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(C) For a first order reaction,the rate constant $K$ is related to half-life $t_{1/2}$ as $K = \frac{0.693}{t_{1/2}}$.At $T_1 = 300 \ K$,$K_1 = \frac{

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(C) For a first order reaction,the rate constant $K$ is related to half-life $t_{1/2}$ as $K = \frac{0.693}{t_{1/2}}$.At $T_1 = 300 \ K$,$K_1 = \frac{0.693}{50} \ s^{-1}$.At $T_2 = 400 \ K$,$K_2 = \frac{0.693}{10} \ s^{-1}$.Using the Arrhenius equation: $\log \frac{K_2}{K_1} = \frac{E_a}{2.303 \ R} \left[ \frac{T_2 - T_1}{T_1 T_2} ight]$.Substituting the values: $\log \left( \frac{0.693 / 10}{0.693 / 50} ight) = \log 5 = 0.70$.$0.70 = \frac{E_a}{2.303 	imes 8.314} \left[ \frac{400 - 300}{300 	imes 400} ight]$.$0.70 = \frac{E_a}{19.147} 	imes \frac{100}{120000} = \frac{E_a}{19.147 	imes 1200}$.$E_a = 0.70 	imes 19.147 	imes 1200 \approx 16083.48 \ J \ mol^{-1} = 16.08 \ kJ \ mol^{-1}$.Rounding to the nearest value,$E_a \approx 16.1 \ kJ \ mol^{-1}$.

The half-life periods of a first order reaction at $300 \ K$ and $400 \ K$ are $50 \ s$ and $10 \ s$ respectively. The activation energy of the reaction in $kJ \ mol^{-1}$ is $(\log 5 = 0.70)$

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