For a long time scientists knew about the invariance of temperature during phase changes, but this phenomenon cannot be explained in terms of the usually observable variables, that is, the macroscopic or thermodynamic variables. To understand the WHY we need to look at the phase change from a microscopic point of view, in other words, we need to explain the observed behavior in terms of the motion of the atoms and molecules that form the substance.

As an example, let's talk about boiling a pure liquid, but the same reasoning can explain freezing/melting. When you start heating a mass of liquid, the heat absorbed by the system increases the kinetic energy of the molecules, they will move faster.

This energy increase is what we can observe as an increase in the temperature. More heat in and temperature will go higher and higher and higher, until there comes a moment when the molecules have acquired enough energy to overcome the forces that keep them together in the liquid phase and they start to escape from every part of the liquid into the gas phase, that means the liquid boils. (The key point here is FROM EVERY PART OF THE LIQUID PHASE, because normally during the heating up, molecules will be escaping from the surface into the gas phase all the time ). When this happens, the molecules that escape to the gas phase take away some energy from the liquid phase, and since we continue heating they will continue to escape.

The effect we had observed before (increase of temperature as we give more heat to the system) is not possible anymore, because the vaporizing molecules are taking away the energy that is being supplied to the system. That means, boiling occurs at constant temperature until all the liquid is gone, in which case, if we continue to heat up, then we will increase the temperature of the gas formed.

All this process has occurred at constant pressure, as when we do our experiments in an open container.

When a substance is provided energy in the form of heat, it's temperature increases. The extent of temperature increase is determined by the heat capacity of the substance. The larger the heat capacity of a substance, the more energy is required to raise its temperature.

When a substance undergoes a FIRST ORDER phase change, its temperature remains constant as long as the phase change remains incomplete. When ice at -10 degrees C is heated, its temperature rises until it reaches 0 degrees C. At that temperature, it starts melting and solid water is converted to liquid water. During this time, all the heat energy provided to the system is USED UP in the process of converting solid to the liquid. Only when all the solid is converted, is the heat used to raise the temperature of the liquid.

This is what results in the flat part of the freezing/melting of condensation/boiling curve. In this flat region, the heat capacity of the substance is infinite. This is the famous "divergence" of the heat capacity during a first order phase transition.

There are certain phase transitions where the heat capacity does not become infinitely large, such as the process of a non-magnetic substance becoming a magnetic substance (when cooled below the so-called Curie temperature).