To determine the mass of [tex]\(1.81 \times 10^{23}\)[/tex] molecules of nitrogen ([tex]\(N_2\)[/tex]), we need to go through a series of steps involving Avogadro's number and the molecular mass of nitrogen.
1. Identify Key Values:
- Avogadro's Number: [tex]\(6.022 \times 10^{23}\)[/tex] molecules/mol. This is the number of molecules contained in one mole of a substance.
- Molecular Mass of Nitrogen ([tex]\(N_2\)[/tex]): 28.02 grams per mole. This is the mass of one mole of [tex]\(N_2\)[/tex] molecules.
2. Calculate the Number of Moles:
- To find the number of moles of [tex]\(N_2\)[/tex] molecules, we use the formula:
[tex]\[
\text{Number of moles} = \frac{\text{Number of molecules}}{\text{Avogadro's number}}
\][/tex]
- Plugging in the given values:
[tex]\[
\text{Number of moles} = \frac{1.81 \times 10^{23} \text{ molecules}}{6.022 \times 10^{23} \text{ molecules/mol}}
\][/tex]
- Solving the fraction:
[tex]\[
\text{Number of moles} \approx 0.3005645964795749 \text{ moles}
\][/tex]
3. Calculate the Mass:
- Once we have the number of moles, we can find the mass using the molecular mass of [tex]\(N_2\)[/tex]:
[tex]\[
\text{Mass} = \text{Number of moles} \times \text{Molecular mass}
\][/tex]
- Plugging in the values:
[tex]\[
\text{Mass} = 0.3005645964795749 \text{ moles} \times 28.02 \text{ grams/mole}
\][/tex]
- Solving the multiplication:
[tex]\[
\text{Mass} \approx 8.421819993357689 \text{ grams}
\][/tex]
Based on our calculations, the mass of [tex]\(1.81 \times 10^{23}\)[/tex] molecules of [tex]\(N_2\)[/tex] is approximately 8.42 grams.
Thus, the correct answer from the given options is:
[tex]\[
\boxed{8.42 \text{ gN}}
\][/tex]
