`NH_(3)` at 10 atm and `CO_(2)` at 20 atm pressure are introduced in an evacuated chamber. If `K_(p)` for the reaction:
`NH_(2)COONH_(4)(s) hArr 2NH_(3) (g) +CO_(2)(g)`
is 2000 `"atm"^(3)` at 400K, then the final pressure in the reaction chamber will be:

Options

A. less than 30 atm
B. more than 30 atm
C. equal to 30 atm
D. can not be predicted

Answer

To solve the problem, we need to find the final pressure in the reaction chamber after introducing `NH3` and `CO2` and allowing the reaction to reach equilibrium. The reaction given is:

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

Given:
- Initial pressure of `NH3` = 10 atm
- Initial pressure of `CO2` = 20 atm
- \( K_p \) for the reaction = 2000 atm³ at 400 K

### Step 1: Write the expression for \( K_p \)

The equilibrium constant \( K_p \) for the reaction is given by the expression:

\[ K_p = \frac{(P_{\text{NH}_3})^2 \cdot (P_{\text{CO}_2})}{1} \]

Since the reactant is a solid, its activity is considered to be 1 and does not appear in the expression.

### Step 2: Substitute the initial pressures into the \( K_p \) expression

Let \( P_{\text{NH}_3} \) be the equilibrium pressure of ammonia and \( P_{\text{CO}_2} \) be the equilibrium pressure of carbon dioxide. Initially, we have:

- \( P_{\text{NH}_3} = 10 \) atm
- \( P_{\text{CO}_2} = 20 \) atm

Substituting these values into the \( K_p \) expression:

\[ K_p = \frac{(10)^2 \cdot (20)}{1} = 2000 \text{ atm}^3 \]

### Step 3: Determine the change in pressures at equilibrium

Let \( x \) be the change in pressure due to the reaction reaching equilibrium. The changes in pressures will be:

- \( P_{\text{NH}_3} = 10 + 2x \) (since 2 moles of NH3 are produced)
- \( P_{\text{CO}_2} = 20 + x \)

### Step 4: Set up the equation using \( K_p \)

Now, we can set up the equation using the expression for \( K_p \):

\[ 2000 = \frac{(10 + 2x)^2 \cdot (20 + x)}{1} \]

### Step 5: Solve for \( x \)

Expanding the equation:

\[ 2000 = (10 + 2x)^2 \cdot (20 + x) \]

This is a quadratic equation in terms of \( x \). Solving this will give us the value of \( x \).

### Step 6: Calculate total pressure at equilibrium

The total pressure at equilibrium will be:

\[ P_{\text{total}} = P_{\text{NH}_3} + P_{\text{CO}_2} = (10 + 2x) + (20 + x) = 30 + 3x \]

### Step 7: Substitute \( x \) back to find \( P_{\text{total}} \)

After calculating \( x \), substitute it back into the equation for total pressure to find the final pressure in the chamber.

### Final Answer

After solving the equations, we find that the total pressure in the reaction chamber is equal to 30 atm.

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