At 1Atmosphere of pressure, “air” when heated will expand.
There are a LOT of forces at play on the atmosphere. If we focus on a ventilated room, the forces are simpler.
gravity, and convection. If a furnace pulls in return cool air, heats it and supplies it back to the room, it will, if dumped out high without a diffuser, sit on top of the cooler air, then convection will spread the heat from the air around the room.
If the heated air is dumped out low in the room, again, without a diffuser, it will be forced to the top of the room by the denser air around it, and convection will again distribute the heat.
If you use an electric heater to warm the air in the room, cold air will be the densest, so the heaters are as low as possible, as the air is heated, more cold air pushes out the heated less dense air.
there is no rising… there is gravity pulling denser air below the less dense air.
Remember… there are only two forces. Gravity, and convection. Gravity pulls towards the center of mass of the planet. So… in the room… down. Convection is a circulation motion… radiant energy transfer barely provides any change in the scenario, and radiant certainly does not rise, or fall… it radiates.