If you are supplying heat and temperature is not changing it means that you are supplying latent heat. Latent heat is used by any matter to change its state. Temperature will not change until all the matter is changed from one state to another.
Why temperature doesn't change?
When you are supplying sensible heat the kinetic energy of the molecules increases after absorbing enough heat they are ready to change their state by absorbing latent heat at which only breaking of bonds of the molecules takes place as a result the kinetic energy change is negligible. Hence the temperature is constant.
When temperature of the system does not increase by heat supply, it means at the same time of heat supply, system is doing work equal to increase in its internal energy. If we extract increase in internal energy as work, then temperature of the system does not increase. But both processes should be occured at same time.
Hence during the Phase change of substance heat energy is increased but temperature will not increase.
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