To determine the number of carbon atoms in a 2.5-carat diamond, follow these steps:
1. Convert carats to grams:
- Given that 1 carat is equal to 0.20 grams, we can calculate the total mass of the diamond in grams.
[tex]\[
\text{Mass in grams} = 2.5 \, \text{carats} \times 0.20 \, \frac{\text{grams}}{\text{carat}} = 0.5 \, \text{grams}
\][/tex]
2. Determine the number of moles of carbon in the diamond:
- Knowing that 12 grams of carbon constitutes 1 mole (as the molar mass of carbon is 12 grams per mole), we can find the number of moles in 0.5 grams of carbon.
[tex]\[
\text{Moles of carbon} = \frac{\text{Mass in grams}}{\text{Molar mass of carbon}} = \frac{0.5 \, \text{grams}}{12 \, \frac{\text{grams}}{\text{mole}}} = 0.0416667 \, \text{moles}
\][/tex]
3. Calculate the number of carbon atoms:
- Using Avogadro's number, which is [tex]\(6.022 \times 10^{23}\)[/tex] atoms per mole, we can convert the moles of carbon to the number of atoms.
[tex]\[
\text{Number of atoms} = \text{Moles of carbon} \times \text{Avogadro's number} = 0.0416667 \, \text{moles} \times 6.022 \times 10^{23} \, \frac{\text{atoms}}{\text{mole}}
\][/tex]
[tex]\[
\text{Number of atoms} = 2.50917 \times 10^{22} \, \text{atoms}
\][/tex]
4. Express the number of atoms to two significant figures:
- Finally, we should express the result to two significant figures.
[tex]\[
\text{Number of atoms} \approx 2.51 \times 10^{22} \, \text{atoms}
\][/tex]
So, the number of carbon atoms in a 2.5-carat diamond, expressed to two significant figures, is approximately:
[tex]\[
2.51 \times 10^{22} \, \text{carbon atoms}
\][/tex]
