Chemistry - How to calculate the rate constant at different temperature for the decomposition of dinitrogen pentoxide?

Solution 1:

When using Arrhenius equation: you have to multiply the activation energy by $1000$, because it must be in $\pu{J}$ and not $\pu{kJ}$. you must also divide by $RT$ and not by $T$ as you did.

Solution 2:

I arrived at the same $k_{\pu{300 K}}$ as you did. I find it a little weird that there's a 3 order of magnitude decrease in rate constant for a $\pu{50 K}$ increase in temperature.

Per the note below, the original equation I put had a mathematical error. The below should be correct:

\begin{align} \frac{k_{\pu{350 K}}}{k_{\pu{300 K}}} &= \exp\left\{\frac{E_\mathrm{a}}{R\left(\frac{1}{T_{\pu{300 K}}} - \frac{1}{T_{\pu{350 K}}}\right)}\right\}\\ \frac{k_{\pu{350 K}}}{k_{\pu{300 K}}} &= \exp\left\{\frac{103000}{8.314\left(\frac{1}{300} - \frac{1}{350}\right)}\right\}=371.14 \end{align}

If you take $k_{\pu{300 K}} = \pu{2.773E-5}$ and multiply by that factor above (as you should) you get your answer, if you take $k_{\pu{300 K}}$ and divide by the factor above, you get the book's answer, which is where I believe their mistake is.