# Answer: Terminal Temperature of Water with Ice Cube

• ChessEnthusiast
In summary: Let's say that the ice shatters and releases its heat. Then the temperature of the water will be lowered by 3.38 degrees.Yes, it would.
ChessEnthusiast
We have a glass containing 0.5 liter (0.5 kg) of water whose temperature 100 degrees Celsius.
We also have an ice cube with mass 0.01 kg and temperature -10 degrees Celsius.

The cube is put into the glass. The glass is then insulated from the outside world, until the cube has melted. What will be the terminal temperature of the water?

I have read a little bit about Thermodynamics and I think this solution shoudl work:

Let's first calculate the amount of energy required to melt the cube. First, we need to raise its temperature by 10 degrees. The specific heat of ice is 2108 J /(kgK) and we have 0.01 kg of it, and so we will need
$$E_1 = 0.01 \cdot 2108 \cdot 10 = 210.8 J$$

Now, the ice needs to transition to the liquid state.
The amount of heat required to melt one kilogram of ice is 333 700 J. Thus, we need
$$E_2 = 0.01 \cdot 333 700 = 3337 J$$

So, we need the total of
$$\Delta E = 3547.8 J$$

And this energy will be sucked away from the water. So now we can sovle for the change in temperature

$$3547.8 = 0.5 \cdot 4180 \cdot \Delta T$$
$$\Delta T = 1.7$$ Degrees Celsius.

Is it the correct answer? It seems to me that the water will be way too hot.

Looks right to me. My only quibble would be that you didn't include the temperature rise of the melted ice, but at this mass ratio it is just small enough not to affect your answer.*

As a gut check, from memory it takes about 80x as much energy to melt ice as to change water's temperature by 1 degree. You have 50x as much water as ice, so that's 1.6C.

* Oops, on re-checking, it is enough to matter just a little bit.

Last edited:
ChessEnthusiast
So far, so good. You have the melted ice (A.K.A. water) at 0C in the same container as the original water at 98.3C.

ChessEnthusiast and Delta2

ChessEnthusiast said:

For a final temperature? Does that make sense to you? If you take a sliver of ice the size of a quarter and put it in half a liter of water that is about to boil, does that cool the water to almost freezing? Does that match your experience?

I assume that he's talking delta-T, as in his original answer (which also didn't respond to the precise question).

Of course, I was talking about delta-T. I'm sorry for writing an ambiguous post.

## 1. What is the terminal temperature of water with an ice cube?

The terminal temperature of water with an ice cube refers to the final temperature that the water and ice cube will reach when they are in thermal equilibrium with each other.

## 2. How is the terminal temperature of water with an ice cube calculated?

The terminal temperature can be calculated using the principle of conservation of energy. This means that the total energy of the system (water and ice cube) before and after they reach thermal equilibrium will be the same.

## 3. Why does the ice cube melt when placed in water?

The ice cube melts when placed in water because the water has a higher temperature than the ice cube. Heat energy flows from the warmer water to the colder ice cube, causing it to melt and reach thermal equilibrium with the water.

## 4. What factors can affect the terminal temperature of water with an ice cube?

The terminal temperature of water with an ice cube can be affected by the initial temperature of the water and ice cube, the size and mass of the ice cube, and the insulating properties of the container the water and ice cube are in.

## 5. How can the terminal temperature of water with an ice cube be used in real-world applications?

The terminal temperature of water with an ice cube can be used in various applications, such as in refrigeration systems and cooling processes. It can also be used to measure the specific heat capacity of a substance or to determine the efficiency of a heating or cooling system.

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