The pressure of two pure liquid A and B which form an ideal solutions are 400 mm Hg and 800 mm Hg respectively at temperature T.A liquid containing 3:1 molar composition pressure can be varied.The solutions is slowly vapourized at temperature T by decreasing the applied pressure starting with a pressure of 760 mm Hg.A pressure gauge (in mm) Hg is connected which give the reading of pressure applied.
The reading of pressure Gauge at which only liquid phase exists.

Options

A. 501
B. 399
C. 299
D. None

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the given data
- Vapor pressure of pure liquid A, \( P^0_A = 400 \, \text{mm Hg} \)
- Vapor pressure of pure liquid B, \( P^0_B = 800 \, \text{mm Hg} \)
- Molar ratio of A to B = 3:1

### Step 2: Calculate the mole fractions
Given the molar ratio of A to B is 3:1, we can denote:
- Moles of A, \( n_A = 3 \)
- Moles of B, \( n_B = 1 \)

Now, we can calculate the total moles:
\[ n_{total} = n_A + n_B = 3 + 1 = 4 \]

Now, we can find the mole fractions:
\[ x_A = \frac{n_A}{n_{total}} = \frac{3}{4} = 0.75 \]
\[ x_B = \frac{n_B}{n_{total}} = \frac{1}{4} = 0.25 \]

### Step 3: Apply Raoult's Law to find the total vapor pressure
According to Raoult's Law, the total vapor pressure \( P_{total} \) of the solution can be calculated as:
\[ P_{total} = P^0_A \cdot x_A + P^0_B \cdot x_B \]

Substituting the values:
\[ P_{total} = (400 \, \text{mm Hg} \cdot 0.75) + (800 \, \text{mm Hg} \cdot 0.25) \]
\[ P_{total} = 300 \, \text{mm Hg} + 200 \, \text{mm Hg} = 500 \, \text{mm Hg} \]

### Step 4: Determine the pressure at which only the liquid phase exists
From the calculated total vapor pressure, we know that:
- Above \( 500 \, \text{mm Hg} \), the solution will exist only in the liquid phase.

### Conclusion
The reading of the pressure gauge at which only the liquid phase exists is any pressure greater than \( 500 \, \text{mm Hg} \). Therefore, the minimum reading at which only the liquid phase exists is \( 501 \, \text{mm Hg} \).

### Final Answer
**The reading of the pressure gauge at which only liquid phase exists is 501 mm Hg.**

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