$\mathrm{A}(\mathrm{g}) \rightarrow \mathrm{B}(\mathrm{g})+\mathrm{C}(\mathrm{g})$ is a first order reaction…
$\begin{array}{|l|l|l|}\hline \text{Time} & T & \infty \\\hline \mathbf{P}_{\text {system }} & \mathrm{P}_{\mathrm{t}} & \mathrm{P}_{\infty} \\\hline\end{array}$
The reaction was started with reactant A only. Which of the following expression is correct for rate constant k ?
- $\mathrm{k}=\frac{1}{\mathrm{t}} \ln \frac{2\left(\mathrm{p}_{\infty}-\mathrm{P}_{\mathrm{t}}\right)}{\mathrm{P}_{\mathrm{t}}}$
- $\mathrm{k}=\frac{1}{\mathrm{t}} \ln \frac{\mathrm{p}_{\infty}}{\mathrm{P}_{\mathrm{t}}}$
- $\mathrm{k}=\frac{1}{\mathrm{t}} \ln \frac{\mathrm{p}_{\infty}}{2\left(\mathrm{p}_{\infty}-\mathrm{P}_{\mathrm{t}}\right)}$
- $\mathrm{k}=\frac{1}{\mathrm{t}} \ln \frac{\mathrm{p}_{\infty}}{\left(\mathrm{p}_{\infty}-\mathrm{P}_{\mathrm{t}}\right)}$
Solution
& \begin{array}{lllll}
& A_{(g)} & \rightarrow & B_{(g)} & +
C_{(g)} \\
t=0 & P^o & & 0 & 0 \\
t=t & P^o-x & & x & x \\
t=\infty & 0 & & P^o & P^o
\end{array} \\
& \mathrm{P}_{\mathrm{t}}=\mathrm{P}^{\mathrm{o}}+\mathrm{x} \Rightarrow \mathrm{x}=\mathrm{P}_{\mathrm{t}}-\mathrm{P}^{\mathrm{o}}=\mathrm{P}_{\mathrm{t}}-\frac{\mathrm{P}_{\infty}}{2} \\
& \mathrm{P}_{\infty}=2 \mathrm{P}^{\mathrm{o}} \Rightarrow \mathrm{P}^0=\frac{\mathrm{P} \infty}{2} \\
& \mathrm{k}=\frac{1}{\mathrm{t}} \ell \ln \frac{\mathrm{P}^{\mathrm{o}}}{\mathrm{P}^{\mathrm{o}}-\mathrm{x}} \\
& k=\frac{1}{t} \ell n \frac{P_{\infty}}{2\left(P_{\infty}-P_t\right)}
\end{aligned}$
Asked in: JEE Main 2025 (07 Apr Shift 2)