I would like some insight on a problem I have been working on. I have a seal container of air where the outside chamber has a slight temperature differential between top and bottom. I understand this causes a density variation in the air between the top and bottom of the chamber. Now I am looking to expand the chamber to a larger volume and get the new pressure inside the chamber relative to the previous pressure. I assume there is no heat loss during this process, so I believe in the uniform case I can use the formula : (P1/P2) =(V2/V1)^1.4 for air. My question then is this formula valid for the case where the density and temperature is not uniform? Does it matter if the expansion happens from the top (lower density air) or bottom (higher density air)? I appreciate any insight of how to think about this problem. Thank you.