Ice in water problem - how much does water cool here?

[tieywhiey]

1. The problem statement, all variables and given/known data

An ice cube melts in a 10oC glass of water (mass of water is 225g). If
the ice is allowed to melt completely, what will the final temperature
of the water be? (we're told that the mass of ice can be ignored)

Ti = 10oC
Latent heat of fusion = Lf = 334000 J/kg
specific heat capacity of water = C = 4180 J/kg/oC
mass of water = Mw = 0.225kg


2. Relevant equations:

QH = sensible heat = mC(T2-T1)
QE = latent heat = m*Lf


3. The attempt at a solution

I have figured out an equation that requires the mass of ice to be
included, however we've not been given the mass of ice (and are told
to ignore it), so I'm not sure how to create an equation that ignores
the mass of the ice.

Since we can assume this to be a closed system, the heat lost by the
water will equal the latent heat going into melting the ice and
heating this resultant water to the final temperature


Q1 = energy req'd to melt ice = Lf*Mice
Q2 = energy req'd to warm resultant water to final temperature = Mice
*C *( 0 - Tf )

Q3 = energy lost by water in the glass = Mwater * C * (Ti - Tf)

Q1 + Q2 = Q3

(Lf*Mice) + (Mice * C * (Tf - 0) = Mwater * C * (Tf - 10)


How can the Mice be ignored here?? the energy is dependent on Mice! Am I missing something?

Edited October 18, 2015 by tieywhiey

[studiot]

Since you have posted this in homework help I assume it is not a trick question from somewhere, but real coursework designed to make you think.

 

Well done for detailing your thoughts, but you haven't actually stated the question as posed.

So I am going to make some assumptions.

 

what will the final temperature
of the water be?

 

The ice melts because heat is transferred from the water to the ice.

 

So what are the conditions required for heat transfer to take place?

Ans: There has to be a temperature difference between the water and the ice.

 

 

Since the ice starts at 0^oC (Assumption number 1 ) and the water starts at 10^oC heat is transferred from the wate to the ice, which cools the water and melts the ice.

 

The cooler water is now at a temperature lower than 10^oC.

 

You are told that the mass of ice is irrelevant but you are not told that the glass is insulated (Assumption number 2)

 

Assumption number 3 is that the glass of water is initially in equilibrium with its surroundings.

 

Since the water is now cooler than its surroundings heat is transferred from the surroundings to the water.

 

So what do you now think is the final temeperature of the water ?

[Nicholas Kang]

Assumptions based on studiot. Good assumptions, studiot. At least you "fill in the gaps required."

 

Solutions: Picture the problem

 

Imagine an ice cube being placed in a glass of water. Ice cube is 0 degree Celsius. Water is 10 degree Celsius. We know that heat is transferred from the water to the ice so that thermal equilibrium is reached somewhere between 0 and 10 degree Celsius.

 

So, we assume: heat absorbed by the ice = heat released by the water, that is,

 

Q_absorbed = Q_released

 

We first assume that all sensible heat is removed from the water before latent heat of water sets in

Q_absorbed Q_released

m_icel_fusion m_waterc_water(10-0)

m_ice(334000 J/kg) 0.225(4180)(10-0)

m_ice(334000 J/kg) 9405 J

 

Now, we know the heat supplied when the water cools but not freeze at a constant temperature is 9405 J. Yet, some 334000 J of heat is needed to melt 1 kg of ice. So, we know the heat supplied supplied by the cooled water isn't enough. But thermal equilibrium is achieved since both the ice and water is 0 degree Celsius. The ice starts off at 0 degree Celsius and the water has cooled to 0 degree Celsius. No net heat flow occurs between 2 bodies. Thus, the temperature remains constant. The answer is 0 degree Celsius.

 

You can check how "light the ice might be" by using the formula Q=ml, that is

9405=m_ice(334000)

m_ice=9405/334000

=0.02816 kg, about 28.16 g of ice would be sufficient to absorb all 9405 J of heat and remains at a constant temperature with the cooled water.

 

*I ignore the surrounding temperature in this case.

 

Before you consider my answer, studiot, is my answer correct? I am a high school student and after reading the question I think I can answer it. Anyway, you should consult studiot first.

[pavelcherepan]

 

But thermal equilibrium is achieved since both the ice and water is 0 degree Celsius. The ice starts off at 0 degree Celsius and the water has cooled to 0 degree Celsius.

 

Nicholas, I like your solution, but why do you think that water has cooled down to 0^oC? What if the ice cube is just 10g? Or 5g? I think as a general idea, studiot's answer is better, to me at least. If the water was in equilibrium with it's surroundings, it will end up at that same equilibrium in the end.

[studiot]

Hello, Nicholas, nice to see you are still with us.

 

I answered as I did in case this actually was a homework question.

Since the OP appears no longer interested it seems reasonable to me to discuss it in more detail.

 

The problem appears ill posed as described, but the OP did make an effort to point out that the mass of ice is irrelevant.

So in theory the result should be true for 0.1 grammes 1 gramme, 10 grammes, 100 grammes 1,000 grammes or 10,000 grammes of ice.

 

You are, however correct to observe that no amount of ice can ever freeze the water and there is a minimum amount of ice below which the ice will all melt without reducing the water to 0^oC.

Can you see why this is?

It may be that this was the intention of the question, however the OP does say that the ice melts completely.

Edited October 20, 2015 by studiot

[Nicholas Kang]

You are told that the mass of ice is irrelevant but you are not told that the glass is insulated (Assumption number 2)

 

Assumption number 3 is that the glass of water is initially in equilibrium with its surroundings.

 

Since the water is now cooler than its surroundings heat is transferred from the surroundings to the water.

 

So what do you now think is the final temperature of the water ?

 

 

If the water was in equilibrium with it's surroundings, it will end up at that same equilibrium in the end.

 

studiot, I can roughly figure out where are you leading me to. Also, based on your last assumptions and what pavelcherepan had said, you must be talking be about thermal equilibrium with the surroundings. So, the final temperature must be equal to the temperature of the surroundings instead of just 10 degree Celsius. Is this what you mean? If so, this question must not be talking about a closed system, it must be an open system if you talk about surroundings.

[puppypower]

The mass of the ice cannot be ignored, because the mass of the ice is proportional to the total heat needed to melt the ice. One gram of ice will take less heat to melt than 100 grams of ice. The mass of ice times the heat of fusion allows us to calculate the total energy that needs to be absorbed for complete melting. We then take that amount of energy away from the water to find the final temperature of the water. This is where you use liquid mass and heat capacity.

[studiot]

The mass of the ice cannot be ignored, because the mass of the ice is proportional to the total heat needed to melt the ice. One gram of ice will take less heat to melt than 100 grams of ice. The mass of ice times the heat of fusion allows us to calculate the total energy that needs to be absorbed for complete melting. We then take that amount of energy away from the water to find the final temperature of the water. This is where you use liquid mass and heat capacity.

 

If the conditions of the question state that the answer is independent of the mass of ice than that is (one of) the conditions of the question.

 

Such a condition leads to certain inescapable conclusions as we have been discussing.