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It makes sense to me that reaction intermediates exist transiently if they are of higher energy than the reactants and the products, my organic chemistry textbook seems to give me information that doesn’t seem to match up.

My knowledge is that the overall activation energy of the reaction is the energy change between reactants and the highest-energy transition state and the overall energy change is he energy difference between the reactants and the final products. If intermediates could be of lower energy levels than either the reactants or products, would this definition change?

I am confused because the solutions manual to one of the problems gives me an energy diagram with an intermediate that is of lower energy than both the reactants (but not the products), and labels the overall activation energy as the energy change between the intermediate state and the highest-energy transition state. The textbook also gives me an energy diagram of a catalyzed reaction that has several intermediates of lower energy than the reactants, but not the products.

(The textbook I’m referring to is Organic Chemistry. 8th edition by John McMurry)

Would it make sense for intermediates to only be lower in energy than the reactants but not the products since the products would still be more favorable and stable than any of the intermediates? Would this only happen if there is a catalyst?

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The intermediate is a proper chemical species but a reactive one under 'normal' conditions and is not a transition state, i.e. it has a short, but measurable, lifetime. The intermediate can change only by crossing one of two transition states, one back to reactants and the other to another intermediate or to product.

Any intermediate must not be lowest energy species, including reactant and product, but can be lower or higher in energy than the reactant (but less than any transition state) and higher than product. The limiting activation energy will be the largest one in the whole scheme, so may be that between reactant & intermediate or between intermediate & product. Which it is depends on the exact species involved.

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