Challenge Level

Imagine that the rate of a chemical reaction only depends on the concentration of a single chemical $A$, with concentration denoted by $[A]$. The reaction is called $m$th order with respect to $[A]$ if it satisfies a rate equation of the following type

$$

-\frac{d[A]}{dt} =k[A]^m

$$

This problem involves trying to find various solutions to rate equations from a purely mathematical perspective. As a rate equation is non-linear in $[A]$ if $m\neq 1$ it is difficult to say how many different sorts of solution it might possess.

To begin with, can $[A] =\lambda+\mu t$ ever be a solution to a rate equation? What would be the order of the reaction? What possible values could $\lambda$ and $\mu$ take?

Can $[A]=(\lambda+\mu t)^n$ ever solve a rate equation when $n$ is a positive whole number other than $1$? A negative whole number? Zero?

What are the possibilities if $n$ is not a whole number?

Which other solutions can you find to a rate equation?