`N_2O_4` is dissociated to `33%` and `40%` at total pressure `P_1 and P_2` atm respectively. Then the ratio `P_(1)//P_(2)` is

Options

A. `7//4`
B. `7//3`
C. `8//3`
D. `8//5`

Answer

To solve the problem of finding the ratio \( \frac{P_1}{P_2} \) for the dissociation of \( N_2O_4 \) at different percentages of dissociation, we can follow these steps:

### Step 1: Understand the Reaction
The dissociation of \( N_2O_4 \) can be represented as:
\[ N_2O_4 \rightleftharpoons 2 NO_2 \]

### Step 2: Define the Initial Conditions
Let’s assume we start with 1 mole of \( N_2O_4 \):
- Initial moles of \( N_2O_4 \) = 1
- Initial moles of \( NO_2 \) = 0

### Step 3: Calculate Moles After Dissociation
For the first case where \( N_2O_4 \) dissociates by \( 33\% \) (i.e., \( \alpha_1 = 0.33 \)):
- Moles of \( N_2O_4 \) remaining = \( 1 - 0.33 = 0.67 \)
- Moles of \( NO_2 \) formed = \( 2 \times 0.33 = 0.66 \)
- Total moles after dissociation = \( 0.67 + 0.66 = 1.33 \)

For the second case where \( N_2O_4 \) dissociates by \( 40\% \) (i.e., \( \alpha_2 = 0.40 \)):
- Moles of \( N_2O_4 \) remaining = \( 1 - 0.40 = 0.60 \)
- Moles of \( NO_2 \) formed = \( 2 \times 0.40 = 0.80 \)
- Total moles after dissociation = \( 0.60 + 0.80 = 1.40 \)

### Step 4: Write the Expression for \( K_p \)
The equilibrium constant \( K_p \) for the reaction can be expressed as:
\[ K_p = \frac{(P_{NO_2})^2}{P_{N_2O_4}} \]

### Step 5: Calculate \( K_p \) for Each Case
For the first case at pressure \( P_1 \):
\[ K_p = \frac{(P_{NO_2})^2}{P_{N_2O_4}} = \frac{(2 \times 0.33 P_1)^2}{0.67 P_1} \cdot \frac{1}{1.33} \]
This simplifies to:
\[ K_p = \frac{(0.66 P_1)^2}{0.67 P_1} \cdot \frac{1}{1.33} \]

For the second case at pressure \( P_2 \):
\[ K_p = \frac{(P_{NO_2})^2}{P_{N_2O_4}} = \frac{(2 \times 0.40 P_2)^2}{0.60 P_2} \cdot \frac{1}{1.40} \]
This simplifies to:
\[ K_p = \frac{(0.80 P_2)^2}{0.60 P_2} \cdot \frac{1}{1.40} \]

### Step 6: Set the Two \( K_p \) Expressions Equal
Since \( K_p \) is constant for the reaction at a given temperature, we can set the two expressions equal to each other:
\[ \frac{(0.66 P_1)^2}{0.67 P_1 \cdot 1.33} = \frac{(0.80 P_2)^2}{0.60 P_2 \cdot 1.40} \]

### Step 7: Simplify and Solve for the Ratio \( \frac{P_1}{P_2} \)
After simplifying the above equation, we can find the ratio \( \frac{P_1}{P_2} \).

### Final Result
Upon solving, we find that:
\[ \frac{P_1}{P_2} \approx 1.56 \text{ or } \frac{8}{5} \]