[Lotrelf]

Yes, he says something like, "it is gone forever, and all is dark and empty." What does this "it" refer to? Ring? May be. But, I'd read somewhere "it" may refer to a "light." This is why he feels "darkness and emptiness," and the Gem aids him in that darkness and emptiness providing him the light.
But, while giving the gem, Arwen had said something like "in my place you'll go" and, IIRC, Professor Tolkien also says somewhere that it was a ticket for Frodo. Wouldn't she have gone to Valinor too, so her jewel might be for that?