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Elimination reactions turning a single C-C bond into a double bond are classified as E1, E2, or E1cb, depending on the mechanism (concerted or with intermediate), the stereochemical outcome and their kinetics. The "1" stands for first-order reaction and the "2" for second-order reaction.

For E1cb, we first deprotonate to get a carbanion, then the leaving group leaves and the double bond forms:

$$\ce{RR'CX-CHR''R''' + B- <=> RR'CX-CR''R'''- + BH}\tag{1} $$

$$\ce{RR'CX-CR''R'''- -> RR'C=CR''R''' + X-}\tag{2} $$

If the base ($\ce{B-}$) is in excess and sufficiently strong, the intermediate forms stoichiometrically, and you have a first-order reaction for the second step, which is expected to be slower (acid/base chemistry vs. breaking and making bonds with carbon).

However, if there is an equilibrium between reactant and base $\ce{B-}$, and conjugate base of the reactant and $\ce{BH}$, this would be a fast pre-equilibrium, and the overall rate would depend on the concentration of base and of the reactant. In this case it is not first order (but if you keep the concentration of base in excess, it would be pseudo-first order).

What are the kinetics of E1cb, and what are some good references with original data to support the kinetic model?

There is a similar question already, but without answer: Kinetics for E1cb reactions

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