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In the IUPAC Gold Book:
elementary reaction
A reaction for which no reaction intermediates have been detected or need to be postulated in order to describe the chemical reaction on a molecular scale. An elementary reaction is assumed to occur in a single step and to pass through a single transition state.concerted reaction
A single-step reaction through which reactants are directly transformed into products, i.e., without involvement of any intermediates.
What is the difference between them?
The difference between SN1 and SN2 reaction mechanisms illustrates this quite well.
An SN1 type of reaction can be decomposed into two elementary steps:
The rate determining step is the formation of the carbocation, but we do have two distinguishable steps and a reaction intermediate - this is where the other mechanism comes in.
Now, when SN1 is not possible due to - for example - the carbon atom in question being primary, we can have a concerted mechanism - both of the steps mentioned before occur simultaneously, thus avoiding the energetically unstable primary carbocation.
This process can not be broken down into single steps, as it is necessary that the formation and the substitution occur simultaneously. Also, there is no reaction intermediate - hence it is a elementary reaction, where two things happen in a 'concerted' way in a single step.
Also, the experimental rate law will show that the rds is different and the rate depends on both reagents.
The language here is weird. We generally refer to the first concept as an "elementary step." By definition, this is "concerted," in the sense that all of the pieces required for this step take place at the same time.
An elementary step (except for weird barrierless cases) should correspond to a single transition state in a reaction coordinate, and therefore, it links two adjacent intermediates (or reactant/product) in the reaction.
A reaction is a process composed of one or more elementary steps. Generally if two things need to happen in a reaction (e.g., bond-forming or bond-breaking), we can ask if these happen via a single elementary step or via multiple steps. If a single step, then we call the corresponding mechanism "concerted." Mechanistic studies help to distinguish and rule out different possibilities.
For example, we think that a simple Diels-Alder is concerted since a multiple-step reaction, where the two bonds are formed in different steps, would be expected to give some mixture of stereochemistry, but we find that Diels-Alder stereochemical outcome is strictly determined by the stereochemistry of the reactants. Here, the assumption is that bond rotation is generally fast relative to elementary steps.
On the other hand, if you study some edge cases of Diels-Alder and find products that are inconsistent with a single step where all bond formation and bond breaking take place at the same time (link), you start to think about how the reaction might take place over multiple steps, i.e., a reaction via a non-concerted mechanism. These reactions would necessarily go through an intermediate where not all of the desired bond-breaking and bond-forming has taken place. If you're clever, you might be able to observe / trap this intermediate to support your hypothesis.
