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$$\ce{e- + H2O -> [H$\cdot$] + HO-}$$

I was a little confused on the notation of the hydrogen in brackets ([H]) in the product side of the reaction. Why is there a dot next to it and what does it mean? Is it saying Hydrogen gas?

Also, what will be the reduction potential of this reaction?

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The notation $\ce{H^.}$ refers to the hydrogen radical (a hydrogen atom). The dot represents the unpaired valence electron.

An equivalent notation for the reaction would be

$$\ce{e- + H2O -> H + OH-}$$

However, adding the dot emphasizes the fate of the electron on the reactant side.

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There is two atypical conventions in the equation. One of them, the dot, is the usual indication that a radical, with a single unpaired valence electron, exists. Elemental hydrogen could be written that way, H although because it will rapidly react and is not stable near STP, other conventions indicating the atom's electronic state (energy level of the electron) are just as, if not more, common. The other unusual convention is the square brackets. Their meaning is context dependent. In fact, it's possible you, the OP, added them to give emphasis to the species in question, with no further meaning attached. OTOH, their use in a chemical equation might indicate that the enclosed species is nominal (for instance used for bookkeeping) or existing transiently or just a theoretical species which hasn't been confirmed. The equation HCl + H2O → H+ + Cl- + H2O could also be written as HCl + H2O → [H+] + Cl- + H2O because H+ doesn't exist in aqueous solution (rather it is shorthand for H+nOm where m = 1, 2, 3 predominantly and n = 2m+1. You'll note that the Cl- isn't in brackets, that is not to suggest it has no water molecules surrounding it, but rather that the distances Cl ←→ HOH are larger than covalent bond distances, and are considered to be simple solvation.

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