by Patrick Ford

Hey guys. So in previous videos, we saw how to use the kalorama tree steps to go ahead and solve the target variable, which was the equilibrium temperature of a mixture of materials. We're gonna have to do this a lot in kalorama tree problems. So what I'm gonna do in this video, I show you one last time, how to navigate through these steps here and we're going to come up with is a general equation for the equilibrium temperature. And this is actually gonna work for any number of materials. So it's gonna work for two or three or four or even more most of the time. We're just gonna be working with two or three materials. So that's really all there is to it. I'm gonna go ahead and show you using this example real quick. So we have this massive water, this quantity of water here at 10 degrees Celsius. And then we have, we're going to add a block of aluminum that's at a hotter temperature. So what I always like to do is I call the colder one. A so this is gonna be the massive A, which is the colder one. This is 0.4 kg. And the temperature for that is going to be 10 degrees Celsius. The block of aluminum on the other hand, that mass is 0.2 and the temperature of that aluminum is 80 degrees Celsius. So we're gonna throw these things in an insulated container. And in the first part of the problem we want to derive an expression for the final equilibrium temperature. So we actually just want to go ahead and use a bunch of letters rather than numbers and we're gonna see what it's really useful here. So that's we're gonna do in part A. So all we have to do here is actually just go ahead and six of the steps, we're gonna start out with our calorie mystery equation, Q. Is negative QB. And then we're just gonna work out and sort of get towards that final equilibrium temperature. So we have Q A -4 equals negative QB. Remember we're just gonna stick with letters here, it's gonna be a little bit annoying. There's a lot of algebra involved but it's actually not very complicated. Let me show you. So all we have to do now in the second step is replace the cues with them cats, right? Those are Q expressions. So this is gonna be M. A. C. A. Times delta T. A. And this is gonna equal negative M. B. C. B. Times delta T. B. Now, just I'm just gonna use the C. A. That's for water and the C. B. That's gonna be for the aluminum and I just have those specific heats right here. So now what I wanna do is I want to calculate the final equilibrium temperature is as I remember those that's gonna be locked up inside of the delta T terms over here. So all we have to do is expand those. So this is going to be M. A. Times C. Eight times. and then remember delta T. Is just final minus initial for both of these objects, their final temperature is gonna be the equilibrium temperature. So it's gonna be T final minus T. A. So then we're gonna have negative M. B. Times C. B. And then again this is gonna be the same thing that's gonna be T final minus T. B. Alright, so now we have to do is basically just isolate this T final here, that is the equilibrium temperature. We just need to figure out what that expression is and then move everything else on the other side. It's a lot of algebra but it's not very complicated. So let me go ahead and walk you through it. So, if we want this T final here, that's on both sides of this equation here, we're actually gonna have to factor everything out. So we're basically gonna have to distribute this and sort of expand this. So this is gonna be M A C A times T A minus M A C A times T A. All right, so I'm sorry, this is gonna be the final that's T. Final. Okay? And then over here we're gonna get negative M B C B T. Final and then we're gonna have a plus because this negative here distributes this minus sign. So it's gonna be plus M B C B T B. All right. So now all we have to do, right, we just expanded everything else. We're just gonna have to group together these T final terms. So what I'm gonna do is I'm going to bring this one over to the other side. So it becomes positive and I'm gonna do the same thing with this term, except I'm gonna move it to the right. So we're basically just trading those two things. So what I end up with is a See 80 final plus MB CBT final equals. And then on the right side I have M A C A T A plus M B C B T B. All right. So now all we have to do here is we just have to sort of isolate this T final notice how they both are involved in those two terms. So we can actually pull those out as a common factor that T final. And what you end up with is a parentheses M A C A plus M B C B. And that's T final equals. And then Emma, I'm actually just gonna go ahead and see if I can copy paste this just to save a little bit of time. Cool, cool. So we can just get this over on the right side. Alright, so now the last thing I have to do is really just divide by this whole entire expression here and then I'll get to the final by itself. So this T final here is just gonna equal and again I have just I'm just gonna copy paste this over here and then we're gonna divide this by and then we're gonna divide this by M A C A plus, M B C B. And that's it. We're done. We got to the finals equals something else. So this is actually the answer to part A. That's the expression which just involves a bunch of numbers here um for the final equilibrium temperature. Now, this actually works just for the two materials that we have involved here, the water and the aluminum. But we can actually sort of generalize this equation to work for any number of materials, whether you have two or three or four, however many you have here, the general equation is going to be, it's going to be M A C A times T A plus M B C B T B plus and then so on and so forth. If you had another third material, you would just keep doing em C times T and then, you know, so on and so forth. Right? And then on the bottom here you're just gonna get M A C A plus MB CB and then plus. And then again, you would just keep on going however many materials that you have. Alright, so this is our equation or this expression for just two materials, let's go ahead and now do part B in part B. Here. We want to actually basically plug in all the values into our expression to find out what that equilibrium temperature is. So we're basically done. All we have to do is just plug and chug. Alright, so what I get here is 0.4. And remember the C. A. Here is gonna be the specific heat for the water. So this is gonna be 41 86. We're pretty familiar with that one times the initial temperature which is 10. And then for the aluminum it's gonna be 0.2. This CB here is gonna be the specifically for aluminum which we're told is 900 then the initial temperature is 80 degrees. So you're gonna multiply all that stuff. Just remember, do it carefully. You got really just two terms here that you can plug into your calculator And on the bottom you're just going to get 0.4 uh times 41 plus this is gonna be 0.2 times. And then this is gonna be the 900. Alright, so it's kind of tedious. That's palatable. But it's actually very straightforward if you go out and plug all of this in, what you're gonna get is a final equilibrium temperature of 16.7 degrees Celsius. Alright, so that's it for this one, let me know if you have any questions

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