For a chemical reaction,the temperature is increased from $25\,^{\circ}C$ to $55\,^{\circ}C$. The rate of reaction will change by a factor of (Assume $\mu = 3$).

Fill in the blanks:
$(1)$ The time required to complete $99.9\%$ of a first-order reaction is ............ times the ${t_{1/2}}$.
$(2)$ The reaction in which the rate is given by $-\frac{d[R]}{dt} = k$ has an order of reaction equal to ........
$(3)$ For the reaction $[R] = [R]_0 e^{-kt}$,the order is .......

For a chemical reaction, half-life period $(t_{1/2})$ is $10 \ minutes$. How much reactant will be left after $20 \ minutes$ if one starts with $100 \ moles$ of reactant and the order of the reaction is $(i)$ zero, $(ii)$ one, and $(iii)$ two?

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 ......... $mol \, L^{-1}$.

The hydrolysis of an ester was carried out separately with $0.1 \ N$ $HCl$ and $0.1 \ N$ $H_2SO_4$. Then for the rate of reaction $(R)$ in the presence of acid,which of the following relations is true?

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.