The rate constant of a reaction at $500 \mathrm{~K}$ and $700 \mathrm{~K}$ are $0.02 \mathrm{~s}^{-1}$ and…

The rate constant of a reaction at $500 \mathrm{~K}$ and $700 \mathrm{~K}$ are $0.02 \mathrm{~s}^{-1}$ and $0.2 \mathrm{~s}^{-1}$ respectively. The activation energy of the reaction (in $\left.\mathrm{kJ} \mathrm{mol}^{-1}\right)$ is $\left(R=8.3 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\right)$
  1. $66.90$
  2. $33.45$
  3. $22.30$
  4. $44.45$

Solution

Given, $k_1=0.02 \mathrm{~s}^{-1}$ $\begin{aligned} & k_2=0.2 \mathrm{~s}^{-1} \\ & E_a=? \\ & R=8.3 \\ & T_1=500 \mathrm{~K} \\ & T_2=700 \mathrm{~K}\end{aligned}$ $\begin{aligned} & \text { In } \frac{k_2}{k_1}=-\frac{E_a}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right) \\ & \text { In } \frac{0.2}{0.02}=-\frac{E_a}{8.3}\left[\frac{1}{700}-\frac{1}{500}\right] \\ & \text { In } 10=\frac{-E_a}{8.3}\left(\frac{500-700}{700 \times 500}\right)\end{aligned}$ $2.302 \times 8.3 \times 700 \times 500=200 E_a$ $\Rightarrow E_a=\frac{2.302 \times 8.3 \times 700 \times 500}{200}$ $\begin{aligned} & =33436 \mathrm{~J} / \mathrm{mol} \\ & \approx 33.44 \mathrm{~kJ} / \mathrm{mol}\end{aligned}$

Asked in: AP EAMCET 2022 (05 Jul Shift 1)

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