ChemistryElectrochemistryA-Level

AQAAPIBAbiturBaccalauréat GénéralBachilleratoCambridgeCAPS

Nernst Equation

Cell potential under non-standard conditions.

Understand the formulaSee the free derivationOpen the full walkthrough

E = E^{θ} - \frac{R T}{n F} ln Q

Open Full Walkthrough Try Calculator

This public page keeps the free explanation visible and leaves premium worked solving, advanced walkthroughs, and saved study tools inside the app.

Core idea

Overview

The Nernst equation defines the relationship between the reduction potential of an electrochemical cell and the activities of the chemical species involved under non-standard conditions. It effectively relates the thermodynamics of a reaction to its voltage output by incorporating the reaction quotient and temperature.

When to use: Apply the Nernst equation when calculating cell voltage for solutions where concentrations are not 1 M or gas pressures are not 1 atm. It is essential when the system is not at standard state or when determining the concentration of ions using a measured potential.

Why it matters: This equation explains why batteries lose voltage as they run out of reactants and allows scientists to calculate the pH of solutions. In biology, it is used to determine the electrical potential across cell membranes, which is vital for nerve signaling.

Symbols

Variables

E = Cell Potential, E^\theta = Standard Potential, R = Gas Constant, T = Temperature, n = Moles of Electrons

E

Cell Potential

V

E^{θ}

Standard Potential

V

R

Gas Constant

J / m o l K

T

Temperature

K

n

Moles of Electrons

V a r iab l e

F

Faraday Constant

C / m o l

Q

Reaction Quotient

V a r iab l e

Walkthrough

Derivation

Formula: Nernst Equation

Relates electrode potential to concentration (or activity) using the reaction quotient Q for the half-equation as written.

Temperature is constant.
Activities are approximated by concentrations for dilute aqueous solutions (A-Level treatment).
z is the number of electrons transferred in the half-equation.

State the General Form:

Q is written from the half-equation as products over reactants (using concentrations/activities).

E = E^{⊖} - \frac{R T}{z F} ln Q

Note: At 298 K, this is often written as $E = E^{⊖} - \frac{0.0592}{z} lo g_{10} Q$ .

Result

E = E^{⊖} - \frac{R T}{z F} ln Q

Source: AQA A-Level Chemistry (Option) — Electrochemistry

Free formulas

Rearrangements

Solve for $E$

Make E the subject

E = E^\theta - \frac{R T \ln\left(Q \right)}}{n F}

Exact symbolic rearrangement generated deterministically for E.

Difficulty: 3/5

Solve for $E^{θ}$

Make E0 the subject

E^\theta = E + \frac{R T \ln\left(Q \right)}}{n F}

Exact symbolic rearrangement generated deterministically for E0.

Difficulty: 3/5

Solve for $R$

Make R the subject

R = \frac{n F \left(- E + E^\theta\right)}{T \ln\left(Q \right)}}

Exact symbolic rearrangement generated deterministically for R.

Difficulty: 3/5

Solve for $T$

Make T the subject

T = \frac{n F \left(- E + E^\theta\right)}{R \ln\left(Q \right)}}

Exact symbolic rearrangement generated deterministically for T.

Difficulty: 3/5

Solve for $n$

Make n the subject

n = - \frac{R T \ln\left(Q \right)}}{F \left(E - E^\theta\right)}

Exact symbolic rearrangement generated deterministically for n.

Difficulty: 3/5

Solve for $F$

Make F the subject

F = - \frac{R T \ln\left(Q \right)}}{n \left(E - E^\theta\right)}

Exact symbolic rearrangement generated deterministically for F.

Difficulty: 3/5

Solve for $Q$

Make Q the subject

Q = e^{- \frac{n F ( E - E ^{θ} )}{R T}}

Exact symbolic rearrangement generated deterministically for Q.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph follows a logarithmic curve where E decreases at a diminishing rate as Q increases, and it is only defined for Q greater than zero. For a chemistry student, this shape shows that small values of Q represent conditions favoring the forward reaction with higher cell potential, while large values of Q indicate the reaction is approaching equilibrium where the potential drops. The most important feature is that the curve never reaches zero, meaning the cell potential only becomes zero when the system reaches

Graph type: logarithmic

Why it behaves this way

Intuition

The Nernst equation can be visualized as a 'concentration gradient' that adjusts the cell's inherent standard potential based on how far the reactant and product concentrations are from their equilibrium balance, much

The actual cell potential (voltage) under non-standard conditions.

This is the measured voltage output of the electrochemical cell, reflecting the driving force for electron flow under the current concentrations and temperature.

E^{θ}

The standard cell potential, measured under standard conditions (1 M concentrations, 1 atm partial pressures, 298.15 K).

This is the benchmark or ideal voltage of the cell, representing its maximum theoretical potential when all components are at their reference states.

