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If water freezes at 32 °F, what happens to that newly formed ice at that temperature? Does it start melting as soon as it’s frozen, or does it require more heat to melt again? If it does require more heat to melt, then how much?

And what would happen if you were in between that melting point and the water’s freezing temperature of 32 °F?

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Melting points equal to freezing points, not limited to water.

Which of the processes takes place depends on if the surrounding provides or takes away thermal energy.

If there is neutral thermal balance at freezing/melting point temperature, the both opposite processes occur, at the same rate, similarly as the opposite chemical reactions at chemical equilibrium.

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Poutnik's answer is spot on, but I think it's worth elaborating a bit. As pointed out, the melting and freezing temperatures are identical. At thermodynamic equilibrium, which means the entire system is at exactly the same temperature, some fraction will be ice and some fraction will be liquid. if heat is removed or added from the system (very slowly), the fraction that is solid will increase or decrease, but as long as there is still some liquid and some solid, the system is still at equilibrium.

When no heat is being added or removed from the system, the ice is constantly melting, but at exactly the same rate as some of the water is freezing, so there is no change in the fraction that is solid. But the specific bit that was frozen at one moment might be liquid the next.

In real life, it can be hard to heat or cool things so slowly that they remain close to equilibrium. If you heat the air around a block of ice, it is easy for the air temperature to get well above the melting point while the temperature within the solid remains low, so it may appear as if you have to heat it above the melting temperature in order to get it to melt.

Likewise, the air temp can get well below freezing while the liquid remains warming, again creating the impression that you need to go below 32 F in order to get freezing to occur. In both cases, overshooting helps shorten the time it takes to get to freezing/melting point, but it isn't a reflection of the equilibrium process.

One unusual case is supercooling, where you actually do get the liquid below the freezing temp without forming ice. This is also just due to the cooling happening more quickly than the rate of ice nucleation in the given system. The system is not at thermodynamic equilibrium even though the temp may be uniform. We call this a meta-stable state.

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