Chilling water problem

let's do another states of matter phase change problem and we'll deal with water again but this one hopefully will will stretch our neurons a little bit further so let's say I have 500 grams of water 500 grams of liquid 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 so how much ice do I need to bring my fived 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 down to zero degrees so the idea I mean if we just imagine a cup here let me draw a cup this is the cup I have some some 60 degree water in there I'm going to plonk a big chunk of ice in there a big chunk of ice and what's going to happen is is that the water heat from the water is going to go into the ISO the the ice wall is going to absorb heat from the water so the same amount of so in order for water to go from sixty 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 water essentially has to be contained by the ice and the ice can't get above zero degrees so essentially that amount of ice has to absorb all the heat to go from minus ten to zero and then also that 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 with the specific so the energy or the heat that out of the water heat out of water is going to be the specific heat of water half here from our previous videos four point one seven eight joules per grams Kelvin so it's four four point one seven eight joules per grams Kelvin and I have to multiply that times the number of grams of water I have to cool down have to take the heat out of and we know that's 500 grams times 500 grams and then I multiply that times the temperature differential that we care about and end it at just a side note I use this specific heat because we're dealing with liquid water right we're dealing with liquid water liquid water going from 60 to zero so the final thing after multiply it by the change in temperature the change in temperature is 60 degrees times 60 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 four point one seven eight times five hundred times 60 change of sixty degrees I could write it could be a change of sixty degrees Kelvin or a change of sixty degrees Celsius it doesn't matter the actual difference is the same whether we're doing in Kelvin or Celsius and that's 125,340 joules so it's 120 so it's equal to 125 thousand three hundred and forty joules so this is the amount of heat that you have to take out of sixty degree water in order to get it down to zero degrees or this would evaporate the amount of heat you have to add to zero degree water to get it to sixty degrees so essentially our our ice has to absorb this much energy without going above zero degrees so how much energy candy candy ice absorb let's well let's set a variable though the question is how much ice so let's set our variable we call it io2 xx is always the unknown variable so we're going to have X grams grams of ice okay and this 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 that the temperature difference so to go from minus 10 degrees Celsius to zero degrees Celsius the heat that's absorbed by the ice is equal to so Heat I'll call it heat absorbed is equal to the specific heat of ice ice water 2.05 joules per gram Kelvin 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 degrees change in Kelvin degrees so it could 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 the heat absorbed to go from minus 10 degree ice to zero Degree ice is C 2 point 0 5 times 10 is 20 point five so it's 20 point 5 times X times X joules joules this is go from minus 10 degrees to zero degrees now once we're at zero degrees the ice can even absorb more more energy before increasing in temperature as it melts right remember when you draw the draw that phase change diagram you guys gain some energy and then it levels out as it gets as it melts as the the bonds the the hydrogen bonds or kind of starts sliding past each other and the crystalline structure breaks down so this amount of energy the ice can also absorb and that so this is zero degree we do it in a different color zero Degree ice to zero I did it again to zero degree water zero degree water well the heat absorb now is going to be the heat of fusion of ice or the heat the melting heat either one that's 333 joules per gram it's equal to 333 0.55 joules per gram times the number of grams you have once again that's X grams they cancel out so the ice will absorb 333 0.55 joules as it goes from zero Degree ice to zero degree water so the total amount or 333 0.55 X joules let me put that X there that's key so the total amount of heat that the ice can absorb without going above zero degrees because once at zero degree water as you put more heat into it it's going to get it's going to start getting warmer again we if the ice gets above zero degrees there's no way it's going to bring the water down to zero degrees so the ice cannot get or the net at this point 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 ten to zero degrees ice and that's 20.5 X 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 point five five X and of course all of this is joules so this is a 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 the amount of ice the energy absorbs or the amount of energy the ice absorbs has to be this 125,340 so that has to be equal to one hundred twenty-five thousand three hundred and forty fools see we can do a little bit of algebra here add these two things twenty point five x plus three hundred thirty three point five five X is what it's three hundred and fifty 350 for 0.05 X is that right yeah three thirty plus 20 is 350 then you have a three with you have a point five day at 350 for 0.05 X that is equal to the amount of energy we take out of the water and you divide both sides so X is equal to 125 340 divided by 350 for 0.05 I'll take out the calculator for this calculate or the calculator tells me to see 125 340 the amount of energy that has to be absorbed by the ice divided by divided by 350 for 0.05 is equal to 354 grams roughly there's a little bit of extra so actually just to be careful they'll take 355 grams of ice because I definitely want my eye water to be chilled so our answer is X is equal to 350 4.0 two grams of ice so if I take so this is interesting I had 500 grams of liquid and you know intuitively I said oh boy 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 350 this is 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 for minus 10 to 0 degrees while the water is going all the way from 60 degrees to zero degrees so you like what you know 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 the heat of fusion so if you talk about all of the energy that the ice absorbing most of it is due to the heat of fusion right there so the ice can abduct you you can actually have zero Degree ice and it can still absorb this huge amount of energy just just by melting it without even changing its temperature anyway hopefully you found that reasonably useful