Standard cell potentials are calculated in standard conditions. The temperature and pressure are at standard temperature and pressure and the concentrations are all 1 M aqueous solutions. In non-standard conditions, the Nernst equation is used to calculate cell potentials. It modifies the standard cell potential to account for temperature and concentrations of the reaction participants. This example problem shows how to use the Nernst equation to calculate a cell potential.

### Problem

Find the cell potential of a galvanic cell based on the following reduction half-reactions at 25 °C

Cd^{2+} + 2 e^{-} → Cd E^{0} = -0.403 V

Pb^{2+} + 2 e^{-} → Pb E^{0} = -0.126 V

where [Cd^{2+}] = 0.020 M and [Pb^{2+}] = 0.200 M.

### Solution

The first step is to determine the cell reaction and total cell potential.

In order for the cell to be galvanic, E^{0}_{cell} > 0.

(Note: Review Galvanic Cell Example Problem for the method to find cell potential of a galvanic cell.)

For this reaction to be galvanic, the cadmium reaction must be the oxidation reaction. Cd → Cd^{2+} + 2 e^{-} E^{0} = +0.403 V

Pb^{2+} + 2 e^{-} → Pb E^{0} = -0.126 V

The total cell reaction is:

Pb^{2+}(aq) + Cd(s) → Cd^{2+}(aq) + Pb(s)

and E^{0}_{cell} = 0.403 V + -0.126 V = 0.277 V

The Nernst equation is:

E_{cell} = E^{0}_{cell} - (RT/nF) x lnQ

where

E_{cell} is the cell potential

E^{0}_{cell} refers to standard cell potential

R is the gas constant (8.3145 J/mol·K)

T is the absolute temperature

n is the number of moles of electrons transferred by the cell's reaction

F is Faraday's constant 96485.337 C/mol )

Q is the reaction quotient, where

Q = [C]^{c}·[D]^{d} / [A]^{a}·[B]^{b}

where A, B, C, and D are chemical species; and a, b, c, and d are coefficients in the balanced equation:

a A + b B → c C + d D

In this example, the temperature is 25 °C or 300 K and 2 moles of electrons were transferred in the reaction.

RT/nF = (8.3145 J/mol·K)(300 K)/(2)(96485.337 C/mol)

RT/nF = 0.013 J/C = 0.013 V

The only thing remaining is to find the reaction quotient, Q.

Q = [products]/[reactants]

(Note: For reaction quotient calculations, pure liquid and pure solid reactants or products are omitted.)

Q = [Cd^{2+}]/[Pb^{2+}]

Q = 0.020 M / 0.200 M

Q = 0.100

Combine into the Nernst equation:

E_{cell} = E^{0}_{cell} - (RT/nF) x lnQ

E_{cell} = 0.277 V - 0.013 V x ln(0.100)

E_{cell} = 0.277 V - 0.013 V x -2.303

E_{cell} = 0.277 V + 0.023 V

E_{cell} = 0.300 V

### Answer

The cell potential for the two reactions at 25 °C and [Cd^{2+}] = 0.020 M and [Pb^{2+}] = 0.200 M is 0.300 volts.