At a given temperature total vapour pressure (in Torr) of a mixture of volatile components A and B is given by `P = 120 - 75 X_(B)` hence vapour pressure of pure A and B respectively (in Torr) are

Options

A. 120,75
B. 120,195
C. 120,45
D. 75,45

Answer

To find the vapor pressures of pure components A and B from the given total vapor pressure equation, we can follow these steps:

### Step 1: Understand the given equation
The total vapor pressure \( P \) of the mixture is given by the equation:
\[ P = 120 - 75 X_B \]
where \( X_B \) is the mole fraction of component B.

### Step 2: Apply Raoult's Law
According to Raoult's Law, the total vapor pressure of a solution containing two volatile components A and B can be expressed as:
\[ P = P^0_A \cdot X_A + P^0_B \cdot X_B \]
where:
- \( P^0_A \) is the vapor pressure of pure A,
- \( P^0_B \) is the vapor pressure of pure B,
- \( X_A \) is the mole fraction of A, and
- \( X_B \) is the mole fraction of B.

### Step 3: Relate mole fractions
Since there are only two components, the mole fractions must satisfy:
\[ X_A + X_B = 1 \]
Thus, we can express \( X_A \) as:
\[ X_A = 1 - X_B \]

### Step 4: Substitute \( X_A \) in the Raoult's Law equation
Substituting \( X_A \) in the Raoult's Law equation gives:
\[ P = P^0_A \cdot (1 - X_B) + P^0_B \cdot X_B \]
Expanding this, we have:
\[ P = P^0_A - P^0_A \cdot X_B + P^0_B \cdot X_B \]
Rearranging terms, we get:
\[ P = P^0_A - (P^0_A - P^0_B) \cdot X_B \]

### Step 5: Compare coefficients
Now, we can compare this equation with the given equation \( P = 120 - 75 X_B \). From this comparison, we can identify:
1. The coefficient of \( X_B \) gives us:
\[ P^0_A - P^0_B = 75 \]
2. The constant term gives us:
\[ P^0_A = 120 \]

### Step 6: Solve for \( P^0_B \)
Now that we have \( P^0_A = 120 \), we can substitute this value into the equation from step 5:
\[ 120 - P^0_B = 75 \]
Rearranging gives:
\[ P^0_B = 120 - 75 = 45 \]

### Step 7: State the final results
Thus, the vapor pressures of pure A and B are:
- Vapor pressure of pure A, \( P^0_A = 120 \) Torr
- Vapor pressure of pure B, \( P^0_B = 45 \) Torr

### Final Answer
The vapor pressures of pure A and B respectively are \( 120 \) Torr and \( 45 \) Torr.

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