The vapour pressures of ethanol and methanol are `44.5` and `88.7 mm Hg`, respectively. An ideal solution is formed at the same temperature by mixing `60 g` of ethanol with `40 g` of `methanol`. Calculate the total vapour pressure of the solution and mole fraction of methanol in the vapour.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Calculate the number of moles of ethanol and methanol.

1. **Molecular Weight of Ethanol (C2H5OH)**:
- C: 12 g/mol × 2 = 24 g/mol
- H: 1 g/mol × 6 = 6 g/mol
- O: 16 g/mol × 1 = 16 g/mol
- Total = 24 + 6 + 16 = 46 g/mol

2. **Moles of Ethanol**:
\[ \text{Moles of Ethanol} = \frac{\text{mass}}{\text{molecular weight}} = \frac{60 \text{ g}}{46 \text{ g/mol}} \approx 1.304 \text{ moles} \]

3. **Molecular Weight of Methanol (CH3OH)**:
- C: 12 g/mol × 1 = 12 g/mol
- H: 1 g/mol × 4 = 4 g/mol
- O: 16 g/mol × 1 = 16 g/mol
- Total = 12 + 4 + 16 = 32 g/mol

4. **Moles of Methanol**:
\[ \text{Moles of Methanol} = \frac{40 \text{ g}}{32 \text{ g/mol}} = 1.25 \text{ moles} \]

### Step 2: Calculate the total number of moles in the solution.

\[ \text{Total Moles} = \text{Moles of Ethanol} + \text{Moles of Methanol} = 1.304 + 1.25 = 2.554 \text{ moles} \]

### Step 3: Calculate the mole fractions of ethanol and methanol.

1. **Mole Fraction of Ethanol (X_E)**:
\[ X_E = \frac{\text{Moles of Ethanol}}{\text{Total Moles}} = \frac{1.304}{2.554} \approx 0.510 \]

2. **Mole Fraction of Methanol (X_M)**:
\[ X_M = \frac{\text{Moles of Methanol}}{\text{Total Moles}} = \frac{1.25}{2.554} \approx 0.490 \]

### Step 4: Calculate the partial vapor pressures of ethanol and methanol.

1. **Given Vapor Pressures**:
- \( P^0_{\text{Ethanol}} = 44.5 \text{ mm Hg} \)
- \( P^0_{\text{Methanol}} = 88.7 \text{ mm Hg} \)

2. **Partial Vapor Pressure of Ethanol (P_E)**:
\[ P_E = P^0_{\text{Ethanol}} \times X_E = 44.5 \text{ mm Hg} \times 0.510 \approx 22.695 \text{ mm Hg} \]

3. **Partial Vapor Pressure of Methanol (P_M)**:
\[ P_M = P^0_{\text{Methanol}} \times X_M = 88.7 \text{ mm Hg} \times 0.490 \approx 43.443 \text{ mm Hg} \]

### Step 5: Calculate the total vapor pressure of the solution.

\[ P_{\text{total}} = P_E + P_M = 22.695 \text{ mm Hg} + 43.443 \text{ mm Hg} \approx 66.138 \text{ mm Hg} \]

### Step 6: Calculate the mole fraction of methanol in the vapor.

1. **Mole Fraction of Methanol in Vapor (Y_M)**:
\[ Y_M = \frac{P_M}{P_{\text{total}}} = \frac{43.443 \text{ mm Hg}}{66.138 \text{ mm Hg}} \approx 0.657 \]

### Final Answers:
- Total vapor pressure of the solution: **66.14 mm Hg**
- Mole fraction of methanol in the vapor: **0.657**