The plots of `(1)/(X_(A))vs.(1)/(Y_(A))` (where `X_(A)` and `Y_(A)` are the mole fraction of liquid `A` in liquid and vapour phase respectively) is linear with slope and intercept respectively are given as:
(a)`(P_A^(@))/(P_B^(@)),(P_B^(@)-P_A^(@))/(P_B^(@))`
(b)`(P_B^(@))/(P_A^(@)),(P_A^(@)-P_B^(@))/(P_A^(@))`
(c)`(P_B^(@))/(P_A^(@)),(P_B^(@))/(P_B^@-P_A^(@))`
(d)`P_A^(@)-P_B^(@),(P_A^(@))/(P_B^(@))`

Options

A. `(P_A^(@))/(P_B^(@)),(P_B^(@)-P_A^(@))/(P_B^(@))`
B. `(P_B^(@))/(P_A^(@)),(P_A^(@)-P_B^(@))/(P_A^(@))`
C. `(P_B^(@))/(P_A^(@)),(P_B^(@))/(P_B^@-P_A^(@))`
D. `P_A^(@)-P_B^(@),(P_A^(@))/(P_B^(@))`

Answer

To solve the problem, we need to analyze the relationship between the mole fractions of component A in the liquid phase \(X_A\) and in the vapor phase \(Y_A\). We will derive the slope and intercept of the plot of \(\frac{1}{X_A}\) versus \(\frac{1}{Y_A}\).

### Step 1: Understand the relationship between partial pressures and mole fractions
The partial pressure of component A in the liquid phase is given by:
\[ P_A = P_A^0 \cdot X_A \]
where \(P_A^0\) is the vapor pressure of pure A.

Similarly, for component B:
\[ P_B = P_B^0 \cdot X_B \]
where \(X_B\) is the mole fraction of B in the liquid phase.

### Step 2: Express the partial pressures in terms of vapor phase mole fractions
In the vapor phase, the partial pressures can be expressed as:
\[ P_A = P \cdot Y_A \]
\[ P_B = P \cdot Y_B \]
where \(P\) is the total pressure and \(Y_A\) and \(Y_B\) are the mole fractions of A and B in the vapor phase, respectively.

### Step 3: Relate the mole fractions
Since the total mole fractions in both phases must equal 1, we have:
\[ X_B = 1 - X_A \]
\[ Y_B = 1 - Y_A \]

### Step 4: Set up the equations
From the equations for the partial pressures, we can write:
\[ \frac{P_A^0 \cdot X_A}{P \cdot Y_A} = 1 \]
\[ \frac{P_B^0 \cdot (1 - X_A)}{P \cdot (1 - Y_A)} = 1 \]

### Step 5: Divide the equations
Dividing the first equation by the second gives:
\[ \frac{P_A^0}{P_B^0} = \frac{Y_A}{1 - Y_A} \cdot \frac{1 - X_A}{X_A} \]

### Step 6: Rearranging the equation
Rearranging this equation, we can express it in the form suitable for linear analysis:
\[ \frac{P_A^0}{Y_A} - P_A^0 = \frac{P_B^0}{X_A} - P_B^0 \]

### Step 7: Identify slope and intercept
This can be rearranged to:
\[ \frac{1}{X_A} = \frac{P_A^0}{P_B^0} \cdot \frac{1}{Y_A} + \frac{P_B^0 - P_A^0}{P_B^0} \]
From this equation, we can identify:
- Slope \(m = \frac{P_A^0}{P_B^0}\)
- Intercept \(c = \frac{P_B^0 - P_A^0}{P_B^0}\)

### Conclusion
Thus, the correct slope and intercept from the options given are:
- Slope: \(\frac{P_A^0}{P_B^0}\)
- Intercept: \(\frac{P_B^0 - P_A^0}{P_B^0}\)

Therefore, the correct answer is option (a):
\[ \left(\frac{P_A^0}{P_B^0}, \frac{P_B^0 - P_A^0}{P_B^0}\right) \]