Problem 45 - Chemistry An Atoms First Approach

Question

The overall reaction in a commercial heat pack can be represented as $$4 \mathrm{Fe}(s)+3 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{Fe}_{2} \mathrm{O}_{3}(s) \quad \Delta H=-1652 \mathrm{kJ}$$ a. How much heat is released when 4.00 moles of iron are reacted with excess \(\mathrm{O}_{2} ?\) b. How much heat is released when 1.00 mole of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) is produced? c. How much heat is released when \(1.00 \mathrm{g}\) iron is reacted with excess \(\mathbf{O}_{2} ?\) d. How much heat is released when \(10.0 \mathrm{g}\) Fe and \(2.00 \mathrm{g} \mathrm{O}_{2}\) are reacted?

Step-by-Step Solution

Answer

a. Heat released when 4.00 moles of iron are reacted with excess O₂: -1652 kJ. b. Heat released when 1.00 mole of Fe₂O₃ is produced: -826 kJ. c. Heat released when 1.00 g iron is reacted with excess O₂: -7.49 kJ. d. Heat released when 10.0 g Fe and 2.00 g O₂ are reacted: -34.39 kJ.

Step 1: Write down the balanced equation and the enthalpy change

The balanced equation: \(4 \textrm{Fe}(s) + 3 \textrm{O}_{2}(g) \rightarrow 2 \textrm{Fe}_{2} \textrm{O}_{3}(s)\) ΔH = -1652 kJ

Step 2: Use stoichiometry to find the heat released

According to the balanced equation, 4 moles of Fe react to produce 2 moles of Fe₂O₃, releasing -1652 kJ of heat. When 4 moles of Fe are reacted, the heat released is the same as the enthalpy change: Heat released = -1652 kJ b. How much heat is released when 1.00 mole of \(\mathrm{Fe}_{2}\mathrm{O}_{3}\) is produced?

Step 1: Use stoichiometry to find the heat released

According to the balanced equation, when 2 moles of Fe₂O₃ are produced, -1652 kJ of heat is released. To find the heat released when 1 mole of Fe₂O₃ is produced, simply divide the enthalpy change by 2: Heat released = -1652 kJ / 2 = -826 kJ c. How much heat is released when \(1.00 \mathrm{g}\) iron is reacted with excess \(\mathbf{O}_{2} ?\)

Step 1: Convert grams of iron to moles

To convert grams to moles, we will use the molar mass of iron(Fe): 55.845 g/mol. Moles of Fe = 1.00 g / 55.845 g/mol = 0.0179 mol

Step 2: Use stoichiometry to find the heat released

The balanced equation tells us that 4 moles of Fe reacts and releases -1652 kJ of heat. We will find the heat released for 0.0179 moles of Fe. Heat released = (0.0179 mol Fe) * (-1652 kJ / 4 mol Fe) = -7.49 kJ (rounded to two decimal places) d. How much heat is released when \(10.0 \mathrm{g}\) Fe and \(2.00 \mathrm{g}\mathrm{O}_{2}\) are reacted?

Step 1: Convert grams of Fe and O₂ to moles

Using the molar mass of Fe(55.845 g/mol) and O₂(32.00 g/mol), convert the grams to moles. Moles of Fe = 10.0 g / 55.845 g/mol = 0.179 mol Moles of O₂ = 2.00 g / 32.00 g/mol = 0.0625 mol

Step 2: Determine the limiting reactant

Based on the balanced equation, we know that 4 moles of Fe react with 3 moles of O₂. We can find the mole ratios of Fe and O₂ in the reaction: Mole ratio (Fe:O₂) = (0.179 mol / 4) / (0.0625 mol / 3) = 1.34 Since this ratio is greater than 1, O₂ is the limiting reactant.

Step 3: Use stoichiometry to find the heat released

Based on the balanced equation and knowing that O₂ is the limiting reactant, we can find the heat released for the given amount of reactants: Heat released = (0.0625 mol O₂) * (-1652 kJ / (3 mol O₂)) = -34.39 kJ (rounded to two decimal places)

Key Concepts

Stoichiometry

Stoichiometry is a fundamental concept in chemistry that involves calculating the quantities of reactants and products in a chemical reaction. It is based on the balanced equation of the reaction.

For example, in the reaction \(4 \mathrm{Fe} + 3 \mathrm{O}_2 \rightarrow 2 \mathrm{Fe}_2\mathrm{O}_3\), stoichiometry helps us understand the relationship between the different moles of reactants and products:

• You need 4 moles of iron (Fe) to completely react with 3 moles of oxygen \(\mathrm{O}_2\).
• This produces 2 moles of iron(III) oxide \(\mathrm{Fe}_2\mathrm{O}_3\).
• The stoichiometric coefficients (4:3:2) show the proportions in which the substances react and are produced.

This basis is critical when calculating heat changes, determining reactant needs, and calculating remaining substances after a reaction.

In given problems, stoichiometry allows you to use mole ratios to find amounts of substances involved at any stage of the reaction. It forms the mathematical backbone of chemical conversions.

Enthalpy Change

Enthalpy change, denoted as \(\Delta H\), represents the heat change in a reaction occurring at constant pressure. It is expressed in units of kilojoules (kJ) per mole of reaction. For the given reaction, \(\Delta H = -1652 \mathrm{kJ}\) indicates that the reaction is exothermic, meaning it releases heat.

Key aspects:

• The negative sign signifies that energy is released, making it an exothermic reaction.
• If you produce 2 moles of \(\mathrm{Fe}_2\mathrm{O}_3\), then 1652 kJ of heat is given off based on the balanced equation.
• To understand individual product formation, such as forming 1 mole of \(\mathrm{Fe}_2\mathrm{O}_3\), you simply halve the enthalpy change, resulting in \(-826 \mathrm{kJ}\).

Enthalpy change is crucial as it helps predict the energy efficiency of chemical processes and understanding reaction behavior under different conditions.

It also assists in the calculation of practical situations like how much heat will be available for purposes like heating a pack as noted in the exercise.

Limiting Reactant

The limiting reactant is the substance that gets completely consumed first during a chemical reaction, thereby limiting the amount of product formed.

Determining it involves these steps:

• Convert all reactants from grams to moles using their molar masses.
• Use the balanced equation to compare the mole ratio of co-reactants.
• Identify which reactant runs out first, thereby halting the reaction.

In the specific scenario involving \(10.0 \mathrm{g}\) of iron and \(2.00 \mathrm{g}\) of oxygen, oxygen enables only a limited amount of reaction due to its lesser availability in mole terms:

- Iron is abundant comparatively, making oxygen the limiting reactant.
- Once you identify the limiting reactant, it's straightforward to use stoichiometry to calculate heat released or products formed based on its moles.

Molar Mass

Molar mass is the mass of one mole of a substance, expressed in grams per mole (g/mol). It provides a critical conversion factor for any calculation involving moles and grams.

Impacts in calculations:

• For iron, the molar mass is 55.845 g/mol. This helps translate grams to moles, such as \(1.00 \mathrm{g} \text{ of } \mathrm{Fe}\) converting to \(0.0179 \text{ moles}\).
• For oxygen, \(\mathrm{O}_2\), the molar mass is 32.00 g/mol. This enables the conversion of grams (like \(2.00 \mathrm{g}\)) into moles.

Having the correct molar mass ensures that stoichiometric calculations are accurate, ultimately impacting predictions of reactant consumption and energy changes.

Without precise molar mass conversions, calculating how much heat is released or how much product is formed would be prone to errors.