How Do You Calculate the Final Temperature in a Mixed Water and Ice System?

[jdstokes]

I'm a conceptual difficulty with this simple problem

In an insulated container, 0.800 kg of water at 40.0 degrees centigrade is mixed with 0.500 kg of ice at -15.0 degrees centigrade. Find the final temperature T_\mathrm{f} of the system. The freezing point of water is 0 degrees centigrade.

I need to know the phases present at equilibrium in order to calculate the final temperature, but these are not given.

 

[~angel~]

Just out of curiosity, how do you determine the final temp. if you know the phases present at equilibrium?

 

[jdstokes]

You set the sum of the heat transfers equal to zero and solve the resulting equation for the final temperature T_\mathrm{f}.

 

[jdavel]

jd.stokes,

Do you know the specific heat of water and ice and the heat of fusion of ice? I think with those, you can find out whether all the ice melts or not. Without them, I don't think you can.

Edit: Ok, I Googled: heat of fusion of ice. One nice site gave me all three values. Unfortunately it looks like there's plenty of enthalpy in the water to melt all the ice, and then some. So it's not just T = 0 But it's not hard.

 

[HallsofIvy]

Certainly you need to know the specific heat of water and ice (it seems odd that the problem would tell you an obvious point (that the melting point of ice is 0) but not tell you the specific heats and heat of fusion (also called "heat of melting"):

What heat does the water give up in going from 40 degrees to 0?
What heat is required to bring the ice from -15 degrees to 0?
(You get those from the specific heats of water and ice.)

If the heat the water gave up was greater than the heat required to raise the temperature of the ice to 0, then at least some of the ice is melted.

How much heat is required to melt all the ice?
(You need the heat of fusion for that.)

If that is more than the heat left after raising the temperature of the ice, then some ice remains. The final temperature would be 0 C.

If that is less than the heat left after raising the termperature of the ice, then all the ice melts and the amount of heat still left now raises the temperature of the total 1.3 kg of water.