Problem 8 - Chemistry and Chemical Reactivity

Question

Iron ore is converted to iron metal in a reaction with carbon. $$ 2 \mathrm{Fe}_{2} \mathrm{O}_{3}(\mathrm{s})+3 \mathrm{C}(\mathrm{s}) \rightarrow 4 \mathrm{Fe}(\mathrm{s})+3 \mathrm{CO}_{2}(\mathrm{g}) $$ If 6.2 mol of \(\mathrm{Fe}_{2} \mathrm{O}_{3}(\mathrm{s})\) is used, what amount of \(\mathrm{C}(\mathrm{s})\) is needed, and what amounts of Fe and \(\mathrm{CO}_{2}\) are produced?

Step-by-Step Solution

Answer

9.3 mol C, 12.4 mol Fe, 9.3 mol CO2

Step 1: Write down the balanced equation

The chemical reaction is given as \( 2 \mathrm{Fe}_{2} \mathrm{O}_{3}(\mathrm{s}) + 3 \mathrm{C}(\mathrm{s}) \rightarrow 4 \mathrm{Fe}(\mathrm{s}) + 3 \mathrm{CO}_{2}(\mathrm{g}) \). This equation is balanced, meaning the number of each type of atom is equal on both sides of the equation.

Step 2: Analyze the stoichiometry of reactants

From the balanced equation, 2 moles of \( \mathrm{Fe}_{2} \mathrm{O}_{3} \) requires 3 moles of \( \mathrm{C} \). Use this relationship to determine the amount of \( \mathrm{C} \) needed.

Step 3: Calculate moles of carbon needed

For 2 moles of \( \mathrm{Fe}_{2} \mathrm{O}_{3} \), 3 moles of \( \mathrm{C} \) are needed. Therefore, for 6.2 moles of \( \mathrm{Fe}_{2} \mathrm{O}_{3} \), the calculation is: \[ 6.2 \text{ moles of } \mathrm{Fe}_{2} \mathrm{O}_{3} \times \frac{3 \text{ moles of } \mathrm{C}}{2 \text{ moles of } \mathrm{Fe}_{2} \mathrm{O}_{3}} = 9.3 \text{ moles of } \mathrm{C} \].

Step 4: Determine products using stoichiometry

The balanced equation indicates that 4 moles of \( \mathrm{Fe} \) and 3 moles of \( \mathrm{CO}_{2} \) are produced per 2 moles of \( \mathrm{Fe}_{2} \mathrm{O}_{3} \).

Step 5: Calculate moles of Fe produced

For 6.2 moles of \( \mathrm{Fe}_{2} \mathrm{O}_{3} \), calculate \( \mathrm{Fe} \) produced: \[ 6.2 \text{ moles of } \mathrm{Fe}_{2} \mathrm{O}_{3} \times \frac{4 \text{ moles of } \mathrm{Fe}}{2 \text{ moles of } \mathrm{Fe}_{2} \mathrm{O}_{3}} = 12.4 \text{ moles of } \mathrm{Fe} \].

Step 6: Calculate moles of CO2 produced

Similarly, calculate \( \mathrm{CO}_{2} \) produced: \[ 6.2 \text{ moles of } \mathrm{Fe}_{2} \mathrm{O}_{3} \times \frac{3 \text{ moles of } \mathrm{CO}_{2}}{2 \text{ moles of } \mathrm{Fe}_{2} \mathrm{O}_{3}} = 9.3 \text{ moles of } \mathrm{CO}_{2} \].

Key Concepts

Balanced chemical equation

A balanced chemical equation is essential in understanding chemical reactions. It accurately represents the quantities of reactants and products involved in a chemical reaction. In this context, **balancing** a chemical equation ensures that the number of atoms of each element is the same on both sides of the equation. This preserves the law of conservation of mass.

In the provided exercise, the chemical equation given is:

• \( 2 \mathrm{Fe}_{2} \mathrm{O}_{3}(\mathrm{s}) + 3 \mathrm{C}(\mathrm{s}) \rightarrow 4 \mathrm{Fe}(\mathrm{s}) + 3 \mathrm{CO}_{2}(\mathrm{g}) \)

This equation is balanced. For every element like iron (Fe), carbon (C), and oxygen (O), the atoms on the reactant side equal the atoms on the product side:

• Iron (Fe): 4 in \(2 \mathrm{Fe}_{2} \mathrm{O}_{3}\) results in 4 in \(4 \mathrm{Fe}\).
• Carbon (C): 3 in \(3 \mathrm{C}\) results in 3 in \(3 \mathrm{CO}_{2}\).
• Oxygen (O): 6 in \(2 \mathrm{Fe}_{2} \mathrm{O}_{3}\) results in 6 in \(3 \mathrm{CO}_{2}\).

Balancing can sometimes be tricky, but keeping track of each type of atom can make it more straightforward.

Mole concept

The mole concept is a fundamental principle in chemistry. It's a way to express amounts of a substance that allow chemists to work with atoms and molecules more conveniently. **One mole** contains Avogadro's number of particles, which is \(6.022 \times 10^{23}\) entities. This makes it useful for converting between atoms/molecules and grams.

In stoichiometry, the mole concept helps to solve problems by relating mass and quantity of a particular substance involved in a chemical equation. For instance, from our balanced equation, we understand that:

• 2 moles of \( \mathrm{Fe}_{2} \mathrm{O}_{3}\) use 3 moles of carbon.
• It also produces 4 moles of iron and 3 moles of \( \mathrm{CO}_{2}\).

Using this information, if we start with 6.2 moles of \( \mathrm{Fe}_{2} \mathrm{O}_{3}\), we utilize moles of carbon based on these proportions. Simply put, the mole concept serves as a bridge between the microscopic world of atoms and the macroscopic observable quantities we manipulate in the lab.

Chemical reactions

Chemical reactions are processes in which substances combine or change to form new substances. **Reactants** are the starting substances and **products** are the substances formed by the reaction. Understanding the nature of chemical reactions involves comprehending how these transformations occur.

In the example given, iron oxide (\( \mathrm{Fe}_{2} \mathrm{O}_{3}\)) reacts with carbon (\( \mathrm{C}\)) to produce iron (\( \mathrm{Fe}\)) and carbon dioxide (\( \mathrm{CO}_{2}\)). This transformation involves breaking bonds in the reactants and forming new bonds to form the products. Here’s what happens:

• The oxygen atoms bonded with iron in \( \mathrm{Fe}_{2} \mathrm{O}_{3}\) are captured by carbon to form \( \mathrm{CO}_{2}\).
• This frees the iron atoms as elemental iron, \( \mathrm{Fe}\).

Chemical reactions can release or absorb energy, typically in the form of heat. This specific reaction is exothermic as it involves the reduction of an oxide by carbon. Understanding these processes is key in fields like metallurgy and environmental science, where control over such reactions is crucial.