For the gaseous reactions:
$A \to B$ $k_1 = 10^{15} e^{-25000 / 8.34 T}$
$C \to D$ $k_2 = 10^{14} e^{-15000 / 8.34 T}$
Calculate the approximate temperature at which $k_1 = k_2$ in $K$.

The equilibrium constant at $27\,^{\circ}C$ and $127\,^{\circ}C$ is $30$ and $40$ respectively. What will be the ratio of activation energy $(E_{a})_{f} / (E_{a})_{b}$?

If we plot a graph between $\log \, K$ and $\frac{1}{T}$ by Arrhenius equation,the slope is

The rate constant of a reaction is increased $4$ times after the addition of a catalyst to the reaction mixture at the same temperature of $27^{\circ} C$. The change in the activation energy of this reaction is (Take $\ln(1/4) = -1.386, R = 8.314 \ J \ K^{-1} \ mol^{-1}$)

For the decomposition reaction of $N_2O_5$,the slope of the graph of $\log K$ versus $1/T$ is $-1.2 \times 10^4 \ K$. Calculate the activation energy $(E_a)$ of the reaction.

For the process $R \rightarrow P$, the energy of the reactant is $50 \ kJ$ and the energy of the transition state is $170 \ kJ$. Determine the activation energy $(E_a)$ for the forward reaction in $kJ$.