What is the difference between adding heat and adding cold?

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The opposite, "heating" with no temperature change is observed quite frequently, though most don't notice. Melting ice can absorb a lot of "heat" -- but just as the ice was 0 degrees C before the state change, the water is 0 degrees after. As long as the heating is slow and even, the ice/water mixture will remain at 0 degrees C until all the ice has melted.