## Wednesday, April 28, 2010 (Day 3)

This narrative$^{1}$ continues the examples of an instructor engaging students in thinking conceptually about the partial derivatives they encounter in thermodynamic contexts. In Name the Experiment sessions, the instructor, David Roundy, invites students to think about how they would measure the quantities represented by a partial derivative.

During the first Name the Experiment session (Day 2 of the thermodynamics paradigm course), David only provided partial derivatives in which entropy was either fixed or not measured. During this second Name the Experiment session (Day 3 of the course), he engaged the students in thinking and talking about partial derivatives in which the entropy was changing.

Of particular interest here is an interaction in which students proposed an experiment that differed from what the instructor intended. Although this was an appropriate experiment, it only works in a very limited context. David tries to reinforce the good points while discussing the limitations in a way that keeps the conversation open. Students start engaging in a conversation in which they propose and discuss several alternatives.

### Overview

Each small group of three or four students had worked on a large white board on which they drew and described an experiment representing a partial derivative in which the entropy was changing. The derivatives were

$$\left(\frac{\partial S}{\partial T}\right)_{V}, \left(\frac{\partial S}{\partial T}\right)_{p}$$

$$\left(\frac{\partial S}{\partial V}\right)_{T},\left(\frac{\partial S}{\partial p}\right)_{T}$$

David views these derivatives as trickier than those considered during the first Name the Experiment session because students are vague about what entropy is. They also are vague about what it means to have entropy change as well as how to measure it or change it intentionally. Keeping it fixed, as they had in several of the partial derivatives considered during the first Name the Experiment session, is easy, that just involves thermal insulation.

Students know that entropy changes when you heat something but are vague on the fact that you can heat something without changing its temperature. They quite naturally think of entropy as being like temperature only different, but they do not understand how it is different.

One way to change entropy without changing temperature is to have something undergo a phase change, like melting ice. Another way involves isothermal compression. In that case, as something is compressed, it would “naturally' heat up, raise its temperature, but if it is in contact with a thermal bath, it will heat its environment and remain at fixed temperature, and therefore will be changing its entropy through heating (or cooling) while at fixed temperature.

This scenario raises the issue of what is the system and what the surroundings, which David had discussed early in the course. Entropy is different from other parameters. One can measure the volume, pressure, and temperature in an isolated system but not entropy. One has to make a change and measure the transfer of energy between the system and surroundings to measure the entropy. During the discussion interpreted below, students considered how entropy changes when the volume changes at constant temperature. A key equation that students make use of is that for the change in entropy

$$\Delta S = \int \frac{dQ_{\text{quasistatic}}}{T}$$

The challenge experimentally is in measuring the small energy transfer $dQ$.

### Initiating the Next Presentation

Several groups had already presented their solutions to various partial derivatives when David initiated the next presentation by asking for a volunteer from several groups who had considered how entropy changes when the volume changes at constant temperature:

${\bf DR:}$ Then [underlines $(dS/dV)_{T}$ on board] say the, the change in entropy with respect to volume. I think we had at least more than, a couple of different approaches on this. So, if someone would like to describe your approach.

${\bf [00:12:05.03]}$

A student from Group 5 offered a solution involving melting ice:

${\bf S1:}$ We, uh, kind of thought of some kind of expandable container like a balloon or something, just some kind of container that would expand. And then we would put ice water in there and then, kind of like our first day of class [the ice calorimetry lab—http://www.physics.oregonstate.edu/portfolioswiki/courses:activities:eeact:eeice], we ran a resistor through it.

${\bf DR:}$ You want to, maybe draw a sketch on the board.

${\bf [00:12:24.09]}$ Student 1 shows work at the board

${\bf S1:}$ This is a balloon, with ice cubes in it [draws balloon with 4 ice cubes], a little resistor [draws resistor] that went through it. And so we ran a current through it and it would add heat energy to this, but the temperature wouldn't change because the heat energy would go to the phase change of ice and so when the ice melts, it's going to change the volume of the balloon.

${\bf DR:}$ And then you measure the volume of the balloon?

${\bf \text{Student 1}:}$ [nods] mmhmm. And versus our power that we put into it.

${\bf [00:12:57.01]}$

${\bf DR:}$ Ok.

Another student, S2, asked about whether this is valid even if you are not using water, because of the phase change issue:

${\bf S2:}$ Does that work because when you change phase in water, uh, like, water is larger as a solid than a liquid, which is not, uh, most things aren't like that, and that's because of the hydrogen bonds?

