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ChemistryEdu Logo Electrochemistry | Nernst Equation#

Nernst Equation#

  • Nernst Equation is used to calculate EMF of a cell.

Nernst Equation derivation

At temperature T, change in Gibbs Free Energy is given by:

\[ ΔG = ΔG^0 + RTlnQ \]

where, Q = Reaction quotient, R = Universal Gas Constant

We know that:

\[ΔG^0 = -nFE^0_{cell}\]
\[ΔG = -nFE_{cell}\]

Putting these two values in above equation:

\[ -nFE_{cell} = -nFE^0_{cell} + RTlnQ \]
\[ E_{cell} = E^0_{cell} - {RT \over {nF}}lnQ \]
\[ E_{cell} = E^0_{cell} - {2.303RT \over {nF}}logQ \]

This equation is known as Nernst Equation.

If T = 298 K and putting the values of F and R in the given equation:

\[ E_{cell} = E^0_{cell} - {0.0591 \over n}logQ \]

Here, n = Number of electrons exchanged during cell reaction

How to apply Nernst Equation?#

  • Calculate E0cell = E0cathode - E0anode
  • Write oxidation half and reduction half reactions.
  • Add oxidation half and reduction half reactions to obtain net cell reaction.
  • Find number of electrons exchanged, n.
  • Calculate reaction quotient of the net cell reaction.
  • Use Nernst Equation to find Ecell:
\[ E_{cell} = E^0_{cell} - {0.0591 \over n}logQ \]

Nernst Equation during Equilibrium#

  • At equilibrium, we know that change in Gibbs Free Energy is zero.
\[ΔG = 0\]
\[-nFE_{cell} = 0\]
\[E_{cell} = 0\]
  • Also, at equilibrium: \(Q = K_c\)

  • Therefore, at equilibrium, Nernst equation can be written as:

\[ E_{cell} = E^0_{cell} - {0.0591 \over n}logQ \]
\[ 0 = E^0_{cell} - {0.0591 \over n}logK_c \]
\[ E^0_{cell} = {0.0591 \over n}logK_c \]

Nernst Equation For Half Cell Reactions#

  • Nernst Equation is applicable for half cell as well as complete cell.

  • For oxidation half cell, we can write:

\[ E_{ox} = E^0_{ox} - {0.0591 \over n}logQ_1 \]
  • For reduction half cell, we can write:
\[ E_{red} = E^0_{red} - {0.0591 \over n}logQ_2 \]
  • Adding oxidation and reduction half cell reaction, we get net cell reaction and Nernst equation of cell can be written as:
\[ E_{ox} + E_{red} = E^0_{ox} + E^0_{red} - {0.0591 \over n}log(Q_1Q_2) \]

We know that: \(E_{ox} + E_{red} = E_{cell}\)

\[E_{cell} = E^0_{cell} - {0.0591 \over n}log(Q)\]

Electrochemical cell and Gibbs Energy of a Reaction#

  • Change in Gibbs Free Energy is given by:
\[ΔG = -nFE_{cell}\]
  • Standard Change in Gibbs Free Energy is given by:
\[ΔG^0 = -nFE^0_{cell}\]

Also,

\[ΔG^0 = -RTlnK_c\]