Question

What is the relationship between Boyle's law and the kinetic theory?

Answer 1

In qualitative terms, the relationship is the following.
According to the Kinetic Theory, the pressure of a gas is due to the elastic collisions of its molecules upon the container's wall. If the temperature is kept constant the average speed of molecules is constant, thus the pressure can change only if the frequency of the collisions changes.
This is what happens by reducing the volume (the molecules become more crowded, thus making the rate of collisions and pressure increasing) or expanding the container (the molecules become more rarefied, so the rate of collisions and pressure decrease).

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(Find PhET simulation for all gas laws, including Boyle's law here: https://phet.colorado.edu/en/simulation/gas-properties)