Name the experiment! Interpreting thermodynamic derivatives as thought experiments

Transcription

1 Name the exeriment! Interreting thermodynamic derivatives as thought exeriments David Roundy, Mary Bridget Kustusch, and Corinne Manogue Deartment of Physics, Oregon tate University, Corvallis, OR We introduce a series of activities to hel students understand the artial derivatives that arise in thermodynamics. tudents construct thought exeriments that would allow them to measure given artial derivatives. hese activities are constructed with a number of learning goals in mind, beginning with heling students to learn to think of thermodynamic quantities in terms of how one can measure or change them. A second learning goal is for students to understand the imortance of the quantities held fixed in either a artial derivative or an exeriment. tudents additionally are given an exerimental ersective articularly when this activity is combined with real laboratory exeriments on the meaning of either fixing or changing entroy. In this aer, we introduce the activities and exlain their learning goals. We also include examles of student work from classroom video and follow-u interviews. I. INRODUCION In thermodynamics, most exeriments are designed to measure how one thermodynamic quantity changes as another is controlled by the exerimenter. his change is reresented by a artial derivative. Understanding this role of exeriment is challenging for students in several ways. 1 8 tudents have been shown to struggle with the oerational definitions of ressure, temerature, and volume, and commonly fail to recognize the covariation among these three variables. 4 At the uer-division level, we add in entroy a articularly challenging quantity as a fourth canonical variable. We go on to exress simle quantities such as ressure or temerature as artial derivatives of the internal energy. Even the notation used to describe artial derivatives in thermodynamics, A, (1 B is rarely used in other subfields of hysics, and is not tyically taught in math classes. Much of thermodynamics involves mathematically deriving relations between different exeriments that measure different artial derivatives. his is a challenge when students struggle to understand these artial derivatives. We have develoed a sequence of activities to address some of the student difficulties with artial derivatives in thermodynamics. his sequence was develoed in the context of Energy and Entroy, the junior-level thermodynamics course at Oregon tate University. 9 his twocredit course teaches basic thermodynamics followed by a brief introduction to statistical mechanics. Energy and Entroy features three laboratory exeriments: a rubber band lab described in a revious aer 10 and two calorimetry exeriments involving ice and water. In this aer, we resent three name-the-exeriment activities, which involve the instructor roviding student grous with a artial derivative, and asking the students to draw a icture of an exeriment that could be used to measure that derivative. We introduce each activity by announcing Name the exeriment! he intent of C this announcement which sounds like a game show is to ta into students intuitive understanding that this is a new eistemological game, with its own rules and victory conditions for them to learn. We assign one artial derivative to each grou of three students and we end the activity by having grous resent to the entire class their solution to each different artial derivative. Each activity tyically requires half an hour of class time, including the wra-u discussion. We do each activity on a searate day, with rogressively more challenging derivatives addressing different learning goals sread throughout the course. here are several overall learning goals for these activities. he rimary objective is for students to be able to identify artial derivatives as descritions of exeriments in which one quantity is changed, while certain others are held fixed. here are also several smaller but still imortant learning goals that san all three activities. One of these goals is for students to understand how to measure all the thermodynamic variables. In the ast, we have observed students failing to recognize that adding weights to a iston will increase the ressure! In these activities, students are reeatedly forced to remember and describe how to control or measure each of the thermodynamic variables. Another aim is for students to discover for themselves that some variables are easier than others to change, to constrain, or to measure. Finally, we desire for students to be able to use canonical thought exeriments such as a gas in a iston or the idea of a heat bath as a big tub of water. Canonical thought exeriments are ubiquitous in hysics they exist in every subfield and allow us to easily aly hysical intuition to roblems. his goal is usually accomlished through the discussions that wra u the activities. One reason for these big icture learning goals is to enculturate these hysics majors into the hysics community and aid their develoment as exert hysicists. As exerts, we smoothly move from a symbolic reresentation of a artial derivative to a descrition of an exeriment that would measure that derivative. For examle, exerts can identify the derivative (/ as

