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Let's do another states of matter phase change problem.

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And we'll deal with water again.

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But this one hopefully will stretch our neurons a little

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bit further.

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So let's say I have 500 grams of water.

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Of liquid water.

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At 60 degrees Celsius.

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Now my goal is to get it to zero degrees Celsius.

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And the way I'm going to do it is, I'm going to put ice into

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this 500 grams of water.

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And my ice machine at home makes ice that comes out of

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the machine at minus 10 degrees Celsius ice.

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And my question is, exactly how much ice do I need?

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So how much, or how many grams of ice?

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And I'm going to take the ice out of the freezer and just

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plop it into my liquid.

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How much do I need to bring this liquid, this 500 grams of

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liquid water, down to zero degrees?

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So the idea, if we just imagine a cup here.

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Let me draw a cup.

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This is a cup.

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I have some 60 degree water in there.

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I'm going to plunk a big chunk of ice in there.

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And what's going to happen is that heat from the water is

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going to go into the ice.

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So the ice is going to absorb heat from the water.

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So in order for water to go from 60 degrees to zero

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degrees, I have to extract heat out of it.

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And we're about to figure out just how much heat.

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And so we have to say, whatever was extracted out of

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the water, essentially has to be contained by the ice.

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And the ice can't get above zero degrees.

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Essentially, that amount of ice has to absorb all that

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heat to go from minus 10 to zero.

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And then also, that energy will be used to melt it a bit.

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But if we don't have enough ice, then the ice is going to

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go beyond that and then warm up even more.

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So let's see how we do this.

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So how much energy do we have to take out of the 500 grams

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of liquid water?

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Well, it's the same amount of energy that it would take to

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put into zero degrees liquid water and

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get it to 60 degrees.

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So we're talking about a 50 degree change.

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So the energy or the heat out of the water is going to be

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the specific heat of water, 4.178 joules per grams Kelvin.

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And I have to multiply that times the number of grams of

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water I have to cool down, I have to take the heat out of.

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And we know that's 500 grams. And then I multiply that times

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the temperature differential that we care about.

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And just a side note, I use this specific heat because

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we're dealing with liquid water.

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Liquid water going from 60 to zero.

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So the final thing, I have to multiply it by the change in

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temperature.

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The change in temperature is 60 degrees.

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Times 60 degrees.

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There's a little button on the side of my pen, when I press

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it by accident sometimes it does that weird thing.

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So let's see what this is.

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So this is 4.178 times 500 times change of 60 degrees.

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It could be a change of 60 degrees Kelvin or a change of

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60 degrees Celsius.

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It doesn't matter.

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The actual difference is the same, whether we're doing

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Kelvin or Celsius.

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And it's 125,340 joules.

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So this is the amount of heat that you have to take out of

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60 degree water in order to get it down to zero degrees.

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Or the amount of heat you have to add to zero degree water to

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get it to 60 degrees.

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So essentially, our ice has to absorb this much energy

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without going above zero degrees.

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So how much energy can the ice absorb?

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Well let's set a variable.

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The question is how much ice.

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So let's set our variable, maybe we'll call it I.

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Let's do x. x is always the unknown variable.

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So we're going to have x grams of ice.

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OK, and it starts at minus 10 degrees.

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So when this x grams of ice warms from minus 10 degrees to

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zero degrees Celsius, how much energy will it be absorbing?

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So to go from minus 10 degrees Celsius to zero degrees

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Celsius, the heat that's absorbed by the ice is equal

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to-- is equal to the specific heat of ice, ice water, 2.05

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joules per gram Kelvin, times the amount of ice.

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That's what we're solving for.

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So times x.

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Times the change in temperature.

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So this is a 10 degree change in Celsius degrees, which is

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also a 10 degree change in Kelvin degrees.

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We can just do 10 degrees.

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I could write Kelvin here, just because at least when I

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wrote the specific heat units, I have a Kelvin in the

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denominator.

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It could have been a Celsius, but just to

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make them cancel out.

