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Why does the Hydrogen proton "prefer" to join the $\ce{H2O}$ molecule instead of remaining bonded to the chlorine atom?

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Chlorine is a very large atom compared to hydrogen, which means the overlap between the hydrogen $1s$-orbital and the chlorine $sp^3$ orbital is poor, which translates into a low bond energy.

Additionally, chlorine is very electronegative, which helps pull the H-Cl bonding pair onto the chlorine atom. The chloride anion solvation in water also plays a big part.

It's simply because it's energetically favourable. Maybe someone knows in greater detail than I do.

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Your answer is already more descriptive than I expected. I was wondering if there was a better answer than simply "being energetically favourable". Saying that water is a stronger base than the chloride ion is just a way to mask that. So is mentioning HOMO/LUMO orbital energies. Of course, all answers boil down to potential energy minimization (which itself boils down to entropy maximization), but the difference is how much feel you can get for why it happens, and how much you can extrapolate to other systems from it. Perhaps someone can still find something to add. –  Nicolau Saker Neto Feb 22 at 16:34

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