2 2 essentially the adiabatic comressibility even though it differs in sign, and by a factor of volume, and has been inverted. he essence of adiabatic comressibility is resent in this derivative. Why is this? In order to find this derivative, one must erform the same measurement that gives the adiabatic comressibility. As was nicely summarized by Jesson et al.,... there is increasing lausibility to the claim that exert scientists formal concetual understanding draws on concrete notions of material substance. 11 In addition to the overall learning goals discussed above, there are others secific to each activity. In this aer, we describe each name-the-exeriment activity, exlain the learning goals for that activity, and discuss some articular student difficulties that are addressed. For more detailed information about how namethe-exeriment fits into our course, see the thermodynamics activities resented on the Paradigms in Physics Activities wiki. 9 his website also includes narratives, which are annotated transcrits of videos of class sessions. hese narratives rovide examles of how these activities can be enacted in the classroom and of what an instructor and students might say and do during such activities. he final reason for all these learning goals is affective we believe that if students understand that the derivatives they are maniulating are hysically measurable quantities, they are likely to be more interested in understanding relationshis between these derivatives. How can students understand the the laws of thermodynamics as real scientific laws if they cannot connect the mathematical exressions involved with exerimental measurements? II. ACIVIY 1A: IMPLE DERIVAIVE We begin the first name-the-exeriment activity with derivatives that relate to exeriments that the students can simly envision. At this stage, we have introduced students to oerational definitions for the thermodynamic quantities, which are descritions of how one could measure these quantities. We have talked about entroy and the concet of adiabatic rocesses as quasi-static rocesses in which there is no heat exchange and the entroy is held fixed. Finally, students have been resented the first law of thermodynamics, which is that the change in the internal energy of a system is the sum of the energy added to it by heating, and energy rovided by doing work on it: du dq + dw. (2 able I lists the derivatives that we give students in this first activity. We grou these derivatives into four categories: easy derivatives for a three-dimensional fluid system (involving only ressure, temerature, and volume V ; easy derivatives for a one-dimensional system (involving temerature, length L, and tension τ; ABLE I. Easy derivatives for the first activity, groued into four categories, according to the hysics concets required. imle 3D: L L imle 1D: τ τ imle adiabatic: U U First law: adiabatic rocesses (with fixed entroy ; and derivatives of the internal energy that require students to use first-law reasoning. he first-law derivatives (discussed in ection III are the most challenging in this set, and are assigned as a second task to grous that quickly finish describing their first exeriment. his first name-the-exeriment activity is designed to address several learning goals. ome of these are general, addressed by all of the artial derivatives in able I; others are secific to only some of these artial derivatives. he challenge of exosing the entire class to these secific learning goals is addressed by having each grou reort on their solution to the class. he first general learning goal for this first activity is to reinforce the oerational definitions of thermodynamic quantities that students have already been shown. For instance, students need to formulate how they will measure or fix the ressure in terms of a force measurement divided by an area, reinforcing the definition of ressure. he second, and rimary, general learning goal is for students to areciate the meaning and imortance of the quantity that is held fixed. Many students exect that this quantity is redundant; they have been taught in their mathematics course, and sometimes in earlier hysics courses, that when taking a artial derivative, everything else is held constant. he belief that the quantity held fixed is redundant is surrisingly ersistent, even in roblems that aren t exlicitly asking about a artial derivative. Research has shown that students also commonly and incorrectly hold fixed any variables that are not currently under consideration in questions involving finite changes. 15 In fact, the idea of measuring the effect of changing one variable while holding everything else constant is taught as early as elementary school, where it is called control of variables and used to teach the scientific rocess. 16 By having students design their exeriment in a way that exlicitly constrains the quantity held fixed, students are given a hysical ersective as to why this matters. One of the secific learning goals is addressed by the adiabatic derivatives, which require students to remember that fixing the entroy corresonds to thermally insulating the system. An examle from the list of adia- V

