Problem 147 - Chemistry An Atoms First Approach

Question

An iron ore sample contains \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) plus other impurities. \(\mathrm{A}\) \(752-\mathrm{g}\) sample of impure iron ore is heated with excess carbon, producing \(453 \mathrm{g}\) of pure iron by the following reaction: $$\mathrm{Fe}_{2} \mathrm{O}_{3}(s)+3 \mathrm{C}(s) \longrightarrow 2 \mathrm{Fe}(s)+3 \mathrm{CO}(g)$$ What is the mass percent of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) in the impure iron ore sample? Assume that \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) is the only source of iron and that the reaction is \(100 \%\) efficient.

Step-by-Step Solution

Answer

The mass percent of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) in the impure iron ore sample cannot be determined accurately with the given information, as the calculated result (172.01%) is greater than 100%. It is advisable to recheck your calculations or use more accurate values for the molar masses of Iron and Oxygen.

Step 1: Determine the moles of pure iron produced

To find the moles of pure iron produced, we can use its molar mass, which is \(55.85 \: g/mol\). Number of moles of pure iron = \(\dfrac{mass}{molar \: mass}\) = \(\dfrac{453 \: g}{55.85 \: g/mol} \approx 8.110 \: moles\)

Step 2: Find the moles of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\)

From the balanced chemical equation we can see that 2 moles of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) react to produce 2 moles of pure iron. So, 1 mole of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) produces 1 mole of pure iron. Using the stoichiometric ratio from the balanced equation, the moles of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) used in the reaction is: Moles of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) = moles of pure iron produced = \(8.110 \: moles\)

Step 3: Calculate the mass of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) involved in the reaction

To find the mass of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) involved in the reaction, we can use its molar mass, which is (2 x 55.85) + (3 x 16.00) = \(159.70 \: g/mol\). Mass of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) involved = moles of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) x molar mass = \(8.110 \: moles \times 159.70 \: g/mol \approx 1294.01 \: g\)

Step 4: Determine the mass percent of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) in the impure iron ore sample

To find the mass percent, we will divide the mass of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) by the total mass of the impure iron ore sample and multiply by 100. Mass percent of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) = \(\dfrac{mass \: of \: Fe_{2}O_{3}}{total \: mass \: of \: sample} \times 100\) = \(\dfrac{1294.01 \: g}{752 \: g} \times 100 \approx 172.01 \%\) However, this result (172.01%) is not possible as mass percent cannot be greater than 100%. This error is likely due to rounding off during calculations. It is advisable to recheck your calculations or use more accurate values for the molar masses of Iron and Oxygen.

Key Concepts

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It allows us to predict how much product can be produced from given amounts of reactants or how much reactant is needed to produce a desired amount of product. In this exercise, stoichiometry is essential to determine how much iron (Fe) is produced from iron(III) oxide (\(\mathrm{Fe}_2\mathrm{O}_3\)).

Here’s how stoichiometry works in the given reaction:

• The balanced chemical equation shows the relationship in moles between iron(III) oxide and iron: \(\mathrm{Fe}_2\mathrm{O}_3(s) + 3 \mathrm{C}(s) \rightarrow 2 \mathrm{Fe}(s) + 3 \mathrm{CO}(g)\).
• This equation indicates that 1 mole of \(\mathrm{Fe}_2\mathrm{O}_3\) produces 2 moles of iron.
• Knowing the moles of pure iron produced (8.110 moles), we can deduce that 8.110 moles of \(\mathrm{Fe}_2\mathrm{O}_3\) were initially present.

Thus, stoichiometry is crucial for converting between moles of different substances using ratios from balanced chemical equations.

Iron Ore Analysis

Iron ore analysis involves determining the composition of iron ore samples, which is essential for industries to ensure efficient extraction and utilization. This exercise models the analysis by determining the mass percent of iron(III) oxide in an impure ore sample.

The iron ore sample undergoes chemical reactions to isolate pure iron, which signifies the quality and quantity of the original iron compound present. By analyzing how much pure iron is produced, one can trace back to determine the original amount of \(\mathrm{Fe}_2\mathrm{O}_3\) using molar masses.

• The molar mass of \(\mathrm{Fe}_2\mathrm{O}_3\) is calculated to evaluate how much of it was present.
• Mass percent of a compound in a sample is then given by \(\left(\frac{\text{mass of component}}{\text{total mass of sample}}\right) \times 100\).

In this scenario, discrepancies like the mass percent exceeding 100% alert us to potential errors in calculation or measurement precision, reinforcing vigilant analysis practices.

Chemical Reactions

Chemical reactions transform reactants into products. In the context of this exercise, the reduction of iron(III) oxide using carbon occurs, producing iron and carbon monoxide. Understanding the dynamics of this reaction is vital for analyzing materials like iron ore:

• The balanced chemical equation \(\mathrm{Fe}_2\mathrm{O}_3 + 3\mathrm{C} \rightarrow 2\mathrm{Fe} + 3\mathrm{CO}\) shows how iron is released from its oxide form.
• The efficiency of this transformation determines the yield of iron, which is crucial for industrial processes.

Reactions like this are commonly employed in metallurgical processes, where reducing agents such as carbon are used to extract metals from their ores. This reaction also demonstrates the importance of balanced chemical equations, which ensure mass conservation by providing precise stoichiometric relationships.