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If I had a beaker of water (i.e. many molecules), at exactly 0 °C and at atmospheric pressure, what phase would be encountered?
And would the answer to this also apply to other substances at their freezing/melting point, and out of interest, would the same effect be observed at their other "phase intercepts" too?

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It depends on how you achieved water at 0 °C. If you cooled water down to 0 °C, then it will remain water, unless you remove more heat, at which point it will start to freeze. Water needs to be at 0 °C to freeze.

If you warm ice up from below freezing to 0 °C, then it will remain ice, unless you add more heat to melt the ice. Ice needs to be at 0 °C to melt.

It will be the same for all substances.

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