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The Nernst Equation is a fundamental equation in electrochemistry that lets us calculate the cell potential of an electrochemical cell under non-standard conditions. It provides a link between the standard electrode potential and the concentrations of the reacting species. This equation is represented as follows: \[ E_{\text{cell}} = E^{\circ}_{\text{cell}} - \frac{RT}{nF} \ln Q \] Where:

• \( E_{\text{cell}} \) is the cell potential under non-standard conditions.
• \( E^{\circ}_{\text{cell}} \) is the standard cell potential.
• \( R \) is the universal gas constant \((8.314 \, J/mol \cdot K)\).
• \( T \) is the temperature in Kelvin (298 K for \( 25^{\circ} \text{C} \)).
• \( n \) is the number of moles of electrons transferred in the cell reaction.
• \( F \) is the Faraday constant \((96485 \, C/mol)\).
• \( Q \) is the reaction quotient.

At room temperature \( 25^{\circ} \text{C} \), the factor \( \frac{RT}{nF} \) simplifies to \( 0.0257 \, V \) when \( n = 2 \). By adjusting the standard potential \( E^{\circ}_{\text{cell}} \) with the logarithm of the reaction quotient \( Q \), the Nernst Equation accommodates changes in concentration, giving us a realistic measure of the cell's potential under given conditions.

The standard electrode potential, denoted as \( E^{\circ} \), is the measure of the potential of a half-cell under standard conditions, which include a solute concentration of 1 M, a gas pressure of 1 atm, and a temperature of 25°C (298 K). It is measured relative to the standard hydrogen electrode, which has an assigned potential of 0 V, acting as the reference point. Standard electrode potentials allow us to predict the direction of electron flow in an electrochemical cell. In half-reactions, a more positive \( E^{\circ} \) indicates a greater tendency to attract electrons, meaning the species is more likely to get reduced. Conversely, a more negative \( E^{\circ} \) reflects a greater tendency to lose electrons, suggesting that the species is more likely to undergo oxidation. For example, in the exercise provided, the nickel half-reaction has a standard potential of \( -0.28 \, V \) and the magnesium half-reaction has \( -2.37 \, V \). These values help establish which reactions will occur at the cathode and anode.

A Galvanic cell, or a voltaic cell, is an electrochemical cell that converts chemical energy into electrical energy through spontaneous redox reactions. It consists of two separate half-cells connected by a salt bridge or a porous membrane to allow the flow of ions and maintain charge balance. Each half-cell contains an electrode and an electrolyte where specific half-reactions occur. The anode is where oxidation takes place, and electrons are generated. These electrons flow through an external circuit to the cathode, where reduction occurs, completing the circuit. Let’s take the example from the exercise:

• The first half-cell involves the reaction \( \mathrm{Ni}^{2+} + 2e^- \rightarrow \mathrm{Ni} \).
• In the second half-cell, the reaction is \( \mathrm{Mg} \rightarrow \mathrm{Mg}^{2+} + 2e^- \).

The magnesium half-cell acts as the anode due to its more negative electrode potential, while the nickel half-cell acts as the cathode. The aforementioned reactions happen spontaneously, allowing the Galvanic cell to generate electrical current.

The Reaction Quotient, denoted as \( Q \), is a mathematical representation of the relative concentrations of reactants and products at any point in a reaction. It helps predict the direction in which a reaction will proceed to reach equilibrium. For redox reactions taking place in a galvanic cell, the reaction quotient is expressed as the ratio of the concentrations of products raised to the power of their coefficients to those of reactants, also raised to the power of their coefficients. In the provided application of the Nernst Equation: \[ Q = \frac{[\mathrm{Mg}^{2+}]}{[\mathrm{Ni}^{2+}]} \]Given that \([\mathrm{Mg}^{2+}] = 0.50 \, M\) and \([\mathrm{Ni}^{2+}] = 1.0 \, M\), the reaction quotient becomes 0.50. The value of \( Q \) affects the calculation of cell potential under non-standard conditions, highlighting any deviations from the equilibrium state or standard setup.

The cell potential, also known as electromotive force (emf), measures the voltage generated by a galvanic cell. It indicates the energy difference between the cathode and anode due to redox reactions. To calculate this potential, follow these steps: - **Identify the half-reactions**: Determine which species is oxidizing and reducing. Align their standard potentials.- **Calculate the standard cell potential**: Subtract the anode potential from the cathode to find \( E^{\circ}_{\text{cell}} \).- **Apply the Nernst Equation**: Adjust \( E^{\circ}_{\text{cell}} \) considering concentrations of involved species to find the actual cell potential.In the exercise example:- Standard potentials were \(-0.28 \text{ V} \) for nickel and \(-2.37 \text{ V} \) for magnesium.- Calculate \( E^{\circ}_{\text{cell}} \):\[ E^{\circ}_{\text{cell}} = -0.28 \text{ V} - (-2.37 \text{ V}) = 2.09 \text{ V} \]- Adjust for concentration using \( Q = 0.50 \) in the Nernst Equation:\[ E_{\text{cell}} = 2.09 \text{ V} - 0.0257 \times \frac{\ln 0.50}{2} \approx 2.0989 \text{ V} \]After adjusting, we achieve a calculated potential very close to the standard potential, illustrating how concentration differences impact the cell’s voltage.