To calculate the number of moles of caffeine in the energy drink containing 40.0 mg of caffeine (C₈H₁₀N₄O₂), we will follow these steps:
1. Convert the mass of caffeine from milligrams to grams:
Since 1 gram is equal to 1000 milligrams, we can convert the mass of caffeine to grams using the following formula:
[tex]\[
\text{mass in grams} = \frac{\text{mass in milligrams}}{1000}
\][/tex]
Given:
[tex]\[
\text{mass of caffeine} = 40.0 \, \text{mg}
\][/tex]
Converting to grams:
[tex]\[
\text{mass of caffeine in grams} = \frac{40.0 \, \text{mg}}{1000} = 0.04 \, \text{g}
\][/tex]
2. Calculate the molar mass of caffeine:
The molecular formula for caffeine is C₈H₁₀N₄O₂. We need to find the molar mass by summing the molar masses of all the atoms in one molecule of caffeine:
- Carbon (C): 12.01 g/mol
- Hydrogen (H): 1.01 g/mol
- Nitrogen (N): 14.01 g/mol
- Oxygen (O): 16.00 g/mol
Using the molecular formula, the molar mass of caffeine can be calculated as follows:
[tex]\[
\text{Molar Mass} = (8 \times 12.01) + (10 \times 1.01) + (4 \times 14.01) + (2 \times 16.00)
\][/tex]
[tex]\[
\text{Molar Mass} = 96.08 + 10.10 + 56.04 + 32.00 = 194.22 \, \text{g/mol}
\][/tex]
3. Calculate the number of moles of caffeine:
We can now use the mass of caffeine in grams and the molar mass to find the number of moles. The formula to calculate the number of moles is:
[tex]\[
\text{moles} = \frac{\text{mass in grams}}{\text{molar mass}}
\][/tex]
Substituting the values we have:
[tex]\[
\text{moles of caffeine} = \frac{0.04 \, \text{g}}{194.22 \, \text{g/mol}} = 0.00020595201318092884 \, \text{mol}
\][/tex]
Thus, the number of moles of caffeine in the energy drink is approximately [tex]\( 0.000206 \, \text{mol} \)[/tex].
