Problem 79 - Chemistry Matter and Change

Question

Iron Production Iron is obtained commercially by the reaction of hematite \(\left(\mathrm{Fe}_{2} \mathrm{O}_{3}\right)\) with carbon monoxide. How many grams of iron is produced when 25.0 \(\mathrm{mol}\) of hematite reacts with 30.0 \(\mathrm{mol}\) of carbon monoxide? $$\mathrm{Fe}_{2} \mathrm{O}_{3}(\mathrm{s})+3 \mathrm{CO}(\mathrm{g}) \rightarrow 2 \mathrm{Fe}(\mathrm{s})+3 \mathrm{CO}_{2}(\mathrm{g})$$

Step-by-Step Solution

Answer

When 25.0 moles of hematite reacts with 30.0 moles of carbon monoxide, 1117 grams of iron is produced.

Step 1: Identify the limiting reactant

To find the limiting reactant, we need to compare the mole ratio of hematite and carbon monoxide. For this, we will use the balanced chemical equation. According to the balanced equation, 1 mole of hematite reacts with 3 moles of carbon monoxide: $$\mathrm{Fe}_2\mathrm{O}_3\ +\ 3\mathrm{CO}\ \rightarrow\ 2\mathrm{Fe}\ +\ 3\mathrm{CO}_2$$ We are given 25.0 moles of hematite and 30.0 moles of carbon monoxide. Divide the moles of each reactant by their stoichiometric coefficients: For hematite: \(25.0\ \mathrm{mol} / 1 = 25.0\) For carbon monoxide: \(30.0\ \mathrm{mol} / 3 = 10.0\) Since \(10.0 < 25.0\), carbon monoxide is the limiting reactant. Step 2: Calculate the moles of iron produced

Step 2: Calculate the moles of iron produced

Since carbon monoxide is the limiting reactant, we will use the mole ratio between carbon monoxide and iron to calculate the moles of iron produced. According to the balanced equation, 3 moles of carbon monoxide produce 2 moles of iron: $$3 \mathrm{CO} \rightarrow 2 \mathrm{Fe}$$ We have 30.0 moles of carbon monoxide, so we'll multiply that by the mole ratio: Moles of iron = \(30.0 \times \frac{2 \, \cancel{\mathrm{mol}\, \mathrm{Fe}}}{3 \, \cancel{\mathrm{mol}\, \mathrm{CO}}}\) Moles of iron = \(20.0\ \mathrm{mol}\) Step 3: Convert moles of iron to mass of iron

Step 3: Convert moles of iron to mass of iron

In order to convert moles of iron to mass, we need to know the molar mass of iron. The molar mass of iron, Fe, is 55.85 g/mol. Now we can multiply the moles of iron by the molar mass: Mass of iron = Moles of iron × Molar mass of iron Mass of iron = \(20.0\ \cancel{\mathrm{mol}}\ \times\ 55.85\ \frac{\mathrm{g}}{\cancel{\mathrm{mol}}}\) Mass of iron = \(1117\ \mathrm{g}\) When 25.0 moles of hematite reacts with 30.0 moles of carbon monoxide, 1117 grams of iron is produced.

Key Concepts

Limiting Reactant

In chemical reactions, finding out which reactant will run out first is a fundamental step. This is known as identifying the limiting reactant. In our iron production exercise, both hematite and carbon monoxide are used to produce iron. To figure out the limiting reactant, we need to use the balanced chemical equation.
We start by looking at the reaction:

• 1 mole of Fe₂O₃ reacts with 3 moles of CO.

Given in the exercise, we have 25.0 moles of hematite (Fe₂O₃) and 30.0 moles of CO. To find the limiting reactant, divide the moles of each reactant by their respective coefficients in the balanced equation. This helps us understand which reactant will be used up first.

• For hematite: 25.0 moles / 1 = 25.0.
• For carbon monoxide: 30.0 moles / 3 = 10.0.

Since 10.0 < 25.0, carbon monoxide becomes the limiting reactant. So, CO will determine the amount of iron produced.

Mole Ratio

The concept of mole ratio is vital in solving problems related to chemical reactions. Mole ratio is derived from the coefficients of substances in a balanced chemical equation. In our problem, we are concerned with how carbon monoxide and iron relate to each other in mole terms.
From the balanced equation:

• 3 moles of CO produce 2 moles of Fe.

Thus, the mole ratio of CO to Fe is 3:2. In this exercise, knowing this ratio helps us convert the amount of limiting reactant (CO) to the amount of iron produced. For every 3 moles of CO used, 2 moles of iron are created.
Given 30.0 moles of carbon monoxide, the following calculation can be made:

• Moles of iron = 30.0 × (2/3) = 20.0 moles of iron.

Using the right mole ratio ensures accurate conversion from moles of reactant to moles of product.

Molar Mass Calculation

Transforming moles into grams is necessary for understanding the tangible quantities of products like iron. Molar mass serves as a bridge in these conversions.
To convert moles to grams, multiply the number of moles by the molar mass of the substance. In this case:

• The molar mass of iron, Fe, is 55.85 g/mol.

By doing the calculation:

• Mass of iron = 20.0 moles × 55.85 g/mol = 1117 g of iron.

This process provides the final answer: 1117 grams of iron can be produced, connecting us from moles to mass, a practical measure in any synthesis and industrial production processes.