Calorimetry: Using q=mdeltaTc to find Temperature + Example INSTRUCTOR: All right. I caught some flak on my Facebook page for not making enough videos. So this one goes out to Maria, Andrea, and Romano for bullying me into it. Check it out. A 10-gram iron bar, which starts out at 80 degrees Celsius, is dropped into 70 milliliters of water, which starts out at 25 degrees Celsius. And we want to know the final temperature. Obviously, if you drop a hot iron bar into room temperature water, in the end, they'll be the same temperature, colder than the iron bar started, at but warmer than the water started at. This formula, q equals m delta Tc, tells you how much heat is exchanged when a certain mass undergoes a certain temperature change. Now, to figure out how much actual heat was transferred, we need something called the specific heat capacity. And it's different for everything. It's higher for substances that hold their heat better. See? Water holds its heat pretty well. It has a heat capacity of 4.184 joules per gram degrees Celsius. But iron is easier to cool off. Its 0.46 joules per gram degrees Celsius. Anyways, this is done in something called Coffee Cup Calorimetry, which is probably a lab you're going to end up doing. But this is how you end up solving a question like this. Because the heat absorbed by the water is the same as the heat given off by the iron, we can just start by writing that the amount of heat absorbed by the water is the amount of heat lost by the iron. Long story short, m delta Tc, for the water, is negative m delta Tc for the iron, because heat gained is heat lost. Now, we need to fill in some numbers. The mass of the water is 70 grams. If you didn't know water was 1 gram per milliliter, well, then, you need to learn yourself something. The change in temperature. Now, we don't know what the final temperature is here. So I'm going to call it x. And it started out at 25 degrees Celsius. Now, notice this is just a surrogate for the change in temperature. If the temperature goes up to 50, then that makes this 50 minus 25, which is 25. Just trust me. This works. And then, your heat capacity of water is 4.184 joules per gram degrees Celsius. Sweet. Now, I'm running out of space. But that's OK. I have 10 grams of iron. Oh, I forgot my negative out front. The change in temperature there is x minus 80. Final temperature, we don't know. Initial temperature, we don't know. And we're going to have to multiply that by 0.46 joules per gram degrees Celsius. Notice, x is going to be somewhere in between 25 and 80, because you're going to end up warmer than the water, but colder than the iron bar, let's say 50. This is going to be 50 minus 25. That's 25. This is going to be 50 minus 80, which is negative 30. But that negative is why we have this negative out front the whole time, to account for it. Plus gaining and losing are opposites. So we need the negative in there too. Here's the problem though. How are we going to solve this? This is a linear equation. And you're going to need a little bit of algebra to solve it. I think it's time to move that up. Check it out. 70 times 4.184 gives me 292.88. And 10 times 0.46 ends up giving me 4.6. I've dropped the units, just so you can see, this is clearly just a linear equation. The way that we end up solving this is to multiply these numbers through the brackets. Let's do that now. Why do I keep switching colors? 292.88x and 292.88 times negative 25 gives me minus 7,322. Negative 4.8x. Negative 4.6 times negative 80 gives me 368. Now, the way you solve this is put all your x's on one side, all your numbers on the other. Collect like terms. And then, divide by whatever's in front of x. Let me repeat that. Put all your x's on one side of the equals sign. There's an x. Going to move my 4.6 over. Notice, I switched the sign. I got my 368 on the right-hand side still. And I'm moving the 7,322 over. I can collect my like terms. This plus this gives me to 297.48. And this gives me 7690. How do you solve for x? Divide both sides by what's in front of x. Divided by 297.48. Divided by 297.48. And the answer, x, which represents my final temperature, ends up being-- 7690 divided by 297.48. Now, if we're right, it's going to be between 25 and 80. 25.85 degrees Celsius. That fits the bill. It's between the two. Why is it so close to 25 degrees? Well, we had 7 times as much water. And the heat capacity of water is 10 times as much as it is for iron. So the water is going to hold on to its heat, or be more resistant to absorbing heat, than the iron is to giving it away. And that's why the temperature change is more in the water [INAUDIBLE] direction. Get what I'm saying? Look. Long story short, if you're asked to mix two things at two temperatures and you're also given their masses, you're going to need their heat capacities, all you have to do is use m delta Tc equals negative m delta Tc, plug in your numbers, solve your linear equation algebraically. Although, I get that the math might be tough for you. Solve for x. Boom. Nicely done. Best of luck.