Complete and balance each of the following half-reactions (steps 2-5 in half- reaction method): (a) \(\operatorname{Sn}^{4+}(a q) \rightarrow \operatorname{Sn}^{2+}(a q)\) (b) \(\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}\right]^{+}(a q) \longrightarrow \mathrm{Ag}(s)+\mathrm{NH}_{3}(a q)\) (c) \(\mathrm{Hg}_{2} \mathrm{Cl}_{2}(s) \longrightarrow \mathrm{Hg}(l)+\mathrm{Cl}^{-}(a q)\) (d) \(\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{O}_{2}(g)\) (in acidic solution) (e) \(\mathrm{IO}_{3}^{-}(a q) \longrightarrow \mathrm{I}_{2}(s)\) (f) \(\operatorname{SO}_{3}^{2-}(a q) \rightarrow \operatorname{SO}_{4}^{2-}(a q)\) (in acidic solution) (g) \(\operatorname{MnO}_{4}^{-}(a q) \rightarrow \mathrm{Mn}^{2+}(a q)\) (in acidic solution) (h) \(\mathrm{Cl}^{-}(a q) \longrightarrow \mathrm{ClO}_{3}^{-}(a q)\) (in basic solution)

Understanding how to balance redox reactions is a fundamental skill in chemistry. Redox reactions involve the transfer of electrons between chemical species, with one species undergoing oxidation (losing electrons) and another undergoing reduction (gaining electrons). The half-reaction method is an effective way to balance these reactions by separately balancing the oxidation and reduction processes.

In this approach, you first divide the overall reaction into two half-reactions, each representing either the oxidation or the reduction process. You then balance the atoms for elements other than hydrogen and oxygen, before ensuring that oxygen and hydrogen atoms are also balanced, often by adding water (H₂O), hydrogen ions (H⁺) in acidic solutions, or hydroxide ions (OH^-) in basic solutions. Charges are balanced by adding electrons, and finally, the half-reactions are adjusted to have the same number of electrons so that they can be added together to give the overall balanced equation. The beauty of this method lies in its systematic approach, which provides a clear pathway from unbalanced equations to balanced chemical reactions.

An essential step in the half-reaction method is identifying the oxidation states of the elements involved in the reaction. Oxidation states, often referred to as oxidation numbers, are hypothetical charges that an atom would have if all bonds to atoms of different elements were completely ionic. They provide insight into the electron transfer during the reaction.

For example, in the half-reaction \(\text{Sn}^{4+}(aq) \rightarrow \text{Sn}^{2+}(aq)\), tin is reduced as its oxidation state decreases from +4 to +2, signifying a gain of two electrons. By understanding the changes in oxidation states, we can determine how many electrons are involved in each half-reaction and thereby balance the overall reaction correctly. The concept of oxidation numbers also helps in identifying the species being oxidized or reduced, which is critical for writing half-reactions. Remember that in a redox reaction, the substance that gets reduced will have a decrease in its oxidation state, while the one that gets oxidized will have an increase in its oxidation state.

The overarching goal of the half-reaction method is to create a balanced chemical equation where the number of atoms for each element, and the total charge, are equal on both sides of the equation. Balancing chemical equations is more than an exercise in arithmetic — it reflects the fundamental law of conservation of mass and charge.

Each half-reaction must be balanced in terms of mass and charge before the redox reaction can be considered balanced as a whole. In the case of the half-reaction \(\text{MnO}_{4}^{-}(aq) \rightarrow \text{Mn}^{2+}(aq)\) under acidic conditions, hydrogen ions and water molecules are added to balance the oxygen and hydrogen atoms, while electrons are included to balance the charge. By adhering to the principle that the number of atoms and the charge must be equal on both the reactant and product side, the final balanced equation accurately represents the stoichiometry of the reaction and ensures that the equation abides by the laws of chemistry. The magic of the half-reaction method is that it systematically leads to a balanced chemical equation, providing clarity and understanding in the complex dance of atoms and electrons that characterizes redox reactions.