The rate constant of a first order reaction was doubled when the temperature was increased from 300 to 310 K . What is its approximate activation energy (in $\mathrm{kJ} \mathrm{mol}^{-1}$ )? ( $\mathrm{R}=8.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1} ; \log 2=0.3$ )

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Aicharya · Accepted Answer

According to Arrhenius theory. $\begin{aligned} & \mathrm{K}=\mathrm{A} \times \mathrm{e}^{-\mathrm{E}_{\mathrm{a}} / \mathrm{RT}} \\ & \log \frac{\mathrm{~K}_2}{\mathrm{~K}_1}=\frac{\mathrm{E}_{\mathrm{a}}}{2.303 \mathrm{R}}\left[\frac{1}{\mathrm{~T}_1}-\frac{1}{\mathrm{~T}_2}\right] \end{aligned}$
Given, $\begin{array}{ll} \mathrm{K}_1=\mathrm{x}, & \mathrm{~T}_1=300 \\ \mathrm{~K}_2=2 \mathrm{x} & \mathrm{~T}_2=310 \\ \log \frac{2 \mathrm{x}}{\mathrm{x}}=\frac{\mathrm{E}_{\mathrm{a}}}{2.303 \times 8.3}\left[\frac{1}{300}-\frac{1}{310}\right] \\ \mathrm{E}_{\mathrm{a}}=\frac{\log 2 \times 2.303 \times 8.3 \times 300 \times 310}{10} \\ \mathrm{E}_{\mathrm{a}}=53.49 \mathrm{~kJ} \mathrm{~mol}^{-1} \end{array}$

The rate constant of a first order reaction was doubled when the temperature was increased from 300 to 310 K…

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