During the cooling of a liquid, if no impurity of site of nuclearation, an appreciable solidification will begin only after the temperature has been lowered to below the equilibrium solidification (or melting) temperature. This phenomenon is termed supercooling. My question is, if supercooling is necessary, what is the meaning of "equilibrium" solidification (or melting) temperature (e.g. 0oC for water)?

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If you draw the phase diagram of water, the phase boundaries will be the points at which the free energy change for phase change is zero, that is the two phases are in thermodynamic equilibrium. However thermodynamic equilibrium is the infinite time limit of the system so at the phase boundary the phase change will only take place if you wait long enough.

It isn't the case that supercooling is necessary to freeze water, only that supercooling is necessary to freeze water quickly. Water at $273 - \delta$K will freeze for arbitrarily small values of $\delta$ provided you are willing to wait long enough.

The reason for this is that there is a potential barrier that has to be overcome for the phase change to take place. In the water there will be some water molecules with enough energy to overcome the barrier but the proportion of those molecules varies roughly as $e^{-\Delta E/kT}$, where $\Delta E$ is the barrier height. The barrier doesn't have to be too many multiples of $kT$ before the rate is effectively zero.

Supercooling increases the rate of freezing because it reduces the height of the potential barrier $\Delta E$.

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In the case of solidification exactly at the equilibrium temperature, does "phase change" mean the liquid becomes a mixture of solid and liquid instead of completely change to liquid? What determines the portion of liquid and solid? –  Kelvin S Aug 20 at 12:57
@KelvinS: If you're precisely at the equilibrium temperature then the system will not change. If you're slightly on the solid side then the whole system will (eventually) freeze, and if you're slightly on the liquid side then the whole system will (eventually) melt. –  John Rennie Aug 20 at 14:38
Experience of putting things in the freezer tells us that we quite often get systems that are half frozen and appear to be stable in the half frozen state. Typically this is because the composition changes with freezing e.g. as you freeze a bottle of salt water the remaining water gets saltier so its freezing point changes. In effect your phase boundaries move as the system freezes. –  John Rennie Aug 20 at 14:39
@KelvinS: In a pure system (so avoiding the binary system John is suggesting), you would add heat to the ice until it reaches the melting temperature. To actually convert some ice to water, you have to add energy to account for the enthalpy of melting. As you add energy, you convert ice to water, but do not change the temperature of the system. Once all the ice is melted, continuing to add energy once again results in an increase in the temperature. This description ignores all kinetics. –  Jon Custer Aug 22 at 20:32