In the Classroom edited by Resources for Student Assessment John Alexander University of Cincinnati Cincinnati, OH 45221 Three Forms of Energy Sigthór Pétursson Department of Natural Resource Science, University of Akureyri, 600 Akureyri, Iceland; sigthor@unak.is Thermodynamics is of fundamental importance to chemistry. Over one hundred papers in this Journal show the appreciation of this importance by the chemical community in the last five years. Reference to only a few of these papers is given here (1–9). The importance of reactions that are primarily carried out for the production of energy and the principles used to work out the quantity of energy released are well known. Of no less importance are the fundamental principles relating to entropy and the free energy of reactions to determine the spontaneity of reactions and even the equilibrium constant (10, 11). The use of the common thermodynamic equations, as presented in general chemistry courses, is relatively straightforward, even if some of the underlying concepts, covered further in more advanced physical chemistry courses, are difficult for the beginner. The modern students and practitioners of chemistry have at their disposal a vast quantity of thermodynamic data; it is therefore essential that the practical aspects of the subject are not obscured by too much theory. The authors of contemporary books on thermodynamics are aware of this as noted in recent reviews (12, 13). Calculations It is probably true that chemists are more familiar with heat energy than mechanical energy. To appreciate fully the important and common transformation of chemical energy into mechanical energy, for example in the internal combustion engine or in our bodies, it is helpful to compare the energy involved in familiar events. Three different transformations that everybody should be able to envisage are examined. The three forms of energy are described and illustrated in Figures 1–3. We may have our own feelings on which of these three events involves the greatest amount of energy, but let us work it out. Heat What is the energy needed to heat 200 g of water from 7.0 ⬚C to 37.0 ⬚C? specific heat capacity of H2O (c) = 4.184 J ∆t = 37.0 ⬚C − 7.0 ⬚C = 30.0 ⬚C mass of H2O (m) = 200 g What is the energy (heat) involved in warming 200 g of water from 7.0 °C to 37.0 °C? Remember that every time you drink a glass of cold water your body expends this quantity of heat to warm the water up to your body temperature. glass of water 200 g of water heated from 7.0 oC to 37.0 oC Figure 1. Heat. Elevation of a body against Earth’s gravitational force. What is the energy (work) needed to lift a 50.0 kg sack of cement to a height of 10.0 m? This is roughly equal to carrying a sack of cement to the third floor of a building. 50.0 kg elevated by 10.0 m against gravity, g = 9.80 m s-2 cement, 50.0 kg Figure 2. Mechanical energy. What is the work performed in expanding a cylinder by 90.0 L against an external pressure of 1.0 atm? external pressure P = 1.00 atm g᎑1 ⬚C᎑1 area = A The heat energy, q, involved is therefore: q = (m)(∆t)(c) = (200 g) × (30.0 °C) × (4.184 J g᎑1 °C᎑1) = 25104 J = 25.1 kJ 776 piston extracted against external pressure ∆V ∆ = 90.0 L Figure 3. Expansion. Journal of Chemical Education • Vol. 80 No. 7 July 2003 • JChemEd.chem.wisc.edu l In the Classroom Mechanical Energy What is the energy needed to lift a 50.0 kg sack of cement to a height of 10.0 m? The gravitational acceleration, g, is 9.80 m s᎑2. The force acting on the sack is therefore: f = (m)(g) = (50.0 kg) × (9.80 m s᎑2) = 490 kg m s᎑2 = 490 N Mechanical energy or work is by definition the product of force and displacement. The work, w, performed by displacing the sack of cement by 10.0 m against gravity is therefore: w = ( f )(l) = (490 N) × (10.0 m) = 4900 N m = 4.90 kJ Expanding Cylinder What is the work, w, performed to expand a cylinder by 90.0 L against a pressure of 1.00 atm? w = ( f )(d) = ( f )(l); where d is the displacement P = f 兾A; pressure is the force per unit area, A rearranging f = P × A, thus w = (P)(A)(l) Since (A)(l) is the change of volume, ∆V, w = (P)(∆V) w =(1.00 atm) × (90.0 L) = 90.0 atm L The unit atm L must be equivalent to energy (work). We can confirm this and convert the unit to joule by representing the pressure in SI units 1 atm = 1.013 × 105 Pa or in fundamental SI units 1 atm = 1.013 × 105 kg m᎑1 s᎑2 Thus atm L = 1.013 × 105 kg m᎑1 s᎑2 L Since L = dm3 = 10᎑3 m3, then atm L = (1.013 × 105 kg m᎑1 s᎑2) × (10᎑3 m3) = 101.3 kg m2 s᎑2 = 101.3 joule as kg m2 s᎑2 is equivalent to joule. Therefore the work done on the surroundings is: w = (90.0 atm L) × [101.3 J兾(atm L )] = 9117 J = 9.12 kJ Discussion It is interesting to compare these results, especially the energy involved in heating the water and elevating the sack of cement. It takes a reasonably fit person to carry a mass of 50.0 kg of cement to the third floor of a house. This activity could be considered a good exercise. Do that a few times every day and you would be justified in feeling that you were getting rid of a few excess Calories (capital C is used to denote nutritional calories, equal to 1000 heat calories)! The energy expended in doing this activity is, however, only 19.5% of the energy expenditure in drinking a glass of cold water. Even if we add the body weight of the person, let us say 75 kg, the energy expenditure is still only about 49% of that involved in drinking the water. We can take a more extreme case regarding the heat energy. Consider the example of a person who ingests, perhaps a bit excessively, 1 L of hot (67 ⬚C) liquid daily (coffee, tea, etc.). If that person changed to drinking 1 L of cold (7 ⬚C) water he would be expending 251 kJ per day, since he is switching from heat gain in cooling 1 L at 67 ⬚C to 37 ⬚C to heat expenditure in heating 1 L at 7 ⬚C to 37 ⬚C. Let us look at the expanding cylinder. The thermodynamic importance of the expansion or compression of gases, when reactions take place at constant pressure, is familiar. Keeping in mind Torricelli’s barometer and remembering that mercury has a density of 13.5 g兾cm3, this is equivalent to lifting [(90.0 dm3) × (1000 cm3兾dm3) × (13.5 g兾cm3)] = 1.22 × 106 g = 1.22 × 103 kg or 1.22 metric tons to a height of 760 mm or 0.760 m. This is best appreciated if we think in terms of extracting a piston with a radius of 19.42 cm or 1.942 dm, by 0.760 m or 7.60 dm. The pressure is 1 atm or 760 mm Hg and the volume swept by the process is (1.942 dm)2 × π × 7.60 dm = 90.0 dm3 or 90.0 L. The weight of the mercury is about 24 times the weight of the sack of cement and a much larger force is needed to perform this task, but the displacement is also much smaller. Nutritional Energy Comparison Since two of the examples above led us into comparisons relevant to nutrition we can develop that theme very briefly. First it must be stressed that these examples are not illustrated to trivialize the need for healthful physical activity because we need to exercise our muscles and drinking cold water is not the best way to accomplish that. However the comparison does draw attention to a worthwhile and economical nutritional fact. Consider that the basal metabolic rate—the energy expenditure when lying at complete rest— of an average middle-aged man (180-cm high, 75 kg) is about 7100 kJ (about 1700 kcal or Calories) per 24 hours. The total energy use of the same man doing light physical work during the day is about 8400 kJ (about 2000 Calories) per 24 hours. Thus the energy requirement for physical activity for the man is about 1300 kJ or 310 Calories (14). The extra 251 kJ of energy the man would utilize by switching from drinking a liter of hot drinks to drinking a liter of cold water is about 20% of the energy needed for light physical activity. Literature Cited 1. Mills, Pamela; Sweeney, William V.; Cieniewicz, Waldemar. J. Chem. Educ. 2001, 78, 1360–1361. 