The equilibrium constant for the reaction $$ \mathrm{Sr}(s)+\mathrm{Mg}^{2+}(a q) \rightleftharpoons \mathrm{Sr}^{2+}(a q)+\mathrm{Mg}(s) $$ is \(2.69 \times 10^{12}\) at \(25^{\circ} \mathrm{C}\). Calculate \(E^{\circ}\) for a cell made up of \(\mathrm{Sr} / \mathrm{Sr}^{2+}\) and \(\mathrm{Mg} / \mathrm{Mg}^{2+}\) half-cells.

The Nernst Equation is a fundamental relationship used in electrochemistry to calculate the electric potential of a cell under non-standard conditions. It correlates the observed cell potential to the standard cell potential, the temperature, and the reaction quotient.

At its core, the equation accounts for the tendencies of ions to move toward reduced concentration and the influence of temperature on this movement. Here's the general form of the Nernst Equation:
\[E = E^{\text{o}} - \frac{RT}{nF} \ln(Q)\]
Where:

• \(E\) is the cell potential under non-standard conditions.
• \(E^{\text{o}}\) is the standard cell potential.
• \(R\) is the universal gas constant (\(8.314\) J/(mol·K)).
• \(T\) is the temperature in Kelvin.
• \(n\) is the number of moles of electrons transferred in the reaction.
• \(F\) is Faraday's constant (\(96485\) C/mol).
• \(Q\) is the reaction quotient, a measure of the relative amounts of products and reactants at any point in time.

The Nernst Equation is especially useful for figuring out cell potentials when the concentrations don't align with standard conditions, which is typically \(1\) M for solutions and \(1\) atm for gases. This equation also links thermodynamics with the electrochemical behavior, as it allows us to calculate potentials that help understand the system's tendency toward equilibrium.

The electrochemical cell standard potential, often denoted as \(E^{\text{o}}\), is a measure of the inherent voltage of an electrochemical cell under standard conditions, which include a temperature of \(298\) K, all solutes at a concentration of \(1\) M, and all gases at \(1\) atm pressure.

\(E^{\text{o}}\) is determined by the difference in potential between two half-cell reactions, serving as a predictor of how spontaneously a redox reaction may occur. The greater the \(E^{\text{o}}\) value, the more likely the reaction will proceed forward, indicating a stronger oxidative power of the anode material or reductive power of the cathode material in a cell.

Example: Standard Cell Potential from Equilibrium Constant

In our exercise, we are asked to calculate \(E^{\text{o}}\) for a cell composed of strontium and magnesium half-cells. To do this, we utilize the relationship between Gibbs free energy (\(\text{Δ}G^{\text{o}}\)) and the Nernst Equation:
\[E^{\text{o}} = -\frac{\text{Δ}G^{\text{o}}}{nF}\]
This relationship also links to the equilibrium constant (\(K\)) because Gibbs free energy for a reaction at equilibrium is related to \(K\) by:

• \(\text{Δ}G^{\text{o}} = -RT \ln(K)\)

Hence, we substitute the equilibrium constant into the Nernst Equation to find out the standard cell potential and understand the cell's efficiency at converting chemical energy into electrical energy.

Gibbs free energy, represented as \(\text{Δ}G\), is a thermodynamic property that predicts the direction of chemical reactions and indicates whether a process will occur spontaneously. In the context of an electrochemical cell, changes in Gibbs free energy can inform us about the cell's ability to perform electrical work. The relationship between \(\text{Δ}G^{\text{o}}\) and the equilibrium constant \(K\) is pivotal in this analysis.

For a reaction at equilibrium, \(\text{Δ}G^{\text{o}} = 0\), reflecting that no net work can be extracted. However, when \(\text{Δ}G^{\text{o}}\) carries a negative value, the reaction is spontaneous, and positive work can be done. Inversely, a positive \(\text{Δ}G^{\text{o}}\) suggests a non-spontaneous process requiring energy input.

Linking Equilibrium to Cell Potential

The equilibrium constant \(K\) is mathematically linked to \(\text{Δ}G^{\text{o}}\) by the expression:
\[\text{Δ}G^{\text{o}} = -RT \ln(K)\]
Subsequently, since cell potential is interrelated with \(\text{Δ}G^{\text{o}}\), the larger the value of \(K\) (indicating a more product-favored reaction), the more negative \(\text{Δ}G^{\text{o}}\) will be, and thus the higher the standard potential \(E^{\text{o}}\) of the cell. This understanding allows us to predict whether a reaction is likely to proceed under standard conditions.