So I know that the coefficients of the species in a reaction reflect the order of the reaction of that species ONLY if it's an elementary reaction, but how can you tell if that is the case?
Well, to know whether something is elementary, you'd have to know its reaction coordinate diagram. Or, specifically, if it reactions directly to yield the product(s) going through a single energy barrier, single transition state, and no intermediates. So you would have to know the reaction profile.
So if it has multiple energy barriers/transition states then it can't be elementary?
Yes, if it has multiple intermediates, it cannot be elementary. For example, you wouldn't call glycolysis (glucose to pyruvate) an elementary reaction because it has multiple intermediates and could depend on any or all of them in theory.
If it is a multistep reaction, then it is not an elementary reaction but if you are given the reaction progress diagram or the activation energy values of each of the individual steps that make up the overall reaction, then you can determine the reaction coefficients.
Then is any catalyzed reaction elementary? Because you have the energy barrier between between the E+S and the ES complex, then another energy barrier between the ES complex and the product
I'm not sure you can determine the reaction coefficients just from the activation energy values though. You can tell what the RDS is, but you can't tell how exactly the intermediates contribute. You would need, on top of that, a rate experiment where you measure the dependence of the rate on each intermediate.
You don't necessarily have two barriers there. Imagine this. E + S makes ESI, which also happens to be the transition state. Then the reaction flows downhill to the products E + P. That's elementary, assuming no intermediate steps occur in the active site.
But the E+S doesn't go directly to transition state. You have overcome the first barrier barrier to get to the ES, which is going to be a valley on your reaction diagram and a relatively stable complex, and then a second barrier from your ES to your transition state followed by the downhill path to your product. You're going to have atleast 2 barriers in a catalyzed reaction, albeit both smaller barriers than the uncatalyzed reaction. Atleast I think this is the case. Correct me if I'm mistaken.
You are describing one possibility but this possibility is not the only one. Since I don't know how to draw something here, I can only direct you to this figure: https://en.wikipedia.org/wiki/Physi.../media/File:Catalytic_reaction_coordinate.jpg
A is the uncatalyzed reaction. B is catalyzed. C is also catalyzed but via another route. You're describing C but B could also be true, in which case it would be elementary.
Oh ok. I see now. Thanks. My biochem class had us under the impression that all catalyzed reactions involved an ES complex and multiple barriers