At `40^(@)C`, the vapour pressures in torr, of methyl alcohol and ethyl alcohol solutions is represented by the equation. `P = 119X_(A) + 135` where `X_(A)` is mole fraction of methyl alcohol, then the value of `underset(x_(A)rarr1)(lim)(P_(A))/(X_(A))` is

Options

A. 254 torr
B. 135 torr
C. 119 torr
D. 140 torr

Explanation

To solve the problem, we need to evaluate the limit of the ratio \( \frac{P_A}{X_A} \) as \( X_A \) approaches 1, where \( P \) is given by the equation:

\[ P = 119X_A + 135 \]

### Step-by-Step Solution:

1. **Understand the Equation**: The equation provided represents the vapor pressure \( P \) of a solution of methyl alcohol and ethyl alcohol as a function of the mole fraction \( X_A \) of methyl alcohol.

2. **Identify \( P_A \)**: According to Raoult's law for an ideal solution, the partial vapor pressure \( P_A \) of component A (methyl alcohol) can be expressed as:
\[ P_A = P \cdot X_A \]
where \( P \) is the total vapor pressure of the solution.

3. **Substitute the Given Equation**: We can substitute the given equation into the expression for \( P_A \):
\[ P_A = (119X_A + 135) \cdot X_A \]

4. **Simplify \( P_A \)**: Expanding this gives:
\[ P_A = 119X_A^2 + 135X_A \]

5. **Set Up the Limit**: We need to find the limit of \( \frac{P_A}{X_A} \) as \( X_A \) approaches 1:
\[ \frac{P_A}{X_A} = \frac{119X_A^2 + 135X_A}{X_A} \]

6. **Simplify the Expression**: This simplifies to:
\[ \frac{P_A}{X_A} = 119X_A + 135 \]

7. **Evaluate the Limit**: Now, we take the limit as \( X_A \) approaches 1:
\[ \lim_{X_A \to 1} (119X_A + 135) = 119(1) + 135 = 119 + 135 = 254 \]

8. **Final Answer**: Therefore, the value of \( \lim_{X_A \to 1} \frac{P_A}{X_A} \) is:
\[ \boxed{254} \]