Nernst Equation
Quick Notes
- Nernst Equation:
- E° is standard cell potential and n is the number of electrons exchanged.
- Q is the reaction quotient (ratio of product to reactant concentrations).
- At equilibrium, E = 0 and E° = (0.0591/n) log K.
- Gibbs energy link: ΔG = −nFE and ΔG° = −nFE°
- Sign of E° helps determine reaction spontaneity.
Full Notes
Introduction to the Nernst Equation
In real-world conditions, concentrations of ions and gases often vary from their standard values. The Nernst equation provides a tool to calculate the actual electrode potential or cell potential under such non-standard conditions.
Nernst Equation (Qualitative Form)
The key idea is:
- As Q increases, the cell potential decreases (reaction slows down in the forward direction).
- As Q decreases, the cell potential increases (reaction is more strongly driven forward).
Recap – What Q Tells Us About the Reaction
What is the Reaction Quotient (Q)?
The reaction quotient (Q) is a snapshot of a reaction’s progress. It is calculated by using concentration values at a specific point in time, which might not be equilibrium values.
- If Q < 1 this means more reactants than products and the reaction is far from equilibrium → E > E°
- If Q > 1 this means more products than reactants and the reaction is closer to equilibrium → E < E°
- If Q = K the system is at equilibrium → E = 0
This explains why a cell generates electricity: it's operating to get to equilibrium, and the further it is from equilibrium, the greater the “push” (E) driving electrons through the circuit.
Equilibrium Constant from Nernst Equation
At equilibrium, the cell stops working, i.e., E = 0. This condition helps derive a link between cell potential and equilibrium constant (K) of the redox reaction.
- At equilibrium: E = 0
- Q = K (equilibrium constant)
(note that this form comes from Converting natural log (ln) to log base 10:
ln Kc = 2.303 × log Kc
Substitute this into the equation:
E = E° − (2.303 × RT / nF) × log Kc
This expression allows us to calculate the equilibrium constant of a redox reaction from its standard electrode potential.
Interpretation:
- If E° > 0 then K > 1 = Products favored
- If E° < 0 then K < 1 = Reactants favored
Electrochemical Cell and Gibbs Energy of the Reaction
The Gibbs free energy change (ΔG) helps us determine whether a reaction can occur without external energy input. This section shows how cell potential relates directly to Gibbs energy.
The energy available to drive an electric current (E°cell) comes directly from the energy change in the chemical reaction (ΔG°).
These two quantities are linked by the equation:
This allows us to connect the electrical world of electrochemical cells with the thermodynamic world of Gibbs energy and predict spontaneity from either.
How to Use the Equation
- Identify n = number of electrons transferred from half-equations
- Use provided E°cell (or from the data booklet)
- Use F = 96,500 C mol⁻¹
- Calculate ΔG° (in joules); divide by 1,000 for kJ mol⁻¹
Interpretation:
- If E > 0, then ΔG < 0 and reaction is spontaneous.
- If E < 0, then ΔG > 0 and reaction is non-spontaneous.
Thus, positive cell potential indicates a thermodynamically favorable process.
We can also use Gibbs free energy change to determine an equilibrium constant using the equation:
Zn + Cu2+ → Zn2+ + Cu
From data booklet:
Zn2+/Zn = −0.76 V
Cu2+/Cu = +0.34 V
F = 96,500 C mol⁻¹
- Calculate E°cell
E°cell = 0.34 − (−0.76) = +1.10 V - Electrons transferred
n = 2 electrons (Zn loses 2 e⁻; Cu2+ gains 2 e⁻). - Calculate ΔG°
ΔG° = −nFE°cell = −(2)(96,500)(1.10) = −212,300 J mol⁻¹ = −212.3 kJ mol⁻¹
Summary
- Nernst equation adjusts cell or electrode potentials for non-standard concentrations and pressures.
- As Q increases E decreases and the cell is closer to equilibrium.
- At equilibrium E = 0 and E° relates to K allowing calculation of K.
- ΔG = −nFE links thermodynamics and electrochemistry for spontaneity checks.