2. Weiss, Hilton M. J. Chem. Educ. 2001, 78, 1362–1364. 3. Bartell, Lawrence S. J. Chem. Educ. 2001, 78, 1059–1067. JChemEd.chem.wisc.edu • Vol. 80 No. 7 July 2003 • Journal of Chemical Education 777 In the Classroom 4. Bartell, Lawrence S. J. Chem. Educ. 2001, 78, 1067–1069. 5. Wadsö, Lars; Smith, Allan L.; Shirazi, Hamid; Mulligan, S. Rose; Hofelich, Thomas. J. Chem. Educ. 2001, 78, 1080– 1086. 6. Jacobson, Nathan. J. Chem. Educ. 2001, 78, 814–819. 7. Howard, Irmgard K. J. Chem. Educ. 2001, 78, 505–508. 8. Jansen, Michael P. J. Chem. Educ. 2000, 77, 1578–1579. 9. Jensen, William B. J. Chem. Educ. 2000, 77, 713–717. 10. Masterton, W. L.; Hurley, C. N. Chemistry, Principles and 778 11. 12. 13. 14. Reactions, 4th ed.; Harcourt College Publishers: Orlando, FL, 2001; Chapters 8, 17. Noggle, J. H. Physical Chemistry, 3rd ed.; Harper Collins: New York, 1996; Chapters 2, 3. Gislason, Eric A. J. Chem. Educ. 2001, 78, 1186. Minderhout, Vicky. J. Chem. Educ. 2001, 78, 457. Passmore, R.; Eastwood, M. A. Davidson and Passmore Human Nutrition and Dietetics; Churchill Livingstone: Edinburgh, Scotland, UK, 1986; Chapter 3. Journal of Chemical Education • Vol. 80 No. 7 July 2003 • JChemEd.chem.wisc.edu Student understanding of the ideal gas law, Part II: A microscopic perspective Christian H. Kautz, Paula R. L. Heron, Peter S. Shaffer, and Lillian C. McDermott Citation: American Journal of Physics 73, 1064 (2005); doi: 10.1119/1.2060715 View online: http://dx.doi.org/10.1119/1.2060715 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/73/11?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Comparing student conceptual understanding of thermodynamics in physics and engineering AIP Conf. Proc. 1513, 102 (2013); 10.1063/1.4789662 Cooling of an ideal gas by rapid expansion Am. J. Phys. 74, 54 (2006); 10.1119/1.2110547 Student understanding of the ideal gas law, Part I: A macroscopic perspective Am. J. Phys. 73, 1055 (2005); 10.1119/1.2049286 Irreversible Adiabatic Compression of an Ideal Gas Phys. Teach. 41, 450 (2003); 10.1119/1.1625202 Student understanding of the first law of thermodynamics: Relating work to the adiabatic compression of an ideal gas Am. J. Phys. 70, 137 (2002); 10.1119/1.1417532 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 68.232.23.162 On: Fri, 12 Feb 2016 17:14:09 Student understanding of the ideal gas law, Part II: A microscopic perspective Christian H. Kautz,a兲 Paula R. L. Heron, Peter S. Shaffer, and Lillian C. McDermott Department of Physics, University of Washington, Seattle, Washington 98195 共Received 6 April 2005; accepted 29 July 2005兲 Evidence from research indicates that many undergraduate science and engineering majors have seriously flawed microscopic models for the pressure and temperature in an ideal gas. In the investigation described in this paper, some common mistaken ideas about microscopic processes were identified. Examples illustrate the use of this information in the design of instruction that helped improve student understanding of the ideal gas law, especially its substance independence. Some broader implications of this study for the teaching of thermal physics are noted. © 2005 American Association of Physics Teachers. 关DOI: 10.1119/1.2060715兴 I. INTRODUCTION This paper is the second of two that report on an investigation of student understanding of the ideal gas law 共PV = nRT兲.1 The emphasis in the first is on the macroscopic variables and their relationship to one another through the ideal gas law.2 We found that many of the student difficulties that we identified at the macroscopic level seem to be rooted in incorrect, or incomplete, microscopic models. 共The use of the term model to characterize some related ideas does not mean that students have the robust, self-consistent conceptual structure that physicists associate with this term.兲 In this paper, we describe some common student models, and discuss how the insights gained from this study have guided the development and assessment of tutorials to improve student learning.3 The research, which was conducted by the Physics Education Group at the University of Washington 共UW兲, involved more than 1000 students. The participants were mostly undergraduate science and engineering majors enrolled in introductory algebra- or calculus-based physics courses at UW and other universities and a sophomore-level thermal physics course at UW. Most of these students had taken, or were concurrently taking, introductory chemistry. We obtained additional information from graduate students who were pursuing a Ph.D. in physics at UW. The ideal gas law and the kinetic theory of gases are covered in many introductory physics and chemistry courses and, at a somewhat more advanced level, in thermal physics courses. Often, relatively little time is spent on a macroscopic perspective. The emphasis is mostly on the microscopic model. The underlying assumptions are often stated explicitly and presented in detail. An expression for pressure is derived in terms of the number of particles, their mass and average speed, and the volume of the gas. Temperature is identified with the average kinetic energy of the particles. For most students, this instructional sequence is not the first exposure to the microscopic view of a gas. They are usually aware of the particulate nature of matter before they take an undergraduate science course. II. METHODS OF INVESTIGATION The present study builds on related research in which we examined the ability of students to apply the first law of thermodynamics and the ideal gas law.2,4 We began the in- vestigation with individual demonstration interviews.5 As these progressed, misinterpretations of microscopic processes seemed to underlie many of the errors that the students made. To verify this impression, we conducted additional interviews in which we asked students from the thermal physics course to respond to tasks specifically designed to elicit their ideas about microscopic processes in an ideal gas. For example, students were asked to compare the number of molecules in three hypothetical identical balloons, each filled with a different ideal gas, but all with the same volume and temperature. Fewer than half of the students realized that the pressure in all three balloons