Determine the theoretical yield of \(\mathrm{C}\) when each of the initial quantities of \(\mathrm{A}\) and \(\mathrm{B}\) is allowed to react in the generic reaction: $$ 2 \mathrm{~A}+3 \mathrm{~B} \longrightarrow 2 \mathrm{C} $$ (a) \(2 \mathrm{~mol} \mathrm{~A} ; 4 \mathrm{~mol} \mathrm{~B}\) (b) \(3 \mathrm{~mol} \mathrm{~A} ; 3 \mathrm{~mol} \mathrm{~B}\) (c) \(5 \mathrm{~mol} \mathrm{~A} ; 6 \mathrm{~mol} \mathrm{~B}\) (d) \(4 \mathrm{~mol} \mathrm{~A} ; 5 \mathrm{~mol} \mathrm{~B}\)

Understanding which reactant limits a chemical reaction is crucial in predicting how much product can be formed. The limiting reactant is the substance that is completely consumed first during the chemical reaction, causing the reaction to stop and determining the maximum amount of product that can be formed.

To determine the limiting reactant, we compare the mole ratios of reactants to the coefficients in the balanced chemical equation. We calculate the molar ratio of each reactant to its coefficient and the smallest of these ratios indicates the limiting reactant. For instance, in the given reaction \(2A + 3B \rightarrow 2C\), we divide the moles of A and B by their respective coefficients. Thus, for case (a), we do \(\frac{2 \text{ mol A}}{2}\) and \(\frac{4 \text{ mol B}}{3}\) to find which is smaller. By consistently applying this method, we can identify which reactant will be exhausted first, and consequently, control the output of the reaction.

Stoichiometry is the method of quantitatively analyzing the proportions of elements and compounds involved in chemical reactions. It is based on the conservation of mass and the law of definite proportions, which state that in a chemical reaction, elements are combined in fixed ratios by mass, equal to the ratios of their mole amounts.

Stoichiometry allows us to predict the results of chemical reactions, including how much product will form from given amounts of reactants. Calculating stoichiometric quantities requires a balanced chemical equation, as it provides the mole ratios we need to convert between reactants and products. For example, according to the balanced reaction in our exercise, 2 moles of A react with 3 moles of B to produce 2 moles of C. This fixed ratio means that the amount of product C can be directly calculated from the amount of the limiting reactant using stoichiometry.

Balancing a chemical reaction is a fundamental skill in chemistry as it ensures that the law of conservation of mass is respected. It means adjusting the coefficients of the reactants and products so that the number of atoms of each element is the same on both sides of the reaction equation.

Using the generic balanced reaction from our exercise \(2A + 3B \rightarrow 2C\), we can see that for every 2 moles of A and 3 moles of B reacted, 2 moles of C are produced. Balancing chemical equations is vital because it lays the groundwork for stoichiometric calculations, allowing us to accurately determine the theoretical yields of products in a chemical reaction. In studying reaction balancing, one also learns to systematically approach problems and apply mathematical reasoning to the physical world.

The mole-to-mole ratio, derived from the coefficients of a balanced chemical equation, serves as a conversion factor between the amount of reactants and products. It's an essential concept in stoichiometry as it quantifies the direct relationship between the substances in a chemical reaction.

Continuing with our reaction \(2A + 3B \rightarrow 2C\), the coefficients indicate the mole-to-mole ratios: for every 2 moles of A, we can produce 2 moles of C, and for every 3 moles of B, 2 moles of C can be produced. These ratios allow us to calculate the amount of products formed from a given amount of reactants. Always remember, it is the limiting reactant's mole-to-mole ratio to the product that informs the maximum theoretical yield of the product in any given chemical reaction.