[hamdani yusuf (OP)]

In a sealed container, put some ice and water at 0 degree C (in atmospheric pressure). Is there a net heat exchange between them?

[Halc]

If the container is insulated (no heat exchange with its environment), then it should all be in equilibrium and there will be no heat exchange with the ice.
If heat is generated somewhere or comes through the incompletely insulated container, then there will be heat transferred from the water to the ice, which allows the ice water to maintain its 0°C temperature.  This is how your ice water maintains constant temperature despite the glass sitting there on a hot day.

[hamdani yusuf (OP)]

Ice and water has different emissivity.

Material   Emissivity Coefficient - ε -
Water   0.993 - 0.998
Ice        0.98
Snow   0.969 - 0.997
Sand   0.949 - 0.962
Granite   0.898
https://www.engineeringtoolbox.com/radiation-heat-emissivity-d_432.html

It means water radiates more energy than ice at the same temperature. Doesn't it mean that they are in a dynamic equilibrium? some water molecules will turn into ice while some ice molecules turn into water?

[Bored chemist]

They are in a dynamic equilibrium (you can prove this with isotope tracing).
But it's not to do with the emissivity.
Conductivity and random thermal fluctuations will do the job.

[Petrochemicals]

In a sealed container, put some ice and water at 0 degree C (in atmospheric pressure). Is there a net heat exchange between them?

By definition in your question, at 0c everything is the same temperature. Unless one changes temperature no. No ice can form or melt without the exchange of energy in the form of kinetic energy of the molecules. Non nil nein nada negagive niet nix.

[hamdani yusuf (OP)]

They are in a dynamic equilibrium (you can prove this with isotope tracing).

Do you mean they exchange phase? How is the isotope tracing done?

But it's not to do with the emissivity.
Conductivity and random thermal fluctuations will do the job.

Do you think that emissivity has no effect here?

[hamdani yusuf (OP)]

Will it make a difference if the ice is put first filling the bottom half of the container, compared to when the water is put first so that te ice is above it?

[evan_au]

How is the isotope tracing done?

One way is to make the ice cubes from "heavy water" and the liquid water from normal "light" water.
- Heavy Water is made with Deuterium instead of Hydrogen; the two are easily distinguished with a mass spectrometer

If you hold the temperature constant, the ice won't all melt into the water, and the water won't all freeze into ice.
- But if you lift out the ice cube after a few minutes in the water, you will find that some of the heavy water has migrated from the ice into the liquid water, and some of the light water has migrated from the liquid water into the ice.

You may want to rerun the experiment with the roles reversed, as heavy water has a freezing point of 3.8C, compared to normal water at 0C.

It looks odd, but heavy water ice cubes don't float...
https://en.wikipedia.org/wiki/Heavy_water

Due to the difference in melting points of heavy water and normal water, it may be better to use heavy water which has normal hydrogen, but a heavier isotope of oxygen...

[Bored chemist]

Do you think that emissivity has no effect here?

Certainly not much effect.
Ice dosn't emit much IR, and nor does cold water.

The emissivity and absorption are identical at any wavelength.
https://en.wikipedia.org/wiki/Kirchhoff%27s_law_of_thermal_radiation

And imagine that I had some water, and some ice (Initially at 0 C) enclosed in a reflector so that any radiation they emitted  was contained until it was absorbed again.
They must both stay at 0C.
If they didn't, I could run a heat engine from the difference and get free energy.

On the other hand, the water and ice are in contact, so thermal conductivity is a good pathway for heat transfer- if there was going to be any.

 

Do you mean they exchange phase? How is the isotope tracing done?

Freeze some tritium labeled water.
Put the radiolabeled ice in normal water at 0C.
Wait, and then measure the activity of the water phase.
It slowly becomes active.

The same thing can be used to show that "insoluble" solids exchance ions with the solution they arein.

By definition in your question, at 0c everything is the same temperature. Unless one changes temperature no. No ice can form or melt without the exchange of energy in the form of kinetic energy of the molecules. Non nil nein nada negagive niet nix.

Fluctuations.
The temperature is a property of the ensemble of molecules, but individual molecules are allowed to have different KE from the average.
So the exchange is possible.

Incidentally, after someone has said the exchange does, in fact, take place, it's not wise to categorically  say that it can't.

[hamdani yusuf (OP)]

In a sealed container, put some ice and water at 0 degree C (in atmospheric pressure). Is there a net heat exchange between them?

Here's an idea. The container has a thermally conductive separator which split the container into two chambers. Put the ice into the left chamber, and water into the right chamber. Will there be any thermal energy transfer through the separator?

[Bored chemist]

Most people have hear of the three laws of thermodynamics.
Not many have heard of the zeroth law of thermodynamics.
https://en.wikipedia.org/wiki/Zeroth_law_of_thermodynamics

Since the ice and water are both in equilibrium with the barrier, they are in equilibrium with eachother.
That means their temperature is the same and there's no temperature gradient for heat to pass"down".

[hamdani yusuf (OP)]

Fluctuations.
The temperature is a property of the ensemble of molecules, but individual molecules are allowed to have different KE from the average.

How thin the barrier can be until energy exchange from fluctuation can no longer take place?

[Bored chemist]

The question has no meaning.
Energy will be loaned and returned by random molecular movements and vibrations regardless of whether there's a barrier there or not.

But temperature is only defined for an ensemble of molecules.

[hamdani yusuf (OP)]

So,  the ice will stay as ice,  while the water stays as water?
At one point on the barrier, energy flows from water side to ice side. What prevents the ice molecules touching the barrier from transferring the energy it just received to another molecules further from the barrier?

[Bored chemist]

What prevents the ice molecules touching the barrier from transferring the energy it just received to another molecules further from the barrier?

Nothing.
Nor will anything prevent energy transfer the other way.
So the net effect will be nothing.

[hamdani yusuf (OP)]

What prevents the ice molecules touching the barrier from transferring the energy it just received to another molecules further from the barrier?

Nothing.
Nor will anything prevent energy transfer the other way.
So the net effect will be nothing.

Ice melting is an endothermic process. The molecule undergoing this phase transition has local temperature lower than its surroundings where no phase transition is occurring. Since ice side has more ice than water side, there will be more endothermic process there. Energy will flow until equilibrium is achieved.

[Origin]

Ice melting is an endothermic process. The molecule undergoing this phase transition has local temperature lower than its surroundings where no phase transition is occurring

Like you said melting is endothermic, but there is no temperature difference so there is no bulk heat flow and hence no bulk melting.  This has been said multiple time in multiple ways, so I am not sure where your problem is.

[Bored chemist]

Energy will flow until equilibrium is achieved.

It was already at equilibrium.

[gem]

Hi all,
 Hamdani you raise a interesting scenario,

BC

Energy will flow until equilibrium is achieved.

It was already at equilibrium.

I am not sure that statement is totally correct, if you consider the energy imbalance between the two sides as described,
for example the latent heat/energy is unbalanced as is the kinetic energy of the molecules, also the density's.   

[hamdani yusuf (OP)]

Ice melting is an endothermic process. The molecule undergoing this phase transition has local temperature lower than its surroundings where no phase transition is occurring

Like you said melting is endothermic, but there is no temperature difference so there is no bulk heat flow and hence no bulk melting.  This has been said multiple time in multiple ways, so I am not sure where your problem is.

I'd like to add that freezing is an exothermic process before I continue. In a mixture of water and ice in equilibrium, melting and freezing happen at the same rate. 
The key concepts here are fluctuation and local temperature difference. In the side where there's more ice, melting occurs more often than freezing. On the other hand, in the side where there's more water,  freezing occurs more often than melting.