3 and other fields of hysics, it is rate to invert artial derivatives in this way. It is hard to imagine the meaning of the derivative ( t/ ψ x,y,z in quantum mechanics, for instance. In their math classes, students are likely to be taught that such a relationshi is, in fact, not true for artial derivatives. Notationally, Eq. 4 is close to meaningless when exressed in standard math notation: 3 FIG. 1. ketch of an exeriment to measure the derivative ( /. he rocedure involves slowly adding weights to the to of the iston to change its ressure, having measured the area. he iston itself is insulated to kee the rocess adiabatic, and we use a thermometer inside to measure the resulting change in temerature. batic derivatives is, (3 which showcases several of the student learning oortunities at this stage. ince entroy is being held fixed, this corresonds to an adiabatic rocess, in which the system is thermally insulated from its environment. We are changing the temerature, while at the same time not heating the system! his is a source of confusion for students, an issue that will be more forcefully addressed by the isothermal derivatives in the second activity (described in ection IV. his exeriment is actually much easier to imagine if we turn the derivative uside down, ( 1. (4 By examining this inverse, we see that we can change the ressure on an insulated iston, for instance by utting weights on it, and measure how much the temerature changes with a thermometer, as illustrated in Fig. 1, which is far easier than trying to directly change the temerature of a thermally insulated iston. Inverting artial derivatives in this way blurs the distinction between the deendent and indeendent variables, and the distinction between controlled and measured variables. Most students (and even some faculty are unsure of the validity of Eq. (4. In both abstract math courses u x 1 x u (5 because u here is a function, while x is not a function but an indeendent variable. If we do understand x to reresent a function, it is not clear what variables it is a function of, which would determine which variables are being held constant. In short, even the simle rocess of inverting a artial derivative, which on its face may seem obvious to exerts, requires assistance for students to gras. III. ACIVIY 1B: HE FIR LAW In the same first activity in which we do simle derivatives, as described in the revious section, we also include derivatives that require the use of the first law, which states that the change in internal energy of a system is found by adding together its changes due to heating and working One of the first-law derivatives from able I is U, (6 which corresonds to another adiabatic rocess and is thus easiest to understand by imagining an insulated iston. We are changing the ressure, which is easy to manage by lacing weights on the iston. However, we don t have a way to directly measure the change in internal energy, so we must relate this change to something else using the first law. ince the change is adiabatic, there is no heating (Q 0, and the change in internal energy is equal to the amount of work done, so U V for small V. hus, U U V, (7 which is a result we could also obtain using the ordinary chain rule together with the definition U. (8

4 4 FIG. 2. ketch of an exeriment to measure the derivative ( U/ in Eq. (6. he iston is the same as that described in Fig. 1. As mentioned in the text, the rocedure involves slowly adding weights to the to of the iston to change the ressure, and measuring the change in volume using a ruler. From this, the work is found, and from that the change in internal energy. Our students seldom arrive at Eq. (7 mathematically, but students are able to exress that they measure the work by seeing how the volume changes, in order to find the change in internal energy as ressure changes. Using Eq. (7, we can design an exeriment to measure the change in volume as we change the ressure adiabatically, as illustrated in Fig. 2. Devising an exeriment to measure derivative (6 is a challenging task, which requires students to recognize that the first law is needed to describe a measurement of work needed to change the volume slightly, to realize that changing the ressure requires that they change the force on a iston, and to realize that the system must be insulated. When this name-the-exeriment question was given on a final exam, over half of the students (16 out of 27 were correctly able to exress either Eq. (7 or the idea that they would measure the work in order to find U. he same number of students although not the same set of students recognized that the system must be insulated in order to measure a rocess at fixed entroy. As a art of a searate study, 17 one of the authors (MBK conducted interviews with six students just after they had comleted the Energy and Entroy course. Part of the interview involved asking them to draw and describe an exeriment to measure ( U/, a derivative that these students