A solution of 36% water and 64% acetaldehyde `(CH_(3)CHO)` by mass. Mole fraction of acetaldehyde is

Options

A. `0.42`
B. `0.2`
C. `4.2`
D. `2.1`

Explanation

To find the mole fraction of acetaldehyde in a solution that is 36% water and 64% acetaldehyde by mass, we can follow these steps:

### Step 1: Assume Total Mass of the Solution
Let the total mass of the solution be \( x \) grams.

### Step 2: Calculate the Mass of Each Component
- Mass of water = \( 36\% \) of \( x \) = \( 0.36x \) grams
- Mass of acetaldehyde = \( 64\% \) of \( x \) = \( 0.64x \) grams

### Step 3: Calculate the Number of Moles of Each Component
To find the mole fraction, we need the number of moles of each component.

**Moles of Acetaldehyde:**
- Molecular weight of acetaldehyde (CH₃CHO) = \( 2 \times 12 + 4 \times 1 + 16 = 44 \) g/mol
- Number of moles of acetaldehyde = \( \frac{\text{mass of acetaldehyde}}{\text{molecular weight}} = \frac{0.64x}{44} \)

**Moles of Water:**
- Molecular weight of water (H₂O) = \( 2 \times 1 + 16 = 18 \) g/mol
- Number of moles of water = \( \frac{\text{mass of water}}{\text{molecular weight}} = \frac{0.36x}{18} \)

### Step 4: Calculate the Total Number of Moles
Total number of moles in the solution = Moles of acetaldehyde + Moles of water
\[ \text{Total moles} = \frac{0.64x}{44} + \frac{0.36x}{18} \]

### Step 5: Calculate the Mole Fraction of Acetaldehyde
The mole fraction of acetaldehyde (denoted as \( X_{acetaldehyde} \)) is given by:
\[ X_{acetaldehyde} = \frac{\text{moles of acetaldehyde}}{\text{total moles}} = \frac{\frac{0.64x}{44}}{\frac{0.64x}{44} + \frac{0.36x}{18}} \]

### Step 6: Simplify the Expression
Since \( x \) is common in the numerator and denominator, we can cancel it out:
\[ X_{acetaldehyde} = \frac{0.64/44}{0.64/44 + 0.36/18} \]

### Step 7: Calculate the Denominator
To simplify the denominator:
\[ \text{Denominator} = \frac{0.64}{44} + \frac{0.36}{18} \]
Finding a common denominator (which is \( 44 \times 18 \)):
\[ \text{Denominator} = \frac{0.64 \times 18 + 0.36 \times 44}{44 \times 18} \]

### Step 8: Calculate the Final Value
Now, we can calculate the mole fraction:
\[ X_{acetaldehyde} = \frac{0.64 \times 18}{0.64 \times 18 + 0.36 \times 44} \]
Calculating the values:
\[ = \frac{11.52}{11.52 + 15.84} = \frac{11.52}{27.36} \approx 0.42 \]

### Conclusion
The mole fraction of acetaldehyde in the solution is approximately \( 0.42 \).

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