Problem 23 - General Chemistry: Principles and Modern Applications

Question

Iron ore is impure \(\mathrm{Fe}_{2} \mathrm{O}_{3} .\) When \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) is heated with an excess of carbon (coke), metallic iron and carbon monoxide gas are produced. From a sample of ore weighing \(938 \mathrm{kg}, 523 \mathrm{kg}\) of pure iron is obtained. What is the mass percent \(\mathrm{Fe}_{2} \mathrm{O}_{3},\) by mass, in the ore sample, assuming that none of the impurities contain Fe?

Step-by-Step Solution

Answer

The mass percent of \(Fe_2O_3\) in the ore sample is about 72.39%.

Step 1: Calculate the amount of \(Fe_2O_3\) that could form from 523 kg Fe

Given the molar mass of Fe is about 55.85 g/mol and \(Fe_2O_3\) is about 159.69 g/mol, the moles of Fe and \(Fe_2O_3\) can be calculated respectively with the formula: \( \text{No.of moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \). After calculating the moles of Fe, multiple it by 1.5 to get the moles of \(Fe_2O_3\), because 2 moles of Fe produce 1 mole of \(Fe_2O_3\). Lastly, convert the moles of \(Fe_2O_3\) back to mass in kg.

Step 2: Calculate mass percent of \(Fe_2O_3\)

Next, calculate the mass percent of \(Fe_2O_3\) by dividing the mass of \(Fe_2O_3\) obtained in step 1 by the total mass of the ore sample (938 kg), and then multiply by 100. The result is the mass percent of \(Fe_2O_3\) in the ore sample.

Key Concepts

Molar Mass

Molar mass is a fundamental concept in chemistry that helps us understand the mass of one mole of a given substance.
It is expressed in grams per mole (g/mol). Knowing the molar mass of a compound allows chemists to convert between the mass of a substance and the number of moles of the substance.

To find the molar mass of a specific compound, you simply add up the molar masses of all the atoms that are present in its chemical formula. Take iron(III) oxide, \(\mathrm{Fe}_{2}\mathrm{O}_{3}\), for example:

• Iron (Fe) has a molar mass of about 55.85 g/mol, and
• Oxygen (O) has a molar mass of approximately 16.00 g/mol.

Adding these up, you get:
\[2(55.85) + 3(16.00) = 159.69\ \text{g/mol}\]

By understanding molar mass, you can better interpret chemical reactions and calculations, turning weight measurements into more usable mole-based data.

Iron Ore

Iron ore is a natural substance found in the Earth's crust. It contains iron in various forms, often mixed with other elements or compounds.
Commonly, iron ore consists of iron oxides like hematite (\(\mathrm{Fe}_{2}\mathrm{O}_{3}\)) and magnetite (\(\mathrm{Fe}_{3}\mathrm{O}_{4}\)), which are used in the extraction of iron.

In practical terms, the iron ore mined from the ground is not pure. It typically includes several impurities such as rocks and other minerals. Therefore, when processing iron ore, it's essential to account for these impurities to calculate the actual iron content.

In our specific example, iron ore is used in a reaction where iron oxide reacts with carbon to produce pure iron. Understanding the composition of iron ore and removing impurities is vital for yielding the maximum quantity of metallic iron.

Mass Percent

Mass percent is a way to express the concentration of a component in a mixture as a percentage of the total mixture mass.
It is calculated by dividing the mass of the component by the total mass of the mixture, then multiplying by 100.

Using this concept, chemists can determine the proportion of a given component within a sample. In the exercise:

• First, the mass of \(\mathrm{Fe}_{2}\mathrm{O}_{3}\) that can form the obtained iron is calculated.
• Then, this is compared to the total mass of the ore sample (938 kg) to establish the mass percent.

The formula used here is:
\[\text{Mass percent} = \left(\frac{\text{mass of } \mathrm{Fe}_{2}\mathrm{O}_{3}}{\text{total mass of ore}}\right) \times 100\]

This calculation helps understand the quality and richness of the ore in terms of its valuable iron content.

Chemical Reactions

Chemical reactions describe the process where substances transform into new substances through the breaking and forming of bonds.
In the case of iron extraction, a specific chemical reaction involves iron(III) oxide reacting with carbon (often in the form of coke) to produce metallic iron and carbon monoxide gas.

The reaction can be simplified and balanced as follows:

• \[ \mathrm{Fe}_{2}\mathrm{O}_{3} + 3\mathrm{C} \rightarrow 2\mathrm{Fe} + 3\mathrm{CO} \]

This equation shows the conservation of mass, where the number of atoms on both sides of the reaction remains equal.

Understanding these reactions provides significant insights into industrial processes like metallurgy, helping efficiently convert raw minerals into valuable materials.
Every step—from determining reactants to calculating formulas—relies on precise chemical knowledge to achieve desired outcomes.