She complained about her friend and her brother has forsaken her
Question

I have a friend whom I have known for 18 years. One day I thought that she was speaking on the phone with a man, and I was afraid for her because she is very respectable. I told my younger brother about that, seeking his advice, but after a while I found out for sure that she was innocent, and I had been too hasty in speaking to my brother. The problem is that my brother has become suspicious of me and he can hardly bear to have anything to do with me. I do not know how to solve this problem. Do I have to offer kafaarah (expiation)? What should I do with my brother?.
Praise be to Allah.If you spoke about your friend out of fear for her and hoping to save her, then there is no sin on you, but you should have waited and should not have been so hasty. You do not have to offer kafaarah or do anything else, unless you got carried away in speaking about her and said something about her that she dislikes, in which case you should tell her and seek her forgiveness for backbiting about her. If she has heard what you said about her, then you have to apologize to her and ask her to forgive you. If she did not hear of what you said about her, then do not tell her, rather make a lot of du’aa’ for her, pray for forgiveness, and speak well of her and mention her good points to those to whom you spoke about her. We ask Allah to forgive us and you. 
See also the answer to question no. 6308. 
With regard to the problem of your brother’s doubts about your behaviour, this may be solved by asking Allah to guide him and to make him stop that. Then you should adhere to Allah’s command of hijab, lowering the gaze, and avoiding non-mahram men. Do a lot of naafil prayers and fasts, and give in charity. Try to speak to him frankly and warn him against being suspicious of you. 
Undoubtedly if your brother sees that you are righteous he will no longer doubt you, and that suspicion will disappear from his heart, in sha Allah. 
And Allah knows best.