To find the number of nitrogen atoms in 137.0 grams of [tex]\(N_2O_3\)[/tex], we need to follow these steps:
1. Determine the molar mass of [tex]\(N_2O_3\)[/tex]:
The molar mass of [tex]\(N_2O_3\)[/tex] is given as 108.02 g/mol.
2. Calculate the number of moles of [tex]\(N_2O_3\)[/tex]:
Using the formula:
[tex]\[
\text{Number of moles} = \frac{\text{Mass (grams)}}{\text{Molar mass (g/mol)}}
\][/tex]
[tex]\[
\text{Number of moles of } N_2O_3 = \frac{137.0 \text{ grams}}{108.02 \text{ g/mol}} = 1.2682836511757083 \text{ moles}
\][/tex]
3. Determine the number of molecules of [tex]\(N_2O_3\)[/tex]:
Using Avogadro's number (approximately [tex]\(6.022 \times 10^{23}\)[/tex] molecules/mol), we calculate:
[tex]\[
\text{Number of molecules of } N_2O_3 = 1.2682836511757083 \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mol} = 7.637604147380116 \times 10^{23} \text{ molecules}
\][/tex]
4. Calculate the number of nitrogen atoms in the sample:
Each molecule of [tex]\(N_2O_3\)[/tex] contains 2 nitrogen atoms. Therefore, we multiply the number of molecules by 2:
[tex]\[
\text{Number of nitrogen atoms} = 7.637604147380116 \times 10^{23} \text{ molecules} \times 2 = 1.5275208294760232 \times 10^{24} \text{ nitrogen atoms}
\][/tex]
5. Identify the closest answer option:
We compare the calculated number of nitrogen atoms [tex]\(1.5275208294760232 \times 10^{24}\)[/tex] with the provided answer choices:
- A. [tex]\(1.085 \times 10^{23}\)[/tex]
- B. [tex]\(1.802 \times 10^{23}\)[/tex]
- C. [tex]\(5.985 \times 10^{23}\)[/tex]
- D. [tex]\(2.171 \times 10^{24}\)[/tex]
- E. [tex]\(3.604 \times 10^{24}\)[/tex]
The closest option to [tex]\(1.5275208294760232 \times 10^{24}\)[/tex] is D. [tex]\(2.171 \times 10^{24}\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{2.171 \times 10^{24}} \][/tex]
