Question

Calculate the theoretical yield of the product (in moles) for each initial amount of reactants. $$ \mathrm{Ti}(s)+2 \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{TiCl}_{4}(l) $$ a. \(4 \mathrm{~mol} \mathrm{Ti}, 4 \mathrm{~mol} \mathrm{Cl}_{2}\) b. \(7 \mathrm{~mol} \mathrm{Ti}, 17 \mathrm{~mol} \mathrm{Cl}_{2}\) c. \(12.4 \mathrm{~mol} \mathrm{Ti}, 18.8 \mathrm{~mol} \mathrm{Cl}_{2}\)

Short answer

a. 2 moles TiCl4, b. 7 moles TiCl4, c. 9.4 moles TiCl4

Step-by-step solution

- Understand the Stoichiometry

The balanced chemical equation is given by \[\mathrm{Ti}(s) + 2\mathrm{Cl}_2(g) \longrightarrow \mathrm{TiCl}_4(l)\].This equation suggests that 1 mole of titanium reacts with 2 moles of chlorine gas to produce 1 mole of titanium tetrachloride.

- Determine the Limiting Reactant for Each Scenario

Use stoichiometry to find the limiting reactant for each ratio of reactants.a. Since there is an equal number of moles (4 mol) of titanium and chlorine gas and titanium needs only half as much chlorine to react completely, chlorine gas will be the limiting reactant.b. For the second reaction, we see that 7 moles of titanium would require 14 moles of chlorine gas to react completely; since there is more chlorine provided, titanium will be the limiting reactant.c. In the last scenario, every mole of titanium requires 2 moles of chlorine. We have 12.4 moles of Ti and 18.8 moles of Cl2, so \[12.4 \times 2 = 24.8\], and since there is less chlorine available than needed, chlorine will be the limiting reactant.

- Calculate the Theoretical Yield for Each Scenario

The theoretical yield is calculated based on the limiting reactant.a. If chlorine is limiting, and we have 4 moles of Cl2, then, according to the stoichiometry (2:1), we can form 2 moles of TiCl4.b. If titanium is limiting, then 7 moles of Ti will produce 7 moles of TiCl4.c. Since we have determined that chlorine is the limiting reactant for the third scenario, 18.8 moles of Cl2 will produce \[18.8 \div 2 = 9.4\] moles of TiCl4.

Key concepts

Stoichiometry

Stoichiometry is the area of chemistry that pertains to the quantitative relationships between the reactants and products in a chemical reaction. When embarking on stoichiometry calculations, it's essential to begin with a balanced chemical equation. This equation is a representation of the conservation of matter; it indicates that atoms are neither created nor destroyed, only rearranged. Each reactant and product in a chemical equation is written with a coefficient that represents the number of moles needed or produced in the reaction.

For example, in the equation \(\mathrm{Ti}(s) + 2 \mathrm{Cl}_{2}(g) \longrightarrow \mathrm{TiCl}_{4}(l)\), the coefficients suggest that for every one mole of titanium (\(\mathrm{Ti}\)), two moles of chlorine gas (\(\mathrm{Cl}_{2}\)) react to form one mole of titanium tetrachloride (\(\mathrm{TiCl}_{4}\)). Understanding these ratios is crucial for predicting the amounts of products formed from given reactants and vice versa. This is especially important if the amounts of reactants provided do not exactly match the stoichiometric ratios, as it will influence the calculation of the theoretical yield.

Limiting Reactant

In chemical reactions, the limiting reactant is the substance that is totally consumed when the chemical reaction is complete. The amount of product formed is limited by this reactant since the reaction cannot proceed without it. Identifying the limiting reactant is a vital step in predicting the amounts of products formed.

In any given chemical reaction, comparing the mole ratio of reactants used to the stoichiometric ratio indicated in the balanced equation helps determine which reactant is limiting. For example, with the exercise given, the mole ratio of titanium to chlorine gas is not always 1:2 as required by the reaction's stoichiometry. In situation (a), with both \(4 \mathrm{~mol} \mathrm{Ti}\) and \(4 \mathrm{~mol} \mathrm{Cl}_{2}\), we observe that titanium has twice as much as needed when compared to chlorine, making chlorine the limiting reactant. Knowing which reactant will run out first allows us to predict how much product can be produced, hence the importance of identifying the limiting reactant.

Chemical Reaction Equations

Chemical reaction equations are symbolic representations that showcase the reactants and products involved in a chemical reaction. They convey meaningful information about the process, including the physical states of substances with symbols (s for solid, l for liquid, g for gas, and aq for aqueous solution) and the necessary conditions for the reaction, such as temperature or catalysts.

A balanced chemical equation adheres to the law of conservation of mass, meaning the number of atoms for each element must be equal on both the reactant and product sides of the equation. Coefficients are used to achieve this balance and provide a direct relationship to the mole concept in stoichiometry. They dictate the proportions of reactants and products, which is essential for all subsequent quantitative analyses of the reaction, like the calculation of theoretical yield. An understanding of how to read and balance chemical reaction equations is a fundamental skill for any student tackling exercises in stoichiometry or predicting yields in chemistry.