Nernst Equation Calculator

Calculate electrode potentials under non-standard conditions using the Nernst equation. Determine cell potential, concentrations, pH, and other electrochemical properties.

Calculation Mode

Standard Cell Potential (E°) in V

Temperature (T) in K

298.15 K = 25°C (standard temperature)

Electrons Transferred (n)

Reaction Quotient (Q)

Q = [products]^coeff / [reactants]^coeff

Example: For Zn(s) + Cu²⁺(aq) → Zn²⁺(aq) + Cu(s):
Q = [Zn²⁺]/[Cu²⁺]

The Nernst Equation

The Nernst equation relates the cell potential at any conditions to the standard cell potential, temperature, and the activities (or concentrations) of the chemical species undergoing reduction and oxidation.

General Form:

E = E° - (RT/nF) ln(Q)

At 25°C (298.15 K):

E = E° - (0.0592/n) log₁₀(Q)

E = E° - (0.0257/n) ln(Q)

Variables:

E: Cell potential at non-standard conditions (V)
E°: Standard cell potential at 25°C, 1 M, 1 atm (V)
R: Universal gas constant = 8.314 J/(mol·K)
T: Temperature in Kelvin (K)
n: Number of moles of electrons transferred in the balanced equation
F: Faraday constant = 96,485 C/mol
Q: Reaction quotient = [products]/[reactants]

Derivation from Thermodynamics

The Nernst equation is derived from the fundamental relationship between Gibbs free energy and the reaction quotient:

Step 1: Gibbs free energy at non-standard conditions:

ΔG = ΔG° + RT ln(Q)

Step 2: Relationship between ΔG and cell potential:

ΔG = -nFE

ΔG° = -nFE°

Step 3: Substitute into the Gibbs equation:

-nFE = -nFE° + RT ln(Q)

Step 4: Solve for E:

-nFE = -nFE° + RT ln(Q)

E = E° - (RT/nF) ln(Q)

This is the Nernst equation!

Practical Applications

pH Meters

Glass electrodes measure pH by responding to H⁺ concentration:

E = E° - (0.0592) log[H⁺]

E = E° + 0.0592 pH

The potential varies linearly with pH (59.2 mV per pH unit at 25°C).

Concentration Cells

Cells with identical electrodes but different concentrations:

E = (RT/nF) ln(C₁/C₂)

Used to measure concentration differences and in some battery designs.

Ion-Selective Electrodes

Measure specific ion concentrations (Na⁺, K⁺, Ca²⁺, F⁻, etc.) in solution. Used in clinical chemistry, environmental monitoring, and process control.

Battery Voltage

Predicts voltage of batteries under load as reactant concentrations change. Explains why battery voltage drops as it discharges.

Example Calculations

Example 1: Non-Standard Cell Potential

Problem: Calculate the cell potential for the Daniel cell at 25°C:
Zn(s) + Cu²⁺(aq) → Zn²⁺(aq) + Cu(s)
Given: [Cu²⁺] = 0.010 M, [Zn²⁺] = 1.0 M, E° = 1.10 V

Q = [Zn²⁺]/[Cu²⁺] = 1.0/0.010 = 100

E = E° - (0.0592/n) log(Q)

E = 1.10 - (0.0592/2) log(100)

E = 1.10 - (0.0296) × (2)

E = 1.10 - 0.059 = 1.041 V

Result: E = 1.04 V (lower than E° due to low [Cu²⁺])

Example 2: pH Measurement

Problem: A pH electrode measures a potential of 0.414 V relative to the standard hydrogen electrode at 25°C. Calculate the pH.

For hydrogen electrode: 2H⁺ + 2e⁻ → H₂, E° = 0.000 V

E = E° - (0.0592/2) log(1/[H⁺]²)

E = 0.000 - (0.0296) × (-2 log[H⁺])

E = 0.0592 log[H⁺]

Since pH = -log[H⁺]: E = -0.0592 pH

pH = -E/0.0592 = -0.414/0.0592 = 7.0

Result: pH = 7.0 (neutral solution)

Example 3: Finding Concentration

Problem: A copper half-cell measures 0.310 V at 25°C. If [Cu²⁺] is unknown and E° = 0.34 V, find [Cu²⁺].

Half-reaction: Cu²⁺ + 2e⁻ → Cu(s)

E = E° - (0.0592/n) log(1/[Cu²⁺])

0.310 = 0.34 - (0.0592/2) log(1/[Cu²⁺])

-0.030 = -0.0296 log(1/[Cu²⁺])

log(1/[Cu²⁺]) = 1.014

1/[Cu²⁺] = 10^1.014 = 10.3

[Cu²⁺] = 0.097 M ≈ 0.10 M

Result: [Cu²⁺] = 0.10 M

Temperature Effects

Temperature significantly affects cell potential through the RT/nF term in the Nernst equation. The coefficient (RT/nF) changes with temperature:

Temperature	T (K)	RT/F (mV)	0.0592 × T/298.15
0°C	273.15	23.5	0.0542
25°C (standard)	298.15	25.7	0.0592
37°C (body temp)	310.15	26.7	0.0615
50°C	323.15	27.8	0.0642

Important: For pH measurements, the slope changes from 59.2 mV/pH at 25°C to 61.5 mV/pH at 37°C. Modern pH meters include automatic temperature compensation (ATC) to account for this effect.

Common Pitfalls and Tips

1. Reaction Quotient Q

• Always write Q with products in numerator, reactants in denominator
• Use activities: pure solids and liquids have activity = 1
• Include stoichiometric coefficients as exponents
• Don't confuse Q with K (equilibrium constant)

2. Natural Log vs Log₁₀

• General form uses ln (natural logarithm)
• Simplified form at 25°C uses log₁₀ (common logarithm)
• Relationship: ln(x) = 2.303 × log₁₀(x)
• 0.0592 = 2.303 × RT/F at 25°C

3. Sign Conventions

• E > E° when Q < 1 (more reactants favors forward reaction)
• E < E° when Q > 1 (more products favors reverse reaction)
• At equilibrium: Q = K and E = 0 (not E°!)

4. Units

• E and E° in volts (V)
• Temperature in Kelvin (K), not Celsius
• R in J/(mol·K), F in C/mol → RT/nF gives volts
• Concentrations in molarity (M) as approximation for activity

Note: The Nernst equation assumes ideal behavior and uses concentrations as approximations for activities. Real systems may deviate from ideal behavior, especially at high ionic strengths. For precise work, activity coefficients should be used. Standard potentials are typically measured at 25°C and may vary slightly with temperature.

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