`NH_4COONH_2 (s) hArr 2NH_3(g)+CO_(2)(g)`
If equilibrium pressure is 12 atm for the above reaction, `K_p` will be :(volume and temperature are constant)

Options

A. 12
B. 64
C. 36
D. 48

Explanation

To solve the problem, we need to find the equilibrium constant \( K_p \) for the reaction:

\[ NH_4COONH_2 (s) \rightleftharpoons 2NH_3(g) + CO_2(g) \]

Given that the equilibrium pressure is 12 atm, we can follow these steps:

### Step 1: Understand the Reaction
The reaction involves the decomposition of solid ammonium carbamate into gaseous ammonia and carbon dioxide.

### Step 2: Identify Moles of Gases
From the balanced equation:
- 1 mole of \( NH_4COONH_2 \) produces 2 moles of \( NH_3 \) and 1 mole of \( CO_2 \).
- Therefore, the total number of moles of gas produced is \( 2 + 1 = 3 \) moles.

### Step 3: Relate Total Pressure to Partial Pressures
At equilibrium, the total pressure \( P_{total} \) is given as 12 atm. The total pressure is the sum of the partial pressures of the gases:

\[ P_{total} = P_{NH_3} + P_{CO_2} \]

Let \( P \) be the partial pressure of \( CO_2 \). Then, the partial pressures can be expressed as:
- \( P_{NH_3} = 2P \) (since there are 2 moles of \( NH_3 \))
- \( P_{CO_2} = P \)

Thus, the total pressure can be written as:

\[ P_{total} = 2P + P = 3P \]

### Step 4: Solve for Partial Pressure \( P \)
Given that \( P_{total} = 12 \) atm, we can set up the equation:

\[ 3P = 12 \text{ atm} \]

Solving for \( P \):

\[ P = \frac{12}{3} = 4 \text{ atm} \]

### Step 5: Calculate Partial Pressures
Now we can find the partial pressures:
- \( P_{NH_3} = 2P = 2 \times 4 = 8 \text{ atm} \)
- \( P_{CO_2} = P = 4 \text{ atm} \)

### Step 6: Write the Expression for \( K_p \)
The equilibrium constant \( K_p \) is defined as:

\[ K_p = \frac{(P_{NH_3})^2 \cdot (P_{CO_2})}{P_{NH_4COONH_2}} \]

Since \( NH_4COONH_2 \) is a solid, its activity is considered to be 1, so:

\[ K_p = (P_{NH_3})^2 \cdot P_{CO_2} \]

### Step 7: Substitute the Values
Substituting the values we found:

\[ K_p = (8 \text{ atm})^2 \cdot (4 \text{ atm}) = 64 \cdot 4 = 256 \]

### Final Answer
Thus, the value of \( K_p \) is:

\[ \boxed{256} \]