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Chapter 3: Problem 102

One of the reactions that occurs in a blast furnace, where iron ore is converted to cast iron, is $$ \mathrm{Fe}_{2} \mathrm{O}_{3}+3 \mathrm{CO} \longrightarrow 2 \mathrm{Fe}+3 \mathrm{CO}_{2} $$ Suppose that \(1.64 \times 10^{3} \mathrm{~kg}\) of Fe are obtained from a \(2.62 \times 10^{3}-\mathrm{kg}\) sample of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\). Assuming that the reaction goes to completion, what is the percent purity of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) in the original sample?

Short Answer

Expert verified
The percent purity of \(\mathrm{Fe}_2\mathrm{O}_3\) in the original sample is approximately 89.7%

Step by step solution

01

Understand the stoichiometry of the reaction.

In the given chemical reaction, one mole of ferric oxide (\(\mathrm{Fe}_2\mathrm{O}_3\)) reacts with three moles of carbon monoxide (CO) to produce two moles of iron (Fe) and three moles of carbon dioxide (CO2). Therefore, theoretically, the mass of \(Fe_2O_3\) needed to produce the obtained amount of Fe can be calculated by applying the law of conservation of mass and the stoichiometry of the reaction.
02

Calculate the moles of \(Fe\) obtained.

The molar mass of iron (\(Fe\)) is approximately 55.85 g/mol. So, the moles of \(Fe\) obtained in the reaction can be calculated by the formula: Number of moles = Mass (g) / Molar mass (g/mol). In this case, with a mass of \(1.64 \times 10^{3} \mathrm{~kg}\) or \(1.64 \times 10^{6} \mathrm{g}\), the moles of \(Fe\) would be \(1.64 \times 10^{6} g / 55.85 g/mol \approx 2.94 \times 10^{4} \mathrm{mol}\).
03

Calculate the theoretical mass of \(Fe_2O_3\).

From the stoichiometry of the reaction, we know that 1 mole of \(Fe_2O_3\) produces 2 moles of \(Fe\). Therefore, the moles of \(Fe_2O_3\) theoretically required to produce the obtained moles of \(Fe\) can be calculated as follows: Moles of \(Fe_2O_3\) required = \(2.94 \times 10^{4} \mathrm{mol} / 2 = 1.47 \times 10^{4} \mathrm{mol}\) . The molar mass of \(Fe_2O_3\) is approximately 159.69 g/mol. Therefore, the theoretical mass of \(Fe_2O_3\) required can be calculated as follows: Theoretical mass = Moles of \(Fe_2O_3\) * Molar mass of \(Fe_2O_3\), or \(1.47 \times 10^{4} \mathrm{mol} * 159.69 g/mol \approx 2.35 \times 10^{6} g\) or \(2.35 \times 10^{3} \mathrm{kg}\).
04

Calculate the percentage purity.

The percent purity of \(Fe_2O_3\) in the original sample can be calculated by the formula: Percent purity = (Theoretical mass / Original sample mass) * 100%. Substituting the given values, we find that the percent purity of the sample is: Percent purity = \((2.35 \times 10^{3} \mathrm{kg} / 2.62 \times 10^{3} \mathrm{kg}) * 100% \approx 89.7%\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Stoichiometry
Stoichiometry is a branch of chemistry that deals with the quantitative relationships between the reactants and products in a chemical reaction. It involves using balanced chemical equations to calculate the masses of reactants and products, ensuring that the atoms are conserved during the reaction.

In our exercise, stoichiometry is used to understand that one mole of ferric oxide (\(\mathrm{Fe}_2\mathrm{O}_3\)) will react with three moles of carbon monoxide (CO) to produce two moles of iron (Fe) and three moles of carbon dioxide (CO2). Knowing this ratio is central to predicting the mass of iron that can be yielded from a given mass of ferric oxide.
Chemical Reaction
A chemical reaction is a process where substances, known as reactants, transform into different substances, called products. Chemical reactions abide by the law of conservation of mass, meaning that the mass of the reactants must be equal to the mass of the products.

In the depicted blast furnace reaction, iron ore (\(\mathrm{Fe}_2\mathrm{O}_3\)) reacts with carbon monoxide (CO) to form cast iron (Fe) and carbon dioxide (CO2). Understanding the process allows us to compute how efficient the reaction is—quantified in our case as percent purity.
Law of Conservation of Mass
The law of conservation of mass states that matter is neither created nor destroyed in a chemical reaction. It is this principle that underpins stoichiometry and allows us to predict the outcome of chemical reactions.

In the exercise, we apply this law to ascertain the mass of pure \(\mathrm{Fe}_2\mathrm{O}_3\) required to produce the known mass of iron. The consistency in mass between reactants and products is foundational to finding the percent purity of our ferric oxide sample.
Mole Concept
The mole concept is fundamental in chemistry for dealing with large quantities of atoms and molecules. A mole corresponds to Avogadro's number (\(6.022 \times 10^{23}\)) of particles, be they atoms, ions, or molecules.

In tackling the exercise, we calculate the moles of iron produced based on its molar mass and then find the moles of \(\mathrm{Fe}_2\mathrm{O}_3\) needed. By understanding the mole concept, we can quantitatively connect the mass of a substance to the amount of matter it contains, which is essential for the percent purity calculation.

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Most popular questions from this chapter

The empirical formula of a compound is \(\mathrm{CH}\). If the molar mass of this compound is about \(78 \mathrm{~g}\), what is its molecular formula?

Consider the reaction $$ \mathrm{MnO}_{2}+4 \mathrm{HCl} \longrightarrow \mathrm{MnCl}_{2}+\mathrm{Cl}_{2}+2 \mathrm{H}_{2} \mathrm{O} $$ If 0.86 mole of \(\mathrm{MnO}_{2}\) and \(48.2 \mathrm{~g}\) of \(\mathrm{HCl}\) react, which reagent will be used up first? How many grams of \(\mathrm{Cl}_{2}\) will be produced?

What is the mass in grams of a single atom of each of the following elements? (a) As, (b) Ni.

What are the empirical formulas of the compounds with the following compositions? (a) 40.1 percent \(\mathrm{C}\) 6.6 percent \(\mathrm{H}, 53.3\) percent \(\mathrm{O}\) (b) 18.4 percent \(\mathrm{C}\) 21.5 percent \(\mathrm{N}, 60.1\) percent \(\mathrm{K}\)

A compound X contains 63.3 percent manganese (Mn) and 36.7 percent O by mass. When \(X\) is heated, oxygen gas is evolved and a new compound \(\mathrm{Y}\) containing 72.0 percent \(\mathrm{Mn}\) and 28.0 percent \(\mathrm{O}\) is formed. (a) Determine the empirical formulas of \(X\) and Y. (b) Write a balanced equation for the conversion of \(X\) to \(Y\).
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