Atmospheric pressure is greater at the base of a mountain than at its peak. A hiker drinks from a water bottle at the top of a mountain. The
bottle is capped tightly. At the base of the mountain, the water bottle has
collapsed slightly. What happened to the gas inside the bottle? Assume
constant temperature.

At the top of the mountain, when he tightens the cap onto the bottole, there is some water and some air inside the bottle.  Then he brings the bottle down to the base of the mountain.

The pressure on the outside of the bottle is greater  than it was when he put the cap on.  If anything could get out of the bottlde, it would. But it can't . . . the cap is on too tight. So all the water and all the air has to stay inside, and anything that can get squished into a smaller space has to get squished into a smaller space.

The water is pretty much unsquishable.

Biut the air in there can be COMPRESSED.  The air gets squished into a smaller space, and the bottle wrinkles in slightly.

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