For the first order decomposition reaction of $N_2O_5$,it is found that -
$(a)$ $2N_2O_5 \rightarrow 4NO_{2(g)} + O_{2(g)}$ ; $-\frac{d[N_2O_5]}{dt} = k[N_2O_5]$
$(b)$ $N_2O_5 \rightarrow 2NO_{2(g)} + 1/2 O_{2(g)}$ ; $-\frac{d[N_2O_5]}{dt} = k'[N_2O_5]$
Which of the following is true?

The initial rates of decrease of $I_2$ in acetone-iodine reaction catalysed by $H^{+}$ are given in the table.

Experiment	Initial $[I_2]$ $(mol \ L^{-1})$	Initial $[H^{+}]$ $(mol \ L^{-1})$	Initial $[CH_3COCH_3]$ $(mol \ L^{-1})$	Initial rate $(mol \ L^{-1} \ s^{-1})$
$1$	$0.01$	$0.1$	$0.1$	$0.096$
$2$	$0.01$	$0.2$	$0.1$	$0.192$
$3$	$0.02$	$0.2$	$0.1$	$0.192$
$4$	$0.01$	$0.2$	$0.2$	$0.384$

The order with respect to $I_2, H^{+}$,acetone and total order of the reaction respectively are:

For the reaction $2NO_2 + F_2 \to 2NO_2F$,the following mechanism has been provided:
$NO_2 + F_2 \xrightarrow{slow} NO_2F + F$
$NO_2 + F \xrightarrow{fast} NO_2F$
Thus,the rate expression of the above reaction can be written as:

The reaction of hydrogen and iodine monochloride is given as :
$H_{2(g)} + 2ICl_{(g)} \rightarrow 2HCl_{(g)} + I_{2(g)}$
This reaction is of first order with respect to $H_{2(g)}$ and $ICl_{(g)}$. The following mechanisms were proposed:
Mechanism $A$ :
$H_{2(g)} + 2ICl_{(g)} \rightarrow 2HCl_{(g)} + I_{2(g)}$
Mechanism $B$ :
$H_{2(g)} + ICl_{(g)} \rightarrow HCl_{(g)} + HI_{(g)}$ ; (slow)
$HI_{(g)} + ICl_{(g)} \rightarrow HCl_{(g)} + I_{2(g)}$ ; (fast)
Which of the above mechanism$(s)$ can be consistent with the given information about the reaction?

The order of the reaction occurring by the following mechanism should be:
$(i)$ $A_2 \to A + A$ (fast)
$(ii)$ $A + B_2 \to AB + B$ (slow)
$(iii)$ $A + B \to AB$ (fast)

The three experimental data for determining the differential rate of reaction $2 NO_{(g)} + Cl_{2(g)} \rightarrow 2 NOCl_{(g)}$ at a definite temperature are given below.
$(a)$ Calculate the order of reaction.
$(b)$ Calculate the value of the rate constant.