# What mass of water freezes?

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?

Mentor
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.

Mentor
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?

Mentor
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.

Mentor

rahiem01
I think im undertsanding the concept. I need to find the amount of energy to go from
10°C of water to -7°C of ice .

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

Mentor
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