Question

Calculate the mass in grams for each of the following: (a) \(1.21 \times 10^{24}\) atoms krypton, \(\mathrm{Kr}\) (b) \(6.33 \times 10^{22}\) molecules of dinitrogen oxide, \(\mathrm{N}_{2} \mathrm{O}\) (c) \(4.17 \times 10^{21}\) formula units of magnesium perchlorate, \(\mathrm{Mg}\left(\mathrm{ClO}_{4}\right)_{2}\)

Short answer

(a) 168.44 g of Kr, (b) 4.62 g of \(\mathrm{N}_{2} \mathrm{O}\), (c) 1.54 g of \(\mathrm{Mg(ClO}_{4})_{2}\).

Step-by-step solution

Understand Avogadro's number

Avogadro's number, which is approximately \( 6.022 \times 10^{23} \), represents the number of units (atoms, molecules, etc.) in one mole of any substance. This number will be used to convert between the number of atoms or molecules and moles.

(a) Convert atoms of krypton to moles

For (a), you have \(1.21 \times 10^{24}\) atoms of krypton. To find the moles: \[ \text{Moles of Kr} = \frac{1.21 \times 10^{24} \text{ atoms}}{6.022 \times 10^{23} \text{ atoms/mol}} \approx 2.01 \text{ mol} \]

(a) Find the mass of krypton in grams

The molar mass of krypton (Kr) is approximately 83.80 g/mol. Thus, the mass is: \[ \text{Mass of Kr} = 2.01 \text{ mol} \times 83.80 \text{ g/mol} \approx 168.44 \text{ g} \]

(b) Convert molecules of dinitrogen oxide to moles

For (b), you have \(6.33 \times 10^{22}\) molecules of \(\mathrm{N}_{2} \mathrm{O}\). To find the moles: \[ \text{Moles of } \mathrm{N}_{2} \mathrm{O} = \frac{6.33 \times 10^{22} \text{ molecules}}{6.022 \times 10^{23} \text{ molecules/mol}} \approx 0.105 \text{ mol} \]

(b) Find the mass of dinitrogen oxide in grams

The molar mass of \(\mathrm{N}_{2} \mathrm{O}\) is 44.01 g/mol. Thus, the mass is: \[ \text{Mass of } \mathrm{N}_{2} \mathrm{O} = 0.105 \text{ mol} \times 44.01 \text{ g/mol} \approx 4.62 \text{ g} \]

(c) Convert formula units of magnesium perchlorate to moles

For (c), you have \(4.17 \times 10^{21}\) formula units of \(\mathrm{Mg(ClO}_{4})_{2}\). To find the moles: \[ \text{Moles of } \mathrm{Mg(ClO}_{4})_{2} = \frac{4.17 \times 10^{21} \text{ formula units}}{6.022 \times 10^{23} \text{ formula units/mol}} \approx 0.00692 \text{ mol} \]

(c) Find the mass of magnesium perchlorate in grams

The molar mass of \(\mathrm{Mg(ClO}_{4})_{2}\) is 223.21 g/mol. Thus, the mass is: \[ \text{Mass of } \mathrm{Mg(ClO}_{4})_{2} = 0.00692 \text{ mol} \times 223.21 \text{ g/mol} \approx 1.54 \text{ g} \]

Key concepts

Molar Mass

The concept of molar mass is fundamental in chemistry, as it provides the link between an amount of substance and its mass in grams. The molar mass is defined as the mass of one mole of a given substance, be it an element or a compound, and is typically expressed in grams per mole (g/mol). For instance, the molar mass of krypton, derived from the periodic table, is approximately 83.80 g/mol, meaning that one mole of krypton weighs 83.80 grams.

A few key points to understand about molar mass include:

• It is specific to each element or compound, reflecting the sum of the atomic masses of all atoms present in a molecule or formula unit.
• For compounds, the molar mass is found by adding together the molar masses of the constituent elements. For example, dinitrogen oxide (\( \mathrm{N}_{2} \mathrm{O} \)) has a molar mass of 44.01 g/mol, calculated from the molar masses of two nitrogen atoms and one oxygen atom.
• Being able to convert between grams and moles using the molar mass is essential for conducting chemical calculations such as stoichiometry.

Understanding molar mass allows chemists and students alike to work backward from a measured mass to ascertain the number of moles, and vice versa, facilitating a deeper understanding of material quantities in chemical reactions.

Moles Conversion

Moles serve as a bridge between the microscopic world of atoms and molecules and the macroscopic world we experience. To perform chemical calculations, such as determining the mass of a substance from the number of its atoms or molecules, you first need to convert the count of individual particles to moles. This conversion uses Avogadro's number, which is approximately \( 6.022 \times 10^{23} \). This is the number of particles in one mole of substance, providing a constant ratio to use in calculations.

Key aspects of moles conversion include:

• To convert atoms, molecules, or formula units to moles, you divide the quantity by Avogadro's number.
• In practice, this means if you have \(1.21 \times 10^{24}\) atoms of krypton, you would calculate the moles as \(\frac{1.21 \times 10^{24}}{6.022 \times 10^{23}} \approx 2.01\) moles.
• This conversion is crucial since most chemical reactions are expressed in terms of moles, not masses.

Mastering this conversion process is critical in solving chemical calculations effectively, as it allows you to convert between quantities of particles and more practical measurements like grams.

Chemical Calculations

Conducting chemical calculations often requires integrating concepts such as molar mass and moles conversion. These calculations help determine the mass of a substance required or produced in a chemical reaction, enhancing our understanding and control over chemical processes.

Here's how chemical calculations commonly work:

• First, convert the number of molecules or atoms to moles using Avogadro's number.
• Next, use the molar mass to convert moles to grams, providing a tangible measure of substance. For example, to find the mass of dinitrogen oxide from \(0.105\) moles, you multiply by its molar mass \(44.01\) g/mol to get approximately \(4.62\) grams.
• This two-step process is applicable across different substances and reactions, serving as a foundational approach in chemistry for determining quantities.

Discerning these relationships is essential for practical applications such as predicting product yields in reactions, optimizing industrial processes, and conducting lab experiments.