735 ILCS 5/9-203

If any tenant gives notice of his or her intention to quit the premises which are held by him or her, at a time mentioned in such notice, at which time the tenant would have a right to quit by the lease, and does not accordingly deliver up possession thereof, such tenant shall pay to the landlord or lessor double the rent or sum which would otherwise be due, to be collected in the same manner as the rent otherwise due should have been collected.