Ice poured into water. Final temperature?

[hattori_hanzo]

I have a kettle of aluminum with the mass of 1,10kg. There's 1,15kg of water in it, with the temperature of 20,2 C. The water is cooled with ice cubes. 150g of ice with a temperature of -25 C are poured in. What is the final temperature?
I'm relatively new to physics, and I haven't solved assignments with as many factors as this. When solving more complex assignments, like this one, where do I start?
I tried solving it, knowing that the amount of heat absorbed = amount of heat given away, but I ended up with a gigantic equation (chaos really), and a final temperature of 36 C (higher than the original temp.)

The question is really: do you have any tips where I should start when working problems like this?

 

[kuruman]

The general idea with problems like this is that the heat lost by one object is gained by the other. Here, the kettle and the water lose heat and the ice gains heat. What happens to ice at -25 C when heat is added to it?

 

[hattori_hanzo]

The general idea with problems like this is that the heat lost by one object is gained by the other. Here, the kettle and the water lose heat and the ice gains heat. What happens to ice at -25 C when heat is added to it?

When heat is added to the -25 C ice, the temperature of the ice will increase until it reaches 0 C. Then there will be a phase change, and the ice will slowly turn into water. And all the energy used to do this will be taken from the 20,2 water, and the 20,2 C kettle. Right?

So I know that the amount of heat from gained to the ice and the ice water = amount of heat lost by the 20,2 C water and kettle.

So I try to put this in an equation, so that I can get the unknown final temperature on one side, and the rest on the other side, but the equation gets very big and complex, and I pretty much lose the overview.

Can I "split" the equation up somehow, so that it won't be so much?

 

[gneill]

First gather together all the constants you might need. Here you're dealing with water in two phases (liquid and ice) and solid aluminum. So you'll need heat capacities for ice, water, and aluminum, and the heat of fusion for water/ice if the ice is going to melt.

Next determine if any phase transitions are going to take place. To do this, first compare the total heat required to raise the temperature of the ice to the melting point with the heat available in the other components (liquid water and kettle). If the available heat exceeds the heat necessary to raise the temperature of the ice, at least some of the ice will melt. You should be able to determine if it's some or all by comparing the heats.

Next write out a step by step list of the things that have to happen in order to reach equilibrium, such as

1. Raise ice temperature to melting point
2. Melt the ice at 0C to water at 0C
3. Distribute all remaining heat amongst the remaining components

The third step is easy to accomplish if you convert all temperatures to absolute (kelvin) and calculate the total heat residing in all the components, and also calculate the total heat capacity of the system. Divide heat by heat capacity to find the overall final temperature in Kelvins, then convert back to C.

 

[hattori_hanzo]

3. Distribute all remaining heat amongst the remaining components

The third step is easy to accomplish if you convert all temperatures to absolute (kelvin) and calculate the total heat residing in all the components, and also calculate the total heat capacity of the system. Divide heat by heat capacity to find the overall final temperature in Kelvins, then convert back to C.

Thanks.
But I'm not sure how to distribute the remaining heat to the remaining components. Because an equal amount of heat is not distributed to the different components? How much heat is distributed depends on the specific heat capacity, mass etc?

And how do I find the total heat capacity of the system? Is that the average heat capacity of all the components?

 

[gneill]

Heat capacities add (just like electrical capacitors in parallel). The heat capacity of an object is equal to its mass multiplied by its Cp value. Suppose you had a system comprised of three substances. Masses are m1, m2, m3; Specific heats Cp1, Cp2, Cp3; initial temperatures T1, T2, T3 in kelvins (*NOT* C, but K !).

Then the total heat is

Q = m1*Cp1*T1 + m2*Cp2*T2 + m3*Cp3*T3

The total heat capacity is

C = m1*cp1 + m2*Cp2 + m3*Cp3

The final temperature will be:

T = Q/C

 

[zorro]

I have a small doubt in such questions related to calorimetry.

Heat gained/lost = mSΔT

Is ΔT= Final temp - Initial temp for both heat lost and heat gained?
or is it that ΔT must be always positive on both sides of the equation (heat lost=heat gained) ?

 

[hattori_hanzo]

Wow, thanks! But is it right to add BOTH the mass of the ice and the water from the ice when adding the total heat? Because that will be 2x0,15kg. Or do I just add that once?

I got a final temperature of 16,2 C. That sounds about right, don't it?

 

[gneill]

I have a small doubt in such questions related to calorimetry.

Heat gained/lost = mSΔT

Is ΔT= Final temp - Initial temp for both heat lost and heat gained?
or is it that ΔT must be always positive on both sides of the equation (heat lost=heat gained) ?

How you account for the 'direction' of the heat flow is up to you. You can put all the heats on one side of the "=" and make the losses negative and gains positive, or you can equate the losses to the gains on either side of the "=", in which case both sides have positive values.

The magnitudes of the change in heats depends upon the amount of temperature change in each case.

 

[gneill]

Wow, thanks! But is it right to add BOTH the mass of the ice and the water from the ice when adding the total heat? Because that will be 2x0,15kg. Or do I just add that once?

I got a final temperature of 16,2 C. That sounds about right, don't it?

No, don't count the same mass twice. In fact, you should make sure that all of the various components of the system are in their final phase (liquid, solid, vapor, etc.) before calculating the total heat and heat capacity. The reason for this stipulation is due to a small quirk in the physics of phase changes.

When a substance is in the solid state it has a certain specific heat. When it's in the liquid state, it has a another specific heat. The difference has to do with the internal energy associated with the inter-atomic bonds of the material. So, not only does it require that the heat of fusion be accounted for to melt (or freeze) the substance, but after the phase change the overall heat content has changed by another less obvious amount associated with the change in specific heat for the substance.

So the simplest strategy is to perform whatever operations and accounting is required to put all the substances into their final states (solid, liquid,..) before tallying the heats and specific heats.

I think that the final result should be closer to 10C than 16C.

 

[zorro]

How you account for the 'direction' of the heat flow is up to you. You can put all the heats on one side of the "=" and make the losses negative and gains positive, or you can equate the losses to the gains on either side of the "=", in which case both sides have positive values.

The magnitudes of the change in heats depends upon the amount of temperature change in each case.

Thanks!

 

[hattori_hanzo]

No, don't count the same mass twice. In fact, you should make sure that all of the various components of the system are in their final phase (liquid, solid, vapor, etc.) before calculating the total heat and heat capacity. The reason for this stipulation is due to a small quirk in the physics of phase changes.

Sorry, didn't really understand this. Since the ice isn't in it's final state here, what do I do when adding up the total energy? Do I just add 0,15kg of WATER with an initial temp. of -25 C? Guess not, since that has a different heat capacity?

 

[gneill]

Turn the ice into water first: raise its temperature from -25C to zero C, then melt it to form liquid water at zero C. That'll leave you with an energy deficit to carry forward into the final sum (m_ice*Cp_ice*25C + m_ice*H_ice). Then, to sum up the total energy you have some water at 0C, some water at 20.2C, some aluminum at 20.2C, and that deficit.

 

[hattori_hanzo]

what is the H_ice? The specific melting heat (L)? Sorry, doing this in another language.

 

[gneill]

what is the H_ice? The specific melting heat (L)? Sorry, doing this in another language.

Yes, the specific heat of fusion for water.

 

[hattori_hanzo]

Thanks a lot, will give this another try tomorrow!