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# Chilling water problem

How much ice at -10 degrees C is necessary to get 500 g of water down to 0 degrees C? Created by Sal Khan.

## Want to join the conversation?

• Why didn't you convert it into Kelvin before multiplying? The 4.18 that you gave us is Joules over Grams multiplied by Kelvin but you ended up multiplying it by Celsius and still this was able to cancel out. I thought that you have to add 273 to it to change it to Kelvin.
• Because it's a change in temperature, there's no need to convert it. A one degree change is equivalent in both scales: 0°C = 273.15K, 1°C = 274.15K, and so on, so the difference between the temperatures will be the same in C or in K.
• Is there a scientific explanation of why ice usually cracks when put into a cup of water?
• when water turns to ice it expands(volume increases) and when ice turns to water it again contracts(volume decreases)
when you put ice in a cup of water the temperature suddenly decreases which due to which volume must also decrease, but there is a large temperature difference so the molecules of ice which are strongly bonded do not separate easily but the ice tries to shrink which gives rise to cracks!
• I was marginally confused by this video because you included the energy of the phase change capacity of the ice when the question was posed to suggest you intended to have 500g of chilled water at the end of the experiment. In your calculations, because you included the potential of the phase change you actually ended with 854. whatever grams of chilled water and no ice, not an answer to the question as it was posed. You would need a larger block of ice to move that 500g of water -10C so that the amount of water would not be amended by a phase change.
• Your argument is a question of semantics. I believe what was implied, was figuring out the quantity of ice required to lower the temperature of the water by 60 degrees, not making any attempt to maintain a certain final quantity of water. A larger quantity of ice would not have made any difference in maintaining the original quantity because the same amount of ice (354.02 grams) would have melted into the water regardless. The total quantity of liquid water would still be 854 grams, but with a large ice cube still remaining in it.
• At Sal said that the temperature difference is 60 degrees. Why is it positive 60? Wouldn't it be -60? Since , Change of temp = Final Temp - Initial Temp. So, F (0) - I (60) = -60 ? Please Explain , Thank you
• I had the same question at first. But this actually is still correct because this is when he is calculating heat out. So it would make sense that technically it would be negative in regards to the system.. I think why the negative doesn't matter as much here is because he already refers to it as heat "out" not just heat transferred where a negative sign would indicate that this is heat leaving the system not being added to it. So if you were doing this in a technical way and were to calculate just the heat transfer in general terms the negative would be necessary for indicating the direction of the change. But in the case of the video this direction is indicated by calling it heat out and a negative sign would just be redundant.
• I don't understand that 'HEAT OUT = HEAT ABS'? I understood up to the point where he got '20.5xJ and 333.55xJ'. But why did he set up the equation like '20.5xJ + 333.55xJ = 125340J'?
How could they be equal to?
Is 0 water -> 60 water = -10 ice -> 0 water?
• Sal set up the equation 20.5(X)j + 333.55(X)j =125340j for the following reasons; 20.5(X) joules is the amount of heat the solid ice can absorb, the 333.55(X) joules is the amount of heat the "melted" ice can absorb (this is the flat spot on the phase diagram), and the 125340 joules is the amount of energy the existing water has to lose. The X was a variable for the grams of ice required. By adding the 20.5(X) and the 333.55(X) together that gives the total amount of absorption of the ice in both the solid phase and the liquid phase. By setting the other side of the equation to 125340 he was able to apply the ices' absorption capacity to that of the water. Finally by solving for X he was able to find the quantity of ice required to chill the water.
• I stopped the video and tried this on my own and assumed that he was talking about turning all the water into solid water at 0˚ C, and obvious enough I got a different answer. Is it right? My answer is 14,000 g of ICE.
• No, he wasn't talking about turning all the water into solid. He states that he just want's to get the water to 0° C.
(1 vote)
• couldnt the water be below freezing but not frozen?
• It is -possible- to cool - very pure- water below the temperature at which it would normally freeze without it freezing. This is called supercooling. I think Sal is assuming for this problem that the water is not that extremely pure so will freeze at 0 Celsius. This is a fair assumption to make, supercooling is a rare phenomenon.
• How does having 0F' absorb all that energy by melting not change it's temperature? I mean, if ice melted it would HAVE to change it's temperature, so how does it work?
• It's very interesting, but you can have either ice or water at 0 degrees Celsius. The difference between the two is not kinetic energy (measured in Temperature) but potential energy (i.e. how far apart the molecules can move from each other). Therefore, when you add heat to turn ice into water, first the energy is used to add potential energy, causing the phase change. Then any remaining energy is used to add kinetic energy, causing a temperature increase.

