Portion of water that is frozen when in equilibrium

[rg2004]

Homework Statement

Supercooled water at minus 5C in a thermally insulated container suddenly transforms into an equilibrium ice-water mixture.
a. What fraction of water has frozen?
b. How much was the entropy change per gram?

Homework Equations

Clapeyron Equation
dP/dT=\lambda/(T*(Vsolid-Vliquid))

Gibb Free Energy
G=U+pV-TS

Msolid/(Mliquid+Msolid)=fraction of water that is frozen

The Attempt at a Solution

I have a ton of equations, but no way to find the solution. I'm just guessing, but I believe that we are supposed to use Gibb free energy since we are holding temperature and pressure constant (right?). Also we are supposed to minimize the gibb free energy, which id love to do, but I'm having trouble seeing how Ill get mass out of it.

I must be stupid, I've been trying to solve this problem for 15 hours. Could anyone please help me at least get started?

 

[Borek]

First part can be done by simple heat balance. What must be the final temperature of the system? What is ice heat of fusion? Specific heat of water?

Hopefully someone else will give you hints about calculating entropy change. My thermodynamics is so rusty it lost its initial shape.

 

[D H]

Regarding part b, solve it in a series of steps.

1. What is the entropy difference between supercooled water at -5C and water at 0C?
2. How much heat must be transferred from water at 0C to supercool it to -5C?
3. If you slowly transfer the same amount heat from water at 0C it will partially freeze and remain at 0C (until completely frozen).
4. What is the entropy difference this partially frozen mix and liquid water at 0C?

What does this tell you about the difference in entropy between the supercooled water and the frozen mix?

 

[rg2004]

I'm sorry, I don't know heat balance equations. I've looked around, but I'm not sure which source to go by. Also I wouldn't know what to put for a change in temperature. Additionally, in equilibrium, I calculated that the pressure for this system is 649.3 atmospheres, and I don't know if any thing like heat fusion or specific heat are pressure dependent. I'm pretty sure that I am supposed to minimize the free energy of the system, but again I have trouble finding the mass. Likely one of the issues I'm having with this class is the fact that I haven't taken chemistry yet (despite not being a prerequisite)..

Sorry again; thank you.

 

[D H]

Presumably the supercooled water is at 1 atmosphere, not 649.3. The question says supercooled, not ultra-pressurized, after all.

 

[rg2004]

but isn't it also in equilibrium along the melting point line? To me, looking at the PT diagram indicates that the pressure must have gone up.

 

[Borek]

You may safely assume pressure is constant and is 1 atm all the time.

You are trying to make it harder than it is. What must be the final temperature if you have water and ice in equilibrium?

 

[D H]

but isn't it also in equilibrium along the melting point line?

Nope. Supercooled means toss that phase transition diagram out the window. A liquid is supercooled when the phase transition diagram would indicate that the substance has to be solid rather than liquid. For example, the phase transition diagram for water says that water at 1 atmosphere pressure and a temperature of -5C will be solid.

Some kind of nucleation site is needed to start the freezing process. This might be a spec of dust, a spec of ice, a molecule other than water, a bump in the container, bumping the container --- almost anything except liquid water will do. Without anything to initialize the freezing process, pure water can be cooled well below the freezing point and remain liquid.

The same thing happens in the making of glass. If you cool some molten sand (silicon dioxide) to just below its freezing point and drop in bit of sand into the supercooled molten sand the sand will freeze into crystalline silicone dioxide. If instead you keep cooling that supercooled molten sand something else will happen: It will turn into glass. Glasses are a rather different beast from crystalline solids.

 

[rg2004]

What must be the final temperature if you have water and ice in equilibrium?

-5 degrees Celsius apparently..

 

[D H]

Try again.

 

[rg2004]

Alright, my best guess is that the system was disturbed and the temperature increased to 0 degrees C where it is in equilibrium.

 

[Borek]

And that's correct - supercooled (or superheated) systems are not in thermodynamical equilibrium, even if they appear to be so.

Now that you know final temperature, can you do the heat balance?