Translation:I applied to the department of chemistry.
"its department" is "bölümü"; the -n- is a buffer consonant between the possessive ending and the case ending. ("Bölümün" would be genitive case, "department's", not possessive, "its department".)
Another way of saying it in English is "the chemistry department" -- perhaps this makes it clearer that this is not a (fully) possessive construction but more of a noun-noun compound.
I suppose kimyanın bölümü might be "the department of the chemistry".