Question

The mineral magnetite is an iron oxide ore that has a density of \(5.20 \mathrm{~g} / \mathrm{cm}^{3}\). At high temperature, magnetite reacts with carbon monoxide to yield iron metal and carbon dioxide. When \(2.660 \mathrm{~g}\) of magnetite is allowed to react with sufficient carbon monoxide, the \(\mathrm{CO}_{2}\) product is found to have a volume of \(1.136 \mathrm{~L}\) at \(298 \mathrm{~K}\) and \(751 \mathrm{~mm} \mathrm{Hg}\) pressure. (a) What mass of iron in grams is formed in the reaction? (b) What is the formula of magnetite? (c) Magnetite has a somewhat complicated cubic unit cell with an edge length of \(839 \mathrm{pm}\). How many \(\mathrm{Fe}\) and \(\mathrm{O}\) atoms are present in each unit cell?

Short answer

(a) 1.935 g of iron is formed; (b) Formula of magnetite is Fe₃O₄; (c) Each unit cell contains 12 Fe atoms and 16 O atoms.

Step-by-step solution

Write the Balanced Reaction Equation

The reaction of magnetite (Fe₃O₄) with CO is as follows:\[ \text{Fe}_3\text{O}_4 (s) + 4\text{CO} (g) \rightarrow 3\text{Fe} (s) + 4\text{CO}_2 (g) \]

Calculate Moles of CO₂ Produced

Use the ideal gas law to determine the moles of CO₂. First, convert the pressure from mmHg to atm: \[ P = 751 \, \text{mmHg} \times \frac{1 \, \text{atm}}{760 \, \text{mmHg}} = 0.9882 \, \text{atm} \]Now apply the ideal gas law: \[ PV = nRT \]Solving for \( n \): \[ n = \frac{PV}{RT} = \frac{(0.9882 \, \text{atm})(1.136 \, \text{L})}{(0.0821 \, \text{L atm/mol K})(298 \, \text{K})} = 0.0462 \, \text{mol} \]

Determine Moles of Fe Produced

According to the balanced equation, 4 moles of CO₂ yield 3 moles of Fe. Therefore, calculate moles of Fe:\[ \text{Moles of Fe} = \frac{3}{4} \times 0.0462 \, \text{mol} = 0.03465 \, \text{mol} \]

Convert Moles of Fe to Mass

Multiply the moles of Fe by its molar mass (55.85 g/mol):\[ \text{Mass of Fe} = 0.03465 \, \text{mol} \times 55.85 \, \text{g/mol} = 1.935 \, \text{g} \]

Determine the Formula of Magnetite

Magnetite is Fe₃O₄, which combines two iron(III) oxide (Fe₂O₃) and one iron(II) oxide (FeO). This matches the stoichiometry of the reaction and confirms its formula.

Calculate Number of Atoms in a Unit Cell

Magnetite's cubic structure has an edge of 839 pm. For the number of Fe and O atoms: - Per unit cell: 4 Fe₃O₄ units yield - 3 (Fe per unit of Fe₃O₄) x 4 = 12 Fe atoms - 4 (O per unit of Fe₃O₄) x 4 = 16 O atoms Thus, each unit cell contains 12 Fe atoms and 16 O atoms.

Key concepts

Magnetite

Magnetite is an intriguing mineral, scientifically known as iron(II,III) oxide with the formula \( \text{Fe}_3\text{O}_4 \). It is one of the most magnetic minerals found in nature. This unique characteristic is due to its chemical composition that includes both ferrous (\( \text{Fe}^{2+} \)) and ferric (\( \text{Fe}^{3+} \)) irons. Each molecule of magnetite is structured with three iron atoms and four oxygen atoms, making it a significant ore of iron used in various industrial processes.
The black hue and metallic luster of magnetite along with its density of \(5.20 \text{ g/cm}^3\) make it identifiable in geological and chemical studies. Its role in producing iron metal through chemical reactions is crucial to understanding both its economic value and chemical behavior.

Chemical Reactions

Chemical reactions involving magnetite are essential to producing iron metal. The specific reaction described in the exercise is between magnetite (\( \text{Fe}_3\text{O}_4 \)) and carbon monoxide (\( \text{CO} \)), which produces iron (\( \text{Fe} \)) and carbon dioxide (\( \text{CO}_2 \)). This is a reduction reaction where the magnetite loses oxygen, creating metallic iron and releasing carbon dioxide as a byproduct.
The balanced equation \( \text{Fe}_3\text{O}_4 (s) + 4\text{CO} (g) \rightarrow 3\text{Fe} (s) + 4\text{CO}_2 (g) \) illustrates the stoichiometry of the process, showing the mole ratio between reactants and products. Typically, this type of reaction is significant in metallurgy, especially in the steel industry.

Stoichiometry

Stoichiometry is the section of chemistry that deals with the relative quantities of reactants and products in chemical reactions. In the context of magnetite reactions, stoichiometry helps us determine how much iron is produced from a given amount of magnetite.
Employing the balanced equation, we can see that 4 moles of carbon monoxide react to produce 3 moles of iron. By knowing the moles of carbon dioxide produced (0.0462 mol), we can calculate the number of moles of iron produced using the mole ratio from the balanced reaction. This calculation is crucial for understanding yield and efficiency in chemical processes and helps us determine the mass of iron formed (1.935 grams).
Clearly outlining each mole relationship and calculation step is vital for mastering stoichiometry and applying it to practical chemical reaction problems.

Unit Cell Calculations

Unit cell calculations provide insight into the microscopic structure of crystalline materials like magnetite. The unit cell is the smallest divisible unit of a crystal lattice, which repeats in space to form the crystal.
In magnetite's cubic unit cell, which has an edge length of 839 picometers, it's important to determine the number of atoms present. Given that each unit cell contains four formula units of magnetite (\( \text{Fe}_3\text{O}_4 \)), it accommodates 12 iron atoms and 16 oxygen atoms per cell. This configuration ensures the magnetite maintains its characteristic properties.
Understanding unit cell compositions is essential for calculating density, determining physical properties, and grasping the molecular and atomic arrangements within a solid. It forms a crucial part of material science, contributing to our knowledge of magnetic materials, ceramics, and metal alloys.