Sure! Here's a step-by-step solution to determine the number of atoms in 1.23 mL of mercury, given that the density of mercury is [tex]\(13.5 \, \text{g/mL}\)[/tex].
### Step 1: Calculate the Mass of Mercury
First, we need to find the mass of mercury in grams using the volume and the density.
[tex]\[
\text{Volume of mercury} = 1.23 \, \text{mL}
\][/tex]
[tex]\[
\text{Density of mercury} = 13.5 \, \text{g/mL}
\][/tex]
[tex]\[
\text{Mass of mercury} = \text{Volume} \times \text{Density} = 1.23 \, \text{mL} \times 13.5 \, \text{g/mL} = 16.605 \, \text{g}
\][/tex]
### Step 2: Calculate the Number of Moles of Mercury
Next, we need to convert the mass of mercury to the number of moles. We use the molar mass of mercury for this conversion.
[tex]\[
\text{Molar mass of mercury (Hg)} = 200.59 \, \text{g/mol}
\][/tex]
[tex]\[
\text{Moles of mercury} = \frac{\text{Mass of mercury}}{\text{Molar Mass}} = \frac{16.605 \, \text{g}}{200.59 \, \text{g/mol}} \approx 0.08278 \, \text{mol}
\][/tex]
### Step 3: Calculate the Number of Atoms
Finally, we convert the number of moles to the number of atoms using Avogadro's number.
[tex]\[
\text{Avogadro's number} = 6.02214076 \times 10^{23} \, \text{atoms/mol}
\][/tex]
[tex]\[
\text{Number of atoms} = \text{Moles of mercury} \times \text{Avogadro's number} = 0.08278 \, \text{mol} \times 6.02214076 \times 10^{23} \, \text{atoms/mol} \approx 4.98 \times 10^{22} \, \text{atoms}
\][/tex]
Therefore, the number of atoms in 1.23 mL of mercury is [tex]\(4.98 \times 10^{22}\)[/tex], which corresponds to the last option in the list.
[tex]\[
\boxed{4.98 \times 10^{22}}
\][/tex]