${\bf DR:}$ Uh huh

${\bf S2:}$ So, it seems like that's adding in an extra factor that doesn't really have anything to do with the heat, necessarily.

${\bf DR:}$ [makes faces as he thinks]

${ \bf S2:}$ Or like, thermodynamics in general

David responded to the latter with a general statement about thermodynamics:

${\bf DR:}$ Well it does certainly does have to do, I mean, thermodynamics is all about what do things actually do.

${\bf S2:}$ But, I mean, I guess what I'm saying is if you use something that wasn't water, like just some other.

${\bf DR:}$ If you use-

At this point, another student, S3, addressed S2’s concern before David had a chance to respond:

${\bf S3:}$ The change of volume would just be different

David thought about that and affirmed that response:

${\bf DR:}$ Mmhmm, yeah. So, your change of volume would be different in that case.

${\bf S2:}$ But, ok.

${\bf [00:13:51.25]}$

At the end of this exchange, it is not clear that S2 is completely convinced by S3's answer. However, both S2's critique of the previous group's methodology and S3's willingness to try to address S2's question demonstrate that they understand that they are a part of a culture that values peer feedback.

Then David summarized the phase change solution and raised the limitation that this only works at the temperature for which the phase change occurs. Although one could change the material used, the problem remains. David invited another group to offer a different solution

${\bf DR:}$ but if you want to measure it for room temperature water, you are now in something of a pickle. And, did anyone have a solution for what, that didn't involve a phase change? I thought [gestures to back right] that you, it sounded like I heard an idea at the back table there, would one of you, um.

${\bf [00:15:11.02]}$

S4, student from another group, gave a solution that did not involve a phase change. Instead she proposed a thermostat, where one has a feedback loop to keep the temperature constant. Then however much heat is being added indicates the entropy change:

${\bf S4:}$ Um, we were just thinking that we could have like um, something, an object that we can change the volume and then we'd have like a resistor and thermometer and um, say like we expand it so it cools off and measure how much heat we have to put into the system to keep the temperature constant.

${\bf [00:15:30.02]}$ David summarized Group 1's approach

${\bf DR:}$ Ok, so that idea was something like [drawing], you had like, maybe a piston type of system and you had a little resistor and thermometer, that's the thermometer. And, so you could, expand it, if it's something like a gas and that would cause it to cool down, but you have a little feedback loop, where you turn on your current when you start detecting a cooling down to keep it from cooling down and measure how much current you need. Which, this is, about as good as I could come up with.

${\bf [00:16:08.12]}$

Another student, S5, raised a relevant issue and suggested doing it really slowly.

${\bf S5:}$ You could also just do it really slowly, so the room temp, you can kind of equi, cause it to equalize.

David affirmed that suggestion:

${\bf DR:}$ So, yes, and certainly you want to do this slowly as well [pointing to drawing of thermostat]. All of these things we'd like to do slowly. So, if you just do it really slowly [starts new drawing], then there's a question of what are you going to measure. How will you find out how much, cause if you do it really slowly it will certainly remain at constant temperature, but you need to find out how much heat went in or out in order to find out the entropy change.

${\bf [00:16:48.12]}$

David then offered a third solution, describing using a very large water bath.

${\bf [00:17:27.27]}$

Another student, S6, asked how to measure entropy change.

${\bf \text{Student 6}:}$ How would you measure the entropy change?

${\bf DR:}$ How would you measure the entropy change? For this picture [pointing to large bath]? So, you would, this thing would change by a delta T, which is small [writes delta T small], uh, but you know the heat capacity of the water and so, you know that $Q$ is $C_{p} \Delta T$ and, yes?

${\bf S6:}$ So, we would say that the water would heat up, but we wouldn't say that the, or it would, so it would heat up maybe temperature change, there would be no corresponding temperature change of the gas itself?

${\bf DR:}$ Well, there would be a corresponding temperature change, but the key thing is, if you have a whole lot of water then the temperature change would be very small.

${\bf S6:}$ Oh.

David brought the discussion of these various ways to measure $(dS/dV)_{T}$ to a close by acknowledging that keeping the temperature constant would not be easy via the feedback loop of the thermostat or via a very large bath. That is the motivation for the next topic, an alternative approach using Maxwell’s relations.

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