To determine the number of Carbon (C) atoms in a [tex]\(10.0\)[/tex] gram sample of the compound [tex]\(C_7H_{12}N_2O_3\)[/tex], follow these steps:
1. Calculate the molar mass of [tex]\(C_7H_{12}N_2O_3\)[/tex]:
- The molar mass of Carbon (C) is [tex]\(12.01\)[/tex] grams per mole.
- The molar mass of Hydrogen (H) is [tex]\(1.008\)[/tex] grams per mole.
- The molar mass of Nitrogen (N) is [tex]\(14.01\)[/tex] grams per mole.
- The molar mass of Oxygen (O) is [tex]\(16.00\)[/tex] grams per mole.
Therefore, the molar mass of [tex]\(C_7H_{12}N_2O_3\)[/tex] is calculated as:
[tex]\[
(7 \times 12.01) + (12 \times 1.008) + (2 \times 14.01) + (3 \times 16.00) = 172.186 \text{ g/mol}
\][/tex]
2. Determine the number of moles in the [tex]\(10.0\)[/tex] gram sample:
- The molar mass of [tex]\(C_7H_{12}N_2O_3\)[/tex] is [tex]\(172.186\)[/tex] grams per mole.
- The mass of the sample is [tex]\(10.0\)[/tex] grams.
The number of moles ([tex]\(n\)[/tex]) in the sample is:
[tex]\[
n = \frac{\text{mass of sample}}{\text{molar mass}} = \frac{10.0 \text{ g}}{172.186 \text{ g/mol}} = 0.058077 \text{ moles}
\][/tex]
3. Calculate the number of molecules in the sample:
- Avogadro's number ([tex]\(N_A\)[/tex]) is [tex]\(6.022 \times 10^{23}\)[/tex] molecules per mole.
The number of molecules in the sample is:
[tex]\[
\text{Number of molecules} = \text{moles of sample} \times N_A = 0.058077 \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mol} = 3.49738 \times 10^{22}
\][/tex]
4. Determine the number of Carbon atoms:
- Each molecule of [tex]\(C_7H_{12}N_2O_3\)[/tex] contains 7 Carbon atoms.
The total number of Carbon atoms in the sample is:
[tex]\[
\text{Number of Carbon atoms} = \text{number of molecules} \times 7 = 3.49738 \times 10^{22} \times 7 = 2.44817 \times 10^{23}
\][/tex]
Thus, the number of Carbon atoms in a [tex]\(10.0\)[/tex] gram sample of [tex]\(C_7H_{12}N_2O_3\)[/tex] is approximately [tex]\(2.448 \times 10^{23}\)[/tex].
