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Today we learned in class through a vivid demonstration that water sticks to glass. We were then taught that water sticks to glass because while glass is mainly composed of silicon dioxide, the surface of glass is actually silanol. Hydrogen bonds form between molecules of water and silanol, which is why water sticks to glass.

Is this correct?

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Yes, it is correct that the surface of glass has silanol ($\ce{Si-O-H}$) at the surface. This is because water will react at the surface to relieve bond strain of a $\ce{Si-O-Si}$ bond to form $\ce{Si-O-H H-O-Si}$ pairs. additionally there is sodium siloxide at the surface ($\ce{Si-O-Na}$) which will release sodium hydroxide on contact with water $(\ce{Si-O-Na + H2O -> Si-O-H + NaOH (aq)})$ and is major reason why normal glass cannot be used for biomedical implants.

Additionally these surface silanols can react with chlorosilanes to produce hydrophobic surfaces $$\ce{Si-O-H + Cl-SiR3 -> Si-O-SiR3 + HCl}$$ where the R is a non-polar organic pendant group that is hydrophobic. This coating causes water to bead off of treated glass that would priory be wetting.

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