Balance each of the following: $$ \begin{array}{l}{\text { a. } \mathrm{H}_{2}+\mathrm{Cl}_{2} \rightarrow \mathrm{HCl}} \\ {\text { b. } \mathrm{Al}+\mathrm{Fe}_{2} \mathrm{O}_{3} \rightarrow \mathrm{Al}_{2} \mathrm{O}_{3}+\mathrm{Fe}} \\ {\text { c. } \mathrm{Ba}\left(\mathrm{ClO}_{3}\right)_{2} \rightarrow \mathrm{BaCl}_{2}+\mathrm{O}_{2}} \\ {\text { d. } \mathrm{Cu}+\mathrm{HNO}_{3} \rightarrow \mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2}+\mathrm{NO}+\mathrm{H}_{2} \mathrm{O}}\end{array} $$

Stoichiometry is like the recipe for a chemical reaction. It tells you how much of each substance you need and how much you'll produce. In the context of balancing chemical equations, stoichiometry requires that you have the same number of atoms of each element on both sides of the equation. It's a pivotal concept in chemistry that ensures the conservation of mass in a reaction.

For instance, in the exercise solution for equation (a) \( H_{2} + Cl_{2} \rightarrow HCl \), it's not immediately obvious how many molecules of \( HCl \) will be produced from one molecule of \( H_{2} \) and one molecule of \( Cl_{2} \). By applying the principles of stoichiometry, we concluded that two molecules of \( HCl \) will form so that each hydrogen from \( H_{2} \) pairs up with a chlorine atom from \( Cl_{2} \).

This step-by-step method ensures each element is conserved, mirroring the exact stoichiometry needed for a balanced equation.

Chemical reactions involve the transformation of one set of chemical substances to another. Balancing chemical equations, as seen in the example problem, is an essential skill because it represents the law of conservation of mass—meaning, matter is neither created nor destroyed, only transformed.

Throughout a chemical reaction, like \( Al + Fe_{2}O_{3} \rightarrow Al_{2}O_{3} + Fe \) from the exercise, the atoms rearrange themselves to form new products. The challenge lies in ensuring that the same number of each type of atom appears on both sides of the equation. Understanding and identifying the reactants and products—as well as the need for their balance—is a core aspect of engaging with chemical reactions in a deeply meaningful way.

Improvement in writing chemical equations comes from recognizing patterns in how substances interact and practicing the balancing act, mirroring the given step-by-step solution for each problem presented.

The principle of conservation of mass states that mass cannot be created or destroyed in a closed system through ordinary chemical reactions or physical transformations. When balancing chemical equations, this principle ensures that the mass of the reactants equals the mass of the products after the reaction.

Take, for example, equation (c) from our problem set: \( Ba(ClO_{3})_{2} \rightarrow BaCl_{2} + O_{2} \). Initially, there are more oxygen atoms on the left side of the equation than on the right. By multiplying \( O_{2} \) by 3, we balance the oxygen atoms without violating the conservation of mass.

Students often struggle with this concept when complex reactions like equation (d) include multiple products and reactants. The meticulous steps provided in the solution demonstrate how to maintain the balance in mass by adjusting coefficients of compounds in the equation \( Cu + HNO_{3} \rightarrow Cu(NO_{3})_{2} + NO + H_{2}O \), thus adhering to the conservation of mass. This strategy is valuable in solving more complicated scenarios and is crucial for a solid understanding of chemical reactions.