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I'm trying to identify the name of every functional group sticking off the sides of tartrazine (FD&C Yellow No. 5), and the one I can't find anywhere is this $\ce{COONa}$ group. Does anyone know how to describe it? Is it really two groups put together?

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This is simply the salt of a carboxylic acid and sodium hydroxide, with the general reaction looking something like this:

The names of carboxylate salts can be derived from that of the original carboxylic acid and the base used, "-oic acid" becoming "[metal] -oate" (e.g. benzoic acid reacting with sodium hydroxide to become sodium benzoate). If part of a more complex molecule, the $\ce{COO-}$ group is treated like any other substituent and given the name "carboxylate".

This can be seen in the name of tartrazine: trisodium [...]-3-carboxylate, as well as the name of the sodium salt of citric acid: trisodium 2-hydroxypropane-1,2,3-tricarboxylate.

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