(a) A certain first-order reaction has a rate constant of $2.75 \times 10^{-2} \mathrm{~s}^{-1}\( at \)20^{\circ} \mathrm{C}\(. What is the value of \)k$ at \(60^{\circ} \mathrm{C}\) if $E_{a}=75.5 \mathrm{~kJ} / \mathrm{mol} ?(\mathbf{b})\( Another first-order reaction also has a rate constant of \)2.75 \times 10^{-2} \mathrm{~s}^{-1}\( at \)20^{\circ} \mathrm{C}$. What is the value of \(k\) at \(60^{\circ} \mathrm{C}\) if $E_{a}=125 \mathrm{~kJ} / \mathrm{mol} ?(\mathbf{c})$ What assumptions do you need to make in order to calculate answers for parts (a) and (b)?

In summary, for a first-order reaction with a rate constant of \(2.75 \times 10^{-2} ~\mathrm{s}^{-1}\) at \(20^{\circ}\mathrm{C}\) and activation energy of \(75.5 \mathrm{~kJ/mol}\), the value of k at \(60^{\circ} \mathrm{C}\) is approximately \(5.35 \times 10^{-2}~\mathrm{s}^{-1}\). For another first-order reaction with the same rate constant at \(20^{\circ}\mathrm{C}\) and an activation energy of \(125 \mathrm{~kJ/mol}\), the value of k at \(60^{\circ} \mathrm{C}\) is approximately \(1.37 \times 10^{-2}~\mathrm{s}^{-1}\). These calculations are based on the assumptions that the Arrhenius equation is valid, the activation energy and frequency factor A remain constant, and the reaction is a first-order reaction.