A reaction has a rate constant $k=8.54\times {10}^{-4}{\mathrm{M}}^{-1}$ ${\mathrm{s}}^{-1}$ at 45 °C and an activation energy, $E_{\alpha }=90.8\mathrm{kJ}$ What is the value of k at 25° C?

Arhenius equation; $k=A\cdot e^{-E_a/RT}$ R  rate Constent; A: pre exponential frotor
T: temperature, Ea: Actination energy
R : universal gas constent

$\left.\begin{array}{l}\frac{{45}^{\circ }c}{{25}^{\circ }C}(45+273)=318K,k_1\\ 25+273)=298K,K_2\end{array}\right\}$

$E_q=90.8\mathrm{kJ}=90.8\times {10}^3\mathrm{J}$

$R=8.314\mathrm{J}/\text{ molek. }$

$k_1=A{e}^{-E_9/R{T}_1}{K}_2=A{e}^{-E_{9/R{T}_2}}$

$\begin{array}{r}\frac{K_1}{K_2}=e^{-\frac{E_2}{R{T}_1}+\frac{E_a}{R{T}_2}}\\ =e^{\frac{E_a}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)}\end{array}$

$=e^{\frac{90.8\times {10}^3}{8.314}}\left(\frac{1}{298}-\frac{1}{318}\right)$

= 2.3

$\begin{array}{r}\frac{K_{1}v}{K_2}=9.97K_2=\frac{k_1}{9.97}\\ =\frac{8.54\times {10}^{-4}}{9.97}\\ =8.54\times {10}^{-5}{\mathrm{M}}^{-1}{\mathrm{se}}^{-1}\end{array}$