the kinetic theory of gases
A theory that explains the large-scale behaviour of a gas, its pressure and temperature, starting from the picture that the gas is countless tiny molecules in constant random motion, forever bouncing off the walls and each other. It turns the macroscopic gas laws into a story about molecular billiards.
Precisely, it derives pressure from the momentum transferred by molecules striking the walls: P = (1/3)(N/V) m <v^2>, where a volume V holds N molecules of mass m and <v^2> is the mean square speed. Comparing this with P V = N k_B T gives (1/2) m <v^2> = (3/2) k_B T, so the temperature is nothing other than the average translational kinetic energy of the molecules.
This is the key and honest insight: temperature is not some kind of hot substance stored in the gas, it is a measure of the average kinetic energy of the molecules. The derivation assumes the gas is dilute and behaves ideally, exactly the conditions where the ideal gas law holds.
The momentum handed to the walls by countless molecular impacts, added up over the whole surface, is exactly the pressure we measure.
Pressure and temperature derived from molecular collisions.
The identification of temperature with average kinetic energy is exact for the translational motion of an ideal gas; storing energy in rotation and vibration needs the equipartition theorem to account for it.