must be the same and, hence, the number of molecules must be equal. Most assumed that the size, mass, and structure of the gas particles required different values for the pressure or number of molecules. The analysis of responses to the interview tasks yielded information about the microscopic models that students commonly use to predict and explain the behavior of ideal gases. We found that these models often are so seriously flawed that they inhibit the development of a functional understanding of important concepts in thermal physics, including operational definitions of pressure and temperature, conservation of energy as expressed by the first law of thermodynamics, and substance independence of the ideal gas law. These findings laid the foundation for the development of written problems that enabled us to explore in greater detail some of the difficulties that we had identified and also to estimate their prevalence. The problems were administered on course examinations or on nongraded written quizzes. All involved qualitative questions for which explanations of reasoning were required. In presenting the data, we have combined the results from multiple sections of the same course, rounded the numbers of students, and given the percentages of correct and incorrect responses to the nearest 5%. Our research methods and justification for this approach are discussed in greater detail in Ref. 2. III. PROBLEMS DESIGNED TO PROBE STUDENT UNDERSTANDING We designed three types of problems to probe student understanding of the ideal gas law from a microscopic perspective. Some involved several tasks. Unless otherwise noted, they were administered after standard instruction, but before 1064 Am. J. Phys. 73 共11兲, November 2005 http://aapt.org/ajp © 2005 American Association of Physics Teachers 1064 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 68.232.23.162 On: Fri, 12 Feb 2016 17:14:09 any research-based instruction. Thus, they can also be considered as pretests for the type of instruction that we will describe. For each problem we give a proper response and note the prevalence of correct answers. A more detailed analysis follows later. A. Isobaric expansion or compression problems In one set of problems, students were asked about microscopic aspects of an isobaric process. In each case, a fixed amount of an ideal gas undergoes a specified change in volume at constant pressure. Usually the students were asked to compare the initial and final equilibrium states in order to avoid the issue of whether the process could be considered quasi-static. Students were sometimes first asked to sketch the process in a PV diagram. Most drew correct diagrams. 1. Kinetic-energy task The students were asked whether the average kinetic energy per molecule increases, decreases, or remains the same as the result of an isobaric expansion. To answer correctly the students could first recognize that the increase in volume at constant pressure must correspond to an increase in temperature. The average kinetic energy per molecule is proportional to the temperature, which therefore also increases. Alternatively they could realize that an expansion results in a decrease in number density. Therefore, the average speed of the gas particles must increase to maintain the same pressure. Consequently, the average kinetic energy 共and hence temperature兲 must increase. About 70% of the introductory students 关N ⬃ 155兴 and about 80% of the thermal physics students 关N ⬃ 40兴 recognized that the average kinetic energy increases. Many gave the right answers for a variety of incorrect reasons. On a similar task involving a compression, about 55% of the introductory students realized that the kinetic energy decreases 关N ⬃ 60兴. 2. Change-in-momentum and particle-flux tasks Students were also asked whether the average change in momentum of a single particle as the result of a collision with a wall would be different after the isobaric expansion. This question, which is admittedly difficult, was not asked primarily to assess understanding, but rather to help the students think in terms of a microscopic model in which gas pressure can be expressed as the product of the average change in momentum per collision of a single particle and the particle flux incident on the wall.6 In the second task, the students were asked whether the number of particles incident on a container wall per unit time interval per unit area 共the particle flux兲 would be greater than, less than, or equal to the number before the expansion.7 To answer this question, students could recognize that in the standard elementary microscopic treatment of an ideal gas, elastic collisions are assumed.8 Therefore, the average change in momentum of a particle in a single collision is proportional to the average initial momentum of the particle, which in turn is a 共monotonically increasing兲 function of the temperature. In the process described, the average change in momentum increases due to the increase in the temperature. Because the product of the average change in momentum and the particle flux remains constant in an isobaric process, the particle flux must be less after the expansion. In the introductory course about 35% of the students gave correct answers with correct explanations for the change-inmomentum task and only 10% gave correct answers with correct explanations for the particle-flux task 关N ⬃ 95兴. In the thermal physics course the corresponding percentages were 65% and 20% 关N ⬃ 80兴. On the particle-flux task, a significant number of students arrived at the correct answer through incorrect reasoning. The particle-flux task also proved to be difficult for graduate students. When a version was asked as part of a basic physics question on a Ph.D. qualifying examination at UW, only 8 of 15 students gave a correct answer and only 7 gave a complete explanation. B. Rebounding-particle task Results from the change-in-momentum task suggested difficulty with basic concepts in mechanics that are necessary for developing a microscopic model of pressure in a gas. This inference was supported by informal interactions with students in the thermal physics course and by our earlier interviews. To separate difficulties with mechanics from others specific to the microscopic model of gases, we posed a simple problem in which a moving ball is normally incident on an immovable wall. The students were told that the ball’s initial and final speeds are equal. They were asked to find its change in momentum, ⌬pជ , during the collision in terms of the mass, m, and initial velocity, vជ i, of the ball. The answer, −2mvជ i, could be found algebraically or by drawing initial and final momentum vectors. We gave the problem to 200 students in the mechanics portion of the UW calculus-based course within weeks of tutorial instruction on the use of vectors in kinematics. About 35% gave a correct response. An additional 20% gave the correct magnitude but the incorrect sign. We also gave this problem to students studying thermal physics in the calculusbased course at another university, where the tutorials on mechanics had not been used. Only 15% 关N ⬃ 60兴 gave a correct response. In the UW thermal physics course 关N ⬃ 105兴, the results were similar to those for the UW calculus-based course. A detailed examination of these results revealed that the thermal physics students who had worked through the tutorials when they had taken introductory mechanics at UW gave correct answers at about twice the rate 共45%兲 as those who had taken this course elsewhere 共20%兲. C. Two-tanks task During the interviews, many students implied that the size or mass of the molecules made a difference in the number contained in the balloons. The two-tanks task was designed to determine the prevalence of such ideas. Students were asked to compare the number of molecules in two rigid gasfilled containers of equal size and shape at the same temperature and pressure. One contains oxygen; the other, hydrogen 共see Fig. 1兲. 