We can often equate purpose with theme, which might help you in your analysis of this excellent novel. To aid you further, I have included a link below to the enotes section of this novel about the themes that there are. You might want to look at this to help you consider other themes.
However, for me, this novel is essentially about an age-old conflict that forms the basis of so many works of literature. The individual vs. society places individual characters with their feelings, dreams and emotions, up against the harshness of society with its norms and established conduct of behaviour. Relating this conflict to the novel, we can see that Newland Archer wants to be with Ellen, but what prevents him from making such a radical decision is the pressure of society. Although he is clearly presented as a torn character, trying to balance his own desire with what is expected of him by society, ultimately society wins as he realises that he can never reject the demands of society upon him. The carefully planned announcement of May's pregnancy, and the new series of responsibilities that it necessarily involves, clearly is May's final trump card that she plays with full awareness of what it will mean for both Ellen and her husband. This is made clear in Chapter 33, when May confesses that she told Ellen she was sure of her pregnancy even before she was:
Her colour burned deeper, but she held his gaze. "No; I wasn't sure then--but I told her I was. And you see I was right!" she exclaimed, her blue eyes wet with victory.
At the end of the day, the demands of society are greater than Newland's own personal will and desire, and this is what ends his relationship with Ellen as his personal passion is sacrificed for the good of his family.
Let us not forget the way in which Ellen faces a similar conflict. Hoping for a divorce, she is disheartened to discover that she will be rejected unless she stays married. Ironically, Newland has to advise her that her own personal desires and happiness are secondary to the way in which her divorce would impact her family. She, like Newland, bows to the power of society by remaining married against her own personal wishes.