Number of Correct statement(s) regarding the following first-order reaction is/are 

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

A) The concentration of the reactant decreases exponentially with time

B) The half life of the reaction decreases with increasing temperature 

C) The half life of the reaction depends on the initial concentration of the reactant.

D) The reaction proceeds to 99.6% completion in eight half life cycles.

According to first – order reaction, $-\text{d}\left[{\text{N}}_{\text{2}}{\text{O}}_5\right]/\text{d}t=k\left[{\text{N}}_{\text{2}}{\text{O}}_5\right]$ . The half life can be found as $t=\left(\frac{2.303}{k}\right)\log \frac{\text{a}}{\text{a}-x}$ 

where, a = 100 and $\text{a}-x=100-99.6=0.4$

Then half-life period for first-order reaction is $t_{\frac{1}{2}}=\frac{\text{In}2}{k}=\frac{0.693}{k}$.  With temperature, k increases (according to Arrhenius equation) and hence, $t_{1/2}$ decreases. ( $t_{1/2}$ is indirectly proportional to k),If the reaction completes to 99.6% completion then,

$t_{99.6}=\left(\frac{2.303}{k}\right)\log \frac{100}{0.4}=\left(\frac{2.303}{k}\right)2.4=8\times \frac{0.693}{k}=8t_{1/2}$