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Why does the temperature of melting ice not rise when energy is added?

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Why does the temperature of melting ice not rise when energy is added?

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To make this clearer, let’s say the ice suddenly stops melting altogether. What will happen to the already liquid water? Its temperature will RISE due to heat transfer FROM the ambient heat (until it reaches equilibrium). If the ice begins to melt again, the heat exchanges will make sure the temperature will NOT rise from the equilibrium point. Until of course, the ice melts completely.

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