“No – I described the derivation and questioned the assumptions.”
Questioning assumptions doesn’t entitle you to assert mutually inconsistent claims. You claimed that under an adiabatic expansion process the average kinetic energy of the molecules is reduced but the total isn’t reduced because (as you alleged) it’s the same amount of energy spread out in a larger volume. But this explanation doesn’t work at all. If the average KE drops while the number of molecules is constant, then so does the total.
As I suggested a couple days ago, the only way I could see for you resolve this inconsistency would be to claim that the van der Waals forces between the molecules of air in the atmosphere are *so* *very* *strong* that real parcels of air hold cohesively together and don’t exert *any* force (and hence no work either) on the surrounding air while expanding. But this is obviously not what happens in the atmosphere.
I think just like Doug C, you are thrown away by the fact that there aren’t solid physical boundaries around the notional boundaries of the parcels of air for you to mentally visualize. Because you can’t picture the boundary as a concrete material object, you believe the motion of this notional boundary can’t have a real effect. But no real physical boundary is required at all for the effect to be produced by the expansion process. While the boundary is notional, its motion isn’t.
Consider the adiabatic expansion of the air in a vertical cylinder as the piston above is allowed to move out. Initially, the piston was held in position by a force equal to the pressure of the air multiplied by the area A of the piston. As the outside force on the piston is reduced, the air in the cylinder expands and the pressure in the cylinder drops accordingly. The temperature of the air drops and the reduction in internal energy must match exactly the work performed on the moving piston, as conservation of energy dictates. This work is W = F*dz = (P*A)*dz = P*dV.
But notice that there is no flow of heat through the piston. The process still is adiabatic. The energy is transferred through mechanical work (force times displacement). If fact there isn’t any flow of heat anywhere. The internal energy density drops uniformly throughout the volume of the expanding gas as its internal pressure also drops uniformly.
Thus we are entitled to consider separately the parcel of air constituted by the lower half of the cylindrical air column and inquire why *its* internal energy is dropping since *it* makes no contact with the piston. How is this energy leaving the parcel? Its notional boundary is the mid cross-section of the total air column below the piston. As the piston moves out some distance dz, this notional boundary moves out half the distance, dz/2. And the work performed by the lower air parcel is consequently W/2, which indeed must match its reduction in internal energy. When some virtual parcel of air is allowed to expand through a reduction of the pressure of the air above (or around) it, then the loss in internal energy only is a function of the pressure change. It doesn’t matter in the least that there isn’t a solid boundary in direct contact with the relevant parcel. The reduction in internal energy always is exactly PdV and doesn’t require a solid wall nearby to push against.
