Need help understanding this concept regarding melting ice

[horsedeg]

Homework Statement

In an insulated vessel, 265 g of ice at 0°C is added to 630 g of water at 17.0°C. (Assume the latent heat of fusion of the water is 3.33

105 J/kg and the specific heat is 4 186 J/kg · °C.)

a) What is the final temperature of the system?
b) is already solved

Homework Equations

The Attempt at a Solution

The answer is:
Since the heat required to melt 265 g of ice at 0°C exceeds the heat required to cool 630 g of water from 17°C to 0°C, the final temperature of the system (water + ice) must be 0°C .

I don't understand this. Is there a better explanation? I think maybe I just lack the knowledge of some sort of fundamental concept around this, because I don't know why this is.

 

[BvU]

Hi,

Pity you didn't add a few relevant equations, for that clarifies a lot.
You 'don't know why this is ' but don't specify. Is it clear to you that - because of the temperature difference - heat will flow from the liquid to the ice ?
Is it clear to you that this transport will stop when there is no temperature difference any more ?
Well then !

 

[horsedeg]

Hi,

Pity you didn't add a few relevant equations, for that clarifies a lot.
You 'don't know why this is ' but don't specify. Is it clear to you that - because of the temperature difference - heat will flow from the liquid to the ice ?
Is it clear to you that this transport will stop when there is no temperature difference any more ?
Well then !

Sorry about that, I just didn't think there were any equations that were relevant.

So I guess I understand now that clearly one would happen first - either the ice would completely melt and the water would be at some temperature above 0 degrees C, OR the ice wouldn't completely melt and thus the whole system would reach 0 degrees.

I suppose now that knowing which one happens would require applying some sort of equation/concept. Could you explain what that is?

 

[haruspex]

I suppose now that knowing which one happens would require applying some sort of equation/concept. Could you explain what that is?

The answer you quoted in post #1 seems to supply just such a concept.
If you prefer you could think in terms of total internal energy. This will be simplest if we pick some reference level as zero, such as ice at 0C. If we start with m_i g of ice at 0C (having zero internal energy) and m_w g of water at T_w then the total internal energy of the system is E_I=m_w(L_f+T_ws_w), where L_f is the latent heat of fusion of water and s_w its specific heat.
There is a unique state for the whole mass to be at a uniform temperature and the same total internal energy as initially. If E_I>(m_w+m_i)L_f then it will all be water (at some temperature).

 

[horsedeg]

The answer you quoted in post #1 seems to supply just such a concept.
If you prefer you could think in terms of total internal energy. This will be simplest if we pick some reference level as zero, such as ice at 0C. If we start with m_i g of ice at 0C (having zero internal energy) and m_w g of water at T_w then the total internal energy of the system is E_I=m_w(L_f+T_ws_w), where L_f is the latent heat of fusion of water and s_w its specific heat.
There is a unique state for the whole mass to be at a uniform temperature and the same total internal energy as initially. If E_I>(m_w+m_i)L_f then it will all be water (at some temperature).

Alright, so clearly I'm a little lacking in the concepts here. So I guess I can try to dissect what you're saying. I think I understand now, maybe.

EI=mw(Lf+Twsw)

At first glance I had no idea where you got this equation, but I guess it would just be the combination of the total internal energy (based off the reference point you're using). The internal energy of the water would be E_int = (mass)_w*(specific heat of water)*T_w. Isn't the energy required to melt the ice E_int = m_iceL_f rather than m_w?

Also, ideally when I'm thinking about this I'd use ΔE_int and ΔT to find how much energy is required to lower the temperature/energy to 0, right? Is that basically the same thing, or am I just thinking wrong? Is that why you used 0 as a reference point? It makes sense like that in my head.

But other than that I think I get it. Basically just get the energies required to do each action (either melt the ice completely or lower the temp of the water completely) and compare them.

EDIT: In addition, it appears I've forgotten the concept of solving b) again after reviewing it, even though I thought I knew it yesterday. The question is: How much ice remains when the system reaches equilibrium?

The answer key states that Q_cold = -Q_hot. If no energy leaves the system this makes sense to an extent. However, what doesn't make sense to me is how this equation comes up. I'm assuming this comes from the equation Q_cold + Q_hot = 0. Why is this true? I guess it's the law of conservation of energy, but could you explain this a bit more?

 

[haruspex]

Isn't the energy required to melt the ice ... miceLf rather than mw?

Yes, but E_int as I defined it is not the energy required to melt the ice; it's the internal energy the system has, above a baseline of "all consists of ice at 0C".

The answer key states that Qcold = -Qhot.

I assume the Qs are defined as transferred heat, one being the heat transferred out of the hot part of the system and the other being the heat out of the cold part of the system (so is negative).
Having decided that not all ice melts, suppose x grams remain frozen. What is the total internal energy now, as a function of x?