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I am confused as to how an order of a reactant (n or m) in the rate law can be negative. This means that increasing the concentration of a reactant would actually decrease its rate of disappearance. How does this make sense? Doesn't a higher concentration always allow for more collisions? Also, when the order is 0, this means that concentration has no effect on the rate of disappearance. How does this make sense chemically/at the atomic level?

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In principle, there are no zeroth-order reaction. Experimentally, however, zeroth-order rate law occurs because of the reaction condition. For example, in catalytic reactions such as hydrogenation of an alkene using platinum as a catalyst, the hydrogenation occurs on the surface of platinum. There is a finite surface area of platinum, so at a some point, it doesn't matter at all how much more hydrogen is put in. This is because no reaction can occur if the hydrogen cannot bind to the platinum.

An example of inverse first order reaction can be found here.

Basically, you have a two-step reaction: $$\ce{A + B <=> C}$$ $$\ce{C <=> D\ (slow)}$$ Increasing concentration of $\ce{C}$ would decrease the rate of reaction.

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