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Why does salt melt ice?
Dissolved salt lowers the freezing point of water, changing the balance between ice and liquid.
What is happening?
Even below 0 °C, a thin liquid layer can exist on ice. When salt dissolves in that water, the solution has a lower freezing point than pure water.
If the temperature is within the useful range, some ice melts to establish the new equilibrium. Melting absorbs heat, so the ice-salt mixture can become colder while more liquid forms.
What changes it?
Salt is not equally effective at every temperature. At sufficiently low temperatures, ordinary sodium chloride becomes a poor de-icer, which is why other salts or methods may be used.