For a binary ideal liquid solution, the total vapour of the solution is given as:

Options

A. `P_("total")=P_A^(@)+(P_A^(@)-P_B^(@))X_(B)`
B. `P_("total")=P_B^(@)+(P_A^(@)-P_B^(@))X_(A)`
C. `P_("total")=P_B^(@)+(P_B^(@)-P_A^(@))X_(A)`
D. `P_("total")=P_B^(@)+(P_B^(@)-P_A^(@))X_(B)`

Answer

To solve the question regarding the total vapor pressure of a binary ideal liquid solution, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Components**:
We have a binary solution consisting of two components, A and B.

2. **Applying Raoult's Law**:
According to Raoult's Law, the total vapor pressure (P_total) of the solution is the sum of the partial pressures of the components:
\[ P_{\text{total}} = P_A + P_B \]
where \( P_A \) is the partial pressure of component A and \( P_B \) is the partial pressure of component B.

3. **Calculating Partial Pressures**:
The partial pressures can be expressed in terms of the mole fractions and the vapor pressures of the pure components:
\[ P_A = x_A \cdot P_{0A} \]
\[ P_B = x_B \cdot P_{0B} \]
where \( x_A \) and \( x_B \) are the mole fractions of components A and B, respectively, and \( P_{0A} \) and \( P_{0B} \) are the vapor pressures of the pure components A and B.

4. **Using Mole Fraction Relationship**:
Since we have a binary solution, the sum of the mole fractions is equal to 1:
\[ x_A + x_B = 1 \]
This allows us to express \( x_B \) in terms of \( x_A \):
\[ x_B = 1 - x_A \]

5. **Substituting into Total Pressure Equation**:
Now, substitute the expressions for \( P_A \) and \( P_B \) into the total pressure equation:
\[ P_{\text{total}} = x_A \cdot P_{0A} + (1 - x_A) \cdot P_{0B} \]

6. **Simplifying the Equation**:
Expanding the equation gives:
\[ P_{\text{total}} = x_A \cdot P_{0A} + P_{0B} - x_A \cdot P_{0B} \]
Combining like terms results in:
\[ P_{\text{total}} = P_{0B} + x_A \cdot (P_{0A} - P_{0B}) \]

7. **Final Expression**:
Thus, the total vapor pressure of the binary ideal liquid solution is:
\[ P_{\text{total}} = P_{0B} + x_A \cdot (P_{0A} - P_{0B}) \]

### Conclusion:
The total vapor pressure of the binary ideal liquid solution is given by the equation:
\[ P_{\text{total}} = P_{0B} + x_A \cdot (P_{0A} - P_{0B}) \]