Mixture of voltaile omponents A and B has total vapoure pressure (in torr) : `p=254-135 x_(A)` where `x_(A)` is mole fraction of A in mixture hence `P_(A)^(@) and P_(B)^(@)` are (in torr)

Options

A. `254,119`
B. `119,254`
C. `2135,254`
D. `154,119`

Answer

To solve the problem, we need to determine the pure vapor pressures \( P^0_A \) and \( P^0_B \) for components A and B, respectively, using the given total vapor pressure equation and Raoult's law.

### Step-by-Step Solution:

1. **Understand the Given Equation**:
The total vapor pressure \( P \) of the mixture is given by:
\[ P = 254 - 135 \cdot x_A \]
where \( x_A \) is the mole fraction of component A.

2. **Apply Raoult's Law**:
According to Raoult's law, the partial vapor pressure of each component in a mixture is given by:
\[ P_A = P^0_A \cdot x_A \quad \text{and} \quad P_B = P^0_B \cdot x_B \]
where \( P_A \) and \( P_B \) are the partial pressures of components A and B, and \( P^0_A \) and \( P^0_B \) are their pure vapor pressures.

3. **Express \( x_B \)**:
Since the sum of the mole fractions of A and B must equal 1:
\[ x_B = 1 - x_A \]

4. **Substitute \( x_B \) in Partial Pressure Equation**:
The total pressure can also be expressed as:
\[ P = P_A + P_B = P^0_A \cdot x_A + P^0_B \cdot (1 - x_A) \]
Expanding this gives:
\[ P = P^0_A \cdot x_A + P^0_B - P^0_B \cdot x_A \]
Rearranging this, we get:
\[ P = (P^0_A - P^0_B) \cdot x_A + P^0_B \]

5. **Equate the Two Expressions for Total Pressure**:
Now we can set the two expressions for total pressure equal to each other:
\[ 254 - 135 \cdot x_A = (P^0_A - P^0_B) \cdot x_A + P^0_B \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ 254 - P^0_B = (P^0_A - P^0_B + 135) \cdot x_A \]
This implies that the coefficient of \( x_A \) on the left side must equal the coefficient on the right side.

7. **Identify Coefficients**:
From the equation, we can identify:
\[ P^0_A - P^0_B = -135 \]
and
\[ P^0_B = 254 \]

8. **Solve for \( P^0_A \)**:
Substituting \( P^0_B \) into the first equation:
\[ P^0_A - 254 = -135 \]
This simplifies to:
\[ P^0_A = 254 - 135 = 119 \]

9. **Final Values**:
Thus, we have:
\[ P^0_A = 119 \, \text{torr} \quad \text{and} \quad P^0_B = 254 \, \text{torr} \]

### Summary of Results:
- \( P^0_A = 119 \, \text{torr} \)
- \( P^0_B = 254 \, \text{torr} \)