To find the mass in milligrams of [tex]\( 4.61 \times 10^{23} \)[/tex] formula units of [tex]\( \text{Bi}( \text{NO}_3)_3 \cdot 5 \text{H}_2 \text{O} \)[/tex], we'll proceed with the following steps:
1. Determine Avogadro's number: Avogadro's number, [tex]\( 6.022 \times 10^{23} \)[/tex], is the number of units (atoms, molecules, formula units, etc.) in one mole of a substance.
2. Calculate the moles of the substance: To find the number of moles, divide the number of formula units by Avogadro's number:
[tex]\[
\text{Moles} = \frac{4.61 \times 10^{23} \text{ formula units}}{6.022 \times 10^{23} \text{ formula units/mol}} = 0.7655264031883096 \text{ mol}
\][/tex]
3. Given the molar mass of the compound: The molar mass of [tex]\( \text{Bi}( \text{NO}_3)_3 \cdot 5 \text{H}_2 \text{O} \)[/tex] is approximately 500.11 grams per mole.
4. Calculate the mass in grams: Multiplying the number of moles by the molar mass gives the mass in grams:
[tex]\[
\text{Mass (grams)} = 0.7655264031883096 \text{ mol} \times 500.11 \text{ g/mol} = 382.8474094985055 \text{ g}
\][/tex]
5. Convert grams to milligrams: Since there are 1000 milligrams in a gram, we multiply the mass in grams by 1000:
[tex]\[
\text{Mass (milligrams)} = 382.8474094985055 \text{ g} \times 1000 \text{ mg/g} = 382847.4094985055 \text{ mg}
\][/tex]
Therefore, the mass of [tex]\( 4.61 \times 10^{23} \)[/tex] formula units of [tex]\( \text{Bi}( \text{NO}_3)_3 \cdot 5 \text{H}_2 \text{O} \)[/tex] is [tex]\( 382847.4094985055 \)[/tex] milligrams.
