Two miscible liquids `A` and `B` having vapour pressure in pure state `P_(A)^(@)` and `P_(B)^(@)` are mixed in mole fraction `chi_(A)` and `chi_(B)` to get a mixtue having total vapour vapour pressure of mixture `P_(M)`. Which of the following relations are correct?

Options

A. `chi_(A)=(P_(M)-P_(B)^(@))/(P_(A)^(@)-P_(B)^(@))`
B. `(chi_(A)(l))/(chi_(A)(V))=P_(M)/P_(A)^(@)`
C. `(chi_(A)(l))/(chi'_(A)(V))=P_(M)/P_(B)^(@)`
D. All of these

Explanation

To solve the problem, we need to understand the relationship between the vapor pressures of the components in a mixture of two miscible liquids A and B. The total vapor pressure of the mixture can be calculated using Raoult's Law.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Vapor pressure of pure liquid A: \( P_A^0 \)
- Vapor pressure of pure liquid B: \( P_B^0 \)
- Mole fraction of liquid A in the mixture: \( \chi_A \)
- Mole fraction of liquid B in the mixture: \( \chi_B \) (where \( \chi_B = 1 - \chi_A \))
- Total vapor pressure of the mixture: \( P_M \)

2. **Apply Raoult's Law:**
According to Raoult's Law, the partial vapor pressure of each component in the mixture is given by:
\[ P_A = P_A^0 \cdot \chi_A \]
\[ P_B = P_B^0 \cdot \chi_B \]

3. **Calculate Total Vapor Pressure:**
The total vapor pressure of the mixture \( P_M \) is the sum of the partial pressures:
\[ P_M = P_A + P_B \]
Substituting the expressions for \( P_A \) and \( P_B \):
\[ P_M = P_A^0 \cdot \chi_A + P_B^0 \cdot \chi_B \]
Replacing \( \chi_B \) with \( 1 - \chi_A \):
\[ P_M = P_A^0 \cdot \chi_A + P_B^0 \cdot (1 - \chi_A) \]

4. **Simplify the Equation:**
Expanding the equation:
\[ P_M = P_A^0 \cdot \chi_A + P_B^0 - P_B^0 \cdot \chi_A \]
Combining like terms:
\[ P_M = (P_A^0 - P_B^0) \cdot \chi_A + P_B^0 \]

5. **Rearranging the Equation:**
To express \( \chi_A \) in terms of \( P_M \):
\[ P_M - P_B^0 = (P_A^0 - P_B^0) \cdot \chi_A \]
Thus,
\[ \chi_A = \frac{P_M - P_B^0}{P_A^0 - P_B^0} \]

6. **Determine the Vapor Phase Composition:**
The mole fraction of component A in the vapor phase \( y_A \) can be given by:
\[ y_A = \frac{P_A}{P_M} = \frac{P_A^0 \cdot \chi_A}{P_M} \]
Substituting \( \chi_A \):
\[ y_A = \frac{P_A^0 \cdot \frac{P_M - P_B^0}{P_A^0 - P_B^0}}{P_M} \]

### Conclusion:
The relationships derived from the above steps are consistent with the principles of Raoult's Law and the behavior of ideal solutions. The correct relations can be summarized as:
1. \( P_M = P_A^0 \cdot \chi_A + P_B^0 \cdot \chi_B \)
2. \( \chi_A = \frac{P_M - P_B^0}{P_A^0 - P_B^0} \)
3. \( y_A = \frac{P_A^0 \cdot \chi_A}{P_M} \)