Solid Ammonium carbamate dissociates as:
`NH_(2)COONH_(4)(s)hArr2NH_(3)(g)+CO_(2)(g).`
In a closed vessel, solid ammonium carbamate is in equilibrium with its dissociation products. At equilibrium, ammonia is added such that the partial pressure of `NH_(3)` at new equilibrium now equals the original total pressure. Calculate the ratio of total pressure at new equilibrium to that of original total pressure.

Options

A. `4`
B. `9`
C. `(31)/(27)`
D. `(2)/(9)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Write the dissociation reaction
The dissociation of solid ammonium carbamate can be represented as:
\[ \text{NH}_2\text{COONH}_4(s) \rightleftharpoons 2\text{NH}_3(g) + \text{CO}_2(g) \]

### Step 2: Define the initial conditions
Let the total pressure at equilibrium before adding ammonia be \( P \).

### Step 3: Determine the partial pressures at equilibrium
From the stoichiometry of the reaction:
- The partial pressure of ammonia (\( P_{\text{NH}_3} \)) is \( \frac{2}{3}P \)
- The partial pressure of carbon dioxide (\( P_{\text{CO}_2} \)) is \( \frac{1}{3}P \)

### Step 4: Calculate the equilibrium constant \( K_p \)
The equilibrium constant \( K_p \) is given by:
\[ K_p = \frac{(P_{\text{NH}_3})^2 \cdot P_{\text{CO}_2}}{1} \]
Substituting the values we found:
\[ K_p = \left(\frac{2}{3}P\right)^2 \cdot \left(\frac{1}{3}P\right) \]
\[ K_p = \frac{4}{9}P^2 \cdot \frac{1}{3}P = \frac{4}{27}P^3 \]

### Step 5: Add ammonia and establish new equilibrium
When ammonia is added, the new partial pressure of ammonia becomes equal to the original total pressure \( P \). Thus:
\[ P'_{\text{NH}_3} = P \]

### Step 6: Use \( K_p \) to find new partial pressure of \( CO_2 \)
At the new equilibrium, we can express \( K_p \) again:
\[ K_p = \frac{(P')_{\text{NH}_3}^2 \cdot (P')_{\text{CO}_2}}{1} \]
Substituting \( P'_{\text{NH}_3} = P \):
\[ K_p = \frac{P^2 \cdot (P')_{\text{CO}_2}}{1} \]
Setting this equal to the previous expression for \( K_p \):
\[ \frac{4}{27}P^3 = P^2 \cdot (P')_{\text{CO}_2} \]
Solving for \( (P')_{\text{CO}_2} \):
\[ (P')_{\text{CO}_2} = \frac{4}{27}P \]

### Step 7: Calculate the new total pressure
The new total pressure \( P_{\text{new}} \) is the sum of the new partial pressures:
\[ P_{\text{new}} = P'_{\text{NH}_3} + (P')_{\text{CO}_2} = P + \frac{4}{27}P \]
\[ P_{\text{new}} = P \left(1 + \frac{4}{27}\right) = P \left(\frac{27 + 4}{27}\right) = P \left(\frac{31}{27}\right) \]

### Step 8: Calculate the ratio of new total pressure to original total pressure
The ratio of the new total pressure to the original total pressure is:
\[ \frac{P_{\text{new}}}{P} = \frac{\frac{31}{27}P}{P} = \frac{31}{27} \]

### Final Answer
Thus, the ratio of total pressure at new equilibrium to that of original total pressure is:
\[ \frac{31}{27} \]

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