JEE Chemistry 2023 Question with Solution

Given: Reaction is $A \text{(g)} \rightarrow 2B \text{(g)} + C \text{(g)}$ and it is first order. Initial pressure is $800 \, \text{mm Hg}$. Total pressure becomes $1600 \, \text{mm Hg}$ after $10 \, \text{min}$.

Find: Total pressure after $30 \, \text{min}$.

For the reaction, if pressure of $A$ decomposed is $x$, then partial pressures are:

$$P_A = 800 - x, \quad P_B = 2x, \quad P_C = x$$

So total pressure at any time is:

$$P_{\text{total}} = (800 - x) + 2x + x = 800 + 2x$$

At $t = 10 \, \text{min}$, total pressure is $1600 \, \text{mm Hg}$. Therefore,

$$800 + 2x = 1600$$ $$2x = 800$$ $$x = 400$$

Hence pressure of $A$ left after $10 \, \text{min}$ is:

$$P_A = 800 - 400 = 400 \, \text{mm Hg}$$

Thus after $10 \, \text{min}$, pressure of $A$ becomes half of its initial value, so the half-life is $10 \, \text{min}$.

After $30 \, \text{min} = 3t_{1/2}$,

$$P_A = 800\left(\frac{1}{2}\right)^3 = 100 \, \text{mm Hg}$$

So decomposed pressure of $A$ is:

$$x = 800 - 100 = 700 \, \text{mm Hg}$$

Then,

$$P_B = 2x = 1400 \, \text{mm Hg}, \quad P_C = x = 700 \, \text{mm Hg}$$

Total pressure is:

$$P_{\text{total}} = 100 + 1400 + 700 = 2200 \, \text{mm Hg}$$

Therefore, the total pressure after $30 \, \text{min}$ is $2200 \, \text{mm Hg}$. The correct option is A.