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Since theoretically hot air moves upwards and the cooler air moves downwards, I was wondering if there is a steady or unsteady tiny increase in temperature of a room as we move from the floor of the room to the ceiling. I mean are the temperatures at the floor and roof different by at least a small fraction?

Ignore the heating of roof by sunlight and thus the heating of ceiling and also ignore the cooling of floor by the earth below the floor and assume the doors, windows and vents in the room are sealed shut.

Somanna
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Yes, there will be temperature differ by a tiny fraction. Hot air being less dense (hence lighter) will move upwards whereas cold air being relatively denser (hense heavier) will sink down.

However the difference will be very tiny, not enough to be noticeable unless the physical conditions are changed (altered). The average temperature of room though will remain almost equal considering you could change the height of the room.