In the reaction $X \to Y$,if the concentration of reactant $X$ is increased by $1.5$ times,the rate of reaction increases by $1.837$ times. Determine the order of the reaction with respect to $X$.

$A$ reaction $2NO + 2H_2 \longrightarrow N_2 + 2H_2O$ has the following mechanism:
Step-$I$: $2NO \longrightarrow N_2O_2$
Step-$II$: $N_2O_2 + H_2 \longrightarrow N_2O + H_2O$
Step-$III$: $N_2O + H_2 \longrightarrow N_2 + H_2O$
Which of the following substances is a reaction intermediate?

For the reaction:
$2NO_{2(g)} + O_{3(g)} \to N_2O_{5(g)} + O_{2(g)}$
The rate law is $R = K[NO_2]^1 [O_3]^1$.
Which of these possible reaction mechanisms is consistent with the rate law?
Mechanism $I$:
$NO_{2(g)} + O_{3(g)} \to NO_{3(g)} + O_{2(g)}$ (slow)
$NO_{3(g)} + NO_{2(g)} \to N_2O_{5(g)}$ (fast)
Mechanism $II$:
$O_{3(g)} \rightleftharpoons O_{2(g)} + [O]$ (fast)
$NO_{2(g)} + [O] \to NO_3$ (slow)
$NO_{3(g)} + NO_{2(g)} \to N_2O_5$ (fast)

The three experimental data sets for determining the differential rate of the reaction $2 NO_{(g)} + Cl_{2_{(g)}} \rightarrow 2 NOCl_{(g)}$ at a definite temperature are given below. (Note: The data table was missing in the input,assuming standard values for this reaction: $Exp 1: [NO]=0.1, [Cl_2]=0.1, Rate=0.18$; $Exp 2: [NO]=0.1, [Cl_2]=0.2, Rate=0.36$; $Exp 3: [NO]=0.2, [Cl_2]=0.1, Rate=0.72$).
$(a)$ Derive the differential rate law of the reaction.
$(b)$ Calculate the order of the reaction.
$(c)$ Calculate the value of the rate constant.

In the reaction sequence $A$ $\xrightarrow{K_1} B$ $\xrightarrow{K_2} C$ $\xrightarrow{K_3} D$,where $K_3 > K_2 > K_1$,which step determines the rate of the reaction?

The rate equation for the reaction $2 A + B \longrightarrow$ products is $\text{rate} = k[A][B]^2$. If $k$ at $T \ K$ is $5.0 \times 10^{-6} \ mol^{-2} \ L^2 \ s^{-1}$,the initial rate of the reaction,when $[A] = 0.05 \ mol \ L^{-1}$ and $[B] = 0.1 \ mol \ L^{-1}$ is: