How do I explain, with some examples, Sophie's humiliating death as the final statement of pathos to her life in The Chrysalids?I want to show that...

As you say, in literature pathos is a scene or passage that is designed to arouse deep and genuine feelings of pity, sympathy and sorrow in the reader. An explanation of how Sophie's death is the final act of pathos in a pathetic life requires that you establish (1) the deeply sorrowful and pitiful nature of Sophie's death and (2) the earlier events that make hers a lifetime of pathos.

I might suggest that in establishing point (1) you address and dispel the conflicting idea that Sophie's death beside the man she loved was an act of ironic mercy in a life that had known no mercy previously. This argument says that since she could have no hope of a happy life following the destruction wrought by the battle between the Fringe and Waknuk, and since she can never have her dream of bearing children, and since she sees the man she loves lying dead in front of her, and since she had no anticipation of being rescued with the telepaths by the Sealanders, her death was a merciful release from an unkind life.

To establish the pathos of Sophie's death, you must establish the irony of her death at the hands of her enemies after all her heroic deeds to save David and help the telepaths escape (regardless of her private motives). After having proved her worth and value—her humanity—she is shot down by the ones who denied recognition of her humanity because her body did not conform to the Definition of Man: “…and each foot five toes.”

Further, after having proved her ability to transcend perceptions by understanding and loving Spider-Man, her life is ended by the ones who caused Spider-Man’s anger and hatred by their limited perceptions and their fears and inability to love all humanity. With this you must also acknowledge that the Waknuk definition of humanity and rigid restrictions on what constitutes acceptable life, though reprehensible now, had a valid foundation in the immediate aftermath of the holocaust that left unidentifiable mutations outside the borders of Waknuk, as Uncle Axel describes to David.

Then, to establish the pathos in Sophie’s life, you must establish that her life did inspire deep sorrow, pity and sympathy. You can do this by describing what her family felt; that she was not free to socialize with her peers; that she lived in such terror that she was willing to suffer injury to her foot--at a very young age--rather than to expose herself to discovery.

Further, you can describe how David felt after talking to Mrs. Wender in Chapter 1 and describe the nightmares that he had as a result. As the son of the presiding Magistrate responsible for purges of Deviations, David’s reaction carries a great deal of weight because the reader is aware that David knows the true import of having a deviation. Additionally, you can describe Sophie’s misery after being dehumanized further by being deprived of her right to bear children through enforced sterilization at the time of her banishment.