The ideal gas constant, relating energy to temperature and amount of substance.

A universal constant that scales the thermal energy available in the system, influencing how temperature affects the cell potential.

Absolute temperature in Kelvin.

Higher temperature means more thermal energy is available, which can increase the kinetic energy of particles and thus influence the cell's driving force.

The number of moles of electrons transferred in the balanced redox reaction.

This represents the stoichiometry of electron flow; more electrons transferred per reaction unit means more charge moved, affecting the potential.

The Faraday constant, representing the magnitude of electric charge per mole of electrons (approximately 96485 C/mol).

A constant that converts the chemical amount of electrons (moles) into the total electrical charge they carry.

The reaction quotient, expressing the relative amounts of products and reactants at any given time.

This term indicates how far the reaction is from equilibrium. If Q is small (more reactants), the reaction has a stronger drive to produce products; if Q is large (more products), the drive is weaker or even reversed.

Signs and relationships

-\frac{RT}{nF} \ln Q: The negative sign indicates that as the reaction proceeds towards products (Q increases from values less than 1), the cell potential 'E' decreases from 'E^ $θ$ '.

Free study cues

Insight

Canonical usage

The Nernst equation is typically used with SI units, where cell potentials are in Volts, temperature in Kelvin, and the gas and Faraday constants have their SI values.

Common confusion

A frequent mistake is using Celsius instead of Kelvin for temperature (T) or incorrectly applying the 2.303 factor when switching between natural logarithm (ln) and base-10 logarithm (log10).

Dimension note

The number of electrons (n) and the reaction quotient (Q) are dimensionless quantities. The reaction quotient is a ratio of activities, which are themselves dimensionless.

Unit systems

$E$ V · Cell potential under non-standard conditions.

$E^{θ}$ V · Standard cell potential.

$R$ J mol^-1 K^-1 · Ideal gas constant.

$T$ K · Absolute temperature.

$n$ dimensionless · Number of moles of electrons transferred in the balanced half-reaction.

$F$ C mol^-1 · Faraday constant.

$Q$ dimensionless · Reaction quotient, expressed in terms of activities or approximate concentrations/pressures.

Ballpark figures

Quantity:
Quantity:
Quantity:

One free problem

Practice Problem

Calculate the cell potential (E) for a Zn-Cu galvanic cell at 298 K where the reaction quotient (Q) is 50. The standard cell potential (E0) is 1.10 V and the reaction involves the transfer of 2 electrons.

Standard Potential1.1 V

Gas Constant8.314 J/mol K

Temperature298 K

Moles of Electrons2

Faraday Constant96485 C/mol

Reaction Quotient50

Solve for: $E$

Hint: Calculate the term (RT/nF) first, then multiply by the natural log of Q before subtracting from E0.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Calculating voltage of a concentration cell.

Study smarter

Tips

Convert the temperature to Kelvin by adding 273.15 to the Celsius value.
The variable 'n' represents the number of moles of electrons transferred in the balanced redox equation.
Pure solids and liquids have an activity of 1 and are omitted from the reaction quotient Q.
At 298.15 K, the term (RT/nF)ln(Q) can be simplified to (0.0592/n)log₁₀(Q) for convenience.

Avoid these traps

Common Mistakes

Using log10 instead of ln.
Forgetting to include n.

Common questions

Frequently Asked Questions

Relates electrode potential to concentration (or activity) using the reaction quotient Q for the half-equation as written.

Apply the Nernst equation when calculating cell voltage for solutions where concentrations are not 1 M or gas pressures are not 1 atm. It is essential when the system is not at standard state or when determining the concentration of ions using a measured potential.

This equation explains why batteries lose voltage as they run out of reactants and allows scientists to calculate the pH of solutions. In biology, it is used to determine the electrical potential across cell membranes, which is vital for nerve signaling.

Using log10 instead of ln. Forgetting to include n.

Calculating voltage of a concentration cell.

Convert the temperature to Kelvin by adding 273.15 to the Celsius value. The variable 'n' represents the number of moles of electrons transferred in the balanced redox equation. Pure solids and liquids have an activity of 1 and are omitted from the reaction quotient Q. At 298.15 K, the term (RT/nF)ln(Q) can be simplified to (0.0592/n)log₁₀(Q) for convenience.

References

Sources

Atkins' Physical Chemistry
IUPAC Gold Book: Nernst equation
Wikipedia: Nernst equation
NIST CODATA
IUPAC Gold Book
Halliday, Resnick, and Walker, Fundamentals of Physics
Atkins' Physical Chemistry, 11th Edition
IUPAC Gold Book (Compendium of Chemical Terminology)

Nernst Equation

Overview

Variables

Derivation

State the General Form:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Standard Cell Potential

Frequently Asked Questions

Sources