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Salt added to ice will lower the freezing point of water. Thus water, which normally freezes at 32F, with salt added will freeze at a lower temperature. So salt doesn't actually "melt" the ice and snow, it just allows the water to exist as a liquid near its freezing point rather than a solid.

Ice is solid water-vapor water system in equlibrium. Adding salt to the system disrupts the equilibrium by increasing the number of particles in the vapor. So the ice melts to re-establish the equilibrium of the vapor. The idea too is that the added particles disrupt the solid water bonds, where the water and salt dissolve into one another, creating a solution with a lower freezing point.

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