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Now the same wheel (with its rim and tire) remove from the car axle, so it is now freely lying on the floor. Hence no car weight is pressing on it. Let's denote the pressure of the air in the tire as [itex]P_{f}[/itex].

Question: Is [itex]P_{c}[/itex] different than [itex]P_{f}[/itex] ? How to show it by using ideal gas law equation? You may assume the weight of the rim as negligible if needed.

My personal guess: [itex]P_{c}[/itex] is more than [itex]P_{f}[/itex] because with the weight of the car pressing on the tire, volume of tire (hence volume of gas) reduces, thus pressure increases. In other words, if a tire pressure is 30psi while installed on car, once take it out from the car axle, its pressure will drop.