Question

Iron metal can be produced from the mineral hematite, \(\mathrm{Fe}_{2} \mathrm{O}_{3}\), by reaction with carbon. How many kilograms of iron are present in \(105 \mathrm{~kg}\) of hematite?

Short answer

73.38 kg of iron is present in 105 kg of hematite.

Step-by-step solution

Write the Balanced Chemical Equation

The chemical reaction between hematite (\(\mathrm{Fe}_2\mathrm{O}_3\)) and carbon to produce iron and carbon dioxide is:\[\mathrm{Fe}_2\mathrm{O}_3(s) + 3\mathrm{C}(s) \rightarrow 2\mathrm{Fe}(s) + 3\mathrm{CO}_2(g)\]

Calculate Molar Masses

Calculate the molar mass of \(\mathrm{Fe}_2\mathrm{O}_3\) and the molar mass of \(\mathrm{Fe}\).- The molar mass of \(\mathrm{Fe}_2\mathrm{O}_3\) is calculated as: \[(2 \times 55.85 \mathrm{~g/mol}) + (3 \times 16.00 \mathrm{~g/mol}) = 159.7 \mathrm{~g/mol}\]- The molar mass of \(\mathrm{Fe}\) is: \[55.85 \mathrm{~g/mol}\]

Calculate Moles of Hematite

Convert 105 kg of \(\mathrm{Fe}_2\mathrm{O}_3\) to grams:\[105 \mathrm{~kg} = 105,000 \mathrm{~g}\]Calculate the moles of \(\mathrm{Fe}_2\mathrm{O}_3\):\[\text{Moles of } \mathrm{Fe}_2\mathrm{O}_3 = \frac{105,000 \text{ g}}{159.7 \text{ g/mol}} \approx 657 \text{ mol}\]

Calculate Moles of Iron Produced

From the balanced equation, 1 mol of \(\mathrm{Fe}_2\mathrm{O}_3\) produces 2 mol of \(\mathrm{Fe}\).Therefore, the moles of \(\mathrm{Fe}\) produced is:\[657 \text{ mol of } \mathrm{Fe}_2\mathrm{O}_3 \times 2 = 1314 \text{ mol of } \mathrm{Fe}\]

Convert Moles of Iron to Kilograms

Convert moles of \(\mathrm{Fe}\) to grams:\[1314 \text{ mol} \times 55.85 \mathrm{~g/mol} = 73,382.1 \mathrm{~g}\]Convert grams to kilograms:\[73,382.1 \mathrm{~g} = 73.3821 \mathrm{~kg}\]

Conclusion: Result

Therefore, the mass of iron produced from 105 kg of \(\mathrm{Fe}_2\mathrm{O}_3\) is approximately 73.38 kg.

Key concepts

Balanced Chemical Equation

Chemical equations describe the process of a chemical reaction. It's important that these equations are balanced. This means that the number of atoms for each element in the reaction is the same on both sides of the equation. Balancing a chemical equation ensures the law of conservation of mass is met. This law states that matter cannot be created or destroyed, only rearranged.

To illustrate, consider the reaction between hematite (\(\mathrm{Fe}_2\mathrm{O}_3\)) and carbon (\(\mathrm{C}\)) to produce iron (\(\mathrm{Fe}\)) and carbon dioxide (\(\mathrm{CO}_2\)). The balanced chemical equation is:\[\mathrm{Fe}_2\mathrm{O}_3(s) + 3\mathrm{C}(s) \rightarrow 2\mathrm{Fe}(s) + 3\mathrm{CO}_2(g)\]This equation is balanced because there are the same number of Fe, O, and C atoms on each side. A balanced equation is fundamental to predicting the quantities of products and reactants.

Molar Mass Calculation

Molar mass is a key concept in stoichiometry. It describes the mass of one mole of a substance. To calculate molar mass, add up the atomic masses of all the atoms in its chemical formula, expressed in grams per mole (g/mol).

For example, the molar mass of hematite \(\mathrm{Fe}_2\mathrm{O}_3\) involves calculating the individual molar masses of iron \(\mathrm{Fe}\) and oxygen \(\mathrm{O}\) in the compound. Iron has an atomic mass of 55.85 g/mol and oxygen has 16.00 g/mol.

• The molar mass of hematite is: \((2 \times 55.85 \text{ g/mol}) + (3 \times 16.00 \text{ g/mol}) = 159.7 \text{ g/mol}\).

Similarly, the molar mass of elemental iron \(\mathrm{Fe}\) is simply 55.85 g/mol. Knowing these values allows the conversion of mass into moles — an essential step in stoichiometric calculations.

Moles Conversion

Moles conversion is central to solving stoichiometry problems. It involves converting between grams of a substance and moles, using the molar mass as a conversion factor.

For instance, converting 105 kg of hematite \(\mathrm{Fe}_2\mathrm{O}_3\) to grams allows us to determine how many moles of hematite we have:

• First, change kilograms to grams: \(105 \text{ kg} = 105,000 \text{ g}\).
• Then, use the molar mass: \(\text{Moles of } \mathrm{Fe}_2\mathrm{O}_3 = \frac{105,000 \text{ g}}{159.7 \text{ g/mol}} \approx 657 \text{ mol}\).

Finally, using the balanced chemical equation, you can find the number of moles of iron produced. The equation shows that each mole of \(\mathrm{Fe}_2\mathrm{O}_3\) makes 2 moles of iron \(\mathrm{Fe}\). Converting these moles back into mass completes the solution. Understanding mole conversions is vital for precise chemical calculations!