Sure, I'd be happy to help you figure this out!
To determine the number of atoms of phosphorus in a sample with a mass of [tex]\(172.90 \, \text{g}\)[/tex], we need to follow these steps:
1. Determine the molar mass of phosphorus:
The atomic mass of phosphorus is [tex]\(30.974 \, \text{g/mol}\)[/tex]. This value is obtained from the periodic table.
2. Calculate the number of moles of phosphorus:
To find the number of moles, we use the formula:
[tex]\[
\text{moles of phosphorus} = \frac{\text{mass of sample}}{\text{molar mass of phosphorus}}
\][/tex]
Plugging in the values, we get:
[tex]\[
\text{moles of phosphorus} = \frac{172.90 \, \text{g}}{30.974 \, \text{g/mol}} \approx 5.582 \, \text{moles}
\][/tex]
3. Calculate the number of atoms of phosphorus:
We know that one mole of any substance contains [tex]\(6.022 \times 10^{23}\)[/tex] atoms (Avogadro's number). So, to find the number of atoms, we multiply the number of moles by Avogadro's number:
[tex]\[
\text{atoms of phosphorus} = \text{moles of phosphorus} \times 6.022 \times 10^{23} \, \text{atoms/mol}
\][/tex]
Substituting the number of moles we estimated:
[tex]\[
\text{atoms of phosphorus} = 5.582 \, \text{moles} \times 6.022 \times 10^{23} \approx 3.362 \times 10^{24} \, \text{atoms}
\][/tex]
4. Simplify to match the given options:
By simplifying the scientific notation to match the closest option:
[tex]\[
\text{atoms of phosphorus} \approx 3.36 \times 10^{24} \, \text{atoms}
\][/tex]
From the choices given:
[tex]\[
3.48 \times 10^{21} \, \text{atoms}, \quad 1.97 \times 10^{22} \, \text{atoms}, \quad 3.36 \times 10^{24} \, \text{atoms}, \quad 1.04 \times 10^{26} \, \text{atoms}
\][/tex]
The correct answer is:
[tex]\[
\boxed{3.36 \times 10^{24}}
\][/tex]
