A molecule A undergoes isomeric conversion to B via first-order kinetics at 1000 K. The activation energy (energy barrier relative to reactants) is 191.48,text kJ,mol ^ -1 and the frequency factor A_0=10^ 20 . Find the time (in picoseconds, nearest integer) for 50% of A to convert to B. [R=8.314,text J,K ^ -1 text mol ^ -1 ]

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EEJ MAIN Chemistry QUESTION #5513

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A molecule A undergoes isomeric conversion to B via first-order kinetics at 1000 K. The activation energy (energy barrier relative to reactants) is \(191.48\,\text{kJ\,mol}^{-1}\) and the frequency factor \(A_0=10^{20}\). Find the time (in picoseconds, nearest integer) for 50% of A to convert to B. \([R=8.314\,\text{J\,K}^{-1}\text{mol}^{-1}]\)

Correct Answer:

Correct Answer Explanation

Using Arrhenius equation: \(k=A_0\,e^{-E_a/RT}\). \(\frac{E_a}{RT}=\frac{191480}{8.314\times1000}=\frac{191480}{8314}=23.03\approx10\ln10\) since \(e^{23.03}\approx10^{10}\). So \(k=10^{20}\cdot e^{-23.03}=10^{20}\cdot10^{-10}=10^{10}\,\text{s}^{-1}\). For first-order reaction, half-life: \(t_{1/2}=\frac{\ln2}{k}=\frac{0.6931}{10^{10}}=6.931\times10^{-11}\,\text{s}=69.31\,\text{ps}\approx69\,\text{ps}\).

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