The lower atmosphere, where we live, is called the troposphere. "Tropos" is ancient Greek for "stirred" or "mixed". The lower atmosphere is in constant motion and for the bottom 15km or so the gases are uniformly mixed. The mixing is primarily driven by heat from the sun.
Water is the only major exception, because it is barely a gas in the atmosphere. Water vapour enters over water, but can fall out as water or ice at some cooler place.
Imagine taking a jar of water from a river of muddy water. If we leave it sitting in a corner for a long time, the mixture will separate. There will be sand/mud at the bottom, some clear water and some bits of leaves/twigs floating on top. To get the atmosphere to become this still so that heavier gases such as CO2 would sink to the bottom, we would need to turn off the sun. We might also need to wait several billion years for the Earth's internal heating to stop.