JEE Chemistry 2025 Question with Solution

Given: $A(g) \rightarrow B(g) + C(g)$ is a first-order reaction. The total pressure at time $t$ is $P_t$ and at completion it is $P_\infty$.

Find: The correct expression for the rate constant $k$.

Let the initial pressure of $A$ be $P_0$. If pressure corresponding to reacted $A$ at time $t$ is $p$, then

$$P_A = P_0 - p, \qquad P_B = p, \qquad P_C = p$$

So the total pressure at time $t$ is

$$P_t = (P_0 - p) + p + p = P_0 + p$$

Hence,

$$p = P_t - P_0$$

and therefore

$$P_A = P_0 - (P_t - P_0) = 2P_0 - P_t$$

At $t = \infty$, the reaction is complete, so

$$P_\infty = 2P_0$$

Thus,

$$P_0 = \frac{P_\infty}{2}$$

Substituting into $P_A$,

$$P_A = 2\left(\frac{P_\infty}{2}\right) - P_t = P_\infty - P_t$$

For a first-order reaction,

$$k = \frac{1}{t} \ln \frac{P_0}{P_A}$$

Now substitute $P_0 = \frac{P_\infty}{2}$ and $P_A = P_\infty - P_t$:

$$k = \frac{1}{t} \ln \frac{P_\infty/2}{P_\infty - P_t}$$ $$k = \frac{1}{t} \ln \frac{P_\infty}{2(P_\infty - P_t)}$$

Therefore, the correct option is C.

Given: Only reactant $A$ is present initially for the reaction $A(g) \rightarrow B(g) + C(g)$.

Find: Expression of $k$ in terms of $P_t$ and $P_\infty$.

The key idea is to relate the partial pressure of unreacted $A$ to the measured total pressure.

At any time $t$, if reacted pressure is $p$, then the species pressures are

$$P_A = P_0 - p, \qquad P_B = p, \qquad P_C = p$$

So,

$$P_t = P_A + P_B + P_C = (P_0 - p) + p + p = P_0 + p$$

which gives

$$p = P_t - P_0$$

Hence,

$$P_A = P_0 - p = P_0 - (P_t - P_0) = 2P_0 - P_t$$

At completion,

$$P_A(\infty) = 0, \qquad P_B(\infty) = P_0, \qquad P_C(\infty) = P_0$$

Therefore,

$$P_\infty = 0 + P_0 + P_0 = 2P_0$$

So,

$$P_0 = \frac{P_\infty}{2}$$

and

$$P_A = P_\infty - P_t$$

Using the integrated first-order law,

$$k = \frac{1}{t} \ln \frac{[A]_0}{[A]}$$

For gases at constant temperature, concentration is proportional to partial pressure, so

$$k = \frac{1}{t} \ln \frac{P_0}{P_A}$$

Substituting the pressure relations,

$$k = \frac{1}{t} \ln \frac{P_\infty/2}{P_\infty - P_t} = \frac{1}{t} \ln \frac{P_\infty}{2(P_\infty - P_t)}$$

This matches option (3), so the correct option is C.