Pouring Hot Water in an Aluminum Cup

by Patrick Ford
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Hey guys, So hopefully got a chance to work this one out on your own. Let's check this problem out here. So we have some amount of water. 150 g of water at 35 Celsius. You're gonna pour it into a cup as cup is 65 g with 11 degrees Celsius. Now, what happens is we're gonna mix these two things together and it's going to reach some final equilibrium temperature. So we know this is a classic kalorama tree problem. You're mixing two things together until they reach some equilibrium temperature. So I can kind of draw this out here. What happens to have some of that out of water? This is 35 degrees Celsius and we have 150 g of water. You're going to add this to some cup. The cup itself, the material of the cup is at 11 degrees Celsius and this is going to be g. Right? So this is the water and this is the aluminum. You mix them together. And basically what you end up with is just a cup that's filled with water. And the final temperature is what you're looking for. All right, so we know this is a calorie mystery type problem, all we have to do is start off with these steps, we're just gonna write our Q equals negative Q. Equation. However, since we already know what type what what target variable we're looking for the final temperature. We can actually just skip the rest of the steps and we're actually gonna go right to this equation over here. That's the equilibrium temperature equation. So what happens here is we have the Q. A. L. Equals the negative Q. Of the water, right? The aluminum gain some heat, the water loses some heat. And the final temperature is gonna be somewhere in the middle between the two initial temperatures. It's gonna be somewhere between 11 and 35. So just go ahead and shortcut all of that. I'm just gonna start plugging stuff into the equation because I've given it to you. Right? So what happens here is this final temperature is going to be according to this format, you're basically gonna go do M C. T. M C. T divided by M. C. Plus EMC. So this is going to be m aluminum, See aluminum and then the temperature of the aluminum plus the m of the water. Sea of the water and then the t of the water. So you're gonna do that and then you're gonna divide this by the M. Of the aluminum C. Of the aluminum without the teas plus m of the water. Sea of the water. Alright, so it's pretty straightforward, it's kind of just plugging and chugging a bunch of stuff in. So the final temperature here is going to be, well let's see this is this is the mass of the aluminum. Now this is given a 65 g but it's really important that you convert this. So this is equal to 0.065 kg. So that's what we have to plug in, 0.065. And the specific heat for the aluminum is 910. Notice how this is pretty small in comparison to water and that makes sense. Aluminum is a metal so it doesn't require as much energy to change its temperature, it changes temperature is much easier than water does. That's 9 10 and the initial temperature here is going to be the 11°C. It actually doesn't really matter in this case whether you use kelvin or Celsius, because remember that this T final here actually comes from a delta T. So it's actually okay if you use Celsius or kelvin, you'll still get the right answer. Alright, we're gonna add this to the mass of the water. Now, just as we converted this will have to convert this to 0.15, I'm sorry, 0.15 kg. That's water. So this is going to be 0.15. The specific heat for water is 41 86 And the initial temperature of the water is 35°C. Alright, so just kind of a bunch of plugging and chugging. So this is the mass of the aluminum, 0.065 times plus 0.15 times 86. So just really just do this kind of carefully. You can do the top part first, then the bottom part, but basically what you'll end up here with you and you plug this in is you'll end up with a temperature of 32.9 degrees Celsius. Notice how what we said was true, It was somewhere in between 11 and 35 and it kind of makes sense that this temperature is much closer to the water because we have more of the water, 150 versus 65 g. And also the specific heat of water is greater than aluminum. So the number should have been way closer to 35 than it was 2 11. Anyway, so that's it for this one. Guys, let me know if you have any questions.