$\mathrm{A}(\mathrm{g}) \rightarrow \mathrm{B}(\mathrm{g})+\mathrm{C}(\mathrm{g})$ is a first order reaction. | Time | T | $\infty$ | |------|---|----------| | $\mathrm{P}_{\text{system}}$ | $\mathrm{P}_{\mathrm{t}}$ | $\mathrm{P}_{\infty}$ | The reaction was started with reactant A only. Which of the following expression is correct for rate constant k ?

(C) $\mathrm{k}=\frac{1}{\mathrm{t}} \ln \frac{\mathrm{p}_{\infty}}{2\left(\mathrm{p}_{\infty}-\mathrm{P}_{\mathrm{t}}\right)}$

(A) $\mathrm{k}=\frac{1}{\mathrm{t}} \ln \frac{2\left(\mathrm{p}_{\infty}-\mathrm{P}_{\mathrm{t}}\right)}{\mathrm{P}_{\mathrm{t}}}$

(B) $\mathrm{k}=\frac{1}{\mathrm{t}} \ln \frac{\mathrm{p}_{\infty}}{\mathrm{P}_{\mathrm{t}}}$

(D) $\mathrm{k}=\frac{1}{\mathrm{t}} \ln \frac{\mathrm{p}_{\infty}}{\left(\mathrm{p}_{\infty}-\mathrm{P}_{\mathrm{t}}\right)}$

JEE Main · 2025 · 07 Apr 2025 · Shift 2 · Chemical Kinetics

A (g) \to B (g) + C (g)

is a first order reaction.

Time	T	$\infty$
$P_{system}$	$P_{t}$	$P_{\infty}$

The reaction was started with reactant A only. Which of the following expression is correct for rate constant k ?

(A) $k = \frac{1}{t} ln \frac{2 ( p _{\infty} - P _{t} )}{P _{t}}$
(B) $k = \frac{1}{t} ln \frac{p _{\infty}}{P _{t}}$
(C) $k = \frac{1}{t} ln \frac{p _{\infty}}{2 ( p _{\infty} - P _{t} )}$ ✓ Correct
(D) $k = \frac{1}{t} ln \frac{p _{\infty}}{( p _{\infty} - P _{t} )}$

Answer: (C)

Begin with pressure

P_{0}

of A alone. At time

t

, if

x

has decomposed, A stands at

P_{0} - x

while B and C are each

x

, so the total pressure is

P_{0} + x

. At completion all of A is gone and both products sit at

P_{0}

, so the final pressure is

2 P_{0}

, giving

P_{0} = \frac{1}{2} P_{\infty}

. Then

x = P_{t} - P_{0}

, and the pressure of A remaining is

P_{0} - x = 2 P_{0} - P_{t} = P_{\infty} - P_{t}

. For a first-order reaction

k = \frac{1}{t} ln \frac{P _{0}}{P _{A}}

, and substituting gives

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} )}

JEE Main · 2025 · 07 Apr 2025 · Shift 2 · Single correct option