# I Does ice really melts if in conctact with reservoir at 0°C?

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1. Jul 12, 2016

### Soren4

I came up with a basic doubt on heat exchange. Consider this example situation.

A cube of ice of mass $m$ and at temperature $\theta <0°C$ is put in contact with a resevoir exaclty at the temperature $T=0°C$.

The question is: does the ice melts, i.e. does the ice pass to liquid state? Or it will stay at solid state, but at new temperature $T=0°C$?

Of course the ice reaches the temperature$T=0°C$ and I'm totally ok with that. But once it reached that temperature?

On the one hand ice (solid state) already has the same temperature of the resevoir, so temperature gradient is zero, therefore there is no heat flow and ice should remain ice (it should not become water). So I'm quite convinced that this is the right one.

But on the other hand I found a similar exercise in which the involved material actually melts when in conctact with a reservoir at the temperature of melting.

Wich of the two is correct?

2. Jul 12, 2016

### Staff: Mentor

On the molecular level, there will be an equilibrium of melting and freezing of individual molecules. In total, the amount of ice stays constant once everything reached 0 degrees unless you have some thermal contact to the environment.

3. Jul 12, 2016

### Andy Resnick

Interesting problem. Heat and temperature are not the same, and heat may flow without a corresponding change in temperature: thermal expansion, phase changes, etc. (latent heat). Let's assume pure water (no dissolved gas or solutes), since that will complicate things. What matters here is the flow of heat from reservoir to ice, and flow of heat within the ice; heat flow is driven by temperature gradients.

If the ice is at a uniform temperature and we assume infinite thermal conductivity, the process is quasistatic and the ice will warm to 0°C in the limit t -> ∞, heat flow will simultaneously drop to 0 in the infinite time limit, and there will be no liquid water.

Since ice has a finite thermal conductivity, then heuristically, the flow of heat will not stop when the ice in direct contact with the reservoir warms to 0°C, resulting in some melting. Once the entire slushy has reached 0°C, equilibrium has been reached and the flow of heat will stop. In practice, the situation can be quite complex: liquid water has a different thermal conductivity than ice and can flow, resulting in non-uniform transfer of heat within the slushy.

I suspect there is a linear approximation to this which provides an estimate of the melted ice- the full solution for a 1-D arrangement composite media is rather complex but available in Mikhailov and Ozisik's "unified analysis and solutions of heat and mass diffusion" (Dover)- apologies to my colleagues for the lack of diacriticals.