Question

When elemental copper is strongly heated with sulfur, a mixture of 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 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 \(40.0 \mathrm{g}\) 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(s) is 47.81 g. The percent yield of CuS when only 40.0 g of CuS can be isolated from the mixture is 83.70%.

Step-by-step solution

Determine the molar masses of the reactants and product

We need to know the molar masses to find the moles of each reactant and the product. Using the periodic table, we find the following molar masses: \(Cu = 63.55 \,\text{g/mol}\), \(S = 32.07 \,\text{g/mol}\), and \(CuS = 95.62 \,\text{g/mol}\).

Convert the given mass of reactants into moles

We need to find the number of moles for Cu and S using the molar masses calculated in Step 1. For Cu: \(\frac{31.8 \,\text{g}}{63.55\, \text{g/mol}} = 0.5 \text{ moles of Cu}\) For S: \(\frac{50.0 \,\text{g}}{32.07 \,\text{g/mol}} = 1.56 \,\text{moles of S}\)

Identify the limiting reactant

Compare the moles of the reactants with the stoichiometric ratio from the balanced chemical equation to find the limiting reactant. Plugging the given values into the balanced chemical equation: \(0.5 \,\text{moles Cu}\) + \(1.56 \,\text{moles S}\) \(→ x \,\text{moles CuS}\) Stoichiometrically, 1 mole of Cu reacts with 1 mole of S (as given in the balanced reaction); so, Cu will react with just 0.5 moles of S completely. Since we have more sulfur than required, Cu is the limiting reactant.

Calculate the theoretical yield of CuS

Use the limiting reactant (Cu) to find the theoretical yield of CuS. From the balanced chemical equation, the stoichiometric ratio of Cu to CuS is 1:1. Since we have 0.5 moles of Cu, we can produce 0.5 moles of CuS. The mass of CuS produced will be the product of moles and its molar mass: \(0.5 \,\text{moles CuS} \times 95.62 \,\text{g/mol} = 47.81 \, \text{g CuS}\) The theoretical yield of CuS is 47.81 g.

Calculate the percent yield

Using the given actual yield (40.0 g) and the theoretical yield (47.81 g) from the calculations, we can find the percent yield. \(\% \,\text{yield} = \frac{\text{actual yield}}{\text{theoretical yield}} \times 100\) \(\% \,\text{yield} = \frac{40.0 \,\text{g}}{47.81 \text{g}} \times 100 = 83.70\%\) The percent yield of CuS is 83.70%.

Key concepts

Theoretical Yield

Theoretical yield is a calculation that chemists perform to predict the maximum amount of product that can be formed in a chemical reaction. This calculation assumes perfect conversion of reactants to products, with no losses or side reactions.
To find the theoretical yield, we start by looking at the balanced chemical equation. In the case of this reaction, copper and sulfur are used to form copper sulfide (CuS). By calculating the moles of copper we have and noting it as the limiting reactant, we determine how much CuS we can theoretically produce.
In this exercise, the theoretical yield of CuS is found by multiplying the moles of copper by the molar mass of CuS. This calculation results in a theoretical yield of 47.81 g of CuS.

Percent Yield

After determining the theoretical yield, a chemist can calculate the percent yield to see how efficient the reaction was. Percent yield tells us the efficiency of a reaction by comparing the actual amount of product obtained to the theoretical maximum amount possible.
To find the percent yield, use the equation:
\(\text{Percent Yield} = \frac{\text{Actual Yield}}{\text{Theoretical Yield}} \times 100\)
In the example given, 40.0 g of CuS is actually obtained from the reaction, while the theoretical yield is 47.81 g. Plugging these values into the formula gives a percent yield of approximately 83.70%. This indicates that the reaction process was reasonably efficient but not perfect, as some product was lost or not formed.

Limiting Reactant

The limiting reactant (or limiting reagent) in a chemical reaction is the substance that is completely consumed first, limiting the amount of product that can be formed. To identify the limiting reactant, compare the mole ratios of the reactants given in the chemical equation to those of the reactants provided.
In this exercise, copper (Cu) acts as the limiting reactant because it is present in the smallest stoichiometric amount necessary to react with sulfur (S). According to the balanced equation, one mole of Cu reacts with one mole of S to form CuS.
With only 0.5 moles of Cu available, compared to 1.56 moles of S, copper will limit the reaction since it is consumed more quickly. Recognizing the limiting reactant is vital because it determines the maximum yield of product in a reaction.