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 in this problem. We have a metallic container with a movable piston filled with 0. moles of argon at 20°C. The temperature is going to be increased to 70°C at constant pressure. And were asked how much heat is used and then were asked to sketch a PV diagram as well. We're told that CV is 12.47 joules per mole kelvin and gamma is 1.67. All right, So we are looking for heat. Okay, let's recall that we can right? The heat Q is equal to N C times delta T. K. Where C is a heat capacity. Now in this case we're in a constant pressure process. And so the heat capacity we use is going to be C P. The heat capacity at constant pressure. Okay, Alright. Well, we're not given C. P. Okay, we know end we're given the number of moles. We can find delta T. But were not given cp were given CV and gamma. So let's recall that gamma is going to be the ratio of the two heat capacities. Okay, so it's equal to the heat capacity of constant pressure divided by the heat capacity at constant volume. So we have 1.67 is equal to CP over one Not 1 12.47. Okay, jewels, thermal Calvin. And this is going to give us a C. P value of .8249 with the same units, jewels or mole Calvin. Okay, Alright, so we found our cp and we can use that in our equation for Q. In order to find B. Heat. Okay let's go ahead and find delta T. As well before we plug into the equation. Just so it's a little bit simpler. Okay delta T. Well this is gonna be the final temperature which is 70 degrees Celsius. And we're gonna convert this into Calvin to 73. Calvin. Okay minus the initial temperature which is 20 degrees Celsius. And again converting to Calvin plus to 73.15 Calvin. Okay and this is going to give a delta T. Of 50 Calvin. Okay so now we have delta T. We have C. P. And we have N. We can plug them into our equation for Q. Let's scroll down a little bit here. Keep those values up and we have Q. Is equal to N. C. P. Delta T. Okay we're using C. P. Because it's a constant pressure situation Now. And is the number of malls which are told to 0.025. Okay cp we found 20.8249 jules per mole Calvin K times delta T. 50 kelvin. Okay, the unit of mole and Calvin is going to cancel with this. Per mole kelvin. We're gonna be left with jewels which is the unit for heat that we want. Okay so our units work out, We get 26. jewels. This is the heat and now we need to figure out what our sketch will look like. So we have a sketch like this. Okay. We have our pressure on the y axis, our volume on the X axis. We know we have a constant pressure process. So this is just going to be a horizontal line and we need to determine which way the arrow should be going. Okay When we go from our 20°C to our 70°C are we going to be increasing the volume or decreasing the volume? Well when we stay at constant pressure and we increase the temperature we're going to increase the volume. Okay and so this is going to be going arrow to the right. Okay we're starting on the left hand side, we keep the pressure constant, we raise the temperature and this is going to cause an increase in volume. Okay, another way you can look at this is if you consider the ideal gas law PV equals N. R. T. We know that pressure is staying constant so it's not changing and we know that the temperature is increasing and and in our constant as well. So in order for these things to remain equal, that means that the volume has to increase as well. Alright, so that gives us that arrow to the right again volume is increasing. Let's go up to our answer choices. We found Q. is around 26 jewels and we have the diagram with the arrow pointing to the right for that increase in volume. So we see that we get answer. F. The heat is 26 jewels. In our diagram, a horizontal line indicating constant pressure with an arrow pointing to the right indicating increasing volume. Thanks everyone for watching. I hope this video helped see you in the next one.