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The gas is shuttled back from each space via displacer, which does so by changing the volume of each space... the spaces aren't sealed from each other (but I'm still assuming a strict temperature boundary between them)

I want to calculate the total pressure of the system with air, forgetting for the moment about taking power out of the system with a piston..

I suppose my first question is, with PV=mRT, when calculating m, do I take into account that the air density is different at the different temperatures, or is this accounted for already in the formula?

And the second, when I have two pressures for each area, can I simply just combine the two to find the resultant pressure?

Thanks