Question

One of the reactions that occurs in a blast furnace, where iron ore is converted to cast iron, is $$ \mathrm{Fe}_{2} \mathrm{O}_{3}+3 \mathrm{CO} \longrightarrow 2 \mathrm{Fe}+3 \mathrm{CO}_{2} $$ Suppose that \(1.64 \times 10^{3} \mathrm{~kg}\) of Fe is obtained from a \(2.62 \times 10^{3}-\mathrm{kg}\) sample of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\). Assuming that the reaction goes to completion, what is the percent purity of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) in the original sample?

Short answer

The percent purity of \( \mathrm{Fe}_2 \mathrm{O}_3 \) in the original sample is approximately 89.5\%.

Step-by-step solution

Calculate Molar Masses

Determine the molar masses of the compounds involved.\[ \text{Molar Mass of Fe}_2 \text{O}_3 = 2 \times 55.85 + 3 \times 16.00 = 159.7 \text{ g/mol} \] \[ \text{Molar Mass of Fe} = 55.85 \text{ g/mol} \]

Use Stoichiometry of the Reaction

The reaction \( \mathrm{Fe}_2\mathrm{O}_3 + 3\mathrm{CO} \rightarrow 2 \mathrm{Fe} + 3\mathrm{CO}_2 \) tells us that 1 mole of \( \mathrm{Fe}_2\mathrm{O}_3 \) produces 2 moles of \( \mathrm{Fe} \).

Calculate Moles of Fe

Calculate the moles of iron obtained.\[ \text{Mass of Fe obtained} = 1.64 \times 10^3 \text{ kg} = 1.64 \times 10^6 \text{ g} \] \[ \text{Moles of Fe} = \frac{1.64 \times 10^6 \text{ g}}{55.85 \text{ g/mol}} \approx 29370 \text{ mol} \]

Calculate Moles of Fe2O3 Needed

Use stoichiometry to find the moles of \( \mathrm{Fe}_2 \mathrm{O}_3 \) that reacted. Since 2 moles of \( \mathrm{Fe} \) come from 1 mole of \( \mathrm{Fe}_2 \mathrm{O}_3 \):\[ \text{Moles of } \mathrm{Fe}_2 \mathrm{O}_3 = \frac{29370}{2} \approx 14685 \text{ mol} \]

Calculate Mass of Pure Fe2O3

Calculate the mass of \( \mathrm{Fe}_2 \mathrm{O}_3 \) required to produce that much iron.\[ \text{Mass of } \mathrm{Fe}_2 \mathrm{O}_3 = 14685 \times 159.7 \approx 2.345 \times 10^6 \text{ g} = 2.345 \times 10^3 \text{ kg} \]

Calculate Percent Purity

Calculate the percent purity of the \( \mathrm{Fe}_2 \mathrm{O}_3 \) in the sample.\[ \text{Percent Purity} = \left( \frac{\text{Mass of Pure } \mathrm{Fe}_2 \mathrm{O}_3}{\text{Total Mass of Sample}} \right) \times 100 \] \[ \text{Percent Purity} = \left( \frac{2.345 \times 10^3}{2.62 \times 10^3} \right) \times 100 \approx 89.5 \% \]

Conclusion

The percent purity of \( \mathrm{Fe}_2 \mathrm{O}_3 \) in the original sample is approximately 89.5\%.

Key concepts

Stoichiometry

Stoichiometry is a key concept in chemistry that involves the calculation of reactants and products in chemical reactions. It helps in understanding the quantitative relationships between the amounts of reactants and products. Stoichiometry is based on the conservation of mass, meaning matter cannot be created or destroyed. It relies on balanced chemical equations which indicate the ratio of moles of each substance involved in a reaction.

• For instance, in a balanced chemical equation \( ext{Fe}_2 ext{O}_3 + 3 ext{CO} \rightarrow 2 ext{Fe} + 3 ext{CO}_2\), it tells us that 1 mole of \( ext{Fe}_2 ext{O}_3\) will produce 2 moles of \( ext{Fe}\), and consume 3 moles of carbon monoxide (CO) producing 3 moles of carbon dioxide (CO\(_2\)).
• The mole ratio is crucial for converting between moles of different substances in a reaction.

Understanding stoichiometry allows chemists to determine the optimal amounts of reactants needed to produce a desired amount of product, minimizing waste and maximizing efficiency.

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is used to convert between the mass of a substance and the amount in moles. The molar mass can be calculated by adding together the atomic masses of the elements in the compound as found on the periodic table.

• For example, the molar mass of \(\text{Fe}_2 \text{O}_3\) is calculated by taking the sum of the masses of 2 iron (Fe) atoms and 3 oxygen (O) atoms. Iron has an atomic mass of approximately 55.85 g/mol and oxygen has an atomic mass of 16.00 g/mol. Thus, the molar mass of \(\text{Fe}_2 \text{O}_3\) is \(2 \times 55.85 + 3 \times 16.00 = 159.7 \text{ g/mol}\).
• Similarly, the molar mass of iron (Fe) alone is 55.85 g/mol.

This information is vital for carrying out stoichiometric calculations such as determining the moles of a reactant needed to produce a given amount of product.

Chemical Reactions

Chemical reactions involve the transformation of reactants into products, following a balanced chemical equation that conserves mass. In these processes, chemical bonds are broken and new bonds are formed, resulting in changes that can be observed through energy release or absorption, color changes, gas production, and precipitation.

• For the reaction \(\text{Fe}_2\text{O}_3 + 3\text{CO} \rightarrow 2\text{Fe} + 3\text{CO}_2\), the iron oxide (Fe\(_2\)O\(_3\)) reacts with carbon monoxide (CO) to produce iron (Fe) and carbon dioxide (CO\(_2\)).
• This specific reaction takes place in a blast furnace and is essential in the manufacture of iron from iron ore.

The stoichiometric coefficients in a chemical equation represent the relative amounts of each substance participating in the reaction and are crucial for determining reactant consumption and product formation.

Iron Production

Iron production is a key industrial process that involves extracting iron from its ores, primarily using a blast furnace method. This method facilitates the reduction of iron oxides using carbon sources, such as coke, in high-temperature conditions.

• In a blast furnace, the \(\text{Fe}_2\text{O}_3\) is reduced by carbon monoxide derived from the coke. The reaction, \(\text{Fe}_2\text{O}_3 + 3\text{CO} \rightarrow 2\text{Fe} + 3\text{CO}_2\), describes this iron extraction process.
• Here, the iron is produced as a molten material that can be tapped from the bottom of the furnace.
• The operation of the blast furnace requires precise control of the chemical reactions and is highly dependent on the purity of the input materials to optimize yield and efficiency.

Understanding the chemistry behind iron production is crucial for improving metallurgical processes and material quality, thereby impacting industries that rely heavily on iron and steel products.