Er... the Han used for Korea and the Han used for Han people / dynasty / language are not the same word. The Chinese word for Han is 汉, pinyin han4, the word for (South) Korea is 韩, pinyin han2. The name derives from three early Korean states which shared a Korean word in their names that was transliterated into Chinese as 韩. They have no relation to one another aside from a shared romanization.

I've removed this question since it's not really a question about history.

However /u/flyingdragon has given you an appropriate response to the question you have: They're different words entirely. They just happen to sound the same in modern Korean. However in Mandarin they are quite distinctly different.

If you have questions about the development of the writing systems or script reforms, /r/AskHistorians is certainly an appropriate place to ask.