Balance the following oxidation-reduction equations. $$\text { (a) } \mathrm{Fe}_{2} \mathrm{S}_{3}(\mathrm{s})+\mathrm{H}_{2} \mathrm{O}+\mathrm{O}_{2}(\mathrm{g}) \longrightarrow \mathrm{Fe}(\mathrm{OH})_{3}(\mathrm{s})+\mathrm{S}(\mathrm{s})$$ $$\begin{aligned} &\text { (b) } \mathrm{Mn}^{2+}(\mathrm{aq})+\mathrm{S}_{2} \mathrm{O}_{8}^{2-}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O} \longrightarrow\\\ &&\mathrm{MnO}_{4}^{-}(\mathrm{aq})+\mathrm{SO}_{4}^{2-}(\mathrm{aq})+\mathrm{H}^{+}(\mathrm{aq}) \end{aligned}$$ $$\begin{aligned} &\text { (c) } \mathrm{Ag}(\mathrm{s})+\mathrm{CN}^{-}(\mathrm{aq})+\mathrm{O}_{2}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O} \longrightarrow\\\ &&\left[\mathrm{Ag}(\mathrm{CN})_{2}\right]^{-}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq}) \end{aligned}$$

Step by step solution

01

Separate half-reactions

Identify the species that is being oxidized (losing electrons) and the species being reduced (gaining electrons) and write down these half-reactions for each equation. For example, for the first equation \(Fe_{2}S_{3}\) is oxidized to \(Fe(OH)_{3}\) and \(O_{2}\) is reduced to \(S\). Hence we obtain:\(Fe_{2}S_{3} \rightarrow 2Fe(OH)_{3}\) and\(O_{2} + H_{2}O \rightarrow S\)

02

Balance mass

First, balance all atoms except for hydrogen and oxygen in each half-reaction. Then, balance oxygen by adding H2O molecules to the side lacking oxygen. After that, balance hydrogen by adding H+ ions to the side lacking hydrogen. For example, for the first equation we obtain:\(2Fe_{2}S_{3} + 12H_{2}O \rightarrow 4Fe(OH)_{3}\) and\(O_{2} + 2H_{2}O \rightarrow S + 4H^{+}\)

03

Balance charges

Add electrons to the side of the half-reaction that has a greater positive charge. For the first equation we get:\(2Fe_{2}S_{3} + 12H_{2}O \rightarrow 4Fe(OH)_{3} + 24e^{-}\) and\(O_{2} + 2H_{2}O + 8e^{-} \rightarrow S + 4H^{+}\)

04

Combine half-reactions

Once the charges and atoms are balanced in each half-reaction, multiply the half-reactions by the smallest possible integers such that the number of electrons in the two half-reactions are equal when added together. Then add the reactions together, simplifying where necessary.As a result, we get the balanced equation for the first reaction:\(3O_{2}(g) + 4Fe_{2}S_{3}(s) + 12H_{2}O \rightarrow 8Fe(OH)_{3}(s) + 3S(s) + 12H^{+} + 12e^{-}\)

05

Repeat for the rest equations

Repeat the same steps for the remaining reactions.\nOne can apply the same methods to balance the equations (b) and (c).

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