Vapour pressure of water at certain temperature is 155 mm Hg and that of the another solvent 'X' is 'p' mm Hg. Molecular weight of 'X' is 128. An aqueous solution of 'X' (64% by wt) has a vapour pressure of 145 mm Hg. What is 'p' ?

Answer

To find the vapor pressure \( p \) of solvent \( X \), we can use Raoult's law, which states that the vapor pressure of a 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:**
- Vapor pressure of water (\( P^0_A \)) = 155 mm Hg
- Vapor pressure of solvent \( X \) = \( p \) mm Hg
- Weight percentage of \( X \) in the solution = 64%
- Weight of water in the solution = 36% (since it is an aqueous solution)

2. **Assume a Total Mass of the Solution:**
- Let's assume we have 100 g of the solution.
- Mass of solvent \( X \) = 64 g
- Mass of water = 100 g - 64 g = 36 g

3. **Calculate the Number of Moles:**
- Molar mass of water = 18 g/mol
- Moles of water (\( n_A \)) = \( \frac{36 \text{ g}}{18 \text{ g/mol}} = 2 \text{ moles} \)
- Molar mass of solvent \( X \) = 128 g/mol
- Moles of solvent \( X \) (\( n_B \)) = \( \frac{64 \text{ g}}{128 \text{ g/mol}} = 0.5 \text{ moles} \)

4. **Calculate Mole Fractions:**
- Total moles = \( n_A + n_B = 2 + 0.5 = 2.5 \text{ moles} \)
- Mole fraction of water (\( X_A \)) = \( \frac{n_A}{n_A + n_B} = \frac{2}{2.5} = 0.8 \)
- Mole fraction of solvent \( X \) (\( X_B \)) = \( \frac{n_B}{n_A + n_B} = \frac{0.5}{2.5} = 0.2 \)

5. **Apply Raoult's Law:**
- According to Raoult's law, the vapor pressure of the solution (\( P \)) is given by:
\[ P = P^0_A \cdot X_A + P^0_B \cdot X_B \]
- Here, \( P^0_A = 155 \text{ mm Hg} \), \( P^0_B = p \text{ mm Hg} \), and \( P = 145 \text{ mm Hg} \).
- Substituting the values:
\[ 145 = 155 \cdot 0.8 + p \cdot 0.2 \]

6. **Solve for \( p \):**
- Calculate \( 155 \cdot 0.8 = 124 \text{ mm Hg} \):
\[ 145 = 124 + 0.2p \]
- Rearranging gives:
\[ 0.2p = 145 - 124 = 21 \]
- Therefore:
\[ p = \frac{21}{0.2} = 105 \text{ mm Hg} \]

### Final Answer:
The vapor pressure \( p \) of solvent \( X \) is **105 mm Hg**.