The value of `K_(P)` for
`NH_(2)COONH_(4(s))hArr 2NH_(3(g))+CO_(2(g))`
if the total pressure at equilibrium is `P` `:-`

Options

A. `(2)/(27)P^(3)`
B. `(4)/(27)P^(3)`
C. `(27)/(2)P^(3)`
D. `(27)/(4)P^(3)`

Explanation

To find the value of \( K_p \) for the reaction

\[ \text{NH}_2\text{COONH}_4(s) \rightleftharpoons 2\text{NH}_3(g) + \text{CO}_2(g) \]

given that the total pressure at equilibrium is \( P \), we can follow these steps:

### Step 1: Understand the Reaction
The reaction involves one solid reactant and two gaseous products. Since the solid does not contribute to the equilibrium expression, we only need to consider the gaseous products.

### Step 2: Set Up the Initial Conditions
Assume that initially, we have 1 mole of the solid reactant, and at equilibrium, let \( x \) be the amount that decomposes. Therefore, the changes in moles will be:

- For \( \text{NH}_2\text{COONH}_4(s) \): \( 1 - x \) (but since it's solid, we consider it as 1)
- For \( \text{NH}_3(g) \): \( 2x \)
- For \( \text{CO}_2(g) \): \( x \)

### Step 3: Calculate Total Moles of Gaseous Products
The total number of gaseous moles at equilibrium will be:

\[ \text{Total moles} = 2x + x = 3x \]

### Step 4: Relate Moles to Total Pressure
Given that the total pressure at equilibrium is \( P \), we can express the partial pressures of the gaseous products in terms of \( P \).

### Step 5: Calculate Partial Pressures
The partial pressure of each gas is given by the mole fraction multiplied by the total pressure \( P \).

1. **Partial Pressure of \( \text{NH}_3 \)**:
\[ P_{\text{NH}_3} = \left(\frac{2x}{3x}\right) P = \frac{2}{3} P \]

2. **Partial Pressure of \( \text{CO}_2 \)**:
\[ P_{\text{CO}_2} = \left(\frac{x}{3x}\right) P = \frac{1}{3} P \]

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

\[ K_p = \frac{(P_{\text{NH}_3})^2 \cdot P_{\text{CO}_2}}{(P_{\text{NH}_2\text{COONH}_4})} \]

Since the solid does not appear in the expression, we have:

\[ K_p = (P_{\text{NH}_3})^2 \cdot P_{\text{CO}_2} \]

### Step 7: Substitute the Partial Pressures into the Expression
Substituting the values we found:

\[ K_p = \left(\frac{2}{3} P\right)^2 \cdot \left(\frac{1}{3} P\right) \]

Calculating this gives:

\[ K_p = \left(\frac{4}{9} P^2\right) \cdot \left(\frac{1}{3} P\right) = \frac{4}{27} P^3 \]

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

\[ K_p = \frac{4}{27} P^3 \]