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At a given temperature, freezing and melting occur at about the same rate, so the snowflake doesn't change much over time. This is called equilibrium.
When the highway department drops a salt crystal onto the snowflake, it dissolves in the liquid part. Because salt has a different size and shape than water, it doesn't freeze into the ice. Instead, the salt gets in the way of some of the liquid water molecules, preventing them from being frozen into ice.
However, salt doesn't keep molecules from leaving the ice and going into the liquid. In other words, salt speeds up the rate of melting without affecting the rate of freezing. Only if the temperature drops can freezing compete again. If the temperature drops a lot, freezing completely takes over and there is no liquid water to dissolve the salt and the whole process doesn't work. That's why when it's really cold in Ithaca they just put down sand to improve tire traction.
But why salt? Shouldn't the above reasoning apply for any solid that dissolves in water? The answer is yes! Sugar could be used too, but salt is cheaper. Properties that depend on the number of dissolved molecules, not their identity, are called colligative properties.
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