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my question is, if the strength of the bond between two atoms increases as the difference in electronegativity increases as well.

so say, a H-F bond would be stronger than a O-H bond.

I'm a little confused because I know that acidity increases from left to right across the periodic table. So H-F is a stronger acid than H-CH3. Wouldn't this mean that H-F has weaker bonds than H-CH3?

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    $\begingroup$ No it doesn't, those two things are independent. H-F bond is stronger but it's only one of factors influencing acidity. $\endgroup$ – Mithoron Oct 26 '15 at 18:01
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Bond strength usually refers to the energy required to break a bond homolytically--with one electron going to one of the atoms and the other electron going to the other atom. In the case of HF this requires more energy than in CH4. Acidity is the tendency for the bond to break heterolytically--with two electrons going to one atom and none going to the other atom. In the case of HF, we get the two ions H+ and F-. This still requires a lot of energy, but in water solution, the ions are stabilized by hydrogen bonding, so ionization is much more favorable. Methane on the other hand would ionize to give a CH3- anion, which is a lot worse than F- due to carbon being less electronegative.

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First off, you're comparing an ionically bonded $\ce{HF}$ with a covalently bonded $\ce{CH4}$. HF is a stronger acid than $\ce{CH4}$ because its able to furnish out $\ce{H+}$ ions due to its ionic bonding.

So I think your question boils down to: Why is ionic bonding in $\ce{HX}$ (X=Halogen) stronger than covalent bonding?

This is because in covalent bonding there is only a mutual agreement to share electrons whereas electrostatic attraction in ionic bonding.

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