How Much Water Turns to Ice When Mixed with Ice at Subzero Temperatures?

[rahiem01]

1. 30 kg of water at 10°C is mixed with 360 kg of ice at -7°C. (The heat capacity of water is 4190 J/(kg * °C), that of ice is 2090 J/ (kg *°C), and the heat of fusion of water is 3.34 * 10^{5} J/kg

1. What mass of water freezes?

Homework Equations

none that I know of.

The Attempt at a Solution

I was wondering if this is a conceptual question, being that the mass of water wouldn't change, only the density?

 

[Borek]

No, this is a quantitative question.

Have you heard about heat balance? q= m c delta T? Latent heat?

 

[rahiem01]

yeah, i know of that. I wasn't sure if I was over thinking it.

 

[Borek]

So build a heat balance and check what happens.

 

[rahiem01]

okay, to make sure I'm on the right track.

Q_{water} + Q_{ice} = 0, where Q = mc delta T.

I then solve for T, plug back into Q_{water} then solve for m?

 

[Borek]

Not that easy - you have to account for latent heat. Ice will either melt or water will freeze before equilibrium. Could be you will end with just water or just ice in the end - although the way question is worded suggests otherwise.

 

[rahiem01]

okay, i understand what you're saying, but I'm not sure how to set up the equation to account for the latent heat.

 

[Borek]

What does your book/notes say about?

 

[rahiem01]

I think I am undertsanding the concept. I need to find the amount of energy to go from
10°C of water to -7°C of ice .

Q_{ice} = mL_{c}, where m = mass and L_{c} = 3.34 X 10^{5}

 

[Borek]

Correct. Now you just have to combine all these pieces.

 

[rahiem01]

okay thanks for the info. i'll put it together and post a solution.

 

[rahiem01]

so do I do

Q_{water} + Q_{ice} = 0

m_{water}c_{water}(T_{2} -T_{1}) + m_{ice}L = 0

 

[RTW69]

I like to think about these problems as "what gets cooled= what get warmer"

The water is getting cooler= the ice getting warmer.

The water is going to go from 10_c to water at 0_c, then some (or all) of the water is going to turn from water at 0_c to ice at 0_c = ice goes from -7_c to ice at 0_c then from ice at 0_c to water at 0_c

 

[rahiem01]

Thank you. that makes a lot more sense to me now. i appreciate the help