Question

The weight of one molecule of a compound \(\mathrm{C}_{60} \mathrm{H}_{122}\) is a. \(1.2 \times 10^{-20} \mathrm{gm}\) b. \(1.4 \times 10^{-21} \mathrm{gm}\) c. \(5.025 \times 10^{25} \mathrm{gm}\) d. \(6.023 \times 10^{23} \mathrm{gm}\)

Short answer

The weight of one molecule of \(\mathrm{C}_{60} \mathrm{H}_{122}\) is \(1.4 \times 10^{-21} \, \text{g}\).

Step-by-step solution

Calculate the Molar Mass

First, calculate the molar mass of the compound \(\mathrm{C}_{60} \mathrm{H}_{122}\). The atomic mass of carbon (C) is approximately 12 g/mol and that of hydrogen (H) is approximately 1 g/mol. Therefore, the molar mass is calculated as follows: \(60 \times 12 + 122 \times 1 = 720 + 122 = 842 \, \text{g/mol}\).

Use Avogadro's Number

To find the weight of one molecule, use Avogadro's number, which is \(6.022 \times 10^{23}\) molecules/mol. The weight of one molecule in grams is obtained by dividing the molar mass by Avogadro's number: \(\frac{842}{6.022 \times 10^{23}}\).

Calculate the Weight of One Molecule

Perform the division: \(\frac{842}{6.022 \times 10^{23}} \approx 1.398 \times 10^{-21} \, \text{g}\). This value is close to \(1.4 \times 10^{-21} \, \text{g}\) from the choices given.

Key concepts

Molar Mass

Molar mass is essentially the mass of one mole of a given substance. It's calculated by summing up the atomic masses of all the atoms in a molecular formula. In the exercise, we dealt with an organic compound, C\( _{60} \)H\( _{122} \). For this compound, the molar mass is determined by multiplying the atomic masses of each type of atom present by the number of those atoms, and then adding everything up:

• Carbon (C) has an atomic mass of approximately 12 g/mol, and there are 60 carbon atoms in the molecule.
• Hydrogen (H) has an atomic mass of approximately 1 g/mol, with 122 hydrogen atoms in the compound.

So, the molar mass of C\( _{60} \)H\( _{122} \) is calculated by \[ 60 \times 12 + 122 \times 1 = 720 + 122 = 842 \, \text{g/mol} \] Understanding molar mass is crucial in chemistry because it allows us to translate between grams and moles, giving insight into the quantity of material we're handling.

Avogadro's Number

Avogadro's number is a fundamental constant used to convert between moles of a substance and number of molecules or atoms in that substance. Its value is about \(6.022 \times 10^{23}\) per mole, meaning one mole of any substance contains exactly \(6.022 \times 10^{23}\) entities, like atoms or molecules.In the context of the exercise, knowing Avogadro's number helps us find the weight of a single molecule of the organic compound C\( _{60} \)H\( _{122} \). Once we've calculated the molar mass as 842 g/mol, we divide this by Avogadro's number to determine the mass of one molecule:\[ \frac{842}{6.022 \times 10^{23}} \approx 1.398 \times 10^{-21} \, \text{g} \] This division allows us to effectively break down the large-scale measurement of molar mass into the individual molecular scale, useful for detailed atomic calculations.

Atomic Mass

Atomic mass is the mass of a single atom and is usually expressed in atomic mass units (amu). It represents the average mass of atoms of an element, calculated using the relative abundance of isotopes in a naturally occurring element.For our exercise, let's consider carbon and hydrogen:

• Carbon has an atomic mass of about 12 amu because its most common isotope, \(^{12}\)C, is prevalent.
• Hydrogen is simpler, with an atomic mass of around 1 amu, as \(^{1}\)H is the more common isotope.

When calculating the molar mass of a compound like \( \text{C}_{60} \text{H}_{122} \), we use these atomic masses as building blocks. By weighing individual atoms in atomic mass units, scientists can derive the larger weights of molecules or compounds in practical measurements such as grams, making this knowledge foundational in the field of chemistry. Atomic mass contributes to forming a bridge between microscopic details and macroscopic measurements.