Question

An iron bar weighed \(664 \mathrm{~g}\). After the bar had been standing in moist air for a month, exactly one-eighth of the iron turned to rust \(\left(\mathrm{Fe}_{2} \mathrm{O}_{3}\right)\). Calculate the final mass of the iron bar and rust.

Short answer

The final mass is 699.57 g.

Step-by-step solution

Understand the Problem

We have an iron bar weighing 664 g, and one-eighth of it has turned into rust, which is iron oxide, noted as \(\mathrm{Fe}_2\mathrm{O}_3\). Our task is to find the final mass of the iron bar including the rust.

Calculate the Mass of Iron that Turned to Rust

First, determine the mass of the iron that turned to rust by dividing the original mass of the iron bar by 8. \[ \text{Mass of iron turned to rust} = \frac{664}{8} = 83 \text{ g} \]

Determine the Mass of Rust Formed

Realize that rust forms from iron and oxygen. The molar mass of iron (Fe) is 56 g/mol and \(\mathrm{Fe}_2\mathrm{O}_3\) is (56*2 + 16*3) = 160 g/mol. Use stoichiometry to find the mass of rust from 83 g of iron. If 1 mole of iron (56 g) converts to part of a mole of rust, 2 moles of iron (112 g) will yield 160 g of \(\mathrm{Fe}_2\mathrm{O}_3\). The proportion is: \[ \frac{160 ext{ g rust}}{112 ext{ g iron}} = \frac{x ext{ g rust}}{83 ext{ g iron}} \] Solve for \(x\): \[ x = \frac{160}{112} \times 83 = 118.57 \text{ g of rust} \]

Calculate the Final Mass

Add the mass of the remaining iron and the rust. The remaining iron is 664 g - 83 g = 581 g. Thus the total mass after rusting is: \[ \text{Total mass} = 581 ext{ g (remaining iron)} + 118.57 ext{ g (rust)} = 699.57 ext{ g} \]

Conclusion

The final mass of the iron bar, including the rust, is 699.57 g. This accounts for the original iron that did not rust and the additional mass from the oxygen that combined with some of the iron to form rust.

Key concepts

Understanding Chemical Reactions

Chemical reactions are processes in which reactants transform into products. Here, the chemical reaction in focus is the rusting of iron. Rusting occurs when iron reacts with oxygen in the presence of moisture to form iron oxide (\(\mathrm{Fe}_2\mathrm{O}_3\)). This process involves the combination of iron and oxygen atoms to form a new compound.
Chemical reactions are often represented by balanced chemical equations, which describe the reactants and products involved. For the rusting of iron, the simplified equation is:\[4\mathrm{Fe} + 3\mathrm{O}_2 \rightarrow 2\mathrm{Fe}_2\mathrm{O}_3\]Meaning, four iron atoms react with three oxygen molecules to yield two units of iron oxide. This helps us understand how much reactant is needed and what is produced.

Understanding Molar Mass

Molar mass is the mass of one mole of a substance, usually expressed in grams per mole. This concept is essential in stoichiometry to test relationships in a chemical reaction.
For example:

• The molar mass of iron (Fe) is 56 g/mol.
• The molar mass of diatomic oxygen (\(\mathrm{O}_2\)) is 32 g/mol (since 16 g/mol per oxygen atom).
• The molar mass of iron oxide (\(\mathrm{Fe}_2\mathrm{O}_3\)) is 160 g/mol, calculated as (56 g/mol \(\times 2\) for iron) plus (16 g/mol \(\times 3\) for oxygen).

Molar mass allows us to convert between grams and moles, facilitating the determination of how much product is formed from given quantities of reactants.

What is Iron Oxide?

Iron oxide, specifically \(\mathrm{Fe}_2\mathrm{O}_3\), is a reddish-brown compound formed from iron and oxygen. It's commonly known as rust and typically forms when iron is exposed to air and moisture over time.
Key Characteristics:

• Composed of two iron atoms and three oxygen atoms.
• Inefficient as a protective layer for iron since it can flake off, exposing new layers to corrosion.
• Used industrially as a pigment and in various applications such as in ceramics and metallurgy.

Understanding iron oxide is crucial in addressing how metals like iron degrade and are preserved over time.

Performing Mass Calculations

Mass calculations are an essential part of working through chemical reactions, determining how much of each substance is involved. In our exercise, we are concerned with the initial, rusted, and final mass of the iron.
Steps in Mass Calculation:

• Start by determining what portion of the iron bar rusted, here one-eighth turning to rust equals 83 g.
• Using stoichiometry and the known molar masses, calculate the mass of iron oxide formed from the rusted portion, here found to be 118.57 g.
• Add the remaining iron to this rust mass for the total mass, resulting in 699.57 g as the final mass.

These steps illustrate the concept of conservation of mass, where mass is neither created nor destroyed but redistributed in chemical reactions.