JEE Main 2025 — Chemistry Physical Chemistry

$\frac{t_1}{2} \propto\left(C_0\right)^{1-\eta}$2t1​​∝(C0​)1−η $\begin{aligned}
& \frac{t_1}{t_2}=\left(\frac{C_1}{C_2}\right)^{1-\eta} \
& \Rightarrow \frac{200}{100}=\left(\frac{0.100}{0.025}\right)^{1-\eta} \
& \Rightarrow 2=(4)^{1-\eta} \
& (1-\eta)=\frac{1}{2} \
& \eta=\frac{1}{2} \
& \text { For } \eta=\frac{1}{2} \
& \frac{-d A}{d t}=k(A)^{\frac{1}{2}} \
& \int_{C_0}^C \frac{d A}{(A)^{\frac{1}{2}}}=-\int_0^t k d t \
& \Rightarrow 2 \mathrm{A}^{\frac{1}{2}}=-\mathrm{kt} \
& \Rightarrow \sqrt{\mathrm{C}}-\sqrt{\mathrm{C}_0}=\frac{-\mathrm{kt}}{2} \
& \sqrt{\mathrm{C}}-\sqrt{\mathrm{C}_0}-\frac{\mathrm{kt}}{2} \
& \text { For } \mathrm{C}0=0.1 \Rightarrow \mathrm{t}{\frac{1}{2}}=200 \mathrm{min} \
& \sqrt{\frac{c_0}{2}}=\sqrt{c_0}-\frac{k t}{2} \
& \frac{\mathrm{kt}}{2}=\sqrt{\mathrm{c}0}-\sqrt{\frac{\mathrm{c}0}{2}} \
& \frac{\mathrm{kt}}{2}=\sqrt{\mathrm{c}0}\left(\frac{\sqrt{2}-1}{\sqrt{2}}\right) \
& t{\frac{1}{2}}=\frac{2 \sqrt{c_0}}{k}\left(\frac{\sqrt{2}-1}{\sqrt{2}}\right) \
& 200=\frac{2 \sqrt{0.1}}{k}\left(\frac{\sqrt{2}-1}{\sqrt{2}}\right)
\end{aligned}$$\mathrm{k}=\frac{\sqrt{0.1}}{100}\left(\frac{\sqrt{2}-1}{\sqrt{2}}\right)$For$For\mathrm{C}0=1 \mathrm{M}$$t{\frac{1}{2}}=\frac{2 \times 100(\sqrt{2})}{\sqrt{0.1}(\sqrt{2}-1)} \times \frac{(\sqrt{2}-1)}{\sqrt{2}}$$\Rightarrow 200 \sqrt{10} \mathrm{~min}$.$.\Rightarrow B$is correct C is incorrect For$iscorrectCisincorrectFor\mathrm{C}0=1.6 \mathrm{M}$$\begin{aligned} & \mathrm{t}{\frac{1}{2}}=\frac{2 \sqrt{1.6}(\sqrt{2})(\sqrt{2}-1) \times 100}{\sqrt{0.1}(\sqrt{2}-1)(\sqrt{2})} \ & \mathrm{t}{\frac{1}{2}}=400 \times 2 \mathrm{~min} \ & \mathrm{t}{\frac{1}{2}}=800 \mathrm{~min}\end{aligned}$