Question

Calculate the theoretical yield of product (in moles) for each initial amount of reactants. $$2 \mathrm{Mn}(\mathrm{s})+2 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{MnO}_{2}(s)$$ a. \(3 \mathrm{mol} \mathrm{Mn}, 3 \mathrm{mol} \mathrm{O}_{2}\) b. \(4 \mathrm{mol} \mathrm{Mn}, 7 \mathrm{mol} \mathrm{O}_{2}\) c. \(27.5 \mathrm{mol} \mathrm{Mn}, 43.8 \mathrm{mol} \mathrm{O}_{2}\)

Short answer

a. 3 moles MnO2, b. 3.5 moles MnO2, c. 21.9 moles MnO2.

Step-by-step solution

- Identify Limiting Reactant for Set a

Calculate the moles of product that can be formed from each reactant. The reactant that produces the fewer moles of product is the limiting reactant. From the balanced reaction, it takes 1 mole of Mn for every 1 mole of MnO2. For O2, it takes 1 mole of O2 for every 1 mole of MnO2. Given 3 moles of Mn and 3 moles O2 both can produce 3 moles of MnO2. Therefore, neither reactant is in excess, and the theoretical yield of MnO2 is 3 moles.

- Identify Limiting Reactant for Set b

Using the stoichiometric ratios, 4 moles of Mn could produce 4 moles of MnO2, while 7 moles of O2 could produce 3.5 moles of MnO2. Here, O2 is the limiting reactant as it produces fewer moles of MnO2, hence the theoretical yield of MnO2 is 3.5 moles.

- Identify Limiting Reactant for Set c

Using the stoichiometric ratios, 27.5 moles of Mn could produce 27.5 moles of MnO2, while 43.8 moles of O2 could produce 21.9 moles of MnO2. Here, O2 is the limiting reactant since it produces fewer moles of MnO2, hence the theoretical yield is 21.9 moles.

Key concepts

Limiting Reactant

One of the most essential concepts in chemical reaction quantification is identifying the limiting reactant. This is the substance that will be completely used up first during a chemical reaction and hence determines the maximum amount of product that can be produced.

In our exercise, for set a, both manganese (Mn) and oxygen (O2) can produce an equal amount of the product, manganese dioxide (MnO2), which makes neither reactant limiting. However, for sets b and c, oxygen (O2) is the limiting reactant. Understanding the concept of the limiting reactant is crucial because it allows us to calculate the maximum yield of a product, which is of vital importance in industries for cost and resource management.

Identifying the Limiting Reactant

To identify the limiting reactant, you compare the amount of product that can be formed from each reactant based on the stoichiometry of the chemical equation. The reactant that produces less product is the limiting one. It's a fundamental step in stoichiometry to avoid the wastage of resources and time in both laboratory and industrial scales.

Stoichiometry

Stoichiometry is the study of the quantitative relationships, or ratios, between substances in chemical reactions. It is based on the conservation of mass and the concept of the mole. The coefficients in a balanced chemical equation tell us the relative amounts of each substance involved, allowing us to predict the quantities needed or produced in a chemical reaction.

For instance, in the exercise, the balanced equation indicates that 2 moles of Mn react with 2 moles of O2 to form 2 moles of MnO2. Stoichiometry helps us to use these ratios to calculate the theoretical yield—the maximum amount of product that could be formed from given amounts of reactants if the reaction goes to completion.

Chemical Reaction Quantification

Chemical reaction quantification involves determining the amount of reactants consumed and products formed in a chemical reaction. This process typically involves calculating mole ratios, identifying limiting reactants, and determining theoretical and actual yields.

Through the calculation of theoretical yield, as in the textbook exercise, one can estimate the efficiency of a reaction. In practical applications, this skill is essential for chemists to optimize the conditions of a reaction to maximize product formation while minimizing cost and waste.

Practical Implications

Accurate chemical reaction quantification is necessary not only in academic settings but also in real-world applications such as pharmaceuticals, manufacturing, and environmental science, where it informs the synthesis of compounds, production processes, and monitoring of reaction byproducts.

Mole-to-Mole Ratio

The mole-to-mole ratio is derived from the coefficients found in the balanced chemical equation and indicates the proportion of reactants that react with each other to form products. It fundamentally guides the stoichiometry calculations needed to determine how much of each reactant is needed and how much product will be formed.

In the provided exercise, the balanced equation shows a simple 1:1 mole ratio between Mn and MnO2, as well as between O2 and MnO2. This means that one mole of Mn or O2 will produce one mole of MnO2. Accurately understanding and applying mole-to-mole ratios are crucial for predicting the outcomes of chemical reactions and for performing successful theoretical yield calculations.