Before going in for his annual physical, a 70.0-kg man whose body temperature is 37.0°C consumes an entire 0.355-L can of a soft drink (mostly water) at 12.0°C. (a) What will his body temperature be after equilibrium is attained? Ignore any heating by the man’s metabolism. The specific heat of the man’s body is 3480 J/kg K.

Video transcript

Welcome back. Everyone. In this problem. A science experiment involves a student with a mass of approximately 75 kg and a body temperature of about 38 °C, drinking half a liter of water at eight °C at the beginning, what will his new body temperature be after reaching thermal equilibrium? Assume that there is no metabolic heating during this period. The specific heat capacity for human bodies is typically around 3480 joules per kilogram per Kelvin. While the specific heat capacity of water is 4186 joules per kilogram per Kelvin A says the new body, his new body temperature will be 37.8 °C. B 41.2 °C C 54.4 °C and D 72.7 °C. Now, if we're going to find the student's new body temperature, let's first make note of the information that we have here. So so far, we're told that his mass, let's call that MS is 75 kg. Next, we're told that the specific heat capacity of the student's body, let's call that CS is 3480 joules per kilogram per Kelvin. And now, since our temperatures are in Celsius, we can rewrite that or we can just change it to joules per kilogram per Celsius. It will still be the same value. Next. We're also told that the initial temperature of the student ts is 38 °C. No, afterwards, we're told that the student drinks half a liter of water at eight °C. So we have the volume of the water. But what is its mass going to be? Well, we think about it since we know that mass equals density multiplied by volume and the density of water is 1 kg per liter. Then that means the mass of the water is going to be 1 kg per liter multiplied by 0.5 L, which is going to give us a mass of 0.5 kg. So essentially that's how much the water weighs. We know that our water has a temperature of eight °C. OK. So tw is eight °C. And we also know that based on our problem, the specific heat capacity of water CW is 4186. We can write that as Jews per kilogram per Celsius. OK. So this is all of what we know so far. And from this information, we want to use it to figure out the new body temperature t of our student. Now, how can we relate body temperature to the mass and the specific heat capacity. Well, recall, OK, recall that the heat or heat is equal to the mass multiplied by the specific heat capacity multiplied by the change in temperature. And we have that information from most of our, our, our the parts of our problem here, we are also told in our problem that we want to find this new body temperature after reaching thermal equilibrium. OK. So by the conservation of energy equation, we know then that or heat the heat of the system since it's at thermal equilibrium, OK, is going to be equal to zero. So that means the heat from the, the heat, sorry for our body and the heat for our water, if we add both of them is going to be equal to zero. So applying that idea then OK, the heat for the body OK is going to be equal to its mass. So MS multiplied by the student specific heat capacity multiplied by the change in temperature, which is the difference between the final temperature. In this case, it's going to be uh T ts sorry minus T, the difference between TS and T OK. Plus for our water, we have its mass MW multiplied by specific heat capacity. CW multiplied by the temperature TW minus T OK. Where T represents the new body temperature and all of that is going to be equal to zero. Now let's go ahead and expand and see if we can eventually solve for T. Ok. So now when we do that, we're gonna have MS CST S minus MS CST ours. Yeah. Minus right plus. Ok. MW. CWT W minus M WCW T OK. And all of that equals zero. Let's group our terms for T on one side. OK. So now when we do that, we're gonna have MS CST plus MWCWT equal to MS CST S plus MWCWTW. Ok. Now, on our right, on our left hand side, we can factor out T, OK. So that's gonna be T multiplied by MS CS plus MWCW equal to our expression that we have over here. Let me just copy it. OK? Makes it a bit easier and know, know that we have that, then we can solve for T by dividing both sides by our expression on the left. In other words, T OK. T is going to be equal to just take it back to make some space, MS CST S plus MWCWTW, all divided by the expression MS CS plus MWCW. Now we have an expression for tea. So we can go ahead and solve for the new body temperature. Remember the mass of the student is 75 kg multiplied by this. Well, I think I need to come down to a new line. OK. Let me do that instead. So as I was saying, that's gonna be 75 kg multiplied by 3480 joules per kilogram per Celsius. OK? Multiplied by 38 °C, the temperature of the body. OK. Plus 0.5 kg multiplied by the specific capacity of water. 4186 joules per kilogram per Celsius multiplied by eight °C, the temperature of the water and all of that is being divided by 75 kg, multiplied by that specific heat capacity for the human body. OK? Plus 0.5 kg multiplied by the specific heat capacity for water. So we have quite a bit of information here to, to calculate, to put in our calculator. Now, when you go ahead and put that expression there, OK, then you should find that you get t to be approximately equal to 37.76 °C. Remember that all of our answers were into three significant figures. And when we do the same, we get t to be 37.8 °C. In other words, this is going to be the student's new body temperature. If we go back to our answer choices that tells us then that a is the correct answer. Thanks a lot for watching everyone. I hope this video helped.