Consider a galvanic cell based on the following theoretical half-reactions: $$\begin{array}{ll} & \mathscr{E}^{\circ}(\mathrm{V}) \\ \mathrm{M}^{4+}+4 \mathrm{e}^{-} \longrightarrow \mathrm{M} & 0.66 \\ \mathrm{N}^{3+}+3 \mathrm{e}^{-} \longrightarrow \mathrm{N} & 0.39 \\ \hline \end{array}$$ What is the value of \(\Delta G^{\circ}\) and \(K\) for this cell?

Understanding the standard reduction potential is crucial when studying galvanic cells, as it determines the voltage that a half-cell can produce during a reduction reaction under standard conditions. It is measured in volts and denoted by the symbol \( \mathscr{E}^{\circ} \). The higher the value of \( \mathscr{E}^{\circ} \), the greater the tendency of the species to gain electrons and, thus, be reduced.

For a galvanic cell consisting of two half-cells, the cell potential \( \mathscr{E}^{\circ}_{cell} \) is calculated by taking the difference between the standard reduction potentials of the cathode and the anode. It's essential to ensure that the reactions are correctly balanced concerning both the number of electrons and the stoichiometric coefficients to obtain the correct cell potential, as shown in the example exercise.

Gibbs free energy, symbolized as \( \Delta G \), is a thermodynamic property that can be used to predict the direction of a chemical reaction and to determine whether a process is spontaneous at constant temperature and pressure. A negative \( \Delta G \) indicates that the reaction is spontaneous, while a positive value suggests a non-spontaneous reaction.

The relationship between Gibbs free energy and the standard cell potential is given by the equation \( \Delta G^{\circ} = -nFE^{\circ}_{cell} \), where \( n \) is the number of moles of electrons exchanged, \( F \) is the Faraday constant (around \( 96485 \ \mathrm{C/mol} \)), and \( \mathscr{E}^{\circ}_{cell} \) is the standard cell potential. This formula is applied to calculate \( \Delta G^{\circ} \) in the given exercise.

The Nernst equation is a foundational equation in electrochemistry that relates the cell potential to the concentration of reacting species. While the standard cell potential \( \mathscr{E}^{\circ} \) is measured under standard conditions, the Nernst equation allows us to adjust the cell potential for non-standard conditions. This is particularly important as most reactions occur under conditions that differ from standard ones.

The equation is given by: \(\mathscr{E} = \mathscr{E}^{\circ} - \frac{RT}{nF} \ln Q\), where \(\mathscr{E}\) is the cell potential at non-standard conditions, \(R\) is the universal gas constant, \(T\) is the temperature in kelvin, \(Q\) is the reaction quotient, and the other terms are as previously defined. In the example exercise, although we haven't used the full Nernst equation, the relation between Gibbs free energy and the standard cell potential stems from the same principles.

The equilibrium constant, represented as \( K \), is a dimensionless quantity that describes the ratio of concentrations of the products to reactants at equilibrium for a reversible reaction. It is related to Gibbs free energy through the equation \( \Delta G^{\circ} = -RT \ln K \). The equation indicates that if the standard free energy change for a reaction is known, the equilibrium constant can be calculated.

When \( \Delta G^{\circ} \) is negative, \( K \) will be greater than 1, suggesting that the reaction will favor the formation of products at equilibrium. Conversely, a positive \( \Delta G^{\circ} \) results in a \( K \) less than 1, indicating that reactants are favored. In the example exercise, by rearranging the equation to \( K = e^{-(\Delta G^{\circ} / RT)} \) and substituting the calculated value of \( \Delta G^{\circ} \) and known constants, we can determine the equilibrium constant for the galvanic cell.