Sometimes students are taught that "air expands as it is heated". and sometimes they are taught that "as air expands it cools". under what conditions could the first statement be true? under what conditions could the second statement be true?

Both are true under specific circumstances. And are related to Boyle's law. volume and pressure in a gas are inversely proportional.

Explanation:

There is a tendency to entropy in our reality, that is, in particular true and visible with gases, they tend to occupy the whole space where they are confined, when we heat a volume of gas, then the movement of the particles and in consequence the pressure of the gas increases and to compensate this the volume tends to be increased too, according to Boyle's law. And the opposite happens when the volume is increased, then the pressure is  relieved and since the particles are further one from each other, then the temperature is lower, and therefore it cools down.

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