Finding equilibrium temperature when there are phase changes

[lorenz0]

Homework Statement

In a box containing $m_{ice}=0.42kg$ of ice at a temperature $T_{ice}=-15°C$, $m_{w}=0.16kg$ of water at a temperature $T_w=12°C$ are added.
Ignore all dispersions of heat in the environment.
Find the equilibrium temperature and how much ice and how much water there is in the equilibrium state.

Relevant Equations

$\sum \Delta Q=0$

If there weren't phase changes occurring I know that the temperature equilibrium would be $T_e=\frac{m_{ice}c_{ice}T_{ice}+m_{w}c_{w}T_{w}}{m_{ice}c_{ice}+m_{w}c_{w}}$.
Now, by repeating the reasoning to get the above formula ($\sum \Delta Q=0$) and adding the phase changes of the water freezing I get $T_e=\frac{ m_{ice}c_{ice}T_{ice}+m_{w}c_{w}T_{w}-\delta m_w L_c }{m_{ice}c_{ice}+m_{w}c_{w}}$, where $L_c$ is the latent heat of fusion and condensation, respectively, but how do I find how much ($\delta m_w$) of the water freezes?

And, is my reasoning correct in general? I would like to understand this process in general. Thanks

 

[Philip Koeck]

Homework Statement:: In a box containing $m_{ice}=0.42kg$ of ice at a temperature $T_{ice}=-15°C$, $m_{w}=0.16kg$ of water at a temperature $T_w=12°C$ are added.
Ignore all dispersions of heat in the environment.
Find the equilibrium temperature and how much ice and how much water there is in the equilibrium state.
Relevant Equations:: $\sum \Delta Q=0$

If there weren't phase changes occurring I know that the temperature equilibrium would be $T_e=\frac{m_{ice}c_{ice}T_{ice}+m_{w}c_{w}T_{w}}{m_{ice}c_{ice}+m_{w}c_{w}}$.
Now, by repeating the reasoning to get the above formula ($\sum \Delta Q=0$) and adding the phase changes of the water freezing I get $T_e=\frac{ m_{ice}c_{ice}T_{ice}+m_{w}c_{w}T_{w}-\delta m_w L_c }{m_{ice}c_{ice}+m_{w}c_{w}}$, where $L_c$ is the latent heat of fusion and condensation, respectively, but how do I find how much ($\delta m_w$) of the water freezes?

And, is my reasoning correct in general? I would like to understand this process in general. Thanks

The second part of the question seems strange.
If T_e is above 0 °C then you should have only water, below only ice.
Only if T_e = 0 °C you can have a mixture.
Well, maybe that is how you have to deal with it.
Otherwise there are too many unknowns.

 

[kuruman]

The second part of the question seems strange.
If T_e is above 0 °C then you should have only water, below only ice.
Only if T_e = 0 °C you can have a mixture.
Well, maybe that is how you have to deal with it.
Otherwise there are too many unknowns.

It may be that the second part is poorly phrased because whoever wrote it does not want to prejudice the thinking of the reader. There are three possibilities for the contents:

1. Only ice at some equilibrium temperature
2. Only water at some equilibrium temperature
3. A mixture of ice and water at 0 °C

In the first two cases the mass of the single phase is 0.58 kg and one has to find its temperature. In the third case the temperature of the mixture is 0 °C and one has to find how much of it is ice and how much is water.
To @lorenz0 : Remember that

• The temperature of the ice must be raised to 0 °C before any of it starts melting and the heat for that to happen can only come from the water. Calculate how much heat.
• The temperature of the water must be lowered to 0 °C before any of it starts freezing and the heat lost for that to happen can only go into the ice. Calculate how much heat.

Once you have the two numbers, you can figure out which of the three possibilities is the case here and proceed from there.