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From what I've gathered in my lectures, enolates are enolised cabonyl compounds. However e.g. a nitrile with an H on the alpha C can undergo enolisation. Is the resulting molecule called an enolate or an enolate equivalent, or something else?

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There are two questions in here.

First enols/enolates. It is correct that all enolates are enolised carbonyl compounds but not all enolised carbonyl compounds are enolates. Take for example pentan-2,4-dione. While in protic solvents this will indeed have two $\ce{C=O}$ groups, if dissolved in unpolar solvents it will enolise to allow for a cyclic intramolecular hydrogen bond for stabilisation (see scheme).

enolisation of pentan-2,4-diol

Obviously, the resulting compound is an enol, as the functional group $\ce{R2C=CR-OH}$ (a hydroxy group bound directly to a vinylic double bound) is present. However, it is not an anion so it is not an enolate.

Most carbonyl compounds prefer the keto-configuration but if there are electronic factors that favour enolisation (e.g. enolisation results in a sufficiently large conjugated system) then an enol will be preferred.

Enols can be easily deprotonated at the oxygen so the generation of an enolate is straightforward. However, since ketone and enols are tautomers of another with the key limiting step being hydrogen transfer ([1,3]-suprafacial hydrogen migration is forbidden), once the proton is removed the difference becomes purely electronic and thus deprotonated ketones are routinely referred to as enolates whether or not the cation seems more attached to the oxygen or the carbon. (Reaction mechanisms differ; sometimes only one attack (often carbon) results in a productive mechanism so it seems as if the other had not reacted or be not present but this is often a hindsight fallacy.)

The term enolate is not limited strictly to ketones. For one of the more synthetically relevant examples, see the Evans enolate used in Evans alkylation or Evans aldol reactions or the Myers enolate used in Myers alkylation reactions; both are amide enolates.

Evans and Myers enolates

However, these compounds are still characterised by the presence of a $\ce{(C-)C=O}$ bond which upon enolisation becomes a $\ce{(C=)C-OH}$ functionality. In the case of a nitrile or an imine, the oxygen is not present so even though a similar reaction may take place it is not correct to call the resulting product enol. For imines, the compound is known as an enamine and happens to be the standard outcome if a secondary amine is reacted with a carbonyl group.

For nitriles, I am not aware of a similar shorthand and would probably simply call it a nitrile anion (as does Wikipedia). (It is worth pointing out that nitrile anions are somewhat different from enamides/enolates as they are essentially an aza-ketene, not an aza-allyl anion.)

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