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When wet, the hydrogen bonds between the cellulose chains in paper are disrupted due to the formation of hydrogen bonds with water. This leads to the paper becoming extremely weak. This is the case with most paper, however, additives such as gelatin and aluminium sulfate are added to the paper to strengthen it.

I can't see how the addition of either of these strengthens the paper. I would guess that it aids the bonding between cellulose chains, perhaps by creating a 'bridge' between them, but I'm not sure if this is correct or how it would happen. Another idea is that the aluminium could form a complex ion, and then perhaps form a chelate to strengthen the paper, though I have little confidence in this.

So, how do the gelatin and aluminium strengthen the paper - particularly filter paper, which is almost pure cellulose.

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The process of adding gelatin and such to paper (cellulose) is called sizing. The idea behind using gelatin is that it has both a hydrophilic end and a hydrophobic end to the molecule. The aluminum sulfate (alum) is added as a hydrogen bonding linker between the cellulose and the hydrophilic end of gelatin (or some other similar molecule). Mostly this is just to prevent the paper from getting wet in the first place.

Your second paragraph is spot on. Typically other reagents are used to cross link ( the cellulose in order to retain strength upon getting wet (wet strength). Those reagents are a bit more aggressive. Like alkyl ketene dimer alkyl ketene dimer

or urea-formaldehyde polymer. urea-formaldehyde

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