共Sometimes, two other gases were used.兲 Students were told to treat the gases as ideal. Because both samples have the same pressure, volume, and temperature, the number of moles 共and molecules兲 must be equal. This task was given after standard instruction on the ideal gas law in several courses.9 Only about half of the introductory students 关N ⬎ 500兴 realized that the number of molecules would be equal. In the thermal physics course, about 75% 关N ⬃ 35兴 gave correct answers. 1065 Am. J. Phys., Vol. 73, No. 11, November 2005 Kautz et al. 1065 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 68.232.23.162 On: Fri, 12 Feb 2016 17:14:09 Fig. 1. The two-tanks task. Students were shown two identical rigid containers, one containing oxygen, the other hydrogen. They were told that the temperatures and pressures are the same in both and asked to compare the number of molecules in the two tanks. IV. ANALYSIS OF INCORRECT RESPONSES Student performance on the problems described above suggested that the development of a correct microscopic model for an ideal gas is not a typical outcome of standard introductory and sophomore courses. The analysis of student responses revealed serious errors in the interpretation at a microscopic level of the variables in the ideal gas law. Only the most common difficulties are discussed. In this section, pressure, temperature, and n 共number of moles兲 are used as headings to organize specific difficulties. The categories are not mutually exclusive. For example, some difficulties with pressure and temperature are due to difficulties with the average kinetic energy of particles in an ideal gas. A. Analysis of incorrect responses related to pressure Student responses to tasks designed to probe their ideas about pressure indicated several specific difficulties. Below are some examples. Using incorrect (or incomplete) microscopic models of pressure. On the isobaric-expansion problem, about 25% of the introductory students who correctly predicted an increase in the average kinetic energy of the gas particles went on to state that the average change in momentum would be the same before and after the expansion. 共A similar fraction of those who gave incorrect answers about the average kinetic energy answered in this way.兲 Almost all of these students supported their answers by referring to the constant pressure. By associating the pressure directly with the average momentum transfer, they failed to consider the incident flux as a factor that affects pressure. The following response illustrates this difficulty: “The momentum transfer for each collision remains the same. Because the number of molecules, as well as the pressure, does not change, neither should the momentum transfer of each collision.” These students recognized that the gas pressure is related to the change in momentum of the particles and to the number of particles in the sample, but lacked a detailed understanding of that relationship. Similar answers were given by about 10% of the thermal physics students. About 25% of the students in the introductory course and about 40% in the thermal physics course answered that the particle flux would be the same before and after the isobaric expansion. More than half of these students supported their answer by referring to the constant pressure. They apparently did not recognize that the incident particle flux alone does not determine the pressure. One student wrote that “关Particle flux兴 remains the same, because a constant pressure means the number of molecules colliding with the walls per unit area remains the same.” Both types of answers suggest that at least half of the students in the introductory courses had not been successful in building a qualitative microscopic model of gas pressure. Other students seemed to have such basic difficulties with momentum and change in momentum that they could not have developed a correct model. Failing to treat change in momentum as a vector quantity. As the following response suggests, some students did not treat the momentum of a particle as a vector on the changein-momentum task.10 They were unaware that the change in momentum of a particle during a single collision cannot be zero. “关Change in momentum兴 remains the same. Assuming that all collisions are perfectly elastic as in an ideal gas, the change in momentum is zero.” This difficulty was even more prevalent on the rebounding-particle task, which had been designed to probe understanding of momentum in a collision of a ball with a wall. About 30% of the introductory students and 25% in the thermal physics course stated that the change in momentum is zero. Most added the initial and final momenta 共instead of subtracting the former from the latter兲 or subtracted the magnitudes without regard to sign or direction. Some drew vectors representing the initial and final momenta, but still concluded that the change in momentum would be zero. A further 30% of the introductory students and 20% of the thermal physics students were unable to arrive at an answer in numerical or diagrammatical terms. Instead, they gave a verbal description of the momentum of the ball before and after the collision, often including statements such as: “the momentum does not change. 