Question

Deterioration of buildings, bridges, and other structures through the rusting of iron costs millions of dollars a day. The actual process requires water, but a simplified equation is $$4 \mathrm{Fe}(s)+3 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{Fe}_{2} \mathrm{O}_{3}(s) \quad \Delta H=-1.65 \times 10^{3} \mathrm{~kJ}$$ (a) How much heat is released when \(0.250 \mathrm{~kg}\) of iron rusts? (b) How much rust forms when \(4.85 \times 10^{3} \mathrm{~kJ}\) of heat is released?

Short answer

(a) -1848.6 kJ. (b) 938 g Fe2O3.

Step-by-step solution

Balance the Chemical Equation

The given equation is already balanced: \(4 \mathrm{Fe}(s) + 3 \mathrm{O}_2(g) \rightarrow 2 \mathrm{Fe}_2 \mathrm{O}_3(s) \).

Identify the Molar Mass of Iron

The molar mass of iron (Fe) is approximately 55.85 g/mol.

Convert Kilograms of Iron Rusting to Moles

Given that \(0.250 \mathrm{~kg}\) of iron rusts, convert to grams: \(0.250 \mathrm{~kg} = 250 \mathrm{~g}\).Next, convert grams to moles: \[ \text{moles of Fe} = \frac{250 \mathrm{~g}}{55.85 \mathrm{~g/mol}} \approx 4.48 \mathrm{~mol} \].

Determine Heat Released Per Mole of Iron

From the balanced equation, 4 moles of iron release \( -1.65 \times 10^3 \mathrm{~kJ} \) of heat.Thus, heat released per mole:\[ \frac{-1.65 \times 10^3 \mathrm{~kJ}}{4 \text{ mols}} = -412.5 \mathrm{~kJ/mol} \].

Calculate Total Heat Released

Given 4.48 moles of iron rusting, the total heat released is: \[ \text{Total heat} = 4.48 \mathrm{~mol} \times (-412.5 \mathrm{~kJ/mol}) \approx -1848.6 \mathrm{~kJ} \].

Identify Molar Mass of Iron(III) Oxide

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

Calculate Iron(III) Oxide Formed

From the balanced equation, 2 moles of \(\mathrm{Fe}_2\mathrm{O}_3\) correspond to \( -1.65 \times 10^3 \mathrm{~kJ} \) of heat released.Calculate the moles corresponding to \( 4.85 \times 10^3 \mathrm{~kJ} \):\[ \text{moles of } \mathrm{Fe}_2\mathrm{O}_3 = \frac{4.85 \times 10^3 \mathrm{~kJ}}{-825 \mathrm{~kJ/mol}} \approx 5.88 \mathrm{~mol} \].Finally, convert moles to grams:\[ \text{mass of } \mathrm{Fe}_2\mathrm{O}_3 = 5.88 \mathrm{~mol} \times 159.7 \mathrm{~g/mol} \approx 938 \mathrm{~g} \].

Key concepts

enthalpy change

Enthalpy change (\( \boldsymbol{\text{Δ}H} \)) refers to the heat exchange in a chemical reaction at constant pressure. When iron rusts, the reaction releases heat, indicating an exothermic process.
In the given equation, the enthalpy change is -1.65 x 10^3 kJ. This negative sign signifies heat release.
For the rusting of iron, if you know how much iron you have, you can calculate the heat released using the enthalpy of the reaction and stoichiometry.
Keeping track of such values is crucial for various practical applications, like estimating energy changes in industrial processes.

chemical equations

A chemical equation represents the reactants and products in a chemical reaction. It must be balanced, meaning the number of atoms of each element is the same on both sides of the equation.
In our rusting example:

• 4 Fe (solid) + 3 O₂ (gas) → 2 Fe₂O₃ (solid)

This balanced equation tells us the ratio in which reactants react and products form.
Understanding this ratio is vital for calculating quantities of substances involved and energy changes, ensuring reactions proceed as intended in practical scenarios.

molar mass

Molar mass is the mass of one mole of a substance, expressed in grams per mole (g/mol).
For iron (Fe), the molar mass is approximately 55.85 g/mol. Knowing the molar mass allows us to convert between mass and moles.
For example, if we have 0.250 kg of iron, converting this to grams gives us 250 g. Using the molar mass, we get: \[ \frac{250 \text{ g}}{55.85 \text{ g/mol}} \approx 4.48 \text{ moles} \] This conversion is essential for reacting quantities and performing stoichiometric calculations.

stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions using balanced equations. It is like a recipe, ensuring everything reacts in correct proportions.
In our iron rusting example, we used stoichiometry to determine:

• How much heat is produced from a given amount of iron
• How much rust forms when a specific amount of heat is released

Using the balanced equation, the enthalpy change, and molar masses, we did the following calculations:

• Calculated moles of iron and heat released per mole
• Determined total heat release for 4.48 moles of iron
• Calculated moles and mass of Fe₂O₃ formed given specific heat release