why doesn't the temperature of an ice cube increase while it is melting?

Answer 1

Well, you might say that to the extent that the frozen volume is less by virtue of the relatively higher ambient temperature, that amount of ice must have experienced an increase in temperature sufficient to result in the change of phase that allowed that volume of ice to flow away as liquid, leaving a smaller cube.

That is to say, your question implies that the ice experiences no change in temperature, which isn't true, strictly speaking. The surface of the cube will change state if the atmospheric temperature rises to above the temperature at which the liquid freezes, and as the ratio of the cube's surface area to its volume increases so long as the ambient temperature remains above freezing, the cube will appear to melt "faster". (There are conditions under which the ice will melt without appearing to have passed through the liquid phase, but that is another story. Think of the shrinkage of cubes that remain unused in your freezer tray for a long time.)

You are really asking about the equalization, or transfer of energy, in this case, heat energy, between two potentials, and the answer is (to put it somewhat circularly) that under normal circumstances, frozen water remains frozen until melted from the outside inward. Once in the liquid phase, of course, a change in ambient temperature is transferred to the liquid, until the point at which the liquid can absorb no further energy without changing phase again, becoming gaseous.

Answer 2

Heat energy must be added to an ice cube to melt it. However, the energy is used to break the bonds of the ice crystals changing the state to liquid from solid. Only after all the bonds are broken (when heated slowly!) will energy be available to cause the free water molecules to vibrate more energetically which is exhibited as temperature increase. It takes about 80 calories to melt a gram (cc) of water but only one calorie to increase the temperature of a gram of water one degree.