关It兴 is simply directed in the opposite direction,” or “it has the same momentum, just the other way.” They seemed unaware that change in momentum is a well-defined physical quantity that can be expressed quantitatively and can yield a nonzero value. Misapplying conservation of momentum. About 15% of the students in the introductory course supported their statements that the momentum of the ball or particle did not change by referring to the principle of conservation of momentum: “the momentum does not change 共except in opposite direction兲 because momentum is conserved.” Frequently, students seemed not to distinguish between energy and momentum. For example, a student in the thermal physics course wrote that “momentum will be the same because no energy was lost.” A student in the introductory course stated “mv = −mv, the momentum doesn’t change at all; the collision is completely elastic.” Misinterpreting particle flux. There was difficulty with the interpretation of “the number of particles incident on a wall per time interval per unit area.” Some students who 共correctly兲 stated that the particle flux decreases in the isobaric expansion did so simply because of the increase in volume 共or, frequently, the surface area兲 of the container. They treated the particle flux as a type of density that is independent of the particles’ speeds. Others suggested that there would be a compensation for an increase in particle speed by a greater volume. Similar arguments, in which changes in two quantities are assumed to compensate so that a third remains constant, have also been noted in the context of the macroscopic variables in the ideal gas law as well as other topics, such as single-particle dynamics and buoyancy.2,11 B. Analysis of incorrect responses related to temperature It was apparent in their responses to the problems that students often misinterpreted the concept of temperature at a microscopic level. Two difficulties were particularly common. 1066 Am. J. Phys., Vol. 73, No. 11, November 2005 Kautz et al. 1066 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 68.232.23.162 On: Fri, 12 Feb 2016 17:14:09 Mistakenly assuming that lower (greater) particle density implies lower (greater) temperature. On the kinetic-energy task, about 20% of the introductory students stated incorrectly that the average kinetic energy of the gas molecules decreases in an expansion at constant pressure. 共For a compression, the corresponding incorrect answer was given by slightly more than 25% of the students.兲 In explaining why the average kinetic energy is lower at the greater volume, students referred to the space allotted to each gas particle 共that is, the reciprocal of the particle density兲. The following response was typical: “The average kinetic energy per molecule will decrease as the volume gets larger. The molecules will have more freedom of motion, therefore they will move less.” When students elaborated on this perceived relationship between particle density and temperature, they often referred to collisions between the gas molecules. In interview situations, they often tried to describe their ideas in greater detail. Sometimes they made analogies to human behavior 共for example, reactions of people who are crowded into a small room兲. For example, in discussing a compression, one student explained why a greater density leads to greater temperature: “Because you’re all kind of cramped in there…, you want to move…, because all that energy is pent-up so to speak…. They’re doing the same thing, they want to get out, and so it’s getting hotter.” As mentioned in Ref. 2, many students seemed to have trouble reasoning with multi-variable equations in addition to their difficulties that were specific to an ideal gas. The following example shows that it is often difficult to separate the two: “The average kinetic energy decreases because PV = nRT. As V increases, T would have to decrease. The amount of ideal gas remained the same, as did the pressure. The molecules lose kinetic energy.” On the basis of similar responses in student interviews, it appears likely that this student had a preconceived notion that the average kinetic energy decreases during the expansion and then used flawed reasoning with the ideal gas law to support this belief. Mistakenly assuming that molecular collisions generate kinetic energy. Some students imagined a mechanism by which the frequency of collisions between gas particles due to their greater 共or lesser兲 number density results in an increase 共or decrease兲 in temperature and suggested that the collisions would produce or release heat. This argument may lead to a correct answer for an adiabatic process but not for an isobaric one.4 “There is more room for the molecules to move around in and hence they won’t collide as often. Therefore the average kinetic energy will be lower.” “Average kinetic energy increases since gas enclosed by smaller volume, so air molecules more likely to come … in contact with each other …. This results in an increase in temperature and an increase in average kinetic energy.” The difference in performance on the compression and expansion versions of the kinetic energy task is consistent with the belief that collisions between particles can account for an increase in kinetic energy. This belief is elicited more frequently in the case of compression than expansion. Students who think that interactions between particles can change the temperature of a gas do not realize that this mechanism does not allow for a steady state in which the temperature does not change. The change in internal energy is attributed to a process inside the system rather than an interaction with its surroundings.12 The role of work and heat in transferring energy between a system and its surroundings is not recognized. C. Analysis of incorrect responses related to the number of moles During interviews and on written problems that involved pressure and temperature at a macroscopic level, students often made irrelevant assumptions about gas particles or referred to incomplete models of pressure. These ideas often misled them to conclude that equal volumes of different gases under the same conditions contain different numbers of particles. Not recognizing the substance independence of the ideal gas law. On the two-tanks task and similar questions, only about half of the introductory students gave answers consistent with the substance independence of the ideal gas law 共Avogadro’s law兲. Most of the others stated that there would be a greater number of particles of the gas with the smaller molar mass. Mistakenly assuming that a greater number of smaller molecules is needed to fill a given volume. Students frequently treated an ideal gas as if each of the particles in it had a non-negligible volume and these were closely spaced. The belief that a larger number of smaller molecules is needed to fill a given volume is illustrated in the following quote: “The two volumes … are the same. The molecule that takes up less space would be present in greater amounts than the other.” None of the students who gave this type of explanation referred explicitly to the distance between particles, or to whether anything occupies the space between them. Research on children’s understanding of the particulate model of matter has revealed that they often resist the idea of a vacuum between particles.13 University students may have the same reaction. Another common claim was that the mass of the molecules is an indication of how many are needed to “fill a given volume.” It is possible that these students may have failed to distinguish clearly between mass and volume.14 Alternatively, they may have inferred from the greater molar or molecular mass that the size of a single particle must be greater. Mistakenly assuming that a greater number of lighter molecules is needed to produce a given pressure. More than one-third of the students who gave incorrect answers to the two-tanks problem or similar tasks focused on the dependence of the