Ice/Water Composition in Basic Thermodynamics

["pi"mp]

Homework Statement

So we are given an insulating container with two compartments between which, heat can flow. In the left one, there exists m_1 of water at temp T_1 and on the right there exists m_2 of ice at temp T_2. What is the final composition and temperature?

Homework Equations

(i) Q=mCT and (ii) Q=mL_f

The Attempt at a Solution

So...I know there are three possibilities:

(1) The final composition is all water at temp. above 0 degrees Celcius. I'm okay with this.
(2) The final composition is all ice at temp. lower than 0 degrees Celcius. I'm okay with this.
(3) The final composition is part water and part ice exactly AT 0 degrees celcius. This is where I'm having issues!

Let's assume xm_1 of the water turns to ice and ym_2 of the ice turns to water. Where x and y are dimensionless fractions b/w zero and one.

==> We apply the "relevant equations" above s.t. ALL the water is taken to 0 degrees, ALL the ice is taken to 0 degrees (these are equation (i) above) and then the corresponding fractions change phase by using equation (ii) above. Then we apply energy conservation but we have two unknowns, x and y! I can't for the life of me figure out simple secondary relationship b/w x and y s.t. my method has a solution. Thanks for any help!

 

[mfb]

There is no unique solution in your case (3). It is probably a reasonable assumption that either one side stays 100% ice or the other side stays 100% water, but other options are possible as well (and it depends on pressure and other details).

You can calculate the total amount of water and ice.

 

["pi"mp]

I actually just saw something that made me consider that. Perhaps we were supposed to assume that the left side remained all water while ym_2 of the ice on the right side turned into water. But that is kind of frustrating, the problem statement was just as I gave above without any mention of pressure or volume, etc. Thanks for your help!

 

["pi"mp]

So in the case of the general composition...even in certain cases of numerical values for the initial masses and temperatures, it's still not solvable without knowing that one side stays all water or something else like that?

 

[gneill]

If you are given all the initial masses and temperatures then you can solve for the end state (no matter what).

If some initial temperature/mass information is missing then you will need some other constraint(s) to take the place of the missing data so that the number of equations equals the number of unknowns. Sometimes these constraints take the form of specifying the amount of liquid water remaining, or the change in volume of the solid portion, or the final temperature, or...! Some variations are simple, some clever, and some diabolical

 

["pi"mp]

But if you're only given the initial temperatures and masses how do you know how much becomes ice and how much becomes water? It seems like that would give you two unknowns.

 

[gneill]

But if you're only given the initial temperatures and masses how do you know how much becomes ice and how much becomes water? It seems like that would give you two unknowns.

Given the initial masses and temperatures, and along with the known values of specific heat and heat of fusion, you then know where the heat energy is in the system and the heat energy available to move around. Heat flow is driven by temperature differences. Heat stops moving when overall temperature is equalized and all changes cease.

The trick is to bring the components to equilibrium in steps. See how much heat energy is available or can be used from the 'hot' sources before a state change needs to happen (water melting, water freezing, water evaporating) or temperature equilibrium is achieved. When a state change is required, check to see how much more energy is available to drive the state change. Spend whatever heat 'currency' you can by taking it from the hotter source and moving it to the cooler 'sink', thus lowering the source temperature and raising the sink temperature or driving a phase change. After each step you'll have a new configuration of masses and temperatures. Rinse, repeat, until no temperature differences are left. You may find that there will be insufficient heat available to complete the state change of all of the available material before the temperature is equalized (at 0C for water, where both water and ice can coexist).