The air that you exhale contains water vapour. When you exhale during a cold day, the relative humidity increases. Relative humidity is actually the percentage of the amount of water vapour in the air.(the maximum amount of water vapour that the air can hold at that temperature) The colder the air, the less water vapour it can carry. When exhaled, air mixes with cold air, the temperature of the exhaled air drops, but there is more water vapour. When the air becomes saturated, (relative humidity is 100%), the extra water vapour will condense, allowing you to see your breathe on cold days.
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