An example of an autocatalytic reaction is:

Under the same reaction conditions, initial concentration of $1.386 \ mol \ dm^{-3}$ of a substance becomes half in $40 \ s$ and $20 \ s$ through first-order and zero-order kinetics respectively. Ratio $\left(\frac{k_{1}}{k_{0}}\right)$ of the rate constants for first-order $\left(k_{1}\right)$ and zero-order $\left(k_{0}\right)$ of the reactions is

Under identical reaction conditions,for a reactant with an initial concentration of $1.386 \, M$,the time taken for the concentration to become half is $40 \, s$ for a first-order reaction and $20 \, s$ for a zero-order reaction. The ratio of the rate constant of the first-order reaction $(K_1)$ to the rate constant of the zero-order reaction $(K_0)$ is ......... $mol \, L^{-1}$.

Which of the following statements is correct?

Match the column $I$ with column $II$ :

$a$. Rate constant for first order reaction	$i$. $mol \ lit^{-1} \sec^{-1}$
$b$. Molarity	$ii$. $\frac{k \times 1000}{M}$
$c$. Rate constant for zero order reaction	$iii$. $second^{-1}$
$d$. Limiting molar conductivity	$iv$. $\frac{\text{moles of solute}}{\text{Volume of solution (lit)}}$

Fill in the blanks:
$(1)$ If $t_{1/2} = \frac{0.693}{k}$,then the order of reaction is .......... .
$(2)$ For a zero order reaction,$t_{1/2}$ is proportional to .......... of the initial concentration.
$(3)$ In a first order reaction,the rate of reaction is proportional to the .......... exponent of the concentration of the reactant.