-K)
T = temperature (K)
n = number of moles of electrons exchanged in the electrochemical reaction (mol)
F = Faraday's constant, 96500 coulombs/mol
Q = reaction quotient, which is the equilibrium expression with initial concentrations rather than equilibrium concentrations

Sometimes it is helpful to express the Nernst equation differently:

E_cell = E⁰_cell - (2.303*RT/nF)logQ

at 298K, E_cell = E⁰_cell - (0.0591 V/n)log Q

Nernst Equation Example

A zinc electrode is submerged in an acidic 0.80 M Zn²⁺ solution which is connected by a salt bridge to a 1.30 M Ag⁺ solution containing a silver electrode. Determine the initial voltage of the cell at 298K.

Unless you've done some serious memorizing, you'll need to consult the standard reduction potential table, which will give you the following information:

E⁰_red: Zn²⁺_aq + 2e^- → Zn_s = -0.76 V

E⁰_red: Ag⁺_aq + e^- → Ag_s = +0.80 V

E_cell = E⁰_cell - (0.0591 V/n)log Q

Q = [Zn²⁺]/[Ag⁺]²

The reaction proceeds spontaneously so E⁰ is positive. The only way for that to occur is if Zn is oxidized (+0.76 V) and silver is reduced (+0.80 V). Once you realize that, you can write the balanced chemical equation for the cell reaction and can calculate E⁰:

Zn_s → Zn²⁺_aq + 2e^- and E⁰_ox = +0.76 V

2Ag⁺_aq + 2e^- → 2Ag_s and E⁰_red = +0.80 V

which are added together to yield:

Zn_s + 2Ag⁺_aq → Zn²⁺_a + 2Ag_s with E⁰ = 1.56 V

Now, applying the Nernst equation:

Q = (0.80)/(1.30)²

Q = (0.80)/(1.69)

Q = 0.47

E = 1.56 V - (0.0591 / 2)log(0.47)

E = 1.57 V