Mole fraction of ethyl alcohol in aqueous ethyl alcohol (`C_2H_5OH`) solution is 0.15. Hence percentage of ethyl alcohol by weight is:

Explanation

To find the percentage of ethyl alcohol (ethanol, \(C_2H_5OH\)) by weight in an aqueous solution where the mole fraction of ethyl alcohol is 0.15, we can follow these steps:

### Step 1: Understand the mole fraction
The mole fraction of ethyl alcohol is given as 0.15. This means that in a mixture of ethyl alcohol and water, 15% of the total moles are ethyl alcohol, and the remaining 85% are water.

### Step 2: Determine the mole fractions
If the mole fraction of ethyl alcohol is 0.15, then the mole fraction of water is:
\[ X_{\text{H}_2\text{O}} = 1 - X_{\text{C}_2\text{H}_5\text{OH}} = 1 - 0.15 = 0.85 \]

### Step 3: Assume a certain number of moles
To simplify calculations, we can assume we have 1 mole of the solution. This means we have:
- Moles of ethyl alcohol = 0.15 moles
- Moles of water = 0.85 moles

### Step 4: Calculate the mass of ethyl alcohol
The molar mass of ethyl alcohol (\(C_2H_5OH\)) is approximately 46.07 g/mol. Therefore, the mass of ethyl alcohol can be calculated as:
\[ \text{Mass of ethyl alcohol} = \text{moles} \times \text{molar mass} = 0.15 \, \text{moles} \times 46.07 \, \text{g/mol} = 6.91 \, \text{g} \]

### Step 5: Calculate the mass of water
The molar mass of water (\(H_2O\)) is approximately 18.02 g/mol. Therefore, the mass of water can be calculated as:
\[ \text{Mass of water} = \text{moles} \times \text{molar mass} = 0.85 \, \text{moles} \times 18.02 \, \text{g/mol} = 15.32 \, \text{g} \]

### Step 6: Calculate the total mass of the solution
The total mass of the solution is the sum of the mass of ethyl alcohol and the mass of water:
\[ \text{Total mass of solution} = \text{Mass of ethyl alcohol} + \text{Mass of water} = 6.91 \, \text{g} + 15.32 \, \text{g} = 22.23 \, \text{g} \]

### Step 7: Calculate the percentage by weight of ethyl alcohol
The percentage by weight of ethyl alcohol in the solution can be calculated using the formula:
\[ \text{Percentage by weight} = \left( \frac{\text{Mass of ethyl alcohol}}{\text{Total mass of solution}} \right) \times 100 \]
Substituting the values we calculated:
\[ \text{Percentage by weight} = \left( \frac{6.91 \, \text{g}}{22.23 \, \text{g}} \right) \times 100 \approx 31.06\% \]

### Final Answer
The percentage of ethyl alcohol by weight in the solution is approximately **31.06%**.

---