had not encountered in class. We resent here an analysis of Bob s resonse as an examle of a somewhat weak (C+ student s resonse. Bob began by identifying that one must isolate the system to maintain constant entroy. First I m just thinking, I guess just reading this out loud in my head to concetually understand what it s asking, measure the change in internal energy as a function of ressure at constant entroy, which [inaudible] entroy, constant entroy is a little bit interesting, although it s not as hard as constant ressure. Alright, so, the exeriment s going to be isolated from its surroundings if you want the entroy to be constant. As Bob roceeded to address the question of how to measure internal energy, he correctly identified measuring the change in height as a way of measuring work, with the imlicit assumtion that this told him something about internal energy. However, he became confused about what is being held constant, and decided to change the height of the iston by heating the gas with the ressure held fixed. At this oint, Bob was describing a measurement of ( /. o, measure the change in internal energy, at constant ressure [long ause]. I imagine I d have some sort of insulated channel [begins drawing Fig. 3] and then, a iston that can be raised freely. It s a little bit harder with real world exeriments, cause you have to think about the friction of the iston. But there s some gas in here and we have measured, we know the weight of the iston, all of that and we can measure the height here [makes a mark and labels h ] and maybe some height it gets raised to [makes second mark and labels h ] when we increase the temerature of the gas. here d be some isolated heating element inside here [draws at the bottom left] and if you measure the heights, you can find the work done to move this weight in the gravitational otential energy, from this oint to this oint. he unusual hrase isolated heating element is telling, as it suggests that Bob is confusing an isolated system with one that is surrounded by insulation, as the combined system of gas and heating element is in this case. As Bob began to summarize his aroach, he returned to the idea of a thermally isolated system with no heat exchange, correctly using the first law to find the change in internal energy under those conditions, but the heating element is still resent. And how I d essentially model [ause]. he work done would be the internal energy because we re not adding or removing heat from the system and so you d essentially be measuring the work done at

5 5 ABLE II. Derivatives for the second activity, in which we change the entroy, groued according to the tye of exeriment required. Heat caacity measurement: Isothermal (challenging: ( V ( FIG. 3. Interviewed student s sketch of an exeriment to measure the derivative ( U/ in Eq. (6 (comare to Fig. 2. constant entroy, as a function of increase in ressure, because from the temerature increase of our monitored heating element, we can know the ressure of the gas. We ut the gas in there, we know what it is, and all about it. Bob is clearly struggling with the concet of thermal isolation. He recognizes that an isentroic rocess means the the system cannot be heated, and therefore adds thermal insulation to isolate the iston from its surroundings. However, he roceeds to add a heating element to the system, failing to recognize that the energy from the heating element comes from the surroundings. hroughout this eisode, Bob is also inconsistent about whether he sees the ressure as changing or held constant. We include a discussion of this eisode because it is tyical of the student reasoning that is addressed both in small grous and in the whole-class discussion. he activities resented in this aer give students the chance to wrestle with these issues through interactions with their eers, as well as the instructors and teaching assistants. Additionally, observing the conversations between students during these activities can hel instructors to better tailor their instruction to the needs and confusions of that articular set of students. IV. ACIVIY 2: CHANGING ENROPY After we have sent some more time in the class talking about entroy, we have a second name-the-exeriment activity, in which students examine derivatives in which the entroy itself is changing, as listed in able II. his set is comosed of two easier derivatives, which corresond to measurements of the heat caacities C V and C, along with two more challenging isothermal derivatives. ince we don t have a direct way to measure entroy itself, all of these derivatives are more challenging than those in the first activity. Instead of measuring entroy directly, we must infer the change in entroy by measuring the energy transferred by heating and using the thermodynamic definition of entroy: dqquasistatic. (9 he energy transferred by heating can be measured by heating the system with a resistor, as our students do in an exeriment measuring the heat caacity of water. One of the rimary goals of the heat caacity derivatives is to rovide a review of the concet of heat caacity. At this stage of the course, students have already measured the heat caacity of water, and many grous recognize that these derivatives corresond to an exeriment that they have already erformed. 