To be fair to GP, it does take some energy to heat the ice from freezer temp to 0C (or the new, depressed freezing point), part of which will come out of the ice cream. It's just that that amount of energy is very small compared to the other energies we're interested in here (as you pointed out elsewhere).

That's true of the surface of the ice, but the core is colder

For a little while, yes. Ice has middling thermal conductivity, it'll eventually homogenously warm to the melting point.

yes - then it wont be ice any more - but while there's still ice the temp of the liquid will stay at the equilibrium point - after that it will start to warm

"then it won't be ice anymore". Not so: the freezing point is the temperature at which ice and water are in equilibrium. If you take a bowl of ice to its freezing point it will still be entirely frozen. Then if you continue adding energy, it will remain at its freezing point while progressively more of it melts, i.e. going from 100% ice to 50% ice/50% water to 100% water. All at one temperature. In other words, if you are at the freezing point, then all of the energy you add goes to melting the ice (and none of it goes to increasing the temperature), until all of the ice is gone. That is the case even if you wait to true equilibrium, e.g. for all temperature gradients to go away.

Right on. It’s the same for boiling, isn’t it? The water “can’t” get above 100 degrees C - added energy just goes into making steam. Right?

That's right (well... assuming constant pressure lower than the critical pressure.)

> and none of it goes to increasing the temperature

You’re clearly more knowledgeable than I am, but this strikes me as probably wrong? The temperature at the surface has raised, which is how some of the surface ice melted. The temperature on average has to raise because thermodynamics. If I understand the article, the core temperature of the remaining ice can decrease and localities can decrease with them. But the energy is increasing temperature in every other area at a higher rate, because it has to (conservation of energy) and because it has to (more ice melts than freezes).

>> If you take a bowl of ice to its freezing point it will still be entirely frozen. Then if you continue adding energy, it will remain at its freezing point while progressively more of it melts [...] if you are at the freezing point, then all of the energy you add goes to melting the ice (and none of it goes to increasing the temperature), until all of the ice is gone.

> this strikes me as probably wrong? The temperature at the surface has raised, which is how some of the surface ice melted. The temperature on average has to raise because thermodynamics.

What do yo you think is wrong in the quoted passage?

The "temperature on average" of the system described is the temperature of every part of it: the temperature where the liquid and solid phases co-exist. When you heat that system (i.e. add energy) the temperature doesn't have to increase "because of thermodynamics". The ice melts and everything remains at the same temperature. Only when the ice is gone the water starts to warm up.

At melting, the energy is going into the state change.

And the state change occurs when the portion of solid ice becoming liquid water warms however slightly, even if there’s a cooling side effect on the remaining ice as more energy is dissipated during that warming. Or am I still misunderstanding?

Yes. It is literally an isotherm. Energy is added without changing the temperature.