Using Chapters 20 - 27 of Jane Eyre, explain Rochester's rationale in saying he has a right to marry Jane. Then, explain Jane's rationale in...

To understand Rochester's rationale, we must look to Chapter 27 when Rochester explains to Jane the circumstances and history of his marriage to Bertha. Once Bertha's madness had become apparent, Rochester says that he transformed himself into a "will-o'-the-wisp;" a ghostly, unreachable figure who began wandering the world in search of a new life. As part of this transformation, Rochester no longer felt bound by the vows of his marriage to Bertha. He courted other women as he searched out a new wife:

Jane: But you could not marry, sir.

Rochester: I had determined, and was convinced that I could and ought.

So, Rochester felt that he had a right to marry Jane because his marriage to Bertha was, for the most part, non-existent. 

As for the second part of this question, it is important to point out that Jane does not leave Rochester because she is mad at him and cannot forgive his betrayal. On the contrary, she loves him as much as she ever has and bears no grudge, as she says in Chapter 27:

Reader! - I forgave him at the moment and on the spot...I forgave him all: yet not in words, not outwardly; only at my heart's core.

We learn Jane's rationale for leaving Rochester later in this chapter when he suggests that she goes to his villa in the South of France. Jane responds:

Sir, your wife is living: that is a fact acknowledged this morning by yourself. If I lived with you as you desire, I should then be your mistress.

The idea of being Rochester's mistress is impossible to Jane for two reasons. First of all, Rochester has had many mistresses in his past, all of whom fell quickly out of his favour. Jane has no desire to be treated in this manner nor to be scorned and shunned by Rochester, as she comments:

That if I were so far to forget myself and all the teaching that had ever been instilled into me, as—under any pretext—with any justification—through any temptation—to become the successor of these poor girls, he would one day regard me with the same feeling which now in his mind desecrated their memory.

Jane has another reason for not wanting to become Rochester's mistress. She recognises the social implications of such a position; namely, that she would be the object of gossip and scandal and would lose her respectability. Jane is not prepared to accept either of these outcomes and so leaves Rochester at the end of Chapter 27.