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A copper calorimeter can with mass 0.100 kg contains 0.160 kg of water and 0.0180 kg of ice in thermal equilibrium at atmospheric pressure. If 0.750 kg of lead at 255$^\circ$C is dropped into the calorimeter can, what is the final temperature? Assume that no heat is lost to the surroundings.

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$21.4^{\circ} \mathrm{C}$

08:03

Zulfiqar Ali

Physics 101 Mechanics

Chapter 17

Temperature and Heat

Section 6

Calorimetry and Phase Changes

Marissa H.

December 13, 2020

Why does he say the problem ends at 47.10 K, but the solution shows21.4 K?

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now in this problem, we have a copper cholera emitter can and it has a mass of is your 0.1 kg. It contains zero points, 16 kg of water and zero point Oh, 18 kg of ice. It's all that atmospheric pressure, so we don't have to worry about pressure considerations in this problem. And then we drop a hot piece of lead into this and the lead weighs 0.75 kg and it is initially at 255 degrees C. Now we're told that the water and the ice are in thermal equilibrium, which means that they're both at zero degrees Celsius and then t two, I'm going to call for the temperature. So we dropped this in. We have this copper, and then we have this water with some ice in here, and then we drop this lead in here, and then we let it all come down to thermal equilibrium, and we want to figure out what the final temperature is. And we're assuming that basically that this is all insulated and we get no energy loss outside of the system here. So basically there's energy balance. It says that you know the energy before we dropped the, um, the internal energy before we drop the copper and has to be the internal energy afterwards and all these internal energies, because we have, you know, solids and liquids. Um, we're just going to say that we have a constant heat capacity, so there's just a constant heat capacity, times the mass times the temperature. And so then, you know, we know the change is, and we wind up with changes in temperature. The other piece that we have to be very careful about and not forget to include is the is the heat of heat of fusion for the ice. So the ice is going to melt before it starts to rise to two for its temperature to rise. So we actually have a lot of energy going into melting this ice as we'll see in a second here, Um, so we need to make sure we keep that. And this is, um I used to you I'm not sure what you know, the notation they used in the book, I think maybe where they use cue for this, um, heat of heat of, uh uh, this light and heat of, uh, consult, Uh, crystallization? No, uh, whatever it is kind of, uh, this latent heat going from a solid to a liquid. I just lost the name of it in my head. Anyway, um, they don't hit a fusion fusion. Yeah, there we go. And so then we have So we have, You know, the change that just is the change in internal energy in the lead in the copper and the water. And then the ice is winds up being well, changing from ice to water and then from water as it becomes as it goes up in temperature. Um, to equilibrium, it's than water. So what we need to do is we need to look up all these, um, heat capacities, and we can do that. They're all listed in the book. Um, and then we can use degrees Celsius for these temperatures because their temperature differences again, it's a little bit. It's probably better to just convert everything to Calvin. But in the end, a change in and Calvin is equal to a change in the greasy because the difference is just an offset, not a factor. So we can use. We can use degrees Celsius in in this problem because we just have changes in temperature. So we don't really need to do the Celsius to Calvin conversion. So we can see here that the heat capacity for lead is 130 killed joules per kilogram. Calvin, um, and let's see here, um, that is that's actually pretty low, right? If you look at it, what it is for water, the capacity of water is actually very high. This is why we use water to cool things a lot. Now, um, so then the change in temperature, it goes, it's t to finally, but it's then it's initially at 2 55 degrees Celsius for the copper. The copper heat capacity is 390 kg. Calvin. So three times that led respect. Yeah. Um, Then again, the killer, the capacity for water is 4000, 190 so, you know, almost, you know, 10 times what it is for copper. And so again, the copper. We're assuming that the copper is also at zero degrees C. So it's an all thermal equilibrium with the water. The you know, the ice water. Um, so those are all at zero initially, and then they end up at T two. And then we can look up the latent heat of fusion for for water. And that's 334 times 10 to the to the third killer jewels per kilogram. So that's, uh, that's, uh, you know pretty high, as it takes a lot of energy to melt ice pour to freeze water, and so that contributes quite a bit here and then as the after the ice melts, then it gets raised up from zero degrees C to whatever the final temperature is. So again we have the mass of the ice and then the heat capacity of water. And so, um, and that's how it has to be zero, because the energy before has to be equal to the energy after. So this whole equation, we know everything in here but t two. So it's just a matter of plugging in numbers and, you know, solving for t two, and we get that it is a temperature of 21.4 degrees Celsius is the final temperature of this, um, of everything of the copper, the water, the ice, the ice, which has turned into water. The led and the led I'm all that 21 point four degrees Celsius again. That is assuming that this whole thing is insulated so that no energy was lost to the environment, as this was called mhm.

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