10 g of steam at the boiling point are combined with 50 g of ice at the freezing point. What is the final temperature of the system?

Video transcript

Hey, everyone. So this problem is dealing with heat transfer. Let's see what they're asking us. Suppose he mixed 200 g of water at 80 degrees Celsius with 100 mL of ethanol at 20 degrees Celsius. We're asked with the final temperature of the mixture is going to be if the specific heat of water is 4.18 joules per kilogram Celsius. And the specific heat of ethanol is 2.44 jules per kilo Jews per gram Celsius. We're also told to assume that there is no heat lost to the surroundings and that the ethanol does not evaporate. They also tell us that the density of ethanol is 789 kg per meter cube. Our multiple choice answers here are a 84. degrees Celsius. B 74.7 degrees Celsius, C 79.0 degrees Celsius or B 90.4 degrees Celsius. So the key to solving this problem is recognizing that because there is no heat lost to the surroundings, we have a conservation of energy. So the heat or Q of the water is equal to that of the ethanol. And we can recall the equation for Q MC delta T. So we'll have the mass of the water multiplied by the specific heat of the water multiplied by the change in temperature of the water. And that will be equal to the mass of the ethanol multiplied by the specific heat of the ethanol multiplied by the change in temperature of the ethanol. And so the final temperature of the mixture, it's going to come into play when we take these delta T terms and expand them out. So we'll have the mass of the water, the specific heat capacity of the water. And then the delta T of the water is going to be our final temperature, which is the same for both the water and the ethanol because it's mixed minus our initial temperature of the water. And that's going to equal the mass of the ethanol multiplied by the specific heat of the ethanol multiplied by again, our final temperature of the ethanol, which is equal to the final temperature of the water minus the initial temperature of the ethanol. And so from there, it's just a plug and chug we have the mass of the water is 200 g. And so we're going to keep everything in standard units. So that's going to be 0.2 kg multiplied by the specific heat capacity of the water. It tells them the problem of 4.18 joules per ram Celsius. And so, actually, because we are keeping everything in standard units. We're going to rewrite that as 4180 Jews per kilogram Celsius multiplied by TF which is ultimately what we're solving for minus our initial temperature water, which is 80 degrees Celsius. And that is equal to the mass of the, all right. One second here, we're given the volume of the ethanol and the density of the ethanol, but not the mass. But luckily, we can recall that our mass is equal to our density multiplied by our volume. And so it is just a short step to use those two values to find our mass. So our density 789 kg per meter cubed multiplied by our volume, which is 100 mL. And we can recall that that is equal to one times 10, the negative four cubic meters. And therefore, we plug that in our mass is 7. times times the negative 2 kg. OK. So now we're back at it again, this is the mac mass of the ethanol. And so we have 7.89 times 10 to the negative kg multiplied by 2440 Jews per kilogram Celsius multiplied by RT F minus our initial temperature of the ethanol, which the problem tells us was 20 degrees Celsius. And so as we plug all of this into our calculator can simplify out each of these terms. We have 836 jewels per degree Celsius multiplied by our final temperature minus 6.69 times 10 to the fourth jewels. It's equal to 193 Jews per degrees Celsius multiplied by TF minus 3.85 times 10. The third, we're going to subtract our 193 joules per degree Celsius multiplied by TF to the left hand side of the equation. We combine our dual terms on the right hand side of the equation. And that leaves us with 643 jewels per degree Celsius multiplied by TF is equal to 630 times 10 to the four jewels divide both sides by 643 jules per degree Celsius. And we're left with a final temperature of 98.0 degrees Celsius. And we look at our multiple choice answers and that aligns with answer choice C so C is the correct answer for this problem. That's all we have for this one. We'll see you in the next video.