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What is the base ionization reaction for LiOH? Usually, the reaction is like this: $$Base + \ce{H2O->H}[Base] + \ce{OH-} $$

But $\ce{LiOH2+}$ makes no sense. So what could the reaction be?

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What you call bases is in fact a subset thereof. Besides that, there is a whole class of inorganic bases, also known as alkali. These are the hydroxides (sometimes called basic salts) of alkaline metals. Their ionization is somewhat different: $$\ce{NaOH->Na+ + OH-}$$

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