1
I prefer studying lonely when I need to concentrate.
I prefer studying alone when I need to concentrate.
Alone means ‘without other people’. Use alone to say that someone does something without anyone else: study alone, live alone and travel alone.
2
He is an alone child.
He is a lonely child.
Lonely means ‘unhappy because you are not with other people’. We can use it before a noun: a lonely child, a lonely person and a lonely life.
A person can be alone without feeling lonely.