Answer :
Sure, let's tackle this problem step-by-step to determine how many atoms of phosphorus (P) are present in a given sample with a mass of 172.90 grams.
### Step 1: Identify the molar mass of phosphorus
From the periodic table, the molar mass of phosphorus (P) is approximately 30.97 grams per mole (g/mol).
### Step 2: Calculate the number of moles of phosphorus
To find the number of moles, we use the formula:
[tex]\[ \text{number of moles} = \frac{\text{mass of sample}}{\text{molar mass}} \][/tex]
Given:
- Mass of phosphorus sample = 172.90 grams
- Molar mass of phosphorus = 30.97 g/mol
Using the formula:
[tex]\[ \text{number of moles} = \frac{172.90 \text{ g}}{30.97 \text{ g/mol}} \approx 5.5828 \text{ moles} \][/tex]
### Step 3: Utilize Avogadro's number to find the number of atoms
Avogadro's number tells us the number of atoms in one mole of a substance, which is:
[tex]\[ 6.022 \times 10^{23} \text{ atoms/mol} \][/tex]
To find the total number of atoms, we multiply the number of moles by Avogadro's number:
[tex]\[ \text{number of atoms} = \text{number of moles} \times \text{Avogadro's number} \][/tex]
Using the moles calculated:
[tex]\[ \text{number of atoms} = 5.5828 \text{ moles} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 3.361975 \times 10^{24} \text{ atoms} \][/tex]
### Step 4: Select the correct answer from the given options
From the calculations, we have approximately [tex]\(3.36 \times 10^{24}\)[/tex] atoms of phosphorus in the sample.
Thus, the correct answer is:
[tex]\[ 3.36 \times 10^{24} \text{ atoms} \][/tex]
### Step 1: Identify the molar mass of phosphorus
From the periodic table, the molar mass of phosphorus (P) is approximately 30.97 grams per mole (g/mol).
### Step 2: Calculate the number of moles of phosphorus
To find the number of moles, we use the formula:
[tex]\[ \text{number of moles} = \frac{\text{mass of sample}}{\text{molar mass}} \][/tex]
Given:
- Mass of phosphorus sample = 172.90 grams
- Molar mass of phosphorus = 30.97 g/mol
Using the formula:
[tex]\[ \text{number of moles} = \frac{172.90 \text{ g}}{30.97 \text{ g/mol}} \approx 5.5828 \text{ moles} \][/tex]
### Step 3: Utilize Avogadro's number to find the number of atoms
Avogadro's number tells us the number of atoms in one mole of a substance, which is:
[tex]\[ 6.022 \times 10^{23} \text{ atoms/mol} \][/tex]
To find the total number of atoms, we multiply the number of moles by Avogadro's number:
[tex]\[ \text{number of atoms} = \text{number of moles} \times \text{Avogadro's number} \][/tex]
Using the moles calculated:
[tex]\[ \text{number of atoms} = 5.5828 \text{ moles} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 3.361975 \times 10^{24} \text{ atoms} \][/tex]
### Step 4: Select the correct answer from the given options
From the calculations, we have approximately [tex]\(3.36 \times 10^{24}\)[/tex] atoms of phosphorus in the sample.
Thus, the correct answer is:
[tex]\[ 3.36 \times 10^{24} \text{ atoms} \][/tex]
