Does the phenomenon of seeing your breath depend on the temperature of the air, the humidity, or both?

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The phenomenon by which you "see your breath" is a form of condensation.  Condensation is the process in which a gas transforms into a liquid.  When you "see your breath" the moisture (gaseous form) in your breath has condensed and become little droplets of water.

The conditions in which this can happen are affected by a number of factors.  Please see the link below for a very good discussion of the impact of the various factors -- it is below the weather map, so please scroll down.  There, you can see that the possibility of seeing your breath depends on such things as the temperature of the outside air, its relative humidity, the temperature of your breath and its relative humidity.

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