The total pressure exerted in ideal binary solution is given by `P=P_(A)^(@)X_(A)+P_(B)^(@)X_(B)` where `P_(A)^(@)&P_(B)^(@)` are the respective vapour pressure of pure components and `X_(A)&X_(B)` are their mole fraction in liquid phase. And composition of the vapour phase is determined with the help of Datton's law partial pressure: `Y_(A)=(P_(A)^(@)X_(A))/(P)`
If total pressure exerted in an ideal binary solution is given by `P=(5400)/(60+30Y_(A))mm` of Hg.
The value of `P_(A)^(@)` is:

Options

A. 40 mm of Hg
B. 30 mm of Hg
C. 60 mm of Hg
D. 90 mm of Hg

Answer

To solve the problem, we need to find the value of \( P_A^{(@)} \), the vapor pressure of pure component A in an ideal binary solution, given the total pressure equation and the relationship between the vapor and liquid phases.

### Step-by-Step Solution:

1. **Understand the Given Equation**:
The total pressure \( P \) in an ideal binary solution is given by:
\[ P = P_A^{(@)} X_A + P_B^{(@)} X_B \]
where \( P_A^{(@)} \) and \( P_B^{(@)} \) are the vapor pressures of pure components A and B, and \( X_A \) and \( X_B \) are their mole fractions in the liquid phase.

2. **Use Dalton's Law of Partial Pressure**:
The composition of the vapor phase can be expressed as:
\[ Y_A = \frac{P_A^{(@)} X_A}{P} \]
where \( Y_A \) is the mole fraction of A in the vapor phase.

3. **Substitute the Given Total Pressure**:
The total pressure is given as:
\[ P = \frac{5400}{60 + 30 Y_A} \text{ mm Hg} \]

4. **Set Up the Equation**:
We can express \( P_A^{(@)} \) in terms of \( P \) and \( Y_A \):
\[ Y_A = \frac{P_A^{(@)} X_A}{P} \]
Rearranging gives:
\[ P_A^{(@)} = Y_A \cdot P \cdot \frac{1}{X_A} \]

5. **Assume \( Y_A = 1 \) for Pure Component A**:
When only component A is present, \( Y_A \) can be assumed to be 1. Thus:
\[ P_A^{(@)} = P \cdot \frac{1}{X_A} \]

6. **Substitute \( P \) into the Equation**:
Substitute the expression for \( P \):
\[ P_A^{(@)} = \left(\frac{5400}{60 + 30 \cdot 1}\right) \cdot \frac{1}{X_A} \]
Simplifying gives:
\[ P_A^{(@)} = \frac{5400}{60 + 30} = \frac{5400}{90} = 60 \text{ mm Hg} \]

7. **Conclusion**:
The value of \( P_A^{(@)} \) is:
\[ P_A^{(@)} = 60 \text{ mm Hg} \]