The density of acetonitrile (CH3CN) is 0.786 g/mL and the density of methanol (CH3OH) is 0.791 g/mL. A solution is made by dissolving 15.7 mL of CH3OH in 92.3 mL of CH3CN. What is the mole fraction of the methano in this solution?
The Correct Answer and Explanation is
The mole fraction of methanol in the solution is 0.180 (rounded to three significant figures).
Explanation:
- Understanding the Problem
We are given a solution of methanol ((\text{CH}_3\text{OH})) and acetonitrile ((\text{CH}_3\text{CN})), with known volumes and densities. The goal is to find the mole fraction of methanol in the solution. - Calculate the Mass of Each Component
Using the density formula:
[
\text{Mass} = \text{Density} \times \text{Volume}
]
- Mass of methanol:
[
15.7 \text{ mL} \times 0.791 \text{ g/mL} = 12.42 \text{ g}
] - Mass of acetonitrile:
[
92.3 \text{ mL} \times 0.786 \text{ g/mL} = 72.57 \text{ g}
]
- Calculate the Number of Moles
Using the formula:
[
\text{Moles} = \frac{\text{Mass}}{\text{Molar Mass}}
]
- Moles of methanol:
[
\frac{12.42 \text{ g}}{32.04 \text{ g/mol}} = 0.3877 \text{ moles}
] - Moles of acetonitrile:
[
\frac{72.57 \text{ g}}{41.05 \text{ g/mol}} = 1.768 \text{ moles}
]
- Calculate the Mole Fraction of Methanol
The mole fraction is given by:
[
X_{\text{methanol}} = \frac{\text{Moles of methanol}}{\text{Total moles}}
]
[
X_{\text{methanol}} = \frac{0.3877}{0.3877 + 1.768} = \frac{0.3877}{2.1557} = 0.180
]
Thus, the mole fraction of methanol is 0.180.