Determine the theoretical yield of \(C\) when each of the initial quantities of \(\mathrm{A}\) and \(\mathrm{B}\) is allowed to react in the generic reaction: $$ \mathrm{A}+2 \mathrm{~B} \longrightarrow 3 \mathrm{C} $$ (a) \(1 \mathrm{~mol} \mathrm{~A} ; 1 \mathrm{~mol} \mathrm{~B}\) (b) \(2 \mathrm{~mol} \mathrm{~A} ; 2 \mathrm{~mol} \mathrm{~B}\) (c) \(1 \mathrm{~mol} \mathrm{~A} ; 3 \mathrm{~mol} \mathrm{~B}\) (d) \(32 \mathrm{~mol} \mathrm{~A} ; 68 \mathrm{~mol} \mathrm{~B}\)

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between the reactants and products in chemical reactions. To fully understand stoichiometry, imagine it as the recipe for a chemical reaction; it tells you the proportions of reactants needed to make specific products, similar to how a cooking recipe tells you how much of each ingredient is required to make a dish.

For example, if the balanced chemical equation is \( \mathrm{A} + 2 \mathrm{B} \longrightarrow 3 \mathrm{C} \), it shows that one mole of substance \( \mathrm{A} \) reacts with two moles of substance \( \mathrm{B} \) to produce three moles of substance \( \mathrm{C} \). Using this information, you can determine how much of any one reactant is needed to react completely with a given amount of another reactant. This process also allows you to predict the amount of product that will be produced from known quantities of reactants, which is referred to as the theoretical yield.

In applying stoichiometry, it's crucial to start with a balanced chemical equation because it provides the mole ratios required for the calculations. Then, using these ratios, you can perform conversions using the mole concept to find out how much of a reactant or product is involved in the reaction.

The concept of the limiting reactant is essential in predicting the amount of product formed in a chemical reaction. The limiting reactant is the substance that is completely used up first and thus limits the extent of the reaction. Once the limiting reactant is consumed, the reaction stops, and no more product is formed, even if other reactants are still present.

To identify the limiting reactant, you compare the mole ratio of the reactants used in the reaction to the mole ratio of the reactants available. This often involves calculating how much product can be formed from each reactant and then determining which reactant produces the least amount of product.

For instance, in the reaction from our exercise (\( \mathrm{A} + 2 \mathrm{B} \longrightarrow 3 \mathrm{C} \)), if you have 1 mole of \( \mathrm{A} \) and 1 mole of \( \mathrm{B} \), \( \mathrm{B} \) would be the limiting reactant because the reaction requires 2 moles of \( \mathrm{B} \) for every mole of \( \mathrm{A} \). Therefore, \( \mathrm{B} \) will run out first, and the amount of \( \mathrm{B} \) will dictate the amount of product \( \mathrm{C} \) that can be formed.

A balanced chemical equation is fundamental to all stoichiometry calculations because it expresses the conservation of mass in chemical reactions. Atoms are neither created nor destroyed in a chemical reaction. Hence, the number of atoms for each element must be the same on both the reactant and product sides of the equation.

In the balanced equation \( \mathrm{A} + 2 \mathrm{B} \longrightarrow 3 \mathrm{C} \), the coefficients indicate the relative amounts of moles of each substance that react or are produced. The coefficient '1' in front of \( \mathrm{A} \) and '3' in front of \( \mathrm{C} \) shows that 1 mole of \( \mathrm{A} \) will produce 3 moles of \( \mathrm{C} \), while the '2' in front of \( \mathrm{B} \) indicates that 2 moles of \( \mathrm{B} \) are necessary for every mole of \( \mathrm{A} \) reacted.

When approaching stoichiometry problems, always verify that the chemical equation is balanced before proceeding with any calculations. A balanced equation ensures that the stoichiometry calculations are accurate, reflecting the correct mole ratios as governed by the Law of Conservation of Mass. It's the critical step that allows chemists to predict the outcomes of reactions quantitatively.