This idea is touched upon in previous videos, and I'm sure the videos do a much better job explaining it than I can. I suggest watching:
States of Matter part I: http://www.khanacademy.org/video/states-of-matter?playlist=Chemistry
Specific Heat, Heat of Fusion and Vaporization: http://www.khanacademy.org/video/specific-heat--heat-of-fusion-and-vaporization?playlist=Chemistry

I hope this helps!
• Can someone help me. Why did Sal do the heat of fusion thing for ice but he didn't do an equivalent for liquid water going to 0 degrees? Doesn't water freeze at 0 degrees celcius?
• the latent heat is used to account for the phase change, in either direction. If there's no phase change, you don't use it.
• Can water be below the freezing point and not be a solid but a liquid?
(1 vote)
• Yep! Since water expands when it freezes, if water is placed under pressure and then drops below the freezing point, it will not be able to expand into ice, and so will remain a liquid. This is called a supercooled liquid, and they often have interesting properties!

## Video transcript

Let's do another states of matter phase change problem. And we'll deal with water again. But this one hopefully will stretch our neurons a little bit further. So let's say I have 500 grams of water. Of liquid water. At 60 degrees Celsius. Now my goal is to get it to zero degrees Celsius. And the way I'm going to do it is, I'm going to put ice into this 500 grams of water. And my ice machine at home makes ice that comes out of the machine at minus 10 degrees Celsius ice. And my question is, exactly how much ice do I need? So how much, or how many grams of ice? And I'm going to take the ice out of the freezer and just plop it into my liquid. How much do I need to bring this liquid, this 500 grams of liquid water, down to zero degrees? So the idea, if we just imagine a cup here. Let me draw a cup. This is a cup. I have some 60 degree water in there. I'm going to plunk a big chunk of ice in there. And what's going to happen is that heat from the water is going to go into the ice. So the ice is going to absorb heat from the water. So in order for water to go from 60 degrees to zero degrees, I have to extract heat out of it. And we're about to figure out just how much heat. And so we have to say, whatever was extracted out of the water, essentially has to be contained by the ice. And the ice can't get above zero degrees. Essentially, that amount of ice has to absorb all that heat to go from minus 10 to zero. And then also, that energy will be used to melt it a bit. But if we don't have enough ice, then the ice is going to go beyond that and then warm up even more. So let's see how we do this. So how much energy do we have to take out of the 500 grams of liquid water? Well, it's the same amount of energy that it would take to put into zero degrees liquid water and get it to 60 degrees. So we're talking about a 50 degree change. So the energy or the heat out of the water is going to be the specific heat of water, 4.178 joules per grams Kelvin. And I have to multiply that times the number of grams of water I have to cool down, I have to take the heat out of. And we know that's 500 grams. And then I multiply that times the temperature differential that we care about. And just a side note, I use this specific heat because we're dealing with liquid water. Liquid water going from 60 to zero. So the final thing, I have to multiply it by the change in temperature. The change in temperature is 60 degrees. Times 60 degrees. There's a little button on the side of my pen, when I press it by accident sometimes it does that weird thing. So let's see what this is. So this is 4.178 times 500 times change of 60 degrees. It could be a change of 60 degrees Kelvin or a change of 60 degrees Celsius. It doesn't matter. The actual difference is the same, whether we're doing Kelvin or Celsius. And it's 125,340 joules. So this is the amount of heat that you have to take out of 60 degree water in order to get it down to zero degrees. Or the amount of heat you have to add to zero degree water to get it to 60 degrees. So essentially, our ice has to absorb this much energy without going above zero degrees. So how much energy can the ice absorb? Well let's set a variable. The question is how much ice. So let's set our variable, maybe we'll call it I. Let's do x. x is always the unknown variable. So we're going to have x grams of ice. OK, and it starts at minus 10 degrees. So when this x grams of ice warms from minus 10 degrees to zero degrees Celsius, how much energy will it be absorbing? So to go from minus 10 degrees Celsius