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The salt is not actually making the ice colder. The internal portions of the ice are already below 0 degrees Celsius. The salt, or NaCl, gets in the way of the intermolecular forces of the water, allowing the ice to melt at lower temperatures and remain in the water phase below 0. In other words, the water molecules are normally rigid enough to freeze at 0 degrees, but the addition of salt makes it more difficult for them to be rigid enough. This lowers the melting point. As the internal portions of the ice are already below 0, when the external ice is melted below 0 also, the internal, very cold portions, can further cool the water/ice mixture. The above answer is easily disproven. The assertion is that the ice core is colder than the surface, and the warmer surface is just being stripped away by the salt, revealing the cooler inner core.To disprove that, do the following experiment: Measure the temperature of your freezer. If your ice has been in there any length of time, the core and surface of the ice will be at the same temperature that you have measured (e.g. 25 F). Take a pile of ice from your freezer, dump a pile of salt on it, stir (for a while), and measure the temperature with a thermometer. It will get WAY colder than your freezer, thus proving the original answer to be impossible.

REAL ANSWER:

Once salt comes into contact with water, NaCl would split into Na(+) and Cl(-). Since these atoms have charges, they "tug" at individual water molecules. This makes the water less rigid and separate from each other. Since they are less rigid, the water molecules are free to move around. This movement is kinetic energy, which is measured by temperature. Due to the law of conservation of energy, the kinetic energy of the water can't just appear, it has to come from it's surroundings. So the air and objects near the ice/salt solution would transfer their heat to the ice/salt.

More powerful salt, like CaCl2 works better than regular salt because when it touches water, it splits into three atoms: Ca(+2), Cl(-), and Cl(-). The extra atoms mean more force when they "tug" at water molecules.

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