Why does the ice on a path melt when sprinkled with salt

Why does the ice on a path melt when sprinkled with salt  

The ice on a path or road is slippery and dangerous, as we saw when considering black ice and ice skaters. One of the simplest ways to make a road or path safer is to sprinkle salt on it, which causes the ice to melt. In practice, rock salt is preferred to table salt, because it is cheap (it does not need to be purified) and because its coarse grains lend additional grip underfoot, even before the salt has dissolved fully. 

The depression of freezing temperature occurs because ions from the salt enter the lattice of the solid ice. The contaminated ice melts at a lower temperature than does pure ice, and so the freezing point decreases. Even at temperatures below the normal melting temperatures of pure ice, salted water remains a liquid - which explains why the path or road is safer. 

We must appreciate, however, that no chemical reaction occurs between the salt and the water more or less, any ionic salt, when put on ice, will therefore cause it to melt. The chemical identity of the salt is irrelevant - it need not be sodium chloride at all. What matters is the amount of the salt added to the ice, which relates eventually to the mole fraction of salt. So, what is the magnitude of the freezing-point depression  

Let the depression of the freezing point be AT, the magnitude of which depends entirely on the amount of solute in the solvent. Re-interpreting Blagden s law gives 

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