18 his name-theexeriment activity highlights the distinction between C V and C, because students are required to exlain how they hold the volume or ressure fixed. We find this an excellent oortunity to discuss the difficulty of keeing the volume fixed and to introduce the term bomb calorimeter for a device to measure C V. tudents areciate the humor of this term. he isothermal derivatives with changing entroy are considerably more challenging, but also rovide several excellent learning oortunities. In articular, these isothermal derivatives address a common student difficulty: many students assume that when a system is held at fixed temerature, it is not being heated and since Q is zero, its entroy must also be held constant. his is comlementary to the issue mentioned in ection II, in which an adiabatic rocess resulted in a changing temerature. During this activity, this difficulty comes u early in student discussions, and many grous are able to overcome it without the need for instructor intervention. In one of these isothermal derivatives,, (10 we need to change the volume at fixed temerature, and measure the energy transferred between the system and environment by heating. here are two mechanisms one can imagine for this, neither of which is very easy. One aroach is to create a thermostat, described by one grou this way:

6 6 FIG. 4. tudent sketch of an exeriment to measure the derivative ( / in Eq. (10. he students assumed that the material being measured was ice water, and ut the ice water in a balloon. he mixture is then heated with a resistor and the resulting change in volume is measured. We were just thinking that we could have like something, an object that we can change the volume and then we d have like a resistor and thermometer and, say like we exand it so it cools off and measure how much heat we have to ut into the system to kee the temerature constant. his aroach is somewhat awkward and does require that the temerature change somewhat, so that the thermostat can resond. he second aroach, roosed by a different grou, would be to use a mixture of ice and water to hold the temerature fixed. Figure 4 shows a student resenting this grou s solution. he material that is being measured is ice water that is held in a balloon. A resistor heats the ice water, and the volume of the balloon is then measured. his student solution has a few disadvantages: it is hard to imagine thermally insulating the balloon, and measuring the change in volume could be tricky. As mentioned in ection I, we desire to aid in the develoment of these students as exert hysicists. A significant art of the culture of hysics involves eer evaluation and critique. Having students resent their solutions to the rest of the class can rovide oortunities for students to engage in this ractice. For examle, consider the discussion between the instructor (IN and two students (1 and 2 from different grous that followed the resentation of Fig. 4: FIG. 5. ketch of exeriment to measure the derivative ( / in Eq. (10. he system of interest is inside a metal cylinder that is immersed in ice water with a known quantity of ice. he cylinder and ice water are inside an insulated container. he volume of the cylinder is slowly changed, and afterwards the mass of the ice is measured. 1: Does that work because when you change hase in water, uh, like, water is larger as a solid than a liquid, which is not, most things aren t like that, and that s because of the hydrogen bonds? IN: Uh huh. 1: o, it seems like that s adding in an extra factor that doesn t really have anything to do with the heat, necessarily. IN: [furrows brow] 1: Or like, thermodynamics in general. IN: Well it does certainly does have to do, I mean, thermodynamics is all about what do things actually do. 1: But, I mean, I guess what I m saying is if you use something that wasn t water, like just some other. IN: If you use... 2: he change of volume would just be different. IN: [nods] mmhmm, yeah. o, your change of volume would be different in that case. 1: But, [ause] ok. IN: [shifts the discussion to other issue related to hase changes] At the end of this exchange, it is not clear that 1 is comletely convinced by 2 s answer. However, both 1 s critique of the revious grou s methodology and 2 s willingness to try to address 1 s question demonstrate that they understand that they are a art of a culture that values eer feedback. In addition, both students recognize that the exeriment should be able to measure this roerty for an arbitrary system. A more satisfactory variant of this solution is dislayed in Fig. 5, which uses a iston such that the volume is eas-

