I think there most likely is a typing error in your question in regard to the word "immortality." This is not a theme developed in the novel, but Fitzgerald definitely does develop themes of immorality which are introduced in Chapter 1 and Chapter 2.
In Chapter 1, Nick learns that Tom Buchanan is having an extramarital affair with "some woman." Nick is shocked by Tom's immorality, but he is even more shocked when he realizes that Daisy is aware of Tom's behavior and tolerates it:
It seemed to me that the thing for Daisy to do was to rush out of the house, child in arms--but apparently there were no such intentions in her head.
Nick feels "confused and a little disgusted" by Tom's and Daisy's behaviors.
Chapter 2 continues to emphasize the immorality of Tom's affair by introducing Myrtle Wilson and developing her relationship with Tom as they sneak off together to the apartment in New York City where they meet regularly. Unlike Daisy, George Wilson (Myrtle's spouse) does not know that his wife is being unfaithful. Tom shows complete contempt for Myrtle's husband:
[George] thinks [Myrtle] goes to see her sister in New York. He's so dumb he doesn't know he's alive.
When Nick attends a drunken party at the apartment, he overhears a conversation that gives him even more insight into Tom's deceit. He has led Myrtle to believe that he cannot divorce Daisy and marry her because Daisy is a Catholic, which she is not. Tom lies to Myrtle as easily and deliberately as he lies to his wife. Clearly, he feels no shame or any sense of morality at all. Tom's immorality is found in Myrtle, as well, as she deceives her husband without hesitation for her own selfish reasons.