Consider a complex reaction taking place in three steps with rate constants $\mathrm{k}_1, \mathrm{k}_2$ and $\mathrm{k}_3$ respectively. The overall rate constant $k$ is given by the expression $k=\sqrt{\frac{k_1 k_3}{k_2}}$. If the activation energies of the three steps are 60, 30 and $10 \mathrm{~kJ} \mathrm{~mol}^{-1}$ respectively, then the overall energy of activation in $\mathrm{kJ} \mathrm{mol}^{-1}$ is _________ . (Nearest integer)

For the thermal decomposition of $\mathrm{N}_2 \mathrm{O}_5(\mathrm{~g})$ at constant volume, the following table can be formed, for the reaction mentioned below.

$$2 \mathrm{~N}_2 \mathrm{O}_5(\mathrm{~g}) \rightarrow 2 \mathrm{~N}_2 \mathrm{O}_4(\mathrm{~g})+\mathrm{O}_2(\mathrm{~g})$$

$\mathrm{x}=$ __________ $\times 10^{-3} \mathrm{~atm}$ [nearest integer]

Given : Rate constant for the reaction is $4.606 \times 10^{-2} \mathrm{~s}^{-1}$.

$\mathrm{A \rightarrow B}$

The molecule A changes into its isomeric form B by following a first order kinetics at a temperature of 1000 K . If the energy barrier with respect to reactant energy for such isomeric transformation is $191.48 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and the frequency factor is $10^{20}$, the time required for $50 \%$ molecules of A to become B is __________ picoseconds (nearest integer). $\left[\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]$

Consider the following first order gas phase reaction at constant temperature $$ \mathrm{A}(\mathrm{g}) \rightarrow 2 \mathrm{B}(\mathrm{~g})+\mathrm{C}(\mathrm{g})$$

If the total pressure of the gases is found to be 200 torr after 23 $$\mathrm{sec}$$. and 300 torr upon the complete decomposition of A after a very long time, then the rate constant of the given reaction is ________ $$\times 10^{-2} \mathrm{~s}^{-1}$$ (nearest integer)

[Given : $$\log _{10}(2)=0.301$$]