$A$ student has studied the decomposition of a gas $AB_3$ at $25^{\circ} C$. He obtained the following data.

$p \ (mm \ Hg)$	$50$	$100$	$200$	$400$
Relative $t_{1/2} \ (s)$	$4$	$2$	$1$	$0.5$

The order of the reaction is

For the inversion of cane sugar,the half-life is $500 \ min$ at all concentrations of sugar when $pH = 5$. However,when $pH = 6$,the half-life becomes $50 \ min$. Determine the rate law for the inversion of cane sugar.

For the reaction $2H_2 + 2NO \to N_2 + 2H_2O$,the following mechanism has been proposed:
$I$. $2NO \rightleftharpoons N_2O_2$ (fast)
$II$. $N_2O_2 + H_2 \xrightarrow{k_2} N_2O + H_2O$ (slow)
$III$. $N_2O + H_2 \to N_2 + H_2O$ (fast)
What will be the rate law of this reaction?

Consider the following reaction,
$2H_{2(g)} + 2NO_{(g)} \rightarrow N_{2(g)} + 2H_2O_{(g)}$
which follows the mechanism given below:
$2NO_{(g)} \underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons}} N_2O_{2(g)}$ (fast equilibrium)
$N_2O_{2(g)} + H_{2(g)} \stackrel{k_2}{\rightarrow} N_2O_{(g)} + H_2O_{(g)}$ (slow reaction)
$N_2O_{(g)} + H_{2(g)} \stackrel{k_3}{\rightarrow} N_{2(g)} + H_2O_{(g)}$ (fast reaction)
The order of the reaction is

The rate constant is the same for $3$ reactions of order $I, II$,and $III$,respectively,with the unit of concentration being in moles per litre. If the concentration of the reactant is unity,the rates of reaction $R_1, R_2, R_3$ will be:

For the reaction $2A + B \rightarrow \text{Product}$,the half-life remains unchanged when the concentration of $A$ is doubled. When the concentration of $B$ is doubled,the rate of reaction increases by a factor of $2$. What is the unit of the rate constant for this reaction?