The vapour pressures of pure liquids A and B are 0.600 bar and 0.933 bar respectively, at a certain temperature. What is the mole fraction of solute when the total vapour pressure of their mixture is 0.8 bar?

Answer

To solve the problem, we need to find the mole fraction of solute B in a mixture of two liquids A and B, given their vapor pressures and the total vapor pressure of the mixture.

### Step-by-Step Solution:

1. **Understand the Given Data:**
- Vapor pressure of pure liquid A, \( P^0_A = 0.600 \, \text{bar} \)
- Vapor pressure of pure liquid B, \( P^0_B = 0.933 \, \text{bar} \)
- Total vapor pressure of the mixture, \( P_{\text{total}} = 0.800 \, \text{bar} \)

2. **Use Dalton's Law of Partial Pressures:**
According to Dalton's law, the total vapor pressure of the mixture is the sum of the partial pressures of each component:
\[ P_{\text{total}} = P_A + P_B \]

3. **Apply Raoult's Law:**
Raoult's law states that the partial pressure of each component in the mixture is equal to the vapor pressure of the pure component multiplied by its mole fraction in the mixture:
\[ P_A = P^0_A \cdot x_A \]
\[ P_B = P^0_B \cdot x_B \]
where \( x_A \) and \( x_B \) are the mole fractions of components A and B, respectively.

4. **Express Mole Fractions:**
Since \( x_A + x_B = 1 \), we can express \( x_A \) in terms of \( x_B \):
\[ x_A = 1 - x_B \]

5. **Substitute into the Total Pressure Equation:**
Substitute \( P_A \) and \( P_B \) into the total pressure equation:
\[ P_{\text{total}} = P^0_A \cdot (1 - x_B) + P^0_B \cdot x_B \]
Plugging in the values:
\[ 0.800 = 0.600 \cdot (1 - x_B) + 0.933 \cdot x_B \]

6. **Expand and Rearrange the Equation:**
Expanding the equation gives:
\[ 0.800 = 0.600 - 0.600 x_B + 0.933 x_B \]
Combining the terms:
\[ 0.800 = 0.600 + (0.933 - 0.600) x_B \]
\[ 0.800 = 0.600 + 0.333 x_B \]

7. **Isolate \( x_B \):**
Rearranging the equation to isolate \( x_B \):
\[ 0.800 - 0.600 = 0.333 x_B \]
\[ 0.200 = 0.333 x_B \]
\[ x_B = \frac{0.200}{0.333} \approx 0.600 \]

8. **Final Result:**
The mole fraction of solute B in the mixture is approximately \( x_B \approx 0.600 \).