Nernst Equation
Understanding the Nernst Equation
The Nernst Equation is a fundamental equation in electrochemistry that relates the reduction potential of a half-cell in an electrochemical cell to the standard electrode potential, temperature, and activities (often approximated by concentrations) of the chemical species undergoing reduction and oxidation. It was named after the German chemist Walther Nernst who formulated the equation in 1889.
The Nernst Equation
The general form of the Nernst Equation for a reaction at 25°C (298 K) is given by:
$$ E = E^0 - \frac{RT}{nF} \ln Q $$
where:
- $E$ is the cell potential (electromotive force) of the reaction under non-standard conditions (in volts).
- $E^0$ is the standard cell potential (in volts).
- $R$ is the universal gas constant, which is $8.314 \text{ J mol}^{-1} \text{ K}^{-1}$.
- $T$ is the temperature in Kelvin.
- $n$ is the number of moles of electrons transferred in the reaction.
- $F$ is the Faraday constant, approximately $96485 \text{ C mol}^{-1}$.
- $Q$ is the reaction quotient, which is a dimensionless number representing the ratio of the activities (or concentrations) of the products to the reactants.
For practical purposes, especially in introductory chemistry, the Nernst Equation at 25°C is often written in the logarithmic form using base 10:
$$ E = E^0 - \frac{0.0592}{n} \log Q $$
This simplification uses the fact that $RT/F$ is approximately $0.0592$ V at 25°C.
Applications of the Nernst Equation
The Nernst Equation is used to:
- Calculate the cell potential under non-standard conditions.
- Determine the equilibrium constant of a redox reaction.
- Predict the direction of an electrochemical reaction.
- Calculate the concentration of ions in a solution using electrochemical cells (potentiometry).
Important Points and Differences
| Aspect | Standard Conditions | Non-Standard Conditions |
|---|---|---|
| Cell Potential ($E$) | $E^0$ is constant and characteristic of the redox couple | $E$ varies with temperature, pressure, and concentration |
| Reaction Quotient ($Q$) | $Q = 1$ (all reactants and products are in their standard states) | $Q$ varies with the actual concentrations of reactants and products |
| Temperature ($T$) | Usually 298 K (25°C) | Can be any temperature; must be in Kelvin for the equation |
| Equation Form | Not applicable (no need for the Nernst Equation) | $E = E^0 - \frac{RT}{nF} \ln Q$ or $E = E^0 - \frac{0.0592}{n} \log Q$ |
Examples
Example 1: Standard Cell Potential
Consider the standard hydrogen electrode (SHE), which has a standard electrode potential ($E^0$) of 0 V by definition. Under standard conditions, the cell potential is 0 V, and there is no need to apply the Nernst Equation.
Example 2: Cell Potential Under Non-Standard Conditions
Suppose we have a half-cell reaction:
$$ \text{Zn}^{2+} (aq) + 2e^- \rightarrow \text{Zn} (s) $$
The standard electrode potential ($E^0$) for this half-cell is -0.76 V. If the concentration of $\text{Zn}^{2+}$ is 0.1 M, we can calculate the cell potential at 25°C using the Nernst Equation:
$$ E = E^0 - \frac{0.0592}{n} \log Q $$
Since $n = 2$ and $Q = [\text{Zn}^{2+}] = 0.1$, we have:
$$ E = -0.76 - \frac{0.0592}{2} \log(0.1) $$ $$ E = -0.76 - \frac{0.0592}{2} \times (-1) $$ $$ E = -0.76 + 0.0296 $$ $$ E = -0.7304 \text{ V} $$
The cell potential under these non-standard conditions is -0.7304 V.
Example 3: Equilibrium Constant Calculation
For a redox reaction at equilibrium, the cell potential ($E$) is 0. Using the Nernst Equation, we can solve for the equilibrium constant ($K$), which is numerically equal to the reaction quotient ($Q$) at equilibrium:
$$ 0 = E^0 - \frac{RT}{nF} \ln K $$
Rearranging for $K$ gives:
$$ \ln K = \frac{nFE^0}{RT} $$ $$ K = e^{\frac{nFE^0}{RT}} $$
If we know $E^0$, $n$, $T$, and $R$, we can calculate the equilibrium constant for the reaction.
The Nernst Equation is a powerful tool in electrochemistry, providing insights into the thermodynamics of redox reactions and the behavior of electrochemical cells under a variety of conditions. Understanding and applying the Nernst Equation is essential for chemists, materials scientists, and engineers working with batteries, fuel cells, and other electrochemical technologies.