Show that in a first order reaction, the time taken for the completion of some fraction of change is independent of the initial concentration of the reactant.

The decomposition of ${\mathrm{N}}_2{\mathrm{O}}_5$ according to the equation

$2{\mathrm{N}}_2{\mathrm{O}}_5\left(\mathrm{g}\right)\to 4{\mathrm{NO}}_2\left(\mathrm{g}\right)+{\mathrm{O}}_2\left(\mathrm{g}\right)$

is a first order reaction. After $30$ minutes from the start of the decomposition in a closed vessel, the total pressure developed is found to be $284.5 \mathrm{mm}$ of $\mathrm{Hg}$ and on complete decomposition, the total pressure is $584.5 \mathrm{mm}$ of $\mathrm{Hg}$. Calculate the rate constant of the reactions.

The ionization constant of ${\mathrm{NH}}_4^{+}$ in water is $5.6\times {10}^{-10}$ at ${25}^{\mathrm{o}}\mathrm{C}$. The rate constant for the reaction of ${\mathrm{NH}}_4^{+}$ and ${\mathrm{OH}}^{-}$ to form ${\mathrm{NH}}_3$ and ${\mathrm{H}}_2\mathrm{O}$ at ${25}^{\mathrm{o}}\mathrm{C}$ is $3.4\times {10}^{-10} \mathrm{lit} {\mathrm{mol}}^{-1} {\sec }^{-1}.$ Calculate the rate constant for proton transfer from water to ${\mathrm{NH}}_3$.