How do I cope with the fact that my mother favors my younger brother?

My entire life, I have desperately sought my mother’s approval and attention. She has blatantly favored my brother, who is five years younger than me. Despite growing up in the same household, my brother and I were raised in two completely different ways. As a little girl, my mother’s disciplinary methods toward me were borderline child abuse (dumping an entire container of salt over my meals, dragging me across the carpet until my skin became rug-burned, smashing eggs onto my head, hitting me with a broomstick, slapping me, etc). She was very verbally abusive towards me as well and managed to almost permanently damage my self-esteem. I was very well behaved as a child—good grades, polite and painfully shy. My punishments were warranted on days where I would accidentally leave an expensive jacket at elementary school or wet my pants as a toddler. My brother, on the other hand, was (and still is) a spoiled, disrespectful, arrogant, entitled delinquent. He never receives grades higher than a D or F, and when he is punished by my mother, it is very lightly. He will be mildly reprimanded and have his phone taken away for mere hours. He skips class, experiments with a myriad of drugs, won’t graduate from high school on time, yet my mother portrays him as the golden child in my immediate family. I’ve tried everything to get her to love me as equally or at least briefly acknowledge me in a positive light. The only time she even seems interested in speaking to me is when she is asking me for money, which is often. I give her however much she needs without question or hesitation, doing whatever it takes to be a good daughter. Yes…I’m willing to pay for my mother’s approval, and she knows this and takes advantage of that fact. I’m 21 years old and was recently accepted to medical school. For years I’ve been working hard and dreaming of the moment I got accepted and my mom would finally be proud of me. I didn’t do this for her; I did it for myself, but I was looking forward to her excited reaction. She wasn’t excited. Not even remotely. At this point, I’m not getting any older and have decided to give up on seeking her approval. I feel so pathetic, hurt, exhausted, and tired of feeling this way. All I’m asking is for advice on how to cope and accept the fact that in my mother’s eyes, I’ll never be enough.

It is best to distance yourself from toxic abusive family. You'll learn all about that in psychiatry when you go to med school.

That is abusive. Nothing borderline about it.

Often grad schools will have some kind of offer of therapy for their students. I suggest you start talking to someone. It's a long uphill climb to deal with the kid of childhood you've had. This is also a highly recommended book:

If anything have some pity for your brother. She is messing him up as well... just in a different way.

I didn't hear a mention of a father. Just wondering if he is around and supportive. My guess is likely not. Your mother sounds like a sad excuse of a parent. I am a mother and I would never do any of the above described things to my children. I agree with the other poster - you should get into therapy to deal with your reality - that there is nothing wrong with you, but your mother sounds mentally ill (not trying to be funny here). Good luck. Chin up. Be a success for yourself and no one else.