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Evaporative cooling can occur in an air-liquid water system initially at the same temperature. If the air is not saturated with water vapor, the liquid water will evaporate until the partial pressure reaches its vapor pressure. In this process, the temperature of the liquid water decreases. This is because there is a distribution of speeds among the water molecules, and only the water molecules moving fast enough with enough kinetic energy to break their hydrogen bonds will escape into the gas phase. This selection based on kinetic energy leaves the slower molecules in the liquid phase, thereby lowering the temperature of the liquid.

My question concerns the parallel process of an ice cube melting. If the ice cube is in thermal equilibrium with a super-cooled salty brine, say -20 degrees C, there will still be a distribution of speeds among the water molecules in the ice lattice. Since only the fastest moving water molecules in the ice lattice will have enough energy to break the hydrogen bonds, would the melting of the ice cube cause a similar cut-off of the speed distribution curve and lower the ice cube temperature? If the volume of the brine solution was large enough so that the brine temperature doesn't rise much, could there be a situation where the ice cube both melts and also get cooler?

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Yes, and this is exactly how freezing mixture works. Mixing ice and calcium chloride can get down to -10C.

However the mechanism is a bit different to the one you describe. The water molecules in ice are held together by a variety of intermolecular interactions, of which the biggest contributor is hydrogen bonding. This means it takes energy to pull a water molecule away from the ice surface. This energy is the latent heat of fusion.

At any temperature above absolute zero the ice lattice is vibrating, and these vibrations hold energy. The average amplitude of the vibrations is related to the property we call temperature. The energy to break a water molecule free of the ice surface comes from this vibrational energy, so each molecule that leaves the surface reduces the vibrational energy in the ice and hence its temperature.

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