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Take the vaporization of water as an example. When we boil water at 100C= 373K the heat source provides energy to break the water molecules' bonds instead of heating up the temperature of the water. Thus, the temperature of the water remains the same during a phase change.
One way to look at it is that the heat isn't contributing to the kinetic energy of the molecules, as the heat is going towards breaking intermolecular forces. Since average kinetic energy of the water molecules is unchanged when the phase change is happening, temperature stays constant. On the other hand, in situations where there is no phase change, the heat goes towards increasing the average kinetic energy of the molecules, thereby increasing temperature.
Just to put it as an example, say you heat up an ice cube, and it undergoes a phase change to water. During that phase change, the heat is being used to break up the hydrogen bonds between the water molecules. So, the heat energy transitions to break up these intermolecular forces before it is used to increase the energy levels of the water molecules themselves (temperature is the distribution of the energy levels of the molecules).
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