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JEE Main 2020 · Chemistry · Chemical Kinetics · hard

A first-order reaction has a half-life of 20 min at 300 K and an activation energy of 60 kJ mol^-1 (use R=8.314 J mol^-1K^-1; assume the Arrhenius…

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A first-order reaction has a half-life of 20 min20\ \mathrm{min} at 300 K300\ \mathrm{K} and an activation energy of 60 kJ mol−160\ \mathrm{kJ\,mol^{-1}} (use R=8.314 J mol−1K−1R=8.314\ \mathrm{J\,mol^{-1}K^{-1}}; assume the Arrhenius equation). The time needed for 87.5%87.5\% of the reactant to decompose at 320 K320\ \mathrm{K} is nearest to
  1. A. 13.3 min13.3\ \mathrm{min}
  2. B. 8.9 min8.9\ \mathrm{min}
  3. C. 17.8 min17.8\ \mathrm{min}
  4. D. 26.7 min26.7\ \mathrm{min}
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Answer: (A) 13.3 min13.3\ \mathrm{min}

ln⁡k2k1=EaR(1300−1320)=600008.314×2.083×10−4=1.503\ln\dfrac{k_2}{k_1}=\dfrac{E_a}{R}\left(\dfrac1{300}-\dfrac1{320}\right)=\dfrac{60000}{8.314}\times2.083\times10^{-4}=1.503, so k2/k1=4.50k_2/k_1=4.50. Decomposition of 87.5%87.5\% equals three half-lives; at 300300 K this takes 6060 min. At 320320 K the time is 60/4.50=13.360/4.50=13.3 min.
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