Mole fraction of component `A` in vapour phase is `chi_(1)` and that of component `A` in liquid mixture is `chi_2`, then (`p_(A)^@`)= vapour pressure of pure A, `p_(B)^@` = vapour pressure of pure B), the total vapour pressure of liquid mixture is

Options

A. `p_(A)^@(chi_(2))/(chi_(1))`
B. `p_(A)^@(chi_(1))/(chi_(2))`
C. `p_(B)^@(chi_(1))/(chi_(2))`
D. `p_(B)@^(chi_(2))/(chi_(1))`

Explanation

To find the total vapor pressure of the liquid mixture, we can follow these steps:

### Step 1: Understand the Definitions
We are given:
- \( \chi_1 \): Mole fraction of component A in the vapor phase.
- \( \chi_2 \): Mole fraction of component A in the liquid mixture.
- \( P_A^0 \): Vapor pressure of pure A.
- \( P_B^0 \): Vapor pressure of pure B.

### Step 2: Write the Expression for Partial Pressure of A
The partial pressure of component A in the vapor phase can be expressed as:
\[ P_A = P_A^0 \cdot \chi_2 \]
This equation states that the partial pressure of A is equal to the vapor pressure of pure A multiplied by the mole fraction of A in the liquid phase.

### Step 3: Relate Mole Fraction in Vapor Phase to Total Pressure
The mole fraction of A in the vapor phase can also be expressed in terms of the total pressure:
\[ \chi_1 = \frac{P_A}{P_{total}} \]
From this, we can rearrange to find the total pressure:
\[ P_{total} = \frac{P_A}{\chi_1} \]

### Step 4: Substitute the Expression for Partial Pressure of A
Now, substitute the expression for \( P_A \) from Step 2 into the equation from Step 3:
\[ P_{total} = \frac{P_A^0 \cdot \chi_2}{\chi_1} \]

### Final Expression
Thus, the total vapor pressure of the liquid mixture is given by:
\[ P_{total} = \frac{P_A^0 \cdot \chi_2}{\chi_1} \]