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Are all polar molecules both nucleophilic and electrophilic, depending on which atom you are talking about? Take water for example. Oxygen bears the partial negative charge. Oxygen is nucleophilic. Because oxygen isolates electron density from the hydrogens, the hydrogens bear partial positive charges. So would the hydrogens be electrophilic?

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Nucleophilicity and Electrophilicity are no absolute measurable quantities - they are based on an per entry basis and usually in reference to some kind of reaction. So basically you are right in assuming that polar molecules have both properties. In the special case of water these properties also stem from $$\ce{H2O <=> {}^{-}OH + H3+O}$$

But let us have a look at some molecules:

  • $\ce{CO}$ pretty good nucleophile especially towards transition metals. It has quite a small dipole towards the carbon, but a massive overload of electron density due to the $\sigma$-type HOMO
  • $\ce{CO2}$ also quite polar bonds, but no (permanent) dipole - still quite a potent electrophile given the right nucleophile
  • $\ce{SO3}$ has no permanent dipole (but a transitional one) although quite polar bonds. However, it is only a strong electrophile.
  • $\ce{NH3}$ quite polar, hence a reasonably good nucleophile. But it is also almost always way too strong a base to act as an electrophile
  • $\ce{NH(Et)2}$ not so different from ammonia, quite polar, but yet a terrible nucleophile, because it is too big
  • $\ce{BH3}$ not very polar (no permanent dipole) but quite a good electrophile
  • $\ce{I-}$ a monopol, very polarisable, good nucleophile, good leaving group.
  • ...

The consideration of the strength of nucleophiles and electrophiles lies in many variables. It might be polarity, local electron density, molecular orbitals, polarisabilities. It is often highly dependent on the solvent and the other reactants. If you add $\ce{I-}$ to a acyl chloride you will certainly have a substitution to some extend: $$\ce{R-COCl + I- <=> R-COI + Cl-}$$ If you add hydroxide ions, then you will certainly end up with the acid: $$\ce{R-COCl + I- + {}^{-}OH <=> R-COOH + I- + Cl-}$$ Look at it nucleophilicity or electrophilicity from a case to case basis.

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