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If you have an ice cube in water and the ice cube does not melt after reaching equilibrium with the water, does the water have temperature 0 degrees Celsius?

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Yes, when ice is present at equilibrium with liquid water (at 1 atm), the system is at 0 °C.

If you continue to remove heat from the water (such as in your freezer), the temperature of the system will remain at 0 °C until all the liquid (water) becomes solid (ice) after which, continuing to remove heat will result in a drop of temperature below 0 °C. Conversely, if heat is added to the system, the temperature of the system will remain at 0 °C until all of the solid (ice) has melted into liquid, after which, the temperature will increase above 0 °C. The amount of (latent) heat required to melt ice is called its enthalpy of fusion (334 J/g). This energy is converted into intermolecular bonds (a change in internal or potential energy of the system) rather than a change in kinetic energy (that is, a temperature change).

Note that at a much lower pressure (0.006 atm) and slightly higher temperature (0.01 °C), pure water will be in three states or phases (gas, liquid and ice). This is known as the triple point of water, as illustrated in the phase diagram of water, shown below:

phase diagram of water

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