Hey, everybody. So welcome back. Hopefully, you got a chance to try this one on your own. So we're going to be mixing 2 cups of water together. Let's get started. I'm going to draw this out first. We have 1 cup of water that's 0.5 kilograms at 15 degrees Celsius. So I've got a cup of water like this. We know it's 15 degrees Celsius and it's going to be 0.5 kilograms. Now we're going to add this to another cup of water except this water is going to be boiling. Right? Right. So we've got boiling water at 100 degrees Celsius. So we've got some amount of water. We don't know how much it is. All we know is that it's at 100 degrees Celsius, so it's really really close to boiling, and then we're going to add them together so that the final mixture is exactly 80 degrees Celsius. When you combine these two things in a larger container, you're going to end up with this amount of water here, something larger, and this final mixture here is going to be 80 degrees Celsius. Now, this is a pretty classic calorimetry type question because we're going to be combining 2 or more materials in a container and they're both going to reach some equilibrium temperature. Alright. So we're going to go ahead and stick to our steps here. The first thing we're going to do is write QA = −QB. Basically, what's going on here is I've got the heat that the 15 Celsius water gains is going to be equal to the heat that the 100 degrees Celsius water loses until they finally reach something that's somewhere in between. So the first step here is writing qa = −qb. Now, really what's going on here is I want to work my way towards how much boiling water I need to add so that this mixture is 80 degrees Celsius. So if you think about what's happening here is this is cup a and this is cup b. The mass of this water here is 0.5 kilograms, and what I'm looking for here is mb, the mass of the boiling water. That's really what I'm looking towards. Alright. So let's move on with the second step now, which is we're going to replace the Q's with mc∆T. Right? That's our Q = m c∆T equation m∆T. So this is going to be ma × c for water, Right? We're using c for water that's 4186 just in case you forget it. And then times delta a, delta t for a. Now this is equal to negative mb × c for water × ∆Tb. So all I have to do here is move on to the last step, which is solving for my target variable and that's this mb in this case. So let's go ahead and start plugging in some numbers. One thing I can do is I can actually just cross off the specific heats for water on both sides of the equation, and I can only do this because it's the same substance. I'm dealing with water and water, so it kind of just goes away from the equation. That's the first thing I can do, and then I'm just going to start plugging in some numbers. Right? So this is going to be 0.5. What about the change in temperature for the water? Well, that's always a final minus initial. So this is going to be the T final for both of them, whereas this is going to be the TA initial for a and this is going to be TB initial for b. So really this is going to be my final temperature of 80 minus my initial temperature of 15. Now I also don't have to convert it to Kelvins because when you're working with delta T's, remember it could be Celsius or Kelvin. So that's the equation. On the right side, what we have is negative mb, and then for delta TB, what we have is final minus initial. So this is going to be 80. That's still the final for both of them minus the initial of 100. So all I can do now is just go ahead and simplify, and let's see. What I've got here is when you combine this, this is going to be 0.5 times, and this is going to be 65 here. And then when you divide this to the other side here, what you're going to get is you're going to get negative 20 over like this. Except what you also have to do is realize, that there's also another negative sign that's out here. So you can also move this out over here. Basically, what happens is the negative signs are going to cancel out since you end up with something positive. Now it makes sense because we're solving for mass, so we shouldn't get a negative number. So we just should get negative, mb and this is going to equal 1.625, and this is going to be kilograms. So this is how much boiling water we need. It kind of makes sense that we have this number here that was kind of bigger than this number because the final temperature ends up being much closer to the 100 degrees Celsius water and that's just because there's more of it. Anyway, so that's the answer guys. Let me know if you have any questions.
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Intro to Calorimetry
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