Free-transit-24hr? Visa-F? This time I am going to add more details about my trip.

Hi, it's me again.
I am going to apply for the F-Visa (single entry) because I am going to do Volunteering Teaching and I have an official invitation from a university. My program ends on July 21.
I wanted to book a roundtrip from LAX-Beijing (July 2-July 28), but since my program ends July 21, I wanted to go to Seoul from July 21 to 28 (another round trip). Once I leave the country, my F-visa is going to be invalid. My plan was to travel Seoul-Beijing on July 28, wait a few hours and take my Beijing-LAX flight that same day. I just want to know what kind of permit I need so I can transfer from one flight to another (on July 28). As mentioned before, I do not plan to leave the airport and I do not expect to stay 24 hours in mainland China. Thank you for reading :)