pressure on the mass of the molecules. They apparently thought that it takes a greater number of the lighter molecules to generate a given pressure 共or exert a given force兲 at a given volume and temperature. They did not seem to realize that the lighter molecules are moving faster 共resulting in the same average kinetic energy per particle兲. The following comment is from the calculus-based course. “The number of hydrogen molecules is greater than that of the nitrogen molecules. The hydrogen molecules have less mass and thus exert less force than the nitrogen, and there must 关be兴 more molecules to equal the force of the nitrogen.” On a related task, another student in the same course used a similar argument when comparing the pressures of two gas samples 共one of helium, the other of argon兲. He explicitly assumed the root-mean-square velocities of the two gases to be the same at equal temperatures and decided that the gas 1067 Am. J. Phys., Vol. 73, No. 11, November 2005 Kautz et al. 1067 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 68.232.23.162 On: Fri, 12 Feb 2016 17:14:09 with a greater molar mass must have a greater pressure: “P = 1 / 3 ␳m具v2典. v ⬃ T, same temperature. 关Thus兴 P ⬃ ␳m . ␳m共Ar兲 ⬎ ␳m共He兲 共because greater molar mass兲. 关Thus兴 PA ⬎ PB.” A few students made the opposite argument, stating that the smaller mass of the hydrogen particles leads to a greater average speed or kinetic energy and, hence, to a larger pressure. The way in which students understand temperature or pressure in microscopic terms can affect their understanding of the other concept. Misinterpreting the quantities n, M, NA, and R. Many students confused the number of moles in a given sample, n, with the molar mass of a type of gas, M. Also common were incorrect interpretations of the quantities NA, Avogadro’s number, and R, the universal gas constant.15 Usually the misinterpretations were not explicit, but were implied by the use of algebraic expressions. Some students seemed to think that n must have different values for different gases. For example, they frequently set n equal to 32 for oxygen and to 2 for hydrogen in the ideal gas law, or used those values for NA when converting the number of moles to the number of molecules for the respective gases. Because misinterpretations of this type 共as well as some algebraic errors兲 almost inevitably led to the same answer—“there are more hydrogen molecules”—we believe that many of these students had an intuitive feeling that there should be fewer molecules of the heavier species and that the algebraic manipulation was merely used to justify this preconceived response. V. RESEARCH-BASED INSTRUCTION TO IMPROVE STUDENT UNDERSTANDING Our group draws on findings from research to develop tutorials to help students overcome important conceptual and reasoning difficulties through guided inquiry.3,16 Although primarily designed for small-group settings, the tutorials on thermal physics have been used at UW as interactive tutorial lectures in which there are many students and one instructor.17 Each tutorial sequence consists of a pretest, worksheet, homework, and post-test. During the lectures, students collaborate with their neighbors on tutorial worksheets. These consist of carefully sequenced questions that guide students through the reasoning needed to develop a sound conceptual understanding. At intervals ranging from 5 to 20 min, the instructor initiates a discussion based on the questions. The research described in this paper underlies part of a published tutorial on the ideal gas law and an unpublished tutorial worksheet on microscopic processes.18 In the first three parts of the ideal gas law tutorial 共described in Ref. 2兲, students are guided to construct operational definitions for the macroscopic variables, examine their relationship to one another, and interpret PV diagrams.2 The fourth part of the tutorial is on the substance independence of the ideal gas law and is summarized below. We then describe a version of a tutorial worksheet that treats temperature from a microscopic perspective. ditions 共identical movable pistons兲 and recognizing that these imply equal pressures, the students compare the number of moles. As a safeguard against unnoticed misunderstandings, they are asked to check their answers for consistency with the ideal gas law. They compare the masses of the hydrogen and oxygen samples and realize that equal numbers of moles of different gases can have the same pressure, volume, and temperature, in spite of different masses. They next consider a fictional dialogue in which a student suggests that there would be more hydrogen molecules because of their smaller size. Another student disagrees, arguing that there would be more oxygen molecules because n is greater for oxygen than for hydrogen, thus making the common error of confusing molar mass with n. The students are asked to identify the flaws in both statements. B. Temperature in the microscopic model of an ideal gas In the unpublished tutorial worksheet on the microscopic model, students considered a collision between two gas particles that move at different speeds. They were asked to compare the initial kinetic energies. From information given about one particle after the collision, the students could infer that the speed of the other particle has decreased. They are then led to generalize that collisions between gas particles have no effect on the average kinetic energy and, thus, on the temperature of the gas. To reinforce this idea, there is a fictional dialogue between two students about an experiment in which a gas is moved from a larger container to one that is smaller. One student mistakenly claims that the temperature of the gas must have increased due to the increased frequency of collisions between particles in the smaller volume. This statement is contradicted by the other student, who says that the temperature cannot be inferred from the given data without knowing how the pressure may have changed. The students are asked with which fictional student, if either, they agree. The dialogue also helps address difficulties with the interdependence of the variables.2 VI. PROBLEMS DESIGNED TO ASSESS STUDENT LEARNING After working through the tutorial and the tutorial worksheet, students in the algebra-based course were given a variety of post-tests on course examinations. We sometimes were able to give the same problems to students who had only standard instruction, thus enabling a direct comparison. Other tasks that were used to probe student understanding provided additional benchmarks for assessing the effect of research-based instruction. In general, we try to ensure that post-tests are likely to elicit the types of incorrect reasoning that the instructional materials are intended to address. The goal is to obtain a realistic measure of the effectiveness of the materials and guidance in revising them if necessary. The first two examples that follow relate to the substance independence of the ideal gas law. The other two focus on microscopic processes in an ideal