Balance each of the following equations according to the half-reaction method: (a) \(\mathrm{Zn}(s)+\mathrm{NO}_{3}^{-}(a q) \rightarrow \mathrm{Zn}^{2+}(a q)+\mathrm{N}_{2}(g)\) (in acid) (b) \(\mathrm{Zn}(s)+\mathrm{NO}_{3}^{-}(a q) \longrightarrow \mathrm{Zn}^{2+}(a q)+\mathrm{NH}_{3}(a q)\) (in base) (c) \(\operatorname{CuS}(s)+\mathrm{NO}_{3}^{-}(a q) \longrightarrow \mathrm{Cu}^{2+}(a q)+\mathrm{S}(s)+\mathrm{NO}(g)\) (in acid) (d) \(\mathrm{NH}_{3}(a q)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{NO}_{2}(g)\) (gas phase) (e) \(\mathrm{H}_{2} \mathrm{O}_{2}(a q)+\mathrm{MnO}_{4}^{-}(a q) \longrightarrow \mathrm{Mn}^{2+}(a q)+\mathrm{O}_{2}(g)\) (in acid) (f) \(\mathrm{NO}_{2}(g) \longrightarrow \mathrm{NO}_{3}^{-}(a q)+\mathrm{NO}_{2}^{-}(a q)\) (in base) (g) \(\mathrm{Fe}^{3+}(a q)+\mathrm{I}^{-}(a q) \longrightarrow \mathrm{Fe}^{2+}(a q)+\mathrm{I}_{2}(a q)\)

Balancing redox reactions is an essential skill in chemistry, as it ensures the conservation of mass and charge. It involves a systematic process known as the half-reaction method.

Initially, one must identify the oxidation and reduction half-reactions. Oxidation refers to the loss of electrons, resulting in an increase of oxidation state, while reduction involves the gain of electrons and a decrease in oxidation state.

Each half-reaction is balanced separately, considering atoms and charge. Atoms of each element must be balanced by possibly adding water molecules or ions such as H⁺ in acidic solutions, or OH^- in basic solutions. After that, charges are balanced by adding electrons. It's important to balance the electrons lost in the oxidation half-reaction with those gained in the reduction half-reaction.

Finally, these half-reactions are recombined into a single balanced equation that accounts for both mass and charge. The stoichiometry of the reaction must reflect the consumption and production of all species involved. This approach ensures a clear and methodical way to balance complex redox reactions.

In an oxidation half-reaction, a substance loses electrons and increases its oxidation state.

This process can be identified in a redox reaction by looking for elements that have a higher oxidation state in the products than in the reactants. For instance, zinc (Zn) in the solid state is oxidized to zinc ion (Zn²⁺) when it loses two electrons.

The balancing of an oxidation half-reaction involves making sure the number of atoms of the oxidized element and the charge are the same on both sides. If oxygen or hydrogen atoms are unbalanced, water (H₂O) and hydrogen ions (H⁺) are used to achieve balance, depending on the reaction conditions. The final step is to add electrons to the products side to balance the increase in oxidation state.

Conversely, a reduction half-reaction is where a substance gains electrons and decreases its oxidation state.

An example from the exercise would be nitrate ions (NO₃^-) being reduced to nitrogen (N₂) gas. During this process, we look for species that accept electrons, thus decreasing their oxidation state.

Just like in oxidation, we balance the reduction half-reaction by accounting for any imbalances in atoms first, followed by balancing oxygen atoms with water, and finally adjusting for hydrogen atoms by adding hydrogen ions (H⁺) in acid solutions. The last step involves adding electrons to the reactants side to achieve charge balance. The amount of electrons added should be able to cancel out against the electrons from the oxidation half-reaction when the complete reaction is compiled.

Stoichiometry is the part of chemistry that involves the quantitative relationships between the reactants and products in a chemical reaction.

Once you've balanced the half-reactions for mass and charge, you need to ensure that the stoichiometric coefficients (the numbers placed before compounds in a chemical equation) are properly assigned so that the number of electrons lost in oxidation equals the number gained in reduction. To solve this, it involves scaling the half-reactions by appropriate factors before adding them together.

In balancing redox reactions, stoichiometry helps maintain the principle of conservation of mass; this means that atoms are neither created nor destroyed during a chemical reaction. Calculating the quantities of reactants and products within a balanced equation is an essential step in predicting yields of the reactions and in scaling up reactions for industrial applications.

Chemical equations are symbolic representations of chemical reactions, showing the reactants on the left side and the products on the right. They must be balanced so that the same number of each type of atom occurs on both sides of the equation, and the sum of the charges on both sides must be equal.

In the context of redox reactions, the balancing of chemical equations becomes more complex due to the transfer of electrons. To accurately display the change, the half-reaction method separates the overall reaction into two parts, allowing a clearer view of the electron transfer and aiding the balancing process.

Remember that the physical state of the reactants and products (solid, liquid, gas, or aqueous) should also be included in a chemical equation. These states influence the balancing steps, especially when adding water, hydrogen ions, or hydroxide ions depending on the reaction conditions. Precise chemical equations are the basis of understanding reaction mechanisms, predicting reaction products, and calculating the amounts of substances required or produced.