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

Hey everyone welcome back in this problem, we have 0.022 moles of argon heated under conditions of constant volume. And constant pressure came from 15°C to 82°C. We're being asked to determine the change in internal energy during each process. Okay, so during the constant volume process or the constant pressure process and to compare and explain the results. Okay, we're given these capacities Cv and gamma. The ratio of heat capacities 1.67. Okay, so let's start with the constant volume process. Okay, When we have a constant volume process, well this is ice. A cork. Okay. In a constant volume process, we know that the work is just equal to zero which tells us that the change in internal energy DELTA U. Is equal to Q. The quantity of heat. Okay. Alright. Well we know we have an equation to find this. So we can say that DELTA U. Is equal to n C V. James delta. Alright, so let's give ourselves some more space. And actually before we do that I'm going to write the values that we have on the right hand side here. So we can refer to them. So we're given the number of moles and is equal to 0.022. And we have this temperature one, 82°C. So temperature too 82°C And we're giving that CV is equal to 12.47, joules per mole Calvin. That gamma is equal to 1.67. Okay, so that's all the information we're given. Now let's move down and give ourselves some more room and calculate this change in internal energy DELTA U. That we were asked to find. Well this is going to be equal to the number of moles. 0.22 moles times the heat capacity at constant volume. 12.47 jewels per mole kelvin. eight times delta T. Well, what is delta T. Okay, well T one is 15 degrees Celsius. Which is going to be 288.15 kelvin. Okay, we add 273.15 to our Celsius temperature to get our Calvin temperature. So similarly for T. two we are going to get 355. Calvin. Okay now we need to put delta T. Into our equation delta T. Is just going to be equal to T two minus T. One. Which is equal to 355.15 Calvin -288.15 Velvet. Which gives us a delta t. 67. Alright, so plugging this into our equation delta T Calvin. The unit of more will cancel. The unit of Calvin will cancel. And we are going to be left with 18. jewels. Okay and this is our change in internal energy delta U. Which we also know is equal to Q. Okay this Q. I'm gonna call it Q. B. Q. At constant volume. Okay now constant pressure is the second process we are giving information about. For the second process we want to find the internal energy in. Okay now, constant pressure. This is an ice a barrick process. Now to find the change in internal energy. Again when we're in constant pressure we have to worry about both. The Q. Value. Okay the quantity of heat and the W. Value the work. Okay but what do we know about delta U. Okay, dealt to you is independent of path. We also know that delta U depends on delta T. Which we just found. Okay, we just use delta T to find you. So delta U. Is independent of path then it doesn't matter if we have a different process, if we're starting with the same thing and we're ending with the same thing. Then we get delta U. And let's add a V. Up here so that our change in internal energy under the constant volume process is going to be equal to the change in internal energy of the constant pressure. Internal energy. Okay so that means that our change in internal energy is going to be 18.38 jewels. Whether we're in the constant volume or the constant pressure process. Now let's go back up mm So we know that our delta U V. Delta U p are both going to be 18.4 jewels. Okay so we're looking at option C. Or D. Option C. They're saying that these are the same because DELTA U depends on DELTA T. An option D. Is saying they're the same because DELTA U. Is path dependent. Now we just talked about this DELTA U. Is path independent, right? That's why we get the same delta U. With a different process. We but delta U. Does depend on delta T. And so the answer is going to be C. Here Change in internal energy is 18.4 jewels. That's the same in the constant volume or the constant pressure process. Because that change in internal energy depends on the change in temperature. Thanks everyone for watching. I hope this video helped see you in the next one.