gas. A. Substance independence of the ideal gas law The students apply the ideal gas law to two different gases 共hydrogen and oxygen兲 with the same volume and temperature sealed in identical vertical cylinders. Values for the molar masses are provided. After considering the external con- A. Flexible-container task Students were shown a flexible container with an unknown, but fixed, number of hydrogen molecules. A similar container 共not shown兲 is described as having an equal num- 1068 Am. J. Phys., Vol. 73, No. 11, November 2005 Kautz et al. 1068 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 68.232.23.162 On: Fri, 12 Feb 2016 17:14:09 Fig. 2. The particle-speed task. Students were shown a PV diagram and asked to compare the temperatures and average particle speeds for the states labeled A and B. ber of nitrogen molecules. Both gas samples are at the same pressure and temperature and are to be considered ideal. Students are asked whether the hydrogen container is bigger than, smaller than, or the same size as the nitrogen container. The problem was given on a mid-term examination. About 85% of the students recognized that the containers would be the same size 关N ⬃ 125兴. In a section of the course with only standard instruction, about 55% of the students gave a correct answer, with most of the others saying that the nitrogen container would be larger 关N ⬃ 60兴. B. Two-tanks task (variation) To check whether the success rate represented a robust improvement in student learning, we gave a problem that we considered more difficult on the final examination in the same course.19 The students were shown two identical rigid tanks that contain helium and neon gas, respectively, and were told that the number of atoms in each tank is the same and that both are at the same temperature. The students were first asked to compare the average kinetic energies and speeds of the gas particles in the two tanks and then to compare the two pressures. More than 95% recognized that the pressures are the same. C. Particle-speed task Students were asked to compare the temperatures and average particle speeds for two states of an ideal gas sample represented as points on a PV diagram. A grid allowed them to compare the values of the respective pressures and volumes 共see Fig. 2兲. They could determine how the temperatures compare using the ideal gas law and the values of P and V obtained from the grid. In other situations, we had seen many students manipulate the ideal gas law 共often incorrectly兲 to support an answer based on microscopic considerations. In order to assess student learning, we wanted to design a post-test task for which this type of approach would lead to an incorrect conclusion. Therefore, the particle-speed task involves a process in which the state with the smaller volume has a greater pressure but a lower temperature. Thus the tendency to associate greater particle density and higher pressure with greater temperature would yield an incorrect answer. About 95% of the students 关N ⬃ 125兴 made a correct Fig. 3. The kinetic energy task 共post-test version兲. The students were shown a PV diagram depicting a cyclic process and asked to compare the average kinetic energy per particle for states X and Y. This task was part of an examination problem that included other questions about the cyclic process. comparison of the temperatures. More than 90% gave correct reasoning. About 85% correctly compared the average particle speeds. About 75% gave correct reasoning. D. Kinetic energy task A version of the kinetic energy task described earlier was also given as a post-test. The percentage of correct answers was about the same as with standard instruction 共70%兲 关N ⬃ 245兴. The percentage with correct reasoning was about 50%, compared to about 40% with standard instruction. There were fewer incorrect associations of temperature with particle density. In interpreting these results, it is important to note that the version given after standard instruction was embedded in a longer examination problem dealing with a cyclic process. A PV diagram showing the process had been drawn for the students, and values for the temperatures at the initial and final states had been given 共see Fig. 3兲. In principle, the students could read the temperature off the graph and conclude directly that the average kinetic energy had increased in this process. This way of solving the problem was not possible on the version given to the students who had completed the tutorial worksheet. VII. CONCLUSION The research described in this paper was motivated by results from our investigation of student understanding of the macroscopic variables in the ideal gas law. During our analysis of the results, we noticed that many of the student difficulties with these variables at a macroscopic level are linked to misinterpretations of microscopic processes. Therefore, we decided to study the ideas that students have about microscopic entities and processes in the context of an ideal gas. In this paper, we illustrated how students often justify incorrect answers about the behavior of an ideal gas by an incorrect or incomplete microscopic model, even on tasks that do not require application of such a model. Guided by findings reported in both papers on the ideal gas law, we have developed a set of tutorials and a tutorial laboratory 1069 Am. J. Phys., Vol. 73, No. 11, November 2005 Kautz et al. 1069 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 68.232.23.162 On: Fri, 12 Feb 2016 17:14:09 experiment to help students deepen their understanding of the macroscopic variables and to help them construct a more correct and useful microscopic model. The results from the post-tests described in this paper indicate that research-based instruction can substantially improve student understanding of Avogadro’s law. In particular, when difficulties with the substance-independence of an ideal gas are explicitly addressed, almost all the students seem able to develop a robust understanding that can withstand some powerful distracters in post-test questions. Although not as successful in helping students understand temperature at a microscopic level, there is evidence that the tutorial worksheet helps them separate the concepts of temperature and particle density. We hope that the insights that we have gained from the iterative process of research and research-based curriculum development will be useful to physics instructors and to faculty in other disciplines, for example, chemistry and engineering. Because thermal phenomena are included at all levels of science instruction, we think that there also are strong implications for the development of instructional materials for K-12 students and for the preparation of their teachers. Below are some reflections based on our experience. To be able to apply concepts from mechanics properly to thermal physics, students need a stronger command of that material than most acquire during a typical introductory course. The difference in results on the rebounding-particle task between those who had and had not worked through