Problem 108 - General Chemistry: Principles and Modern Applications

Question

A 5.585-kg sample of iron (Fe) contains (a) \(10.0 \mathrm{mol} \mathrm{Fe}\) (b) twice as many atoms as does \(600.6 \mathrm{g} \mathrm{C}\) (c) 10 times as many atoms as does \(52.00 \mathrm{g} \mathrm{Cr}\) (d) \(6.022 \times 10^{24}\) atoms

Step-by-Step Solution

Answer

To know if the statements are correct, the calculations explained in steps 1 through 4 have to be done. That is, conversion of kg to moles, and then moles to number of atoms. Then compare these calculated values with the given quantities.

Step 1: Convert mass to moles

First, start by converting the mass of iron to moles. The molar mass of iron is 55.85 g/mol. But, the mass we have is in kilograms, so we first need to convert it to grams. Then, the moles of iron can be calculated by dividing the mass of iron by its molar mass. This will give us the number of moles present in the given sample of iron.

Step 2: Compare moles

For part a, simply equate the moles of iron calculated in step 1 with 10.0 moles as specified in the exercise.

Step 3: Compare atoms

For part b and c, calculate the number of atoms by multiplying the moles of iron (from step 1) by Avogadro's number (\(6.022 \times 10^{23} \) atoms/mol), then compare this with the number of atoms in the given amount of carbon and chromium. For carbon, the number of atoms can be calculated by dividing its given mass by its molar mass (12.01 g/mol) and then multiplying by Avogadro's number. For chromium, make a similar calculation using its molar mass (52.00 g/mol).

Step 4: Compare atoms

In part d, do a direct comparison of the total number of atoms calculated for the iron sample with \(6.022 \times 10^{24}\) atoms, as given in the problem.

Key Concepts

Avogadro's Number

Avogadro's Number is a fundamental constant in chemistry. It represents the number of atoms, ions, or molecules in one mole of a substance. To put it simply, if you have one mole of any element or compound, you have exactly:

• 6.022 x 10²³ particles
• This applies to atoms for elements, molecules for molecules, and even to more complex structures like ions in a salt.

This number helps us relate the atomic scale to the macroscopic world. For example, in the given exercise, you calculate the moles of iron and use Avogadro's Number to determine how many individual iron atoms are in the sample. Knowing Avogadro's Number makes it possible to transition from talking about vast numbers of microscopic particles to a more manageable molar amount, the mole, which is a fundamental unit in chemistry.

Moles and Atoms

Moles are a convenient unit in chemistry for dealing with very large numbers of atoms or molecules. The concept of the mole allows chemists to "count" atoms by weighing them, considering the atomic mass of each element. Here's how to work with moles effectively:

• First, identify the molar mass of the element from the periodic table, measured in grams per mole.
• Convert the mass of your sample (typically in grams) into moles by dividing by the molar mass.

In this exercise, we had to convert the mass of iron and compare it with moles of other substances. By calculating the number of moles, and then using Avogadro's number, you can determine the number of atoms present in a sample. It bridges the gap between the mass you measure and the number of atoms involved, allowing for direct comparison between different substances, like iron, carbon, and chromium in this exercise.

Elemental Iron

Elemental Iron (Fe) is often used in chemistry examples due to its significance in both industry and biological systems. A few important things to remember about iron include:

• Its molar mass is 55.85 g/mol, which is vital for converting between grams and moles.
• Iron is a key component of hemoglobin, which is why it's crucial in biology for oxygen transport.
• It's used extensively in construction and manufacturing, as steel is an alloy dominated by iron.

In the exercise, you start with a mass of 5.585 kg of iron. The first step is turning this mass into moles, using the molar mass as a conversion factor. After calculating the moles, determining the number of iron atoms involves using Avogadro's number. Understanding these conversions is essential for comparing the iron atoms to those in carbon and chromium, seeing how they interact, and grasping their scale. By mastering these fundamentals, you'll be well-equipped for countless applications in both labs and real-world scenarios.