A cylinder contains 0.0100 mol of helium at T = 27.0°C. (a) How much heat is needed to raise the temperature to 67.0°C while keeping the volume constant? Draw a pV-diagram for this process. (b) If instead the pressure of the helium is kept constant, how much heat is needed to raise the temperature from 27.0°C to 67.0°C? Draw a pV-diagram for this process. (c) What accounts for the difference between your answers to parts (a) and (b)? In which case is more heat required? What becomes of the additional heat? (d) If the gas is ideal, what is the change in its internal energy in part (a)? In part (b)? How do the two answers compare? Why?

Video transcript

Everyone in this problem. We have a metallic container with a movable piston filled with 0. moles of argon at 18°C. Okay, The temperature has increased to 78°C at constant volume and were asked how much heat is used. And were also asked to sketch a PV diagram for this process. We're told that C. V. The heat capacity at constant volume is 12.47 jewel per mole Calvin and gamma is 1.67. Okay, so we're asked to find heat. Well, let's recall that the heat Q can be given by N. C delta T. Okay. And there's a number of moles. C as the key capacity and delta T. Is a change in temperature. And in this case the heat capacity we're going to use because we're in a constant volume process is C. V. The heat capacity for constant volume. Okay, now we're given the number of moles and in the question were given cV in the question, let's go ahead and find delta T. We can substitute in to find Q delta T. Is going to be T final, it's T. Initial. The final temperature is 78°. We're Gonna Add 273.15. Okay, to convert this into Calvin And then we're going to subtract the initial temperature which is 18°C and again adding 273.15. So that we can write this in Calvin. This gives us a delta t of 60 Calvin. Cool. Alright. So getting back to Q. Now Q is equal to N. C. V. Delta T. And the number of moles were given 0.025 0.025 mol Okay, CV were given 12.47. The unit here is jewel per mole kelvin times delta two U. Which we just found to be 60 Calvin. Okay, the unit of mole times Calvin is gonna cancel with this per mole kelvin. We're gonna be left with the unit of jewel which is exactly what we want for heat. So our units checked out there and we get the heat Q. Is equal to 12.705 or sorry? Not 12 18.705 jewels. So that is the heat we were looking for and now we need to go ahead and do a sketch. Okay so let's just scroll down a little bit, give ourselves some room to sketch. So we're asked to sketch a PV diagram. So we're gonna have P along the Y axis and V. Along the X axis. Now we're told that this is a constant volume process. So what that means is we're going to have a vertical line. Okay, because the volume isn't going to be changing. Now what we need to figure out is whether the arrow should go up or the arrow should go down. Okay if the arrow goes up that means we're increasing pressure. If the arrow goes down that means we're decreasing pressure while keeping that volume the same. Well what we're doing, we're keeping the volume constant and we're increasing the temperature. That means that we're gonna have an increase in pressure. So our error was going to go up, the pressure is going to increase. Okay? And if you want to think about this, if you think about this related to the ideal gas law PV is equal to NRT. Okay now our volume is going to be held constant. We know that N and R. Those are constant. Okay our temperature is increasing and so the only way for this equation to hold is for p the pressure to also increase. So our pressure is increasing. We have this arrow up And if we look at the possible solutions, we see that we found a heat q. 18.7 jewels. And our pressure volume sketch is a vertical line with an arrow pointing up indicating increasing pressure. And so we have answer C. Thanks everyone for watching. I hope this video helped see you in the next one