Negative or fractional order of reaction

Let $\ce{A}$ be the reactant and $\ce{P}$ the product at the imaginary elementary reaction $\ce{A->P}$.
Could it possibly have a net total order of $-1?$


$$-\frac{\mathrm d[\ce{A}]}{\mathrm dt} = k[\ce{A}]^{-1} \tag{1}$$


Or even more, a reaction whose rate depends on the concentration of the product as it follows:


$$-\frac{\mathrm d[\ce{A}]}{\mathrm dt} = k[\ce{P}] \tag{2}$$


Is there any example of an actual reaction that behaves like that?


On the other hand, can we think about non-integer orders of reaction like $1.5$, $0.5$, $-0.5$, $2.368$, $\mathrm e$ or $\pi$ for instance? Any examples?




Wikipedia has you covered for both parts of the question in their article on Rate equation.


They give an example of a [negative order of reaction](https://en.wikipedia.org/wiki/Rate_equation#Negative_order):


$$\ce{2 O3(g) -> 3 O2(g)}$$


Under certain conditions, this reaction is of -1 order in $[\ce{O2}]$. This covers your question about a reaction being dependent on the product concentration (equation 2 in OP's question) as well as having a negative order (equation 1 in OP's question).


They also give an example of a [fractional order of reaction](https://en.wikipedia.org/wiki/Rate_equation#Fractional_order):


$$\ce{COCl2(g) -> CO(g) + Cl2(g)}$$


This has an order of 0.5 in chlorine.


#### Elementary or one-step reaction?


If the reaction is elementary, the reaction orders are equal to the the stoichiometric coefficients of the reactants. I'm not sure what is meant with a one-step reaction. For the examples given in this answer, there are radical intermediates in multi-step mechanisms ([What does a fractional order of reaction mean for the mechanism?](https://chemistry.stackexchange.com/q/49088/72973)) that explain the unusual kinetic features.


 6