What is the composition of last droplet of liquid remaining in equilibrium with vapour?

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 question regarding the composition of the last droplet of liquid remaining in equilibrium with vapor, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Concept of Equilibrium**:
- In a system where a liquid is in equilibrium with its vapor, the composition of the liquid and vapor phases will be related. The mole fractions of the components in both phases will be equal at equilibrium.

2. **Setting Up the Problem**:
- Let’s denote component A and component B. We are given that the mole fraction of A in the vapor phase (YA) is 0.6. Therefore, the mole fraction of B in the vapor phase (YB) will be 1 - YA = 0.4.

3. **Using Raoult's Law**:
- According to Raoult's Law, the partial vapor pressure of a component in the vapor phase can be expressed as:
\[ Y_A = \frac{P^0_A \cdot X_A}{P^0_A \cdot X_A + P^0_B \cdot (1 - X_A)} \]
- Here, \(P^0_A\) and \(P^0_B\) are the vapor pressures of pure components A and B, respectively, and \(X_A\) is the mole fraction of A in the liquid phase.

4. **Substituting Known Values**:
- We are given:
- \(P^0_A = 500 \, \text{mmHg}\)
- \(P^0_B = 800 \, \text{mmHg}\)
- \(Y_A = 0.6\)
- Substituting these values into the equation gives:
\[ 0.6 = \frac{500 \cdot X_A}{500 \cdot X_A + 800 \cdot (1 - X_A)} \]

5. **Solving the Equation**:
- Cross-multiplying to eliminate the fraction:
\[ 0.6 \cdot (500 \cdot X_A + 800 \cdot (1 - X_A)) = 500 \cdot X_A \]
- Expanding and simplifying:
\[ 300 \cdot X_A + 480 - 480 \cdot X_A = 500 \cdot X_A \]
\[ 480 = 500 \cdot X_A + 480 \cdot X_A - 300 \cdot X_A \]
\[ 480 = (500 - 300 + 480) \cdot X_A \]
\[ 480 = 680 \cdot X_A \]
\[ X_A = \frac{480}{680} = 0.7059 \approx 0.70 \]

6. **Finding the Composition of B**:
- Since \(X_B = 1 - X_A\):
\[ X_B = 1 - 0.70 = 0.30 \]

7. **Conclusion**:
- The composition of the last droplet of liquid remaining in equilibrium with vapor is:
- Mole fraction of A (\(X_A\)) = 0.70
- Mole fraction of B (\(X_B\)) = 0.30

### Final Answer:
- The composition of the last droplet of liquid is:
- \(X_A = 0.70\)
- \(X_B = 0.30\)