Which cup of water is hotter?

Question

Two styrofoam cups of water are left in the shade. One in Philly, 90 F ( 32.2 C) at 90 percent humidity and one in Palm Springs, 115 F ( 46.1)

Answer 1

Neither cup of water will ever reach equilibrium since evaporation will proceed quite rapidly in both.   Thus the best answer is that equilibrium has been reached when all water has been evaporated.  Now, as far as the intermediate (nonequilibrium) temperature of each is concerned, it is not certain that the the cup in Palm Springs will be warmer.  Due to the significantly lower humidity its evaporation rate will be so much higher than that of the other cup, that in principle it may cool to a lower temperature.  If that will actually happen, I'm not sure, because the difference is air temperature is also large.

Answer 2

Hi Remo,

Good question! Since there are several factors that influence evaporation I would like to take the liberty of tying a few variables down.

1. Surface area of the water is constant regardless of location. The larger the surface area the higher is the rate of evaporation.

2. Atmospheric pressure is the same regardless of location. Lesser pressure will insure higher rate of evaporation.

3. The air movement is about the same in both location. Water vapor present close to the surface of the water will slow evaporation.

4. The water has reach thermal steady state at each location.

The Palm Springs water will evaporate faster cooling the water down. Since the rate of heat transfer between the air and heat of evaporation are in a tug of war the heat of evaporation of water is formidable and wins the tug. My guess the temperature of water in Philly will be less than 90F ( say 88.5F ) while the Palm Springs water will be much cooler than 115F . Probably around 87.5F. (LOL, I hope you're impressed by my computations)

If you want to chill some wine place a wet towel over it. In Palm Springs of course. In Philly with 90% RH it just won't work.

Answer 3

what's the humidity in palm springs?

Answer 4

The cup at 115 F is obviously hotter than the cup at 90 F. Why the question?

Answer 5

they are the same