10 g of aluminum at 200°C and 20 g of copper are dropped into 50 cm^3 of ethyl alcohol at 15°C. The temperature quickly comes to 25°C . What was the initial temperature of the copper?

Video transcript

Hey, everyone. So this problem is dealing with the conservation of energy, specifically heat energy. So let's see what they're asking us. We're asked to determine the initial temperature of iron when 15 g of brass at 250 degrees Celsius and 25 g of iron are placed into 75 cubic centimeters of isopropyl alcohol, which is initially at a temperature of 10 degrees Celsius. This mixture eventually stabilizes at 30 degrees Celsius. They also tell us a few key pieces of information here that we'll use to solve the problem. The specific heat capacities of iron, brass and alcohol and then the density of that isopropyl alcohol. So our multiple choice answers here are a 320 degrees Kelvin B 410 degrees Kelvin C 525 degrees Kelvin or D degrees Kelvin. OK. So the conservation of energy tells us that the heat from the iron plus the heat from the brass plus the heat from the alcohol equal zero. And that heat energy is given by the equation mass multiplied by the specific heat capacity multiplied by the change in temperature. Or MC delta T. So when we expand all of these terms, all these Q terms remember that the problem is asking us for the initial temperature of iron. So we're going to isolate the heat term for iron subtract the brass of the alcohol to the other side to the red anxiety equation. So that looks like the mass of iron multiplied by the specific heat capacity of iron multiplied by our T final minus our T T iron initial. And I'm not denoting this T final as a T iron final because everything comes to the same final temperature. So it's all the same T F and that equals negative mass of the brass multiplied by the specific heat capacity of the brass multiplied by T F minus the initial temperature of the brass minus the mask of the alcohol. Specific heat capacity of the alcohol. And then T F minus initial temperature of the alcohol. OK. So then when we go through all of the terms kind of just figure out what we have or what we need to find mass of the iron was given in the problem specific heat capacity of iron given in the problem we're solving for initial temperature of the iron. And final temperature is given in the problem. Let's see, mass of the brass is given in the problem heat capacity of the brass is given in the problem. We still have final temperature and our initial temperature of grass is given in the problem, our specific heat capacity of alcohol is given our specific or sorry. Our initial out initial temperature of alcohol is given. So the one thing that we don't have, the one term we don't have in this problem to solve for our initial temperature of iron is the mass of the alcohol. But we do have the volume of the alcohol and the density. So this becomes a very simple mass equals density multiplied by volume density is 786 kg per meter cubed. And the volume is 75 cubic centimeters is equal to 75 times 10 to the minus six cubic meters because we need to keep this in standard units. And so that equals 0.59 kg. And so now it really is just a plug and play to solve for this or this initial temperature of the iron. So this looks like 0. kg because we were told 25 g in the problem to keep that in standard units multiplied by the specific heat capacity given in the problem was 450 Jews per kilogram. OK. And then we have our T F which is 30 degrees Celsius. So we can actually just solve this problem in Celsius and then um convert to Kelvin or you could convert to Kelvin initially. And then so for Celsius, it's, it's kind of up to you. I'm gonna just keep everything in Celsius here and convert at the end. And so that's all of that is equal to negative the mass of our brass. Again, 15 g. We're gonna right 150.15 kg. Our specific heat capacity of the brass was given in the problem Jews per kilogram. Kelvin multiplied by 30 C minus. Our initial brass temperature is 250 C minus our alcohol terms. So the mass of the alcohol, we found 0.59 kg, specific heat capacity of the alcohol was given in the problem as 2440 Jews per kilogram K. And then our final temperature again, 30 C minus the initial temperature of the alcohol, which is 10 C. And so when we saw this out, this looks like 30 C minus, our initial temperature of iron is equal to negative 144 C or our initial temperature of iron is equal to 174 degrees Celsius. And then to bind our degrees Kelvin, we simply add 232 173 degrees and we get 447 degrees Kelvin. And so I just want to point out or just kind of remind everyone that a degree Celsius is the same kind of magnitude as a degree Kelvin. So that's why in the problem, the degree Celsius and the, the degree Kelvin canceled. Um So that's how we solve this one. We look at our answers and it aligns with answer choice D so that's all we have for this problem. We'll see you in the next video.