Within The Scarlet Letter, what is Roger Chillingworth's salvation from sin?
Roger Chillingworth, the estranged husband of Hester Prynne, has as an ultimate goal: to see who is the man with which Hester Prynne committed the sin of adultery. He aims to find out who is this mysterious character for whom Hester has rendered herself as a pariah just for the sake of keeping his identity safe.
This quest for the identity of Hester's lover leads Chillingworth to enter Hester's settlement as a "practitioner" of medicine, ending up in the care of Reverend Arthur Dimmesdale; the man who is actually Hester's lover although this is only suspected, not known, by Chillingworth.
Once Chillingworth becomes aware of who Dimmesdale really is, the psychological punishment under which he puts the man is incalculable. He literally weakens Dimmesdale both mentally and physically until the man's untimely death at the scaffold, where he declares himself as a sinner.
Yet, the story tells us how, shortly after Dimmesdale's death, came the death of Chillingworth, himself. Seems as if, once Arthur finally ceases to be, Chillingworth's existence has no further meaning. He does not love nor want Hester, he has no friends, no family, and he is not even living in his country of origin; he is an add-on to Boston.
Therefore, it is perhaps a combination of guilt, the fear for perishing in Hades, and the final realization that his deeds were, indeed, quite evil, that motivate Chillingworth to leave all of his money and properties to Hester's daughter, Pearl.
Although it is not clear whether he intended to do so or not, helping Pearl does as much good as helping Hester; Pearl is able to take care of Hester and leave her set up comfortably in genteel modesty within the settlement. This is because, thanks to her new-found fortune, she has higher social chances to be matched to a good, wealthy marriage- which is what ultimately happens.
Chillingworth basically solidifies the prospects for Pearl to have a better life, and to leave her past behind. Equally, his assets help reinvent Hester into society and make her independent, and even, perhaps, happy. His leaving the money to Pearl is the best event to correlate to his possible salvation from sin.