a chair to sit on;a pen to write with.这种怎么判断最后要不要加介词?

a chair to sit on;a pen to write with.这种怎么判断最后要不要加介词?
从句中，比如说：Although the lecture had already been on for five minutes ,I still was not able to find a chair_________.
这是to sit 还是to sit on .为什么？