For the reaction shown, calculate the theoretical yield of the product in moles for each of the initial quantities of reactants. $$ \mathrm{Ti}(s)+2 \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{TiCl}_{4}(s) $$ (a) \(2 \mathrm{~mol} \mathrm{Ti} ; 2 \mathrm{~mol} \mathrm{Cl}_{2}\) (b) \(5 \mathrm{~mol} \mathrm{Ti} ; 9 \mathrm{~mol} \mathrm{Cl}_{2}\) (c) \(0.483 \mathrm{~mol} \mathrm{Ti} ; 0.911 \mathrm{~mol} \mathrm{Cl}_{2}\) (d) \(12.4 \mathrm{~mol} \mathrm{Ti} ; 15.8 \mathrm{~mol} \mathrm{Cl}_{2}\)

Stoichiometry is the section of chemistry that involves quantitative relationships between the reactants and products in a chemical reaction. Think of stoichiometry as the recipe for a chemical reaction, where the balanced chemical equation provides the quantities of each reactant that reacts and the amount of product that will form. Just like baking a cake requires specific amounts of each ingredient, a chemical reaction requires specific ratios of reactants to produce products.

To perform stoichiometric calculations, you need to start with a balanced chemical equation. This equation tells you the molar ratio between reactants and products. For instance, in the given exercise, the balanced chemical equation is \(\mathrm{Ti}(s) + 2 \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{TiCl}_{4}(s)\), which signifies 1 mole of titanium (Ti) reacts with 2 moles of chlorine gas (\(\mathrm{Cl}_{2}\)) to produce 1 mole of titanium tetrachloride (\(\mathrm{TiCl}_{4}\)).

Using this ratio, we can calculate how much product can be formed from given amounts of reactants. This is an essential tool in chemistry because it helps predict the quantities needed for a reaction to take place fully or to find out how much product we can expect to produce.

The concept of the limiting reactant is crucial in predicting the amount of product formed in a chemical reaction. The limiting reactant is the substance that is totally consumed when the chemical reaction is complete. Once this reactant is used up, the reaction stops, and no more product can be formed, even if other reactants are still available. It's the bottleneck of the reaction, limiting the amount of product that can be generated.

In the exercise provided, we determine the limiting reactant by comparing the mole ratio of the reactants to the ratios in the balanced equation. For instance, in scenario (a), we have 2 moles of titanium and 2 moles of chlorine gas. However, the titanium requires twice as much chlorine gas to react completely, making chlorine gas the limiting reactant, as only 1 mole of titanium can react with the 2 moles of chlorine gas present.

Identifying the limiting reactant is an essential step in stoichiometry. It not only determines the theoretical yield of a reaction but can also be utilized in industry to minimize waste and optimize resource usage. Students often struggle with this concept, so it's important to practice with various examples to become comfortable in finding the limiting reactant in chemical reactions.

Balanced chemical equations are foundational in chemistry—they represent the principle of conservation of mass, stating that matter is neither created nor destroyed in a typical chemical reaction. A balanced equation must have the same numbers of each type of atom on both sides of the reaction arrow.

The balanced chemical equation provided in the exercise, \(\mathrm{Ti}(s) + 2 \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{TiCl}_{4}(s)\), shows a simple 1:2:1 ratio among titanium, chlorine gas, and titanium tetrachloride respectively. This tells us how many moles of each reactant are needed to produce a certain amount of product, and vice versa. To solve stoichiometric problems, the first critical step is always to write and confirm the balanced chemical equation. This step sets the stage for all subsequent calculations, such as identifying the limiting reactant and calculating theoretical yield.

Understanding how to balance chemical equations is an essential skill for any student studying chemistry. It not only is crucial for laboratory work but also helps to reinforce the understanding of atomic structure and conservation laws in chemical reactions.