2008 1500 5.7L Hemi
What is the purpose of the hose that runs from the airbox to the lower part of the oil fill receptacle?

 

fresh air into the into the crankcase for pcv system.

 

It is one of the main hoses for the PCV system. When the engine is running, the airbox is a vacuum, sucking fuel and oil vapours from the crankcase and passing them into the engine intake to be burned through.

 

Actually, that hose allows filtered air to sucked into the crankcase by the PCV valve where the oil vapor is introduced to the intake manifold to be burned in the cylinders. If you are getting oil into the air intake/filter box, you have a problem.