Tenant gave 30 day notice mid month. Is tenant responsible for full Feb rent or for prorated amount?

It appears the tenant was on a month to month tenancy. If so, the tenant is liable for rent through the following month, assuming the tenancy was by the calendar month. For example, if a tenant gives one month's notice on January 15, the tenant is responsible for rent through February 28. If you are reasonably certain the tenant has abandoned, you can change the lock. And don't forget to comply with the security deposit statute if the tenant had made a security deposit.