To calculate the mole fraction of caffeine in the solution, let's walk through the steps in detail:
### Step 1: Understand the Given Data
Given:
- Mass of caffeine ([tex]\( m_{0,\text{caffeine}} \)[/tex]): 7.30 grams
- Mass of water ([tex]\( m_{0,\text{water}} \)[/tex]): 358 grams
- Molar mass of caffeine ([tex]\( M_{\text{caffeine}} \)[/tex]): 194.19 g/mol
- Molar mass of water ([tex]\( M_{\text{water}} \)[/tex]): 18.015 g/mol
### Step 2: Calculate the Number of Moles
1. Number of moles of caffeine:
Using the formula:
[tex]\[
n_{\text{caffeine}} = \frac{m_{0,\text{caffeine}}}{M_{\text{caffeine}}}
\][/tex]
Substituting the given values:
[tex]\[
n_{\text{caffeine}} = \frac{7.30 \, \text{g}}{194.19 \, \text{g/mol}} \approx 0.0376 \, \text{mol}
\][/tex]
2. Number of moles of water:
Using the formula:
[tex]\[
n_{\text{water}} = \frac{m_{0,\text{water}}}{M_{\text{water}}}
\][/tex]
Substituting the given values:
[tex]\[
n_{\text{water}} = \frac{358 \, \text{g}}{18.015 \, \text{g/mol}} \approx 19.87 \, \text{mol}
\][/tex]
### Step 3: Calculate the Total Number of Moles
[tex]\[
\text{Total moles} = n_{\text{caffeine}} + n_{\text{water}}
\][/tex]
Substituting the calculated values:
[tex]\[
\text{Total moles} = 0.0376 \, \text{mol} + 19.87 \, \text{mol} \approx 19.91 \, \text{mol}
\][/tex]
### Step 4: Calculate the Mole Fraction of Caffeine
The mole fraction ([tex]\( x_{\text{caffeine}} \)[/tex]) is given by:
[tex]\[
x_{\text{caffeine}} = \frac{n_{\text{caffeine}}}{\text{Total moles}}
\][/tex]
Substituting the calculated values:
[tex]\[
x_{\text{caffeine}} = \frac{0.0376 \, \text{mol}}{19.91 \, \text{mol}} \approx 0.00189
\][/tex]
### Step 5: Round the Result
Round the mole fraction of caffeine to three significant figures:
[tex]\[
x_{\text{caffeine}} \approx 0.002
\][/tex]
### Final Answer
Thus, the mole fraction of caffeine in the solution, expressed to three significant figures, is [tex]\( 0.002 \)[/tex].
