The fog that you’re seeing is water that condenses from the gas to liquid state. Much like what you see when you breathe out on a cold day and just like clouds.
So how did the fog form? Well, when you take a hot shower, most of the water comes out in the liquid form, but there is also a small amount of water gas that is released as well. This might be surprising since water’s boiling temperature is 100C and I’m sure no one showers with boiling water. However, even at temperatures below the boiling temperature, liquid water can still evaporate and release water gas molecules. If you’ve ever left a glass of water out on the table, you’ll see that some of it evaporates away over time despite the temperature being below room temperature.
So back to your question, there is a lot more water in the gas state that comes out of the shower head because of the higher temperatures of hot showers. These gas molecules can diffuse away from the shower towards the other areas of your bathroom, where it’s colder. Since it's colder in the other parts of your bathroom, the gas molecules that collide together will remain stuck together to form droplets of liquid water, which we can see lingering in the air. The mirror is one area that the water molecules can condense together, which is why it is fogged up and you can draw a heart on it.
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