To determine the mass of [tex]\(1.81 \times 10^{23}\)[/tex] molecules of nitrogen ([tex]\( \text{N}_2 \)[/tex]), we can follow these steps:
1. Calculate the number of moles of [tex]\(\text{N}_2\)[/tex] molecules:
The number of molecules can be converted to moles using Avogadro's number, which states that 1 mole of any substance contains [tex]\(6.02 \times 10^{23}\)[/tex] molecules.
Number of moles ([tex]\( n \)[/tex]) of [tex]\( \text{N}_2 \)[/tex] is given by:
[tex]\[
n = \frac{\text{Number of molecules}}{\text{Avogadro's number}}
\][/tex]
So,
[tex]\[
n = \frac{1.81 \times 10^{23}}{6.02 \times 10^{23}}
\][/tex]
2. Substitute the numbers into the equation:
[tex]\[
n = \frac{1.81 \times 10^{23}}{6.02 \times 10^{23}} = 0.30066445182724255 \text{ moles}
\][/tex]
3. Calculate the mass of [tex]\( \text{N}_2 \)[/tex]:
The molar mass of [tex]\( \text{N}_2 \)[/tex] is given as [tex]\(28.02 \text{ g/mol}\)[/tex]. The mass ([tex]\( m \)[/tex]) of [tex]\( \text{N}_2 \)[/tex] can be calculated by multiplying the number of moles by the molar mass:
[tex]\[
m = n \times M
\][/tex]
where [tex]\( M \)[/tex] is the molar mass. Thus,
[tex]\[
m = 0.30066445182724255 \times 28.02 \text{ g/mol}
\][/tex]
4. Perform the multiplication:
[tex]\[
m \approx 8.424617940199337 \text{ g}
\][/tex]
Therefore, the mass of [tex]\(1.81 \times 10^{23}\)[/tex] molecules of nitrogen ([tex]\(\text{N}_2\)[/tex]) is approximately [tex]\( 8.42 \text{ g} \)[/tex].
Thus, the correct answer is:
D. [tex]\(8.42 \text{ g} \text{ N}_2\)[/tex].