7 7 ABLE III. Derivatives for the third activity, in which we use Maxwell relations to find an easier exeriment. In the uside down cases, the Maxwell relation required involves the inverse of the derivative requested. Right side u: Uside down: ( V ( ily controlled, and the system is taken to be searate from the ice water. In this case, the challenge is to measure how much ice remains after changing the volume of the system. One could use a sieve to ull out the ice and measure its mass. Both of these solutions (Figs. 4 and 5 have the disadvantage of requiring that we make our measurement at 0 C, but the advantage of reinforcing the concet that fixed temerature does not mean zero heating, and they build on the ice-water calorimetry exeriments that we do earlier in the course. During this second name-the-exeriment activity, students encounter their first really hard derivatives, and exerience the difference between quantities that are concetually easy to measure and those that are inherently challenging. In this case the heat caacities are easy to measure, while the isothermal derivatives of entroy are challenging, since it is hard to fix the temerature while changing the entroy by a measured amount. Understanding this distinction is one of the rimary learning goals of this second activity. Allowing students time to struggle with this challenging task that has no articularly elegant solution hels them to areciate the difference between an easily measurable quantity and one that is less so, an areciation that will ay off in the next activity. V. ACIVIY 3: MAXWELL RELAION Before the final name-the-exeriment activity, we have shown students the Legendre transforms, and asked them find the total differentials for enthaly, Helmholtz free energy, and Gibbs free energy. We discuss how each of these total differentials gives us a new set of exressions for the thermodynamic variables, V,, and, and then remind students of Clairaut s theorem regarding the equality of mixed artial derivatives. We then introduce Maxwell relations to our students and have students find a given Maxwell relation in small grous. At this oint, we use a final name-the-exeriment activity. he students are given a artial derivative to measure. hey then use a Maxwell relation to find a second (ideally easier exeriment that is equivalent to their given artial derivative. We actually ask students to find two exeriments for their derivative, one easy exeriment and one hard exeriment. able III lists the derivatives that may be assigned in this activity, each FIG. 6. ketch of an exeriment to directly measure the derivative ( / V in Eq. (11. he rocedure involves heating the contents of an insulated cylinder with a resistor, and measuring how much weight needs to be added to the iston in order to return the system to its original volume. exressed as a derivative of entroy. In two of the four cases, this results in a derivative that is the inverse of the derivative that occurs in a Maxwell relation, which creates an additional challenge for students. his idea of turning derivatives uside down can be used at any stage in the name-the-exeriment sequence, to add one more ste for students to consider in analyzing a derivative. wo of the derivatives in able III are also resent in able II. One of these, ( /, was discussed in detail in the revious section. his derivative required a hard exeriment to measure, erformed with good thermal insulation and either a thermostat or some ice water, but we can find a Maxwell relation involving ( / from the Helmholtz free energy. From the total differential df d dv we have ( 2 ( F. (11 V his gives us a simle derivative to measure, which involves changing the temerature of a system and measuring the change in ressure required to kee the volume fixed, as illustrated in Fig. 6. his exeriment is far easier than the difficult exeriment shown in Fig. 5 in which the quantity of ice melted must be measured. In the case above, we were able to reuse the hard exeriment that we had discussed earlier. When assigning derivatives to students, however, we refer to avoid assigning a derivative to the same students who have already tackled it during a revious name-the-exeriment activity. his olicy rovides the oortunity to reinforce their learning by designing a new exeriment, instead of merely attemting to recall a solution that they reviously created. In many cases students will have seen an alicable exeriment described by another grou during an earlier wra-u discussion, so the derivative

8 8 A In this case, the simle derivative is the inverse of the first derivative we discussed in ection II, which could be measured with a simle insulated iston with a thermometer and a set of weights, as illustrated in Fig. 1. hrough this activity, students encounter unfamiliar derivatives that can be related to much more familiar exeriments. his final activity allows students to gain exerience in the lessons of the revious activities, while at the same time demonstrating how a seemingly obscure relationshi between derivatives is actually a owerful exerimental tool. We follow this activity with a laboratory in which we use a Maxwell relation to enable us to measure the entroy change when isothermally stretching a rubber band without resorting to calorimetry. 