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Why the Temperature of a Gas Increases when it is Compressed
The diagram below represents a quantity of gas in a cylinder with a moveable piston.
If a molecule experiences an elastic collision with a stationary wall it will rebound at the same speed. However, a molecule colliding with the surface of the piston as it is moving so as to compress the gas into a smaller volume will rebound moving faster than before the collision. Thus the average speed of the molecules will increase. This means the temperature will increase.
Similarly, if the piston is moving the opposite way, the average speed (and therefore the temperature) will decrease.