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If specific heat of solids decreases with decrease in temperature then does it mean that at absolute 0 K no energy is needed to increase the temperature of a unit amount of substance by 1 degrees.? If no energy is needed to raise its temperature then can the substance be raised by 1 degrees without requiring any kind of energy. If this is so then how the substance at 0 K remains stable at 0 K?

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You are correct, for (most) solids this is the case. And yes, the item would not be stable at 0 K, one more reason that 0 K is unattainable. I suggest reading up on the 3rd Law of Thermodynamics for more info on this topic. It is the least talked about law, but addresses your situation.

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