What is happening when water boils or an ice cube melts?

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These changes represent a change in the physical state of matter, namely water.  It means that enough heat as been transferred into or out of the water to cause it to change to another physical state.  When you heat up liquid water in a pan on the stove, the heat from the stove is being absorbed by the water, causing its molecular bond rotations, vibrations, and collisions to increase in speed and frequency.  Once a certain threshold of heat is crossed, the liquid water can absorb no more heat and the liquid molecules change into gas molecules.  Enough energy has been put into the system to cause the molecules to change physical states.

Similarly, when frozen water (ice) is exposed to enough heat, the molecules get enough energy to break free of the solid matrix and exist in the higher energy liquid state.  Of course these processes are also completely reversible.  So the change in the physical states of water really has to do with the amount of energy in the form of heat that the system has gained or lost.

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