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Which oxygen will form a stronger hydrogen bond , the one in water or the one in hydrogen peroxide ??

According to me , the one in peroxide will form the stronger H bond as the O here being less electronegative will be more willing to donate lone pair to an incoming dipole .

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Electronegativity is a property of the element. All oxygens are equally electronegative, so your line of reasoning is not correct.

A better argument would probably involve looking at the $\alpha$ effect and how lone pairs on vicinal atoms destabilize each other (generally used to justify why the peroxide oxygen-oxygen bond is weak). One might imagine then that each oxygen could form a stronger hydrogen bond, since this would help to stabilize the repulsion.

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