Consider a ideal solution of component 1 and component 2 component 1 being more volatile `x_(1)` and `x_(2)` represent the respective liquid phase compositions at equilibrium while `y_(1),y_(2)` denote the respective vapour phase compositions at equilibrium. `p_(1),p_(2)` are respective partial pressure in vapour phase at equilibrium which of the following is/are necessarly are?

Options

A. `p_(1)gtp_(2)`
B. `y_(1)lty_(2)`
C. `y_(1)gtx_(1)`
D. `x_(2)gty_(2)`

Explanation

To solve the problem regarding the ideal solution of components 1 and 2, we need to analyze the relationships between their liquid and vapor phase compositions at equilibrium, as well as their partial pressures. Here’s a step-by-step solution:

### Step 1: Understand the Components
We have two components in an ideal solution:
- Component 1 (more volatile)
- Component 2

Let:
- \( x_1 \) = mole fraction of component 1 in the liquid phase
- \( x_2 \) = mole fraction of component 2 in the liquid phase
- \( y_1 \) = mole fraction of component 1 in the vapor phase
- \( y_2 \) = mole fraction of component 2 in the vapor phase
- \( P_1 \) = partial pressure of component 1
- \( P_2 \) = partial pressure of component 2
- \( P_{total} \) = total pressure of the system

### Step 2: Apply Raoult's Law
According to Raoult's Law, the partial pressure of each component in the vapor phase is given by:
- \( P_1 = P_{1}^{0} \cdot x_1 \)
- \( P_2 = P_{2}^{0} \cdot x_2 \)

Where \( P_{1}^{0} \) and \( P_{2}^{0} \) are the vapor pressures of the pure components 1 and 2, respectively.

### Step 3: Apply Dalton's Law of Partial Pressures
According to Dalton's Law, the total pressure is the sum of the partial pressures:
\[ P_{total} = P_1 + P_2 \]

### Step 4: Relate Vapor Phase Compositions to Partial Pressures
From Dalton's Law, we can express the mole fractions in the vapor phase as:
- \( y_1 = \frac{P_1}{P_{total}} \)
- \( y_2 = \frac{P_2}{P_{total}} \)

### Step 5: Analyze the Volatility
Since component 1 is more volatile, we know:
\[ P_{1}^{0} > P_{2}^{0} \]

This implies that \( P_1 \) will be greater than \( P_2 \) when the liquid phase compositions are considered.

### Step 6: Establish Relationships
From the equations derived:
1. \( y_1 = \frac{P_{1}^{0} \cdot x_1}{P_{total}} \)
2. \( y_2 = \frac{P_{2}^{0} \cdot x_2}{P_{total}} \)

Since \( P_{1}^{0} > P_{2}^{0} \) and given that \( P_{total} \) is a constant, we can conclude:
- \( y_1 > x_1 \) (because \( P_{1}^{0} \) is greater)
- \( y_2 < x_2 \) (because \( P_{2}^{0} \) is less)

### Step 7: Final Conclusions
From the analysis:
- \( y_1 > x_1 \) implies that the composition of component 1 in the vapor phase is greater than in the liquid phase.
- \( y_2 < x_2 \) implies that the composition of component 2 in the vapor phase is less than in the liquid phase.

### Summary of Results
- \( y_1 > x_1 \)
- \( x_2 > y_2 \)