# The dissociation of ammonium carbamate occurs as follows - Ammonium carbamate(s) ⇌ 2NH₃(g) + CO₂(g) Ammonia is added to ammonium carbamate kept at equilibrium in a vessel so that the partial pressure of ammonia becomes equal to the total initial pressure. The ratio of the partial pressure of CO₂ in the new equilibrium to the partial pressure of CO₂ at the beginning is: A 4 B 9 C 4/9 D 2/9

## Question

The dissociation of ammonium carbamate occurs as follows -

$$ \mathrm{NH_2COONH_4}(s) \Leftrightarrow 2\mathrm{NH_3}(g) + \mathrm{CO_2}(g) $$

Ammonia is added to ammonium carbamate kept at equilibrium in a vessel so that the partial pressure of ammonia becomes equal to the total initial pressure. The ratio of the partial pressure of $\mathrm{CO}_2$ in the new equilibrium to the partial pressure of $\mathrm{CO}_2$ at the beginning is:

A 4

B 9

C $\frac{4}{9}$

D $\frac{2}{9}$

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## Answer

The correct answer is:

**C**
$$
\mathrm{NH}*{2} \mathrm{COONH}*{4}(s) \Leftrightarrow 2 \mathrm{NH}*{3}(g) + \mathrm{CO}*{2}(g)
$$

Let's assume that **the partial pressure of $\mathrm{CO}_{2}$ (carbon dioxide) at equilibrium is** $P$. Thus, **the partial pressure of $\mathrm{NH}_{3}$ (ammonia) would be** $2P$.

At equilibrium:
$$
P_{\mathrm{NH}*3} = 2P
$$
and the total pressure would be:
$$
P*{\mathrm{total}} = 3P
$$

The equilibrium constant for the reaction in terms of pressure ($K_p$) can be expressed as: $$ K_p = (2P)^2 \times P = 4P^3 $$

Now, **if additional $\mathrm{NH}_3$ is added** such that **the equilibrium partial pressure of $\mathrm{NH}_3$ becomes $3P$**, we can set up the equation as follows:

$$
K_p = 4P^3 = (P_{\mathrm{NH}*3})^2 \times P*{\mathrm{CO}_2}
$$

Given the new partial pressure of $\mathrm{NH}*3$:
$$
P*{\mathrm{NH}_3} = 3P
$$

So, substituting this back into the $K_p$ equation:
$$
K_p = (3P)^2 \times P_{\mathrm{CO}*2} = 9P^2 \times P*{\mathrm{CO}_2}
$$

Since $K_p = 4P^3$: $$ 9P^2 \times P_{\mathrm{CO}_2} = 4P^3 $$

Solving for $P_{\mathrm{CO}*2}$:
$$
P*{\mathrm{CO}_2} = \frac{4P^3}{9P^2} = \frac{4P}{9}
$$

Thus, the ratio of the initial partial pressure of $\mathrm{CO}_2$ to the new partial pressure of $\mathrm{CO}*2$ is:
$$
\frac{P*{\mathrm{CO}*2(new)}}{P*{\mathrm{CO}_2(initial)}} = \frac{\frac{4P}{9}}{P} = \frac{4}{9}
$$

Therefore, **the final answer is:**

**C. $\frac{4}{9}$**

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