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If ice contained in a beaker starts melting, then why does temperature remains constant. Is there a way to understand this phenomenon using the concept of specific heat.

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Let us start with another phase transition that is closer to our daily experience - boiling water. The water temperature can not exceed 100°C (at standard temperature and pressure; ie STP). With a pot on a hot stove, heat is introduced from the bottom into the water, but the water can not become hotter than 100°C because it will turn into steam bubbles, which will rise to the top. So, the body of water remains close to 100°C until it is all boiled off.

The situation with the melting ice is similar. Ice can not exist at a temperature beyond its melting point. But getting heat into the ice is a bit more difficult. In practice, the heat to melt the ice has to come from the liquid phase.

So, if you introduce heat very slowly into the pot that contains water and ice, any temperature increase of the liquid beyond the melting temperature will be used to melt more ice and create more liquid at the melting temperature. As a result, the water temperature will remain very close to the melting temperature of the ice. The amount of energy needed to melt a certain amount of ice (ie the specific heat of melting) has not entered this argument at all.

You can also see that the discussion holds true only as long as the system is in near equilibrium. To see that, let us pour ice cubes and water into a pot and put it on a stove on high heat. Soon the water will be boiling while there are still unmelted ice cubes.