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Does it have something to do with the pkb of the alcohol group at ph 7 which makes it not protonate?

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That is precisely the reason why. Zwitterions are defined as molecules that contain both a positive and a negative charge within the molecule. Alcohol groups won't deprotonate at neutral pH = 7. The pKa of a typical alcohol is about 16, so you need a pH of 16 to deprotonate an alcohol (or a very strong base). Keep in mind a pH of 16 is difficult to achieve in water since for water, pKw = 14.

There are some molecules with an alcohol sidechain that can deprotonate at a reasonable pH. These are typically phenols. Take tyrosine as an example

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The pKa for the carboxyl group is 2.2, the amino group is 9.21, and the hydroxyl group's pKa is 10.46. At neutral pH, the carboxyl group will be deprotonated (negatively charged), the amino group will be protonated (positively charged), and the hydroxyl group will be in its normal protonated form, making the molecule a zwitterion. A pH of 11 will be enough to deprotonate the hydroxyl group, as well as the amino and carboxyl groups to give the molecule an overall negative two charge.

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As you surmised, zwitterions contain groups that are proton ($\ce{H+}$) donors and groups that are proton acceptors. The $\ce{C-O-H}$ group in alcohols, though, is covalently bonded, and is close to neutral. Zwitterions are usually defined by groups with stronger basic or acidic properties.

That said, alcohols can be amphoteric, i.e. act as either base or alkali. For example, 2-methyl-2-butanol reacts with Lucas' reagent as if alkaline, losing its $\ce{OH-}$ group, but reacts as if it were an acid, surrendering $\ce{H+}$ with potassium to form potassium 2-methyl-2-butoxide.

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