Nernst Equation, EMF, and Electrochemical Cell

1. Introduction to Electrochemical Cells

An electrochemical cell converts chemical energy and electrical energy through redox reactions. Two primary types are:

Galvanic (Voltaic) cell — spontaneous redox reaction produces electrical energy.
Electrolytic cell — electrical energy drives a non-spontaneous reaction.

A cell has two electrodes connected by an external circuit and a salt bridge: Anode (oxidation) and Cathode (reduction).

2. Electromotive Force (EMF)

The EMF (open-circuit cell potential) is the maximum potential difference between electrodes under standard, zero-current conditions. It relates to the Gibbs free-energy change by:

ΔG = − n F E_cell

ΔG: Gibbs free energy change (J)
n: moles of electrons transferred
F: Faraday constant (96,485 C·mol⁻¹)
E_cell: cell EMF (V)

Galvanic: E_cell > 0 → ΔG < 0 (spontaneous)

Electrolytic: E_cell < 0 → ΔG > 0 (non-spontaneous)

3. Nernst Equation

The Nernst equation relates the cell potential at non-standard conditions to activities (or concentrations) via the reaction quotient Q:

E_cell = E_cell^° − (R T ⁄ n F) ln Q

E_cell^°: standard cell potential
R: gas constant (8.314 J·mol⁻¹·K⁻¹)
T: temperature (K)
n: electrons transferred; F: Faraday constant; Q: reaction quotient

At 298 K (25°C), using base-10 logarithm:

E_cell = E_cell^° − (0.0591 ⁄ n) log Q

4. Applications of the Nernst Equation

EMF at non-standard conditions — compute E_cell for given concentrations/pressures.
pH determination — using hydrogen electrode potentials (glass electrode calibration).
Solubility product (K_sp) — derive from EMF of cells involving sparingly soluble salts.
Equilibrium constant (K) — at equilibrium, E_cell=0: K = exp(n F E_cell^° ⁄ R T).
Concentration cells — identical electrodes with differing ion activities.

5. Example: Daniell Cell

Overall reaction:

Zn(s) + Cu²⁺(aq) → Zn²⁺(aq) + Cu(s)

Standard EMF:

E_cell^° = E_cathode^° − E_anode^° = (+0.34 V) − (−0.76 V) = 1.10 V

Nernst at 298 K:

E_cell = 1.10 − (0.0591 ⁄ 2) log ( [Zn²⁺] ⁄ [Cu²⁺])

6. Summary Table

Term	Definition / Relation
EMF (E_cell)	Open-circuit potential difference between electrodes
E_cell^°	Cell EMF under standard conditions (1 M, 1 atm, 298 K)
ΔG relation	ΔG = −n F E_cell
Nernst equation	E_cell = E_cell^° − (R T ⁄ n F) ln Q ; at 298 K: E_cell = E_cell^° − (0.0591 ⁄ n) log Q
Cell types	Galvanic (spontaneous, E_cell>0) ; Electrolytic (non-spontaneous, E_cell<0)

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Study Notes