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 .......

(N/A) $(1)$ For a first-order reaction,$t = \frac{2.303}{k} \log \frac{[R]_0}{[R]}$. For $99.9\%$ completion,$[R] = [R]_0 - 0.999[R]_0 = 0.001[R]_0$. Thus,$t = \frac{2.303}{k} \log \frac{[R]_0}{0.001[R]_0} = \frac{2.303}{k} \log(10^3) = \frac{2.303 \times 3}{k} = \frac{6.909}{k}$. Since $t_{1/2} = \frac{0.693}{k}$,we have $t = 10 \times t_{1/2}$.
$(2)$ The rate law $-\frac{d[R]}{dt} = k$ indicates that the rate is independent of the concentration of the reactant,which corresponds to a zero-order reaction.
$(3)$ The integrated rate equation $[R] = [R]_0 e^{-kt}$ is characteristic of a first-order reaction.