A binary solution contains `x_1 and x_2` mile fraction of two components having vapour pressure `p_1^@ and p_2^@` in this pure states. The total vapour pressure above the solution is

Options

A. `(p_1^@-P_2^@)x_1+P_2^@`
B. `(p_2^@-P_1^@)x_1+P_2^@`
C. `(p_1^@-P_2^@)x_1+P_1^@`
D. `(p_2^@-P_1^@)x_1+P_1^@`

Answer

To solve the problem of finding the total vapor pressure above a binary solution with given mole fractions and vapor pressures, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Components**: We have a binary solution consisting of two components, which we will denote as Component 1 and Component 2. The mole fractions of these components in the solution are \( x_1 \) and \( x_2 \), respectively.

2. **Define the Relationship of Mole Fractions**: Since we are dealing with a binary solution, the sum of the mole fractions must equal 1:
\[ x_1 + x_2 = 1 \]
From this, we can express \( x_2 \) in terms of \( x_1 \):
\[ x_2 = 1 - x_1 \]

3. **Identify the Vapor Pressures**: The vapor pressures of the pure components are given as \( P_1^0 \) for Component 1 and \( P_2^0 \) for Component 2.

4. **Calculate Partial Pressures**: The partial vapor pressure of each component in the solution can be calculated using Raoult's Law, which states that the partial pressure of each component is equal to the product of its mole fraction and its pure vapor pressure:
- For Component 1:
\[ P_1 = x_1 \cdot P_1^0 \]
- For Component 2:
\[ P_2 = x_2 \cdot P_2^0 = (1 - x_1) \cdot P_2^0 \]

5. **Calculate Total Vapor Pressure**: The total vapor pressure \( P_{total} \) above the solution is the sum of the partial pressures of both components:
\[ P_{total} = P_1 + P_2 \]
Substituting the expressions for \( P_1 \) and \( P_2 \):
\[ P_{total} = x_1 \cdot P_1^0 + (1 - x_1) \cdot P_2^0 \]

6. **Simplify the Expression**: We can simplify the total vapor pressure expression:
\[ P_{total} = x_1 \cdot P_1^0 + P_2^0 - x_1 \cdot P_2^0 \]
\[ P_{total} = P_2^0 + x_1 \cdot (P_1^0 - P_2^0) \]

### Final Answer:
Thus, the total vapor pressure above the solution is given by:
\[ P_{total} = P_2^0 + x_1 \cdot (P_1^0 - P_2^0) \]