Hygiene thermodynamics 2 Two kinda related questions here:

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*Is evaporation rate and temperature difference related? There is an experiment of pouring cup of hot water out the window during winter. The water evaporates almost instantly forming a vapor cloud.


*I was showering during winter (outside: ~15 degrees Celsius below the Celsius' zero). I opened the window and as the cold air came in the room filled with vapor. It seemed almost like the vapor was coming in through the window. What happened there? My guesses:
a. It's the question 1 thing.
b. I got in a lot of dry air which allowed more water to evaporate.
c. I lowered the air temperature which caused the air humidity to condense forming this fog.
d. or something else did happen?
 A: *

*Regarding your first question, yes, the evaporation rate is related to both the temperature of the liquid and that of the air. There is no single, universal equation describing the evaporation rate at an interface, because so many factors come into play. The simplest equation describing evaporation rate is probably that established by Langmuir:


$$\frac{\mathrm dM}{\mathrm dt}=(p_\mathrm v-p_\mathrm p)\sqrt{\frac m{2\pi kT}}.$$
You can find various dependencies of the evaporation rate discussed here.


*Secondly, a warning. Your questions says “water evaporates almost instantly forming a vapor cloud”. You've got to remember that water vapor is transparent and what you see when you describe a cloud is not the vapor, but condensed droplets suspended in the air:


*Regarding your second question, the main effect is the dependence of water vapor pressure as a function of temperature. If you shower at 40°C, you have Psat = 7.4 kPa, while at 12°C it goes down to 1.4 kPa. So, a large part of the water vapor will condense into liquid water, some of it as air-suspended droplets, some of it on the walls of your shower.