10 FIG. 7. ketch of an exeriment to directly measure the derivative ( / in Eq. (12. he rocedure involves heating the contents of an insulated cylinder with a resistor, while measuring the current (and thus ower and the temerature, and measuring the change in volume. VI. CONCLUION will still feel somewhat familiar to them. Let us consider another of the derivatives from able III, which we have not reviously discussed: (. (12 Interreted directly, this derivative requires us to change the volume at fixed ressure, and measure the change in entroy. Practically, it is easier to change the entroy and measure the change in volume, which we can do by heating a thermally insulated system with a resistive heating element, keeing track of the amount of ower dissiated and the temerature. his direct exeriment is illustrated in Fig. 7. We can construct an alternative exeriment that does not require a heat measurement by seeking a Maxwell relation that involves ( /. As discussed above, there is no Maxwell relation that exlicitly uses this derivative, since Maxwell relations come from mixed artial derivatives between two thermodynamic variables that are not conjugate airs as and V are. We resent students with these uside down derivatives in order to encourage them to think about how else to look at any given derivative. In this case, we seek a Maxwell relation involving ( /, which we can find using the enthaly. From the total differential dh d + V d we have ( 2 H ( (. (13 We have introduced a sequence of three activities in which students describe an exeriment corresonding to a given artial derivative. hese activities rovide students the oortunity to think of thermodynamic derivatives as descritions of exeriments. tudents also gain ractice with the oerational definitions of thermodynamic quantities, and exerience with constructing canonical thought exeriments. hese concrete ways of thinking about abstract concets further enculturate students into ways of thinking like exert hysicists. Finally, these activities exlicitly address student difficulties with artial derivatives and thermodynamics that have been reviously documented in the literature. ACKNOWLEDGMEN We thank Emily van Zee for helful discussions, and for her work in creating the narrative descritions of these activities, which are available, together with more information regarding the Paradigms in Physics curriculum, on the roject webage. 9 We also wish to acknowledge significant contributions from an anonymous referee. he funding for this roject was rovided, in art, by the National cience Foundation under Grant Nos. DUE , DUE , and DUE Ana Raquel Pereira de Ataíde and Ileana Maria Greca, Eistemic views of the relationshi between hysics and

9 9 mathematics: Its influence on the aroach of undergraduate students to roblem solving, ci. & Educ. 22, ( Brandon R Bucy, John R homson, and Donald B Mountcastle, tudent (mis alication of artial differentiation to material roerties, AIP Conf. Proc. 883, ( W. Christensen and J. homson, Investigating student understanding of hysics concets and the underlying calculus concets in thermodynamics, in Proceedings of the 13th Annual Conference on Research in Undergraduate Mathematics Education (Mathematical Association of America ( Christian H. Kautz, Paula R. L. Heron, Michael E. Loverude, and Lillian C. McDermott, tudent understanding of the ideal gas law, Part I: A macroscoic ersective, Am. J. Phys. 73, ( David E. Meltzer, Observations of general learning atterns in an uer-level thermal hysics course, AIP Conf. Proc. 1179, ( Evan B. Pollock, John R. homson, and Donald B. Mountcastle, tudent understanding of the hysics and mathematics of rocess variables in -v diagrams, AIP Conf. Proc. 951, ( John R. homson, Brandon R. Bucy, and Donald B. Mountcastle, Assessing student understanding of artial derivatives in thermodynamics, AIP Conf. Proc. 818, ( John R. homson, Corinne A. Manogue, David J. Roundy, and Donald B. Mountcastle, Reresentations of artial derivatives in thermodynamics, AIP Conf. Proc 1413, ( Webage of the Paradigms in Physics roject, <htt:// hysics.oregonstate.edu/ortfolioswiki>, contains a descrition of the Energy and Entroy course, a summary of the course content, and detailed descritions of the activities used in the course. 10 David Roundy and Michael Rogers, Exloring the thermodynamics of a rubber band, Am. J. Phys. 81, ( Fredrik Jesson, Jeser Haglund, amer G. Amin, and Helge trömdahl, Exloring the use of concetual metahors in solving roblems on entroy, Journal of the Learning ciences 22, ( R. H. Romer, Heat is not a noun, Am. J. Phys. 69, ( J. W. Jewett, Jr., Energy and the confused student III: Language, Phys. each. 46, ( R. Newburgh and H.. Leff, he Mayer-Joule Princile: he foundation of the First Law of hermodynamics, Phys. each. 49, ( Michael E. Loverude, Christian H. Kautz, and Paula R. L. Heron, tudent understanding of the first law of thermodynamics: Relating work to the adiabatic comression of an ideal gas, Am. J. Phys. 70, ( Richard Alan Duschl, Heidi A. chweingruber, and Andrew W. house, aking science to school: Learning and teaching science in grades K-8 (National Academy Press, Mary Bridget Kustusch, David Roundy, evian Dray, and Corinne Manogue, An exert ath through a thermo maze, AIP Conf. Proc. 1513, ( ee the Ice calorimetry lab on the Paradigms in Physics Activities wiki, Ref. 9.