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heat remains constant during a phase change because heat is being supplied to the phase change mechanism itself, not to the actual heating of the subject being heated. heat is being used in creating the phase change at the same rate that it's being applied to the actual material, so the heat remains constant.
The change of a substance from a phase of a lower energy to a phase of a higher energy requires heat to break intermolecular interactions between molecules, thereby making the substance have higher potential energy between molecules (usually, the volume of a substance: solid state < liquid state < gas state).
When heating a substance, for example, water, the heat energy is used to first raise the temperature to the boiling point. When the temperature gets to that point, the heat energy is then used to complete the phase change instead of raising the temperature further. So, the boiling water would stay at 100 C while using the heat energy to convert its molecules to the gas phase instead.
An increase in heat energy is not the same thing as an increase in temperature (delta H is different than T). Therefore it is possible for heat energy to be introduced to the system and for the temperature to not increase.
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