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After a hot shower, the mirror in my bathroom steams up. When I try to clear it with a towel, it immediately refogs. Yet once I use my hair-dryer, it will clear the fog and the mirror will stay clear.

I'd like an answer to the mechanics behind this difference. I suppose it got something to do with the heat, and maybe the refogging is only delayed due to it.

But that is just me speculating, and I'd like a concrete answer on this phenomenon.

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The water vapor in the bathroom will condense preferentially on the coldest spots around the bathroom. When you wipe the mirror with a towel the water vapor readily condenses on it again because you have not changed the mirror temperature. When you use the hair dryer, the temperature of the mirror is now higher and the water vapor will prefer to condense on other colder spots around the bathroom. Of course this will only last until the mirror thermalizes and reaches the same temperature of other objects in the bathroom.

This reminds my of trying to defog the windows inside a car on a cold but humid day. If you turn hot air onto your windshield you will clear it up. If you turn your AC on at the coldest setting, the water vapor will condense on the cooling coil of the AC and not only your windshield but all the other windows in your car will be clear. Try this next time.

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Beyond the principle of the car defogger example, you may notice in some 5-star hotels, that the mirrors are always fog free regardless of how much steam there is in the washroom. They have put a heater plate behind the mirror, and the surface is warm and prevents fogging.

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