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According to Wikidata, anisole has a dipole moment of 1.38D and according to this page on Researchgate, phenol has a dipole moment of 1.70D.

Resonance in anisole: enter image description here

Resonance in phenol:enter image description here

According to my reasoning, $\ce{-CH3}$ can show a good +I effect and hence the positive charge on oxygen is more stabilized in the canonical forms and so the negative charge must be more delocalized and hence anisole should have a higher dipole moment. My concern also extends to the reaction of both anisole and phenol with Lewis acids like $\ce{AlCl3}$ in case of the Friedel-Crafts reaction. This paper states that phenol forms a complex with $\ce{AlCl3}$ and consequently its activity is decreased. But this states that anisole prefers C-acylation. Now, with some hints from this answer, I have come to believe that the reason for the more easy Friedel-Crafts reaction by anisole is due to its less dipole moment causing it to react less with the Lewis Acids and activate the ring more. But it still doesn't make sense to me, and I cannot explain to myself how even the dipole moment of anisole is less than that of phenol. Please explain in regard to Friedel-Crafts also. Thanks!

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The answer lies in the fact that OH group is more activating than OR group that is, it increases the electron density inside the benzene ring to a greater extent.

This can be explained by Bent's rule. Since carbon is more electronegative than hydrogen, the lone pair bearing orbital of Oxygen in OR gets enriched in the percentage s character, which decreases its tendency to donate electrons via mesomeric effect (as the s character in a hybrid orbital increases, the electrons in that orbital become more tightly bound with the atom).

Indeed it is true as you pointed out, that the alkyl group exerts a +I effect, although it is not as significant as compared to the mesomeric effect.

Now once we have established this, the dipole moments can be easily compared. As the donating tendency of the group attached to the ring increases, the electron density of the ring increases and that on the attached group decreases, thus increasing the dipole moment.

The explanation about the Friedel Crafts acylation can be given by the same reasoning. The Oxygen in Anisole does not donate its electrons as effectively and hence does not form an effective complex.

I hope that cleared your doubt.

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Based on Pauling's assigned electronegativites (EN), oxygen is more electron demanding (EN: 3.44), than carbon (EN: 2.55), or hydrogen (EN: 2.20). Hence the bonds $\ce{O-C}$ and $\ce{O-H}$ are polarized towards oxygen.

Conversely, hydrogen (in phenol), or the methyl group (in anisole) are electron donating toward oxygen though this inductive effect by the methyl group exceeds the one by a hydrogen alone. Hence – already more satisfied by the electron donation by the methyl group than by the hydrogen – the polarization between the phenyl group and oxygen is attenuated. This yields a less pronounced molecular dipole moment for anisole, and retains a higher (remaining) pi-electron density in the arene which can react as a Lewis base, e.g. in the course of a Friedel-Crafts reaction.

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