GATE 2023 BT – Question 47
Temperature of a reaction with activation energy $15\ \mathrm{kcal\ mol^{-1}}$ is increased from 300 K to 310 K. With $R=1.9872\ \mathrm{cal\ mol^{-1}K^{-1}}$, the ratio $k_{310}/k_{300}$ (two decimal places) is ___________.
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Correct answer: 2.24 to 2.26
Explanation
**Arrhenius equation:** $k=A\,e^{-E_a/RT}$.
**Ratio of rate constants at two temperatures:**
$$\ln\frac{k_2}{k_1}=\frac{E_a}{R}\left(\frac1{T_1}-\frac1{T_2}\right).$$
**Numbers.** $E_a=15\,000$ cal/mol, $R=1.9872$ cal mol⁻¹ K⁻¹, $T_1=300$ K, $T_2=310$ K:
$$\frac{E_a}{R}=7548.4\text{ K},\qquad\frac1{300}-\frac1{310}=1.0753\times10^{-4}\text{ K}^{-1}$$
$$\ln\frac{k_2}{k_1}=0.8117\;\Rightarrow\;\frac{k_{310}}{k_{300}}=e^{0.8117}=\mathbf{2.25}.$$