The time until the reaction complete for $n^{th}$ order reaction
So there's something that i was wondering about how to determine the reaction time until all the reactan run out, for all $n$ order? so here's my approach about it,
we have the kinetic differentiatial equation
$-\frac{d[A]}{dt}$ $= k{[A]^n}$.
where n is equal the reaction order itself, that is possible for any number ($0,1,2,$ sometimes even fraction and negative number like $-1,$$\frac{1}{2}$)
suppose we have a chemical reaction simular to this formula
$ \ce{A -> B} $
so in this problem i'll take an example $5$ equation with simular form with the equation above, and different order $n$.
assuming the $n$ value of the $5$ example is $1,2,0,-1$ and $\frac{1}{2}$.
we will get these equations for the five example after we input the $n$ value and integrate the kinetic differential equation.
(reaction with ${1^{st}}$ order) $ ln[A] = -kt + ln[A]_0 $
(reaction with ${2^{nd}}$ order) $ \frac{1}{[A]} $ $=$ $\frac{1}{[A]_0} +kt$
(reaction with ${0^{th}}$ order) $ [A] = [A]_0 -kt$
(reaction with $\frac{1}{2}$ order) $ 2 (\sqrt {[A]} - \sqrt {[A]_0}) = -kt$
(reaction with $-1$ order) $\frac{1}{2}[A]^2 = -kt + \frac{1}{2}([A]_0)^2$
what i was thinking is, if i want to determine the time until the reaction complete, then i will have to define t when $[A]=0$ (note : i will call the time that needed to complete the reaction $t_{complete}$) which is not defined for reaction with ${1^{st}}$ and ${2^{nd}}$ order because it will involve $\infty$ value, in fact as much as i know, $t_{complete}$ will not be defined for $n \ge 1$, but we can calculate $t_{complete}$ for $n \lt 1$.
so for $0^{th}$ order, the $t_{complete} = \frac{[A]_0}{k}$
for $\frac{1}{2}$ order, $t_{complete} = \frac{2\sqrt{[A]_0}}{k}$
for $-1$ order, $t_{complete} = \frac{([A]_0)^2}{2k}$.
my question. is all the reaction with $n \ge 1$ never complete? and is all the reaction with $n \lt 1$ is the only reaction that can complete? and i know what i've done is just manipulating the rate law equation, but if this is correct, can the $t_{complete}$ that i got above determine the time that needed for order $n \lt 1$ to complete? 374