Balance the following redox equations by the halfreaction method: (a) \(\mathrm{H}_{2} \mathrm{O}_{2}+\mathrm{Fe}^{2+} \longrightarrow \mathrm{Fe}^{3+}+\mathrm{H}_{2} \mathrm{O}\) (in acidic solution) (b) \(\mathrm{Cu}+\mathrm{HNO}_{3} \longrightarrow \mathrm{Cu}^{2+}+\mathrm{NO}+\mathrm{H}_{2} \mathrm{O}\) (in acidic solution) (c) \(\mathrm{CN}^{-}+\mathrm{MnO}_{4}^{-} \longrightarrow \mathrm{CNO}^{-}+\mathrm{MnO}_{2}\) (in basic solution) (d) \(\mathrm{Br}_{2} \longrightarrow \mathrm{BrO}_{3}^{-}+\mathrm{Br}^{-}\) (in basic solution) (e) \(\mathrm{S}_{2} \mathrm{O}_{3}^{2-}+\mathrm{I}_{2} \longrightarrow \mathrm{I}^{-}+\mathrm{S}_{4} \mathrm{O}_{6}^{2-}\) (in acidic solution)

The half-reaction method is a systematic approach to balancing redox equations, which are equations that involve electron transfer. This approach focuses on separating the oxidation and reduction processes, often referred to as 'half-reactions'.

In simple terms, the method involves breaking down the overall chemical reaction into two separate parts: one in which a species is losing electrons (oxidation), and one in which another species is gaining electrons (reduction). Each half-reaction is balanced independently for mass and charge. Then, these balanced half-reactions are added together to form the balanced redox equation.

Through the process, certain rules apply: balance all atoms except hydrogen and oxygen first, then balance the oxygen atoms using water molecules (H2O), next balance hydrogen atoms using hydrogen ions (H+ in acidic solutions or OH- in basic solutions), and finally balance the charges by adding electrons (e-).

Understanding oxidation states is crucial for the half-reaction method as it tells us about the transfer of electrons during a reaction. An oxidation state, often referred to as oxidation number, essentially indicates the degree of oxidation of an atom in a compound.

An atom's oxidation state is a hypothetical charge it would have if electrons in bonds were distributed according to certain rules, with more electronegative atoms taking the electrons. In the context of redox reactions, when the oxidation state of an element increases, it has undergone oxidation (loss of electrons), and when it decreases, it has undergone reduction (gain of electrons).

By comparing the changes in these states in a reaction, we can identify which species are oxidized and which are reduced, thereby setting the stage for applying the half-reaction method.

In the context of redox reactions, the term 'acidic solution' implies that an excess of H+ ions is present in the environment where the reaction is occurring. This has implications on how the half-reactions are balanced.

For half-reactions in acidic solutions, hydrogen ions (H+) or water (H2O) are used to balance the oxygen and hydrogen atoms respectively. To balance oxygen atoms, add H2O to the side lacking oxygen; for hydrogen atoms, add H+ to the side lacking hydrogen. Once balanced for atoms, electrons are added to balance the charge. This makes balancing redox reactions in acid solutions distinct from those in basic solutions, which use OH- instead of H+.

When dealing with redox reactions in basic solutions, it is essential to recognize that OH- ions are abundant in the reaction environment.

Balancing redox reactions in these solutions requires a different approach from acidic solutions. After balancing for atoms other than oxygen and hydrogen, we balance oxygen by adding water molecules (H2O) to the side needing oxygen and then balance hydrogen by adding OH- ions to the side needing hydrogen. Finally, electrons are added to balance the electrical charge. This distinct procedure accommodates the presence of OH- ions, characteristic of a basic or alkaline environment.

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It is grounded in the conservation of mass and the consistent ratios called 'mole ratios' derived from the coefficients of a balanced chemical equation.

In redox reactions, stoichiometry not only involves balancing atoms but also managing the electrons that are transferred between species. Once the half-reactions are balanced separately, their stoichiometry must be reconciled by ensuring that the same number of electrons are lost in the oxidation half-reaction as are gained in the reduction half-reaction. This is achieved by multiplying the half-reactions by appropriate factors so that the electrons cancel out when they are combined to give the final balanced equation.