to zero degrees Celsius, the heat that's absorbed by the ice is equal to-- is equal to the specific heat of ice, ice water, 2.05 joules per gram Kelvin, times the amount of ice. That's what we're solving for. So times x. Times the change in temperature. So this is a 10 degree change in Celsius degrees, which is also a 10 degree change in Kelvin degrees. We can just do 10 degrees. I could write Kelvin here, just because at least when I wrote the specific heat units, I have a Kelvin in the denominator. It could have been a Celsius, but just to make them cancel out. This is, of course, x grams. So the grams cancel out. So that heat absorbed to go from minus 10 degree ice to zero degree ice is 2.05 times 10 is 20.5. So it's 20.5 times x joules. This is to go from minus 10 degrees to zero degrees. Now, once we're at zero degrees, the ice can even absorb more energy before increasing in temperature as it melts. Remember, when I drew that phase change diagram. The ice gains some energy and then it levels out as it melts. As the the bonds, the hydrogen bonds start sliding past each other, and the crystalline structure breaks down. So this is the amount of energy the ice can also absorb. Let me do it in a different color. Zero degree ice to zero-- I did it again-- to zero degree water. Well the heat absorbed now is going to be the heat of fusion of ice. Or the melting heat, either one. That's 333 joules per gram. It's equal to 333.55 joules per gram times the number of grams we have. Once again, that's x grams. They cancel out. So the ice will absorb 333.55 joules as it goes from zero degree ice to zero degree water. Or 333.55x joules. Let me put the x there, that's key. So the total amount of heat that the ice can absorb without going above zero degrees. Because once it's at zero degree water, as you put more heat into it, it's going to start getting warmer again. If the ice gets above zero degrees, there's no way it's going to bring the water down to zero degrees. The water can't get above zero degrees. So how much total heat can our ice absorb? So heat absorbed is equal to the heat it can absorb when it goes from minus 10 to zero degrees ice. And that's 20.5x. Where x is the number of grams of ice we have. Plus the amount of heat we can absorb as we go from zero degree ice to zero degree water. And that's 333.55x. And of course, all of this is joules. So this is the total amount of heat that the ice can absorb without going above zero degrees. Now, how much real energy does it have to absorb? Well it has to absorb all of this 125,340 joules of energy out of the water. Because that's the amount of energy we have to extract from the water to bring it down to zero degrees. So the amount of energy the ice absorbs has to be this 125,340. So that has to be equal to 125,340 joules. We can do a little bit of algebra here. Add these two things. 20.5x plus 333.55x is 354.05x. Is that right? Yeah, 330 plus 30 is 350. Then you have a 3 with a 0.5 there. 354.05x and that is equal to the amount of energy we take out of the water. You divide both sides. So x is equal to 125,340 divided by 354.05. I'll take out the calculator for this. The calculator tells me 125,340, the amount of energy that has to be absorbed by the ice, divided by 354.05 is equal to 354 grams. Roughly; there's a little bit extra. So actually, just to be careful maybe I'll take 355 grams of ice. Because I definitely want my water to be chilled. So our answer is x is equal to 354.02 grams of ice. So this is interesting. I had 500 grams of liquid. And you know, intuitively I said, well if I have to bring that down to zero degrees, I'd have to have a ton of ice. But it turns out, I only need, what was the exact number? So the liquid is 500 grams. About, roughly half a pound, if you want to get a sense for how much 500 grams is. A kilogram is 2.2 pounds. So this ice is 354. So you actually have to have less ice than water. Which is interesting, because it seems like the ice isn't making that big of a temperature change. The ice is only going from minus 10 to zero degrees. While the water is going all the way from 60 degrees to zero degrees. So you're like, how does that work out? And the reason is because so much energy can be absorbed by the ice in the form of potential energy as it melts. So if you talk about all the energy that the ice is asborbing, most of it is due to the heat of fusion right there. You can actually have zero degree ice and it can still absorb this huge amount of energy just by melting it. Without even changing its temperature. Anyway, hopefully you found that reasonably useful.