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The substance can change phases at a constant temperature. This is why ice can melt into water at room temperature, and if you check on the ice at different times you will see the ratio of ice to water change.
Temperature doesn't change during a phase change. Heat added to a substance either raises the temperature or causes a phase change, but not both simultaneously. If a solid is heated at its melting point, its temperature remains the same until it has completely melted because melting requires energy. After the solid melts, heating the liquid raises its temperature and the slope is no longer a flat line, until the boiling point is reached.
During the flat part of a phase diagram, all the heat is being used to separate the molecules and thus change the phase. There is no heat being contributed to the random motion of the particles and thus the temperature remains constant.
It takes energy to change a phase from one to another. For example, if you're heating Ice into liquid water, it'll take a certain amount of energy to get it from a solid to a liquid. This is the reason why steam at 100 degrees Celsius causes harsher burns than liquid water at 100 degrees Celsius (because the steam has more energy)
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