Mole fraction of ethyl alcohol in aqueous ethyl alcohol (`C_2H_5OH`) solution is 0.50. Hence percentage of ethyl alcohol by weight is:

Options

A. 25.74%
B. 71.87%
C. 32%
D. 23%

Answer

To find the percentage of ethyl alcohol by weight in an aqueous ethyl alcohol solution where the mole fraction of ethyl alcohol is 0.50, we can follow these steps:

### Step 1: Understand the mole fraction
The mole fraction of ethyl alcohol (C₂H₅OH) is given as 0.50. This means that in a mixture, the ratio of moles of ethyl alcohol to the total moles (ethyl alcohol + water) is 0.50.

### Step 2: Determine the mole fraction of water
Since the mole fraction of ethyl alcohol is 0.50, the mole fraction of water (H₂O) will also be 0.50 because the total mole fraction must equal 1.

### Step 3: Assume a total number of moles
For simplicity, assume we have 1 mole of the solution. This means:
- Moles of ethyl alcohol = 0.50 moles
- Moles of water = 0.50 moles

### Step 4: Calculate the weight of ethyl alcohol
The molar mass of ethyl alcohol (C₂H₅OH) is 46 g/mol. Therefore, the weight of ethyl alcohol can be calculated as:
\[ \text{Weight of ethyl alcohol} = \text{Moles} \times \text{Molar Mass} = 0.50 \, \text{moles} \times 46 \, \text{g/mol} = 23 \, \text{grams} \]

### Step 5: Calculate the weight of water
The molar mass of water (H₂O) is 18 g/mol. Therefore, the weight of water can be calculated as:
\[ \text{Weight of water} = \text{Moles} \times \text{Molar Mass} = 0.50 \, \text{moles} \times 18 \, \text{g/mol} = 9 \, \text{grams} \]

### Step 6: Calculate the total weight of the solution
The total weight of the solution is the sum of the weights of ethyl alcohol and water:
\[ \text{Total weight of solution} = \text{Weight of ethyl alcohol} + \text{Weight of water} = 23 \, \text{grams} + 9 \, \text{grams} = 32 \, \text{grams} \]

### Step 7: Calculate the percentage of ethyl alcohol by weight
The percentage of ethyl alcohol by weight in the solution can be calculated using the formula:
\[ \text{Percentage of ethyl alcohol} = \left( \frac{\text{Weight of ethyl alcohol}}{\text{Total weight of solution}} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage of ethyl alcohol} = \left( \frac{23 \, \text{grams}}{32 \, \text{grams}} \right) \times 100 \approx 71.87\% \]

### Final Answer
The percentage of ethyl alcohol by weight in the solution is approximately **71.87%**.

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