Clarifying Latent Heat: Cooling 1kg of Water to 0C?

In summary, the conversation discussed a problem regarding the cooling of water using ice and the specific heat involved. The solution involved calculating the energy required to melt the ice and the heat extracted from cooling the water to 0C. The initial temperature was set at 0C because the heat had to go somewhere in order for the ice to freeze. The calculation also took into account the energy needed to warm the ice to 0C. After considering all factors, it was determined that 176g of ice was melted in the process.
  • #1
PrudensOptimus
641
0
There was this problem regarding Latent Heat that I would like further clarification:

Problem:

200g of ice at -10C is added to 1kg of water at 15C in an insulated container. Is there enough ice to cool the water to 0C? If so, how much ice and water are present once equilibrium is reached?

In the book it demonstrated the solution by finding the Energy required to melt all 200g of ice Q1:

Q1 = mc(dT) + mL = 70.9kJ -- I understand that.

Then it says "Cooling the water to 0C extracts an amount of heat given by Q2":

Q2 = mc(dT) = 1kg(4.184kJ/kg*K)(15K) = 63kJ. -- Why did they set initial Temperature to be 0C?
 
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  • #2
They're calculating a negative (the heat extracted), so the equation has an extra negative sign.

In particular, [itex]- \Delta T = -(T_f - T_0) = T_0 - T_f[/itex]
 
  • #3
OK, when they calculate the mass of the water at 0C after apply ice, they used this:


m = Q/L = 58.7kJ/334 = 176g.

I understand they subtracted 4.1kJ(Specific heat of Water) from 62.8kJ... But why did they do that?
 
  • #4
Because in order to freeze, to pass from the liquid state to the solid state without any change in temperature, the sample had to LOSE that much heat to the environment (by conduction or radiation).

Think of it as representing the amount of kinetic energy lost by the molecules between being able to move around, as in the liquid, and being fixed in a crystilline lattice, in the solid.

The heat has to go somewhere, and it goes into the environment. That is why you get the paradoxical result that when, say, droplets in the air freeze to ice crystals, they have a warming effect on their part of the atmosphere. Conversely when ice crystals melt they take up that specific heat from the environment, resulting in a cooling effect on the surrounding air.

Remember that these exchanges of specific heat don't in themselves change the temperature of the water; this heat goes only into the change of state.
 
  • #5
Is there another explanation for Q2?

And perhaps someone can work this problem in a slight different, clear, simple way?
 
  • #6
Originally posted by PrudensOptimus
I understand they subtracted 4.1kJ(Specific heat of Water) from 62.8kJ... But why did they do that?
Think of the process in steps. Before you can melt the ice, you must first warm it up to the melting temperature. You know that the energy needed to:
1) warm the ice to 0 = 4.2kJ
2) melt all the ice (at 0) = 66.8kJ

The available energy in the warm water (compared to 0 degrees) is: 63kJ

That's more than enough to warm the ice, but not enough to melt it all. (So we know the final temperature will be 0.) After warming the ice, there will be 63-4.2=58.8kJ left to melt ice. So a fraction equal to 58.8/66.8 (= 0.88) of the ice is melted (= 176g).
 
  • #7
Originally posted by Doc Al
Think of the process in steps. Before you can melt the ice, you must first warm it up to the melting temperature. You know that the energy needed to:
1) warm the ice to 0 = 4.2kJ
2) melt all the ice (at 0) = 66.8kJ

The available energy in the warm water (compared to 0 degrees) is: 63kJ

That's more than enough to warm the ice, but not enough to melt it all. (So we know the final temperature will be 0.) After warming the ice, there will be 63-4.2=58.8kJ left to melt ice. So a fraction equal to 58.8/66.8 (= 0.88) of the ice is melted (= 176g).


I see, all previous responses in combine with your response there made me understood the problem completely. I am very greatful and I will cogitate on the procedures now. Thanks.
 

1. What is latent heat and how does it relate to cooling water?

Latent heat is the amount of heat required to change the state of a substance without changing its temperature. In the case of cooling water to 0C, the latent heat is the amount of heat needed to change the water from a liquid to a solid state.

2. How much latent heat is required to cool 1kg of water to 0C?

The amount of latent heat required to cool 1kg of water to 0C is 334,000 joules.

3. Why does water have a high latent heat compared to other substances?

Water has a high latent heat due to its unique molecular structure. The hydrogen bonds between water molecules require a large amount of energy to break, resulting in a high latent heat.

4. Does the cooling process of water to 0C involve a change in temperature?

Yes, the cooling process of water to 0C involves a change in temperature. As heat is removed from the water, its temperature decreases until it reaches 0C and changes state from a liquid to a solid.

5. Can the latent heat of water be used for practical applications?

Yes, the high latent heat of water has several practical applications. It helps regulate the Earth's climate by absorbing and releasing heat over long periods of time. It is also used in refrigeration and air conditioning systems to remove heat from a space. Additionally, the melting and freezing of water is used in some power plants to generate electricity.

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