For a mixture of two volatile , completely miscible liquids A and B , with `P_(A)^(@)=500 " torr and " P_(B)^(@)=800` torr , what is the composition of last droplet of liquid remaining in equilibrium with vapour ? Provided the initial ideal solution has a composition of `x_(A) = 0.6 and x_(B)=0.4`

Options

A. `x_A=0.6,x_B=0.4`
B. `x_A=0.5,x_B=0.5`
C. `x_A=0.7,x_B=0.3`
D. `x_A=0.3,x_B=0.7`

Answer

To solve the problem step by step, we will use Raoult's Law and the concept of vapor-liquid equilibrium.

### Step 1: Write down the given data
- Vapor pressure of component A, \( P_A^0 = 500 \, \text{torr} \)
- Vapor pressure of component B, \( P_B^0 = 800 \, \text{torr} \)
- Initial mole fraction of A, \( x_A = 0.6 \)
- Initial mole fraction of B, \( x_B = 0.4 \)

### Step 2: Calculate the total vapor pressure
Using Raoult's Law, the total vapor pressure \( P_{total} \) can be calculated as:
\[ P_{total} = P_A + P_B = P_A^0 \cdot x_A + P_B^0 \cdot x_B \]
Substituting the values:
\[ P_{total} = (500 \, \text{torr} \cdot 0.6) + (800 \, \text{torr} \cdot 0.4) \]
\[ P_{total} = 300 \, \text{torr} + 320 \, \text{torr} = 620 \, \text{torr} \]

### Step 3: Calculate the mole fractions in the vapor phase
Using Raoult's Law again, we can express the mole fractions in the vapor phase \( y_A \) and \( y_B \):
\[ y_A = \frac{P_A}{P_{total}} = \frac{P_A^0 \cdot x_A}{P_{total}} = \frac{500 \cdot 0.6}{620} \]
Calculating \( y_A \):
\[ y_A = \frac{300}{620} \approx 0.4839 \]

Similarly for \( y_B \):
\[ y_B = \frac{P_B}{P_{total}} = \frac{P_B^0 \cdot x_B}{P_{total}} = \frac{800 \cdot 0.4}{620} \]
Calculating \( y_B \):
\[ y_B = \frac{320}{620} \approx 0.5161 \]

### Step 4: Determine the composition of the last droplet
As the last droplet of liquid remains in equilibrium with the vapor, we can assume that the composition of the liquid will approach the composition of the vapor as the liquid evaporates. Thus, we can set:
\[ x_A = y_A \quad \text{and} \quad x_B = y_B \]

From our calculations:
\[ x_A \approx 0.4839 \quad \text{and} \quad x_B \approx 0.5161 \]

### Step 5: Final composition of the last droplet
Since \( x_B = 1 - x_A \):
\[ x_B \approx 0.5161 \]

Thus, the composition of the last droplet of liquid remaining in equilibrium with vapor is:
- \( x_A \approx 0.4839 \)
- \( x_B \approx 0.5161 \)

### Conclusion
The composition of the last droplet of liquid remaining in equilibrium with vapor is approximately:
- \( x_A \approx 0.484 \)
- \( x_B \approx 0.516 \)