Similarly, you may ask, why is Arwen tied to the ring?
The logic is that Arwen chose to be mortal but she was not tied to Aragorn yet cos of the War. So since she had nothing to keep her alive, she was dying. It was Elrond's plea because Arwen was dying that drove him. In movieverse, the 'evil that spreads from Mordor' was in some way affecting the Elves.
Secondly, what happens to Aragorn and Arwen in the end? Fourth Age. In the year 121 of the Fourth Age, after Aragorn's death, Arwen died of a broken heart at Cerin Amroth in Lórien, and was buried there one year after the death of Aragorn, to whom she had been wedded for 122 years.
Regarding this, does Arwen become mortal?
Yes, Arwen dies a mortal death; eventually. Because she is half-elven, her life is very long even as a mortal.
Why did Arwen lose her immortality?
She fell in love with the human Beren, and despite much tragedy that would put Romeo and Juliet to shame, Beren dies (since he is human), and Lúthien petitions Mandos, the Valar in charge of the fates of all mortals, and she gives away her immortality so that they could live together, for as long as they have.
