Two voltaic cells are assembled in which the following...

Two voltaic cells are assembled in which the following reactions occur. $$\begin{aligned} \mathrm{V}^{2+}(\mathrm{aq})+\mathrm{VO}^{2+}(\mathrm{aq})+2 \mathrm{H}^{+}(\mathrm{aq}) & \longrightarrow2 \mathrm{V}^{3+}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(1) & \quad E_{\mathrm{cell}}^{\circ}=0.616 \mathrm{V} \end{aligned}$$ $$\mathrm{V}^{3+}(\mathrm{aq})+\mathrm{Ag}^{+}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(1) \longrightarrow \mathrm{VO}^{2+}(\mathrm{aq})+2 \mathrm{H}^{+}(\mathrm{aq})+\mathrm{Ag}(\mathrm{s}) \quad E_{\mathrm{cell}}^{\circ}=0.439 \mathrm{V}$$ Use these data and other values from Table 19.1 to calculate $E^{\circ}$ for the half-cell reaction $\mathrm{V}^{3+}(\mathrm{aq})+\mathrm{e}^{-} \longrightarrow \mathrm{V}^{2+}(\mathrm{aq}).$

Two voltaic cells are assembled in which the following reactions occur.
$$\begin{aligned}
\mathrm{V}^{2+}(\mathrm{aq})+\mathrm{VO}^{2+}(\mathrm{aq})+2 \mathrm{H}^{+}(\mathrm{aq}) & \longrightarrow2 \mathrm{V}^{3+}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(1) & \quad E_{\mathrm{cell}}^{\circ}=0.616 \mathrm{V}
\end{aligned}$$
$$\mathrm{V}^{3+}(\mathrm{aq})+\mathrm{Ag}^{+}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(1) \longrightarrow \mathrm{VO}^{2+}(\mathrm{aq})+2 \mathrm{H}^{+}(\mathrm{aq})+\mathrm{Ag}(\mathrm{s}) \quad E_{\mathrm{cell}}^{\circ}=0.439 \mathrm{V}$$
Use these data and other values from Table 19.1 to calculate $E^{\circ}$ for the half-cell reaction $\mathrm{V}^{3+}(\mathrm{aq})+\mathrm{e}^{-} \longrightarrow \mathrm{V}^{2+}(\mathrm{aq}).$

Key Concepts

Hess’s Law in Electrochemistry

Hess’s Law indicates that the overall change in free energy for an electrochemical reaction is the sum of the free energy changes of its individual steps. In the context of electrode potentials, this means that combining or manipulating half-reactions can allow one to calculate unknown standard potentials by ensuring that the sum of the potential contributions, when correctly adjusted for electron transfer, reflects the overall reaction energetics.

Standard Electrode Potential

This is a measure of the inherent tendency of a chemical species to be reduced, expressed as the voltage difference between the electrode and a standard hydrogen electrode under standard conditions. It is an intensive property, meaning it does not depend on the scale or extent of the reaction, and it provides a crucial basis for comparing the redox abilities of various half-reactions.

Half-Reaction

This is a redox process split into two parts: one representing oxidation and the other reduction. In half-reactions, only the reactants and products involved in either the loss or gain of electrons are represented. This separation allows chemists to study electron transfer processes independently.

Cell Potential Calculation

The cell potential is determined by the difference between the standard reduction potentials of the cathode and anode half-cells. It is calculated by subtracting the anode potential from the cathode potential, and this difference drives the overall electrochemical reaction. This concept is key in applications such as battery design and the calculation of reaction spontaneity.

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