This is because of the “heat of vaporization”. A liquid such as water for example doesn’t smoothly absorb heat and rise in temperature until it gets to boiling temperature and then suddenly becomes vapor. Instead there is an amount of energy that the water absorbs without changing in temperature before it makes the phase change. For water this is actually five times more energy than heating it from 0 to 100 Celsius!
Alcohol in the hand sanitizer will evaporate easily at room temperature but it needs to absorb that heat energy required to change into a vapor from the surrounding area. That heat being pulled out of your hands feels cold.
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