An ideal solution is formed by mixing of 460 g. of ethanol with xg. of methanol. The total vapour pressure of the solution is 72 mm of Hg. The vapour pressure of pure ethanol and pure methanol are 48 and 80 mm of Hg respectively. Find the value of x. [Given: Atomic mass H = 1, C = 12, O = 16]

Explanation

To solve the problem, we will use Raoult's Law, which states that the total vapor pressure of an ideal solution is equal to the sum of the partial vapor pressures of each component in the solution.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Mass of ethanol (A) = 460 g
- Vapor pressure of pure ethanol (P°A) = 48 mmHg
- Vapor pressure of pure methanol (P°B) = 80 mmHg
- Total vapor pressure of the solution (P_total) = 72 mmHg
- Molar mass of ethanol (C2H5OH) = 46 g/mol
- Molar mass of methanol (CH3OH) = 32 g/mol

2. **Calculate Moles of Ethanol:**
\[ \text{Number of moles of ethanol (n_A)} = \frac{\text{mass of ethanol}}{\text{molar mass of ethanol}} = \frac{460 \text{ g}}{46 \text{ g/mol}} = 10 \text{ moles} \]

3. **Let x be the mass of methanol. Calculate Moles of Methanol:**
\[ \text{Number of moles of methanol (n_B)} = \frac{x \text{ g}}{32 \text{ g/mol}} = \frac{x}{32} \text{ moles} \]

4. **Calculate Mole Fractions:**
- Mole fraction of ethanol (X_A):
\[ X_A = \frac{n_A}{n_A + n_B} = \frac{10}{10 + \frac{x}{32}} = \frac{10 \cdot 32}{320 + x} \]
- Mole fraction of methanol (X_B):
\[ X_B = 1 - X_A = \frac{\frac{x}{32}}{10 + \frac{x}{32}} = \frac{x}{32 \cdot \left(10 + \frac{x}{32}\right)} \]

5. **Apply Raoult's Law:**
\[ P_{\text{total}} = P°_A \cdot X_A + P°_B \cdot X_B \]
Substituting the known values:
\[ 72 = 48 \cdot X_A + 80 \cdot X_B \]

6. **Substitute X_A and X_B in the equation:**
\[ 72 = 48 \cdot \frac{320}{320 + x} + 80 \cdot \frac{x}{32 \cdot (10 + \frac{x}{32})} \]

7. **Simplify the Equation:**
- Substitute \(X_A\) and \(X_B\) into the equation:
\[ 72 = \frac{48 \cdot 320}{320 + x} + \frac{80x}{32 \cdot (10 + \frac{x}{32})} \]

8. **Cross-Multiply and Solve for x:**
- After simplifying and solving the equation, you will get:
\[ 72(320 + x) = 48 \cdot 320 + 80x \]
- Rearranging gives:
\[ 72 \cdot 320 + 72x = 48 \cdot 320 + 80x \]
- Solving for x yields:
\[ 960 = 8x \implies x = 120 \text{ g} \]

### Final Answer:
The mass of methanol (x) is **120 g**.