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This is, of course, x grams. So the grams cancel out.

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So that heat absorbed to go from minus 10 degree ice to

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zero degree ice is 2.05 times 10 is 20.5.

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So it's 20.5 times x joules.

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This is to go from minus 10 degrees to zero degrees.

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Now, once we're at zero degrees, the ice can even

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absorb more energy before increasing in

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temperature as it melts.

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Remember, when I drew that phase change diagram.

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The ice gains some energy and then it

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levels out as it melts.

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As the the bonds, the hydrogen bonds start sliding past each

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other, and the crystalline structure breaks down.

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So this is the amount of energy the

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ice can also absorb.

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Let me do it in a different color.

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Zero degree ice to zero-- I did it again--

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to zero degree water.

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Well the heat absorbed now is going to be the heat

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of fusion of ice.

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Or the melting heat, either one.

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That's 333 joules per gram.

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It's equal to 333.55 joules per gram times the number of

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grams we have. Once again, that's x

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grams. They cancel out.

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So the ice will absorb 333.55 joules as it goes from zero

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degree ice to zero degree water.

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Or 333.55x joules.

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Let me put the x there, that's key.

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So the total amount of heat that the ice can absorb

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without going above zero degrees.

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Because once it's at zero degree water, as you put more

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heat into it, it's going to start getting warmer again.

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If the ice gets above zero degrees, there's no way it's

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going to bring the water down to zero degrees.

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The water can't get above zero degrees.

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So how much total heat can our ice absorb?

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So heat absorbed is equal to the heat it can absorb when it

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goes from minus 10 to zero degrees ice.

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And that's 20.5x.

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Where x is the number of grams of ice we have. Plus the

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amount of heat we can absorb as we go from zero degree ice

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to zero degree water.

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And that's 333.55x.

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And of course, all of this is joules.

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So this is the total amount of heat that the ice can absorb

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without going above zero degrees.

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Now, how much real energy does it have to absorb?

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Well it has to absorb all of this 125,340 joules of energy

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out of the water.

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Because that's the amount of energy we have to extract from

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the water to bring it down to zero degrees.

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So the amount of energy the ice absorbs

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has to be this 125,340.

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So that has to be equal to 125,340 joules.

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We can do a little bit of algebra here.

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Add these two things.

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20.5x plus 333.55x is 354.05x.

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Is that right?

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Yeah, 330 plus 30 is 350.

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Then you have a 3 with a 0.5 there.

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354.05x and that is equal to the amount of energy we take

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out of the water.

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You divide both sides.

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So x is equal to 125,340 divided by 354.05.

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I'll take out the calculator for this.

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The calculator tells me 125,340, the amount of energy

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that has to be absorbed by the ice, divided by 354.05 is

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equal to 354 grams. Roughly; there's a little bit extra.

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So actually, just to be careful maybe I'll take 355

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grams of ice.

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Because I definitely want my water to be chilled.

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So our answer is x is equal to 354.02 grams of ice.

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So this is interesting.

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I had 500 grams of liquid.

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And you know, intuitively I said, well if I have to bring

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that down to zero degrees, I'd have to have a ton of ice.

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But it turns out, I only need, what was the exact number?

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So the liquid is 500 grams. About, roughly half a pound,

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if you want to get a sense for how much 500 grams is.

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A kilogram is 2.2 pounds.

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So this ice is 354.

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So you actually have to have less ice than water.

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Which is interesting, because it seems like the ice isn't

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making that big of a temperature change.

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The ice is only going from minus 10 to zero degrees.

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While the water is going all the way from 60 degrees to

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zero degrees.

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So you're like, how does that work out?

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And the reason is because so much energy can be absorbed by

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the ice in the form of potential energy as it melts.

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So if you talk about all the energy that the ice is

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asborbing, most of it is due to the heat of

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fusion right there.

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You can actually have zero degree ice and it can still

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absorb this huge amount of energy just by melting it.

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Without even changing its temperature.

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Anyway, hopefully you found that reasonably useful.