Question

When clemental copper is strongly heated with sulfur, a mixture of \(\mathrm{CuS}\) and \(\mathrm{Cu}_{2} \mathrm{~S}\) is produced, with CuS predominating. $$ \begin{array}{c} \mathrm{Cu}(s)+\mathrm{S}(s) \rightarrow \mathrm{CuS}(s) \\ 2 \mathrm{Cu}(s)+\mathrm{S}(s) \rightarrow \mathrm{Cu}_{2} \mathrm{~S}(s) \end{array} $$ What is the theoretical yield of CuS when \(31.8 \mathrm{~g}\) of \(\mathrm{Cu}(s)\) is heated with \(50.0 \mathrm{~g}\) of \(\mathrm{S} ?\) (Assume only CuS is produced in the reaction.) What is the percent yield of CuS if only 40.0 g of CuS can be isolated from the mixture?

Short answer

The theoretical yield of CuS when 31.8 g of Cu(s) is heated with 50.0 g of S is 149 g. The percent yield of CuS when only 40.0 g can be isolated from the mixture is approximately 26.8%.

Step-by-step solution

Find the limiting reactant

Calculate the moles of copper and sulfur: To find which reactant limits the formation of CuS, we first need to calculate the number of moles of Cu and S present. Use the molar masses: Molar mass of Cu = 63.5 g/mol Molar mass of S = 32.1 g/mol Moles of Cu: \(\frac{31.8\ \text{g}}{63.5\ \text{g/mol}} = 0.501\ \text{mol}\) Moles of S: \(\frac{50.0\ \text{g}}{32.1\ \text{g/mol}} = 1.56\ \text{mol}\) Now, compare the mole ratios: Cu:S = 1:1 from the reaction Cu(s) + S(s) → CuS(s) (\$0.501\ \text{mol}\) Cu : \(1.56\ \text{mol}\) S) = 0.321 Since the mole ratio of Cu to S is less than 1:1, Cu is in excess and sulfur is the limiting reactant.

Calculate the theoretical yield of CuS

Determine the maximum moles of CuS that can form: As we determined that sulfur is the limiting reactant, we will calculate the theoretical yield of CuS according to the moles of sulfur present. The stoichiometry of the reaction is 1:1 for CuS formation, Moles of CuS = Moles of S = 1.56 mol Calculate the theoretical yield of CuS: Theoretical yield = Moles of CuS * Molar mass of CuS Molar mass of CuS = 63.5 g/mol (Cu) + 32.1 g/mol (S) = 95.6 g/mol Theoretical yield: \(1.56\ \text{mol}\) * \(95.6\ \text{g/mol}\) = 149\ \text{g} (rounded to nearest whole number)

Determine the percent yield

Calculate the percent yield of CuS: Percent yield = \(\frac{\text{Actual yield}}{\text{Theoretical yield}}\) * 100 Percent yield = \(\frac{40.0\ \text{g}}{149\ \text{g}}\) * 100 = 26.8% The percent yield of CuS in the reaction is approximately 26.8%.

Key concepts

Limiting Reactant

Understanding the concept of limiting reactants is crucial when predicting the outcomes of chemical reactions. A limiting reactant is the substance in a chemical reaction that is totally consumed first, thus determining the maximum amount of product that can be formed. The other reactants may have leftover quantities and are thus considered to be in excess. To identify the limiting reactant, one must first calculate the moles of each reactant. Once the moles are known, the stoichiometry of the reaction—that is, the ratio in which the substances react—can be used to determine which reactant will run out first.

For example, in the reaction of copper and sulfur to produce copper sulfide, by comparing the mole ratio of copper to sulfur, we discovered that sulfur is the limiting reactant. This is a crucial step because all subsequent calculations, such as determining the theoretical yield, are based on the amount of the limiting reactant.

Stoichiometry

In any chemical process, stoichiometry is the mathematical relationship between the quantities of reactants and products. It is based on the balanced chemical equation and the conservation of mass principle. Stoichiometry allows us to predict the amounts of substances consumed and produced in a reaction, provided we know the starting amounts. The key to mastering stoichiometry is to practice converting between mass, moles, and particles using the molar mass of the substances involved as the conversion factor.

In the exercise, stoichiometry is used to calculate the theoretical yield of copper sulfide (CuS) by employing the 1:1 molar ratio between copper and sulfur indicated by the balanced equation. With stoichiometry, the reaction tells us that for each mole of sulfur, one mole of CuS will be produced. Therefore, knowing the moles of the limiting reactant directly leads to the quantity of product expected—the theoretical yield.

Percent Yield

The percent yield is a measure of the efficiency of a chemical reaction, expressed as a percentage. It is calculated by dividing the actual yield (the amount of product actually obtained from a reaction) by the theoretical yield (the amount of product predicted by stoichiometry), and then multiplying by 100. The percent yield gives us insight into the practical limitations and success of a reaction. Factors such as incomplete reactions, side reactions, and product loss during recovery all contribute to a percent yield that is less than 100%.

For the given exercise, the percent yield of CuS is calculated based on the actual yield of 40.0 grams and the theoretical yield of 149 grams. The result of 26.8% indicates that only about a quarter of the predicted amount of CuS was actually produced, which could prompt further investigation into why the yield was lower than expected. Understanding the percent yield helps chemists to optimize reactions and scale up processes accurately.