`NH_4COONH_2 (s) hArr 2NH_3(g)+CO_(2)(g)`
If equilibrium pressure is 7 atm for the above reaction, `K_p` will be :(volume and temperature are constant)

Options

A. 21.7
B. 27
C. 23
D. 78

Explanation

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 7 atm, we can follow these steps:

### Step 1: Identify the reaction components and their moles
In the reaction:
- 1 mole of \( NH_4COONH_2 \) (solid) does not contribute to the equilibrium expression.
- 2 moles of \( NH_3 \) (gas) are produced.
- 1 mole of \( CO_2 \) (gas) is produced.

### Step 2: Determine the total number of moles of gas at equilibrium
At equilibrium, the total number of moles of gas is:
\[ n_{total} = 2 \text{ (from } NH_3\text{)} + 1 \text{ (from } CO_2\text{)} = 3 \text{ moles} \]

### Step 3: Relate the total pressure to the partial pressures
Given that the total equilibrium pressure \( P_{total} \) is 7 atm, we can express the partial pressures in terms of a variable \( P \):
- The partial pressure of \( NH_3 \) is \( 2P \) (since there are 2 moles).
- The partial pressure of \( CO_2 \) is \( P \) (since there is 1 mole).

The total pressure can be expressed as:
\[ P_{total} = P_{NH_3} + P_{CO_2} = 2P + P = 3P \]

### Step 4: Solve for \( P \)
Since \( P_{total} = 7 \) atm, we can set up the equation:
\[ 3P = 7 \implies P = \frac{7}{3} \text{ atm} \]

### Step 5: Write the expression for \( K_p \)
The equilibrium constant \( K_p \) is given by:
\[ K_p = \frac{(P_{NH_3})^2 \cdot P_{CO_2}}{P_{NH_4COONH_2}} \]
Since \( NH_4COONH_2 \) is a solid, it does not appear in the expression. Thus:
\[ K_p = \frac{(2P)^2 \cdot P}{1} = 4P^2 \]

### Step 6: Substitute \( P \) into the \( K_p \) expression
Now substituting \( P = \frac{7}{3} \) atm into the equation:
\[ K_p = 4 \left(\frac{7}{3}\right)^2 \]
Calculating this:
\[ K_p = 4 \cdot \frac{49}{9} = \frac{196}{9} \]
Calculating \( \frac{196}{9} \) gives:
\[ K_p \approx 21.78 \]

### Final Answer
Thus, the value of \( K_p \) is approximately \( 21.78 \).

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