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What happens if we lift a box of an ideal gas? Work is done to the box but no heat gets into it. Does its internal energy increase by the amount of work done? Or is lifting it not counted as work done on the system?

Or, is it the case that work turns into heat in a box which quickly flows out to the surrounding, thereby resulting in no net internal energy change?

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When you lift a box of an ideal gas you are not doing any work on the gas, so its internal energy remains constant. However you are increasing the gravitational potential energy of the box and the gas. That's where the energy you put goes.

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