Question

Calculate the number of grams of iron that contain the same number of atoms as \(2.24 \mathrm{g}\) of cobalt.

Short answer

The mass of iron that contains the same number of atoms as $2.24\ \mathrm{g}$ of cobalt is \(2.12\ \mathrm{g}\).

Step-by-step solution

Find the molar mass of cobalt and iron

Check the periodic table and find the molar mass of Co (cobalt) and Fe (iron). The molar mass of cobalt is 58.93 g/mol and that of iron is 55.85 g/mol.

Calculate the moles of cobalt

Use the given mass of cobalt (2.24 g) and its molar mass (58.93 g/mol) to find the number of moles of cobalt: Moles of cobalt = Mass of cobalt / Molar mass of cobalt Moles of cobalt = \( \frac{2.24\ \mathrm{g}}{58.93\ \mathrm{g/mol}} \) Moles of cobalt = 0.038 mol

Calculate the number of atoms in the given amount of cobalt

To find the number of atoms in the given mass of cobalt, multiply the number of moles by Avogadro's number (6.022 x 10^23 atoms/mol): Number of atoms = Moles of cobalt × Avogadro's number Number of atoms = 0.038 mol × \( 6.022 \times 10^{23}\ \mathrm{atoms/mol} \) Number of atoms = \( 2.29 \times 10^{22}\ \mathrm{atoms} \)

Calculate the number of moles of iron that contains the same number of atoms

Divide the number of atoms calculated in step 3 by Avogadro's number to find the number of moles of iron: Moles of iron = \( \frac{2.29 \times 10^{22}\ \mathrm{atoms}}{6.022 \times 10^{23}\ \mathrm{atoms/mol}} \) Moles of iron = 0.038 mol

Calculate the mass of iron that contains the same number of atoms

Finally, use the molar mass of iron (55.85 g/mol) and the number of moles of iron to find the mass of iron with the same number of atoms: Mass of iron = Moles of iron × Molar mass of iron Mass of iron = 0.038 mol × 55.85 g/mol Mass of iron = 2.12 g Thus, 2.12 grams of iron contain the same number of atoms as 2.24 grams of cobalt.

Key concepts

Molar Mass

The concept of molar mass is one of the cornerstones of stoichiometry in chemistry. Molar mass refers to the mass of one mole of a substance, typically expressed in grams per mole (g/mol). This figure is equivalent to the atomic or molecular weight of the substance, and it allows chemists to convert between the mass of a substance and the number of moles.
For instance, to find the molar mass of an element, you would look at the atomic weight of the element as listed on the periodic table. For cobalt, the molar mass is 58.93 g/mol, while iron has a molar mass of 55.85 g/mol.

• The molar mass is crucial when calculating the number of atoms or molecules in a given mass of a substance.
• The use of molar mass ensures that calculations are consistent and accurate when determining the proportions of elements in reactions.

Understanding and applying molar mass helps in performing stoichiometric calculations, such as finding the number of grams of a substance containing a specified number of atoms or molecules.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry with a value of approximately 6.022 x 10^23. This number represents the quantity of atoms or molecules in one mole of a substance. Named after the Italian scientist Amedeo Avogadro, this constant is essential when dealing with particles at the atomic or molecular level.
By using Avogadro's number, we can convert between the number of moles and the total number of atoms or molecules. For example, to find out how many atoms are in a particular mass of cobalt, you can multiply the number of moles by Avogadro's number:

• This allows you to calculate the exact number of atoms in a sample, which in our case for cobalt was found to be \( 2.29 \times 10^{22} \) atoms.
• Understanding Avogadro's number is crucial for the study of chemical reactions, where the exact number of participating particles is often necessary.

By knowing this constant, it becomes straightforward to translate between atomic mass and real-world quantities.

Moles Calculation

Moles are a fundamental unit in chemistry used to measure the amount of a substance. Calculating the number of moles involves dividing the mass of the substance by its molar mass. This conversion is necessary because reactions occur at the molecular level, and using moles aligns calculations with molecular and atomic quantities.
In our specific example with cobalt and iron, we started with 2.24 g of cobalt:

• The number of moles of cobalt was calculated by dividing the mass (2.24 g) by its molar mass (58.93 g/mol), resulting in 0.038 mol of cobalt.
• Doing the same for iron ensures the same number of atoms as the cobalt sample, affirming that 0.038 mol of iron equates to \(2.12\) grams, both representing the same atomic count due to their respective molar masses.

This calculation process allows chemists to predict and measure the outcome of chemical reactions accurately.