Show that for a combination of half-cell reactions that...

Show that for a combination of half-cell reactions that produce a standard reduction potential for a half-cell that is not directly observable, the standard reduction potential is $$E^{\circ}=\frac{\sum n_{i} E_{i}^{\circ}}{\sum n_{i}}$$ where $n_{i}$ is the number of electrons in each half-reaction of potential $E_{i}^{\circ} .$ Use the following half-reactions: $$ \begin{array}{c} \mathrm{H}_{5} \mathrm{IO}_{6}(\mathrm{aq})+\mathrm{H}^{+}(\mathrm{aq})+2 \mathrm{e}^{-} \longrightarrow \mathrm{IO}_{3}^{-}(\mathrm{aq})+ \\ 3 \mathrm{H}_{2} \mathrm{O}(1) \quad E^{\circ}=1.60 \mathrm{V} \\ \mathrm{IO}_{3}^{-}(\mathrm{aq})+6 \mathrm{H}^{+}(\mathrm{aq})+5 \mathrm{e}^{-} \longrightarrow \frac{1}{2} \mathrm{I}_{2}(\mathrm{s})+3 \mathrm{H}_{2} \mathrm{O}(1) \\ E^{\circ}=1.19 \mathrm{V} \\ 2 \mathrm{HIO}(\mathrm{aq})+2 \mathrm{H}^{+}(\mathrm{aq})+2 \mathrm{e}^{-} \longrightarrow \mathrm{I}_{2}(\mathrm{s})+2 \mathrm{H}_{2} \mathrm{O}(1) \\ E^{\circ}=1.45 \mathrm{V} \\ \mathrm{I}_{2}(\mathrm{s})+2 \mathrm{e}^{-} \longrightarrow 2 \mathrm{I}^{-}(\mathrm{aq}) \quad \quad E^{\circ}=0.535 \mathrm{V} \end{array} $$ Calculate the standard reduction potential for $$ \mathrm{H}_{6} \mathrm{IO}_{6}+5 \mathrm{H}^{+}+2 \mathrm{I}^{-}+3 \mathrm{e}^{-} \longrightarrow $$ $$ \frac{1}{2} \mathrm{I}_{2}+4 \mathrm{H}_{2} \mathrm{O}=2 \mathrm{HIO} $$

Show that for a combination of half-cell reactions that produce a standard reduction potential for a half-cell that is not directly observable, the standard reduction potential is
$$E^{\circ}=\frac{\sum n_{i} E_{i}^{\circ}}{\sum n_{i}}$$
where $n_{i}$ is the number of electrons in each half-reaction of potential $E_{i}^{\circ} .$ Use the following half-reactions:
$$
\begin{array}{c}
\mathrm{H}_{5} \mathrm{IO}_{6}(\mathrm{aq})+\mathrm{H}^{+}(\mathrm{aq})+2 \mathrm{e}^{-} \longrightarrow \mathrm{IO}_{3}^{-}(\mathrm{aq})+ \\
3 \mathrm{H}_{2} \mathrm{O}(1) \quad E^{\circ}=1.60 \mathrm{V} \\
\mathrm{IO}_{3}^{-}(\mathrm{aq})+6 \mathrm{H}^{+}(\mathrm{aq})+5 \mathrm{e}^{-} \longrightarrow \frac{1}{2} \mathrm{I}_{2}(\mathrm{s})+3 \mathrm{H}_{2} \mathrm{O}(1) \\
E^{\circ}=1.19 \mathrm{V} \\
2 \mathrm{HIO}(\mathrm{aq})+2 \mathrm{H}^{+}(\mathrm{aq})+2 \mathrm{e}^{-} \longrightarrow \mathrm{I}_{2}(\mathrm{s})+2 \mathrm{H}_{2} \mathrm{O}(1) \\
E^{\circ}=1.45 \mathrm{V} \\
\mathrm{I}_{2}(\mathrm{s})+2 \mathrm{e}^{-} \longrightarrow 2 \mathrm{I}^{-}(\mathrm{aq}) \quad \quad E^{\circ}=0.535 \mathrm{V}
\end{array}
$$
Calculate the standard reduction potential for
$$
\mathrm{H}_{6} \mathrm{IO}_{6}+5 \mathrm{H}^{+}+2 \mathrm{I}^{-}+3 \mathrm{e}^{-} \longrightarrow
$$
$$
\frac{1}{2} \mathrm{I}_{2}+4 \mathrm{H}_{2} \mathrm{O}=2 \mathrm{HIO}
$$

Key Concepts

Weighted Average of Potentials

When combining multiple half-cell reactions to form an overall reaction, the overall standard reduction potential can be determined as a weighted average of the individual standard potentials. The weights are given by the number of electrons involved in each half-reaction, reflecting how each step contributes proportionally to the total free energy change of the reaction.

Electron Stoichiometry

Electron stoichiometry involves the balancing of electrons transferred in the half-reactions in a redox process. Each half-cell reaction is associated with a specific number of electrons (n) exchanged, and these values are crucial for correctly combining reactions and calculating overall reaction parameters like the net potential.

Relationship Between Gibbs Free Energy and Reduction Potential

The reduction potentials of half-reactions are related to the free energy changes (?G) of the processes through the equation ?G = ?nFE°. This relationship underpins the additivity of free energy changes in combined reactions and explains why the overall reduction potential is a weighted average of the individual potentials based on the number of electrons transferred.

Standard Reduction Potential

The standard reduction potential (E°) measures the tendency of a chemical species to acquire electrons under standard conditions. It is an intrinsic property of a redox couple and is used to compare the relative strengths of oxidizing agents. This concept is central in electrochemistry as it allows prediction of the direction of electron flow in voltaic cells.

Half-Cell Reaction

A half-cell reaction represents the oxidation or reduction component of an overall redox reaction. Each half-cell is written to show either the gain or loss of electrons, and its corresponding standard potential indicates how readily that electrode reaction occurs. Understanding half-cell reactions is essential for dissecting complex redox processes into simpler parts.

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