Hey, guys. So in previous videos, we saw how to use the calorimetry steps to go ahead and solve for the target variable, which was the equilibrium temperature of a mixture of materials. We're going to have to do this a lot in calorimetry problems. So what I'm going to do in this video is show you one last time how to navigate through these steps here, and what we're going to come up with is a general equation for the equilibrium temperature. And this is actually going to work for any number of materials. So it's going to work for 2 or 3 or 4 or even more. Most of the time, we're just going to be working with 2 or 3 materials. So that's really all there is to it. I'm going to go ahead and show you, using this example real quick. So, we have this mass of water, this quantity of water here at 10 degrees Celsius, and then we are 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 the A, so this is going to be the mass of A, which is the colder one. This is 0.4 kilograms 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 going to 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 going to see why it's really useful here. So that's what we're going to do in part A. So all we have to do here is actually just go ahead and stick to the steps. We're going to start out with our calorimetry equation Q_A=-Q_B and then we're just going to work out and sort of get towards that final equilibrium temperature. So we have Q_A=-Q_B. Remember, we're just going to stick with letters here. It's going to 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 Qs with MCΔTs. Right? Those are our Q expressions. So this is going to be m_ac_aΔt_a and this is going to equal -m_bc_bΔt_b. Now I'm just going to use the c_a, that's for water, and the c_b, that's going to be for the aluminum, and I just have those specific heats right here. So now what I want to do is I want to calculate the final equilibrium temperature. So remember those that's going to be locked up inside of the ΔT terms over here. So all we have to do is expand those. So this is going to be m_ac_a×, and then remember ΔT is just final minus initial. For both of these objects, their final temperature is going to be that equilibrium temperature, so it's going to be T_final minus T_a. So then we're going to have -m_bc_b, and then again this is going to be the same thing. It's going to be T_final minus T_b. Alright? So now what 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 off to 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 going to have to factor everything out. So you're basically going to have to distribute this and sort of expand this. So this is going to be m_a c_a times t_a minus m_ac_a times t_a. Alright. Oh, sorry. This is going to be t_final. That's t_final. Okay. And then over here, we're going to get -m_bc_bt_final, and then we're going to have a plus because this negative here distributes this minus sign. So this is going to be plus m_b c_b t_b. Alright. So now all we have to do, right, we just expanded everything else. We're just going to have to group together these t_final terms. So what I'm going to do is I'm going to bring this one over to the other side, so it becomes positive, and I'm going to do the same thing with this term except I'm going to move it to the right. So we're basically just trading those two things. So wha