Find the final temperature of this system

[~angel~]

Please help.

In an insulated container, 0.800kg of water at 40.0C is mixed with 0.500kg of ice at -15.0C. Find the final temperature T_f of the system. The freezing point of water is 0C.

I found the heat released by bringing all the water to freezing point - 134000J
The amount of heat to bring the ice to melting pt- 15700J
The amount of heat needed to melt all of the ice as well as bringing it to melting pt- 182000J

I'm not sure where to go from here. Any help would be great,

 

[whozum]

I think once you bring all the ice to water, you can use the regular
$$q_1 = q_2 = q_x = m_xc_x(\Delta T_x)$$ with T in kelvin solved for final temp.

 

[learningphysics]

I'd approach the problem like this...we have two parts to the system...

1) The 0.800kg of water.
It goes from 40.0C to the final temperature T_f
So what is the total change in the heat content of this part (in terms of T_f)?

2) The 0.500kg of ice.
It goes from -15.0C to 0C. Then it turns to water. Then it goes to the final temperature T_f. so we have 3 parts here.
What is the total change in the heat content of this part.

The sum of the answers to 1) and 2) add to 0.

 

[~angel~]

I'd approach the problem like this...we have two parts to the system...

1) The 0.800kg of water.
It goes from 40.0C to the final temperature T_f
So what is the total change in the heat content of this part (in terms of T_f)?

2) The 0.500kg of ice.
It goes from -15.0C to 0C. Then it turns to water. Then it goes to the final temperature T_f. so we have 3 parts here.
What is the total change in the heat content of this part.

The sum of the answers to 1) and 2) add to 0.

But you have to take into account the heat of fusion of ice to water and all.

 

[learningphysics]

But you have to take into account the heat of fusion of ice to water and all.

Right that goes into 2)... it will have 3 parts... first the ice goes to 0. Then there's the fusion turning the ice into water...then that water water goes to T_f

 

[~angel~]

My final answer is -14.5c. This doesn't seem right. it's too close to the initial temp of the ice.

 

[learningphysics]

My final answer is -14.5c. This doesn't seem right. it's too close to the initial temp of the ice.

Ah... I believe that the final answer is 0c. Sorry. I think my approach was unnecessary and time consuming.

Like you worked out... assuming all the ice turns to water leads a contradiction... a temperature below zero... likewise assuming all the water turns to ice will similarly lead a contradiction with a temperature above 0.

I think the best way to look at this is see how much heat is released when the 0.800kg water reaches freezing point (Hw)... Then look at the amount of heat required to bring the ice up to melting point(Hi)... Find Hw-Hi (should be positive). That's the amount of energy left for melting of the ice. Find how much energy is required to melt all the ice (Ht).

If Hw-Hi<Ht, then there isn't enough energy left to turn all the ice to water...
Some of the ice turns to water but not all of it.

So what we have is some ice and some water forming an equilibrium at 0c.