The rate constant of a first order reaction is $3.46 \times 10^{-2} \mathrm{~s}^{-1}$ at 298 K . What is the…

The rate constant of a first order reaction is $3.46 \times 10^{-2} \mathrm{~s}^{-1}$ at 298 K . What is the rate constant of the reaction at 350 K if its activation energy is $50.1 \mathrm{~kJ} \mathrm{~mol}^{-1} ?(\mathrm{R}=8.314 \mathrm{~J}$ $\mathrm{K}^{-1} \mathrm{~mol}^{-1}$ ) $(\log 2=0.3010)$
  1. $0.592 \mathrm{~s}^{-1}$
  2. $0.692 \mathrm{~s}^{-1}$
  3. $0.792 \mathrm{~s}^{-1}$
  4. $0.892 \mathrm{~s}^{-1}$

Solution

$\begin{aligned} & \text { } \mathrm{K}_1=3.46 \times 10^{-2} \\ & \mathrm{~T}_1=298 \mathrm{~K} \\ & \mathrm{~K}_2=? \\ & \mathrm{~T}_2=350 \mathrm{~K} \\ & \mathrm{E}_{\mathrm{a}}=50.1 \mathrm{~kJ} \mathrm{~mol}^{-1} \\ & \mathrm{R}=8.314 \mathrm{JK}^{-1} \mathrm{~mol}^{-1} \end{aligned}$
According to Arrhenius theory, $\begin{aligned} & \log \frac{\mathrm{K}_2}{\mathrm{~K}_1}=\frac{\mathrm{E}_{\mathrm{a}}}{2.303 \mathrm{R}}\left[\frac{1}{\mathrm{~T}_1}-\frac{1}{\mathrm{~T}_2}\right] \\ & \log \frac{\mathrm{K}_2}{3.46 \times 10^{-2}}=\frac{50.1 \times 10^3}{2.303 \times 8.314}\left[\frac{1}{298}-\frac{1}{350}\right] \\ & \log \frac{\mathrm{K}_2}{3.46 \times 10^{-2}}=1.30 \\ & \frac{\mathrm{~K}_2}{3.46 \times 10^{-2}}=10^{1.30}=20 \end{aligned}$ $\begin{aligned} \mathrm{K}_2 & =3.46 \times 10^{-2} \times 20 \\ \mathrm{~K}_2 & =69.03 \times 10^{-2} \\\end{aligned}$ $\therefore \quad \mathrm{K}_2=0.69 \mathrm{~s}^{-1}$

Asked in: AP EAMCET 2024 (21 May Shift 1)

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