the mechanics tutorials suggests that a thorough review of the relevant mechanics can promote a better understanding of the behavior of gases. As discussed in Ref. 2, we found that students often needed explicit help in relating the variables in the ideal gas law to the real world, for example, pressure and temperature to the devices that measure and/or control them. Many students did not seem to understand the role of idealization in the construction and interpretation of microscopic models. They often treated ideal gas particles like macroscopic objects to which a volume and temperature could be ascribed. Sometimes they referred to collisions between particles in trying to account for changes in temperature during expansion or compression. These concerns, which are supported by our findings in other investigations, have led us to question the advisability of emphasizing microscopic models early in the study of thermal physics before students have become sufficiently familiar with the macroscopic phenomena that the models have been developed to explain.20 It is vitally important to ensure that the ideas that students internalize at the microscopic level are correct and sufficiently complete. Without such intervention, it is very likely that the models that they develop will have a negative effect, not only on their understanding of microscopic processes, but also of related macroscopic phenomena. ACKNOWLEDGMENTS The investigation described in this paper has been a collaborative effort by many members of the Physics Education Group. Substantive contributions were made by Michael E. Loverude. Mark N. McDermott made valuable suggestions that contributed to the development of the tutorial worksheet on the microscopic model. The authors deeply appreciate the cooperation of faculty whose classes participated in the study at UW, the University of Cincinnati, the University of Mary- land, and Syracuse University. They also gratefully acknowledge support from the National Science Foundation by the Division of Undergraduate Education and the Physics Division. a兲 Present address: Hamburg University of Technology, 21071 Hamburg, Germany. In most of the courses in this study, the ideal gas law is expressed as PV = nRT, where n is the number of moles. In a few cases, the law was expressed as PV = NkT, where N is the number of molecules. 2 C. H. Kautz, P. R. L. Heron, M. E. Loverude, and L. C. McDermott, “Student understanding of the ideal gas law, Part I: A macroscopic perspective,” Am. J. Phys. 73, 1055–1063 共2005兲. 3 L. C. McDermott and P. S. Shaffer, and the Physics Education Group at the University of Washington, Tutorials in Introductory Physics 共Prentice Hall, Upper Saddle River, NJ, 2002兲. 4 See M. E. Loverude, C. H. Kautz, and P. R. L. Heron, “Student understanding of the first law of thermodynamics: Relating work to the adiabatic compression of an ideal gas,” Am. J. Phys. 70, 137–148 共2002兲. 5 For a succinct description of this research method, see L. C. McDermott, “Millikan Lecture 1990: What we teach and what is learned—Closing the gap,” Am. J. Phys. 59, 301–315 共1991兲. This article includes references to papers that give specific examples. 6 Some of the course instructors state this qualitative model explicitly before or after going through the formal mathematical derivation of pressure in the microscopic model. 7 In some cases, the term incident particle flux or the symbol ⌽ were explicitly used; in other cases, only a verbal description of the quantity was given. The results did not seem to differ. 8 Although inter-molecular collisions play no role in the derivation of the ideal gas equations of state, collisions between molecules are referred to in the postulates presented in many introductory textbooks, including those used in courses in which the tutorials described in this article were tested. 共These texts stipulate that the collisions are elastic and of negligible duration.兲 Moreover, students clearly believe that such collisions are important. Therefore, we think it is appropriate to deal with them in the tutorial. 9 In one section of the algebra-based course, the problem was given before the relevant lectures. In an informal poll, almost all students indicated that they had previously studied the ideal gas law, usually in a college chemistry course. The results did not differ much from those obtained in other sections after standard instruction. 10 The tendency to treat vectors as scalars in finding the difference of two vectors is also discussed in the context of collisions in P. S. Shaffer and L. C. McDermott, “A research-based approach to improving student understanding of kinematical concepts,” Am. J. Phys. 73, 921–931 共2005兲. 11 For other examples of compensation reasoning, see R. A. Lawson and L. C. McDermott, “Student understanding of the work-energy and impulsemomentum theorems,” Am. J. Phys. 55, 811–817 共1987兲; T. O’Brien Pride, S. Vokos, and L. C. McDermott, “The challenge of matching learning assessments to teaching goals: An example from the work-energy and impulse-momentum theorems,” ibid. 66, 147–156 共1998兲; M. E. Loverude, C. H. Kautz, and P. R. L. Heron, “Helping students develop an understanding of Archimedes’ principle, Part I: Research on student understanding,” ibid. 71, 1178–1187 共2003兲. 12 A situation in which the temperature of a gas is changed as a result of processes in the interior of the gas is a chemical reaction between different substances. In that case, one form of internal energy 共that is, chemical兲 is changed to another 共that is, thermal兲. Students may fail to distinguish between the two cases and interpret an adiabatic compression as a process similar to a chemical reaction. 13 S. Novick and J. Nussbaum, “Junior high school pupils’ understanding of the particulate nature of matter: An interview study,” Sci. Educ. 62, 273–281 共1978兲; S. Novick and J. Nussbaum, “Pupils’ understanding of the particulate nature of matter: A cross-age study,” ibid. 65, 187–196 共1981兲. 14 Difficulties with mass and volume are frequently seen at the pre-college level. At the college level, such difficulties still occur. See, for example, the last article in Ref. 11. 15 In a few instances, even the subscripts describing the stoichiometric composition of a given compound 共as in H2O兲 were confused with the molar mass or the number of moles. 1 1070 Am. J. Phys., Vol. 73, No. 11, November 2005 Kautz et al. 1070 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 68.232.23.162 On: Fri, 12 Feb 2016 17:14:09 16 For a brief discussion of this instructional approach as implemented by the Physics Education Group, and a description of the tutorials and their implementation, see L. C. McDermott, Oersted Medal Lecture 2001: “Physics education research—The key to student learning,” Am. J. Phys. 69, 1127–1137 共2001兲. 17 For a description of interactive tutorial lectures, see P. R. L. Heron, M. E. Loverude, P. S. Shaffer, and L. C. McDermott, “Helping students develop an understanding of Archimedes’ Principle, Part II: Development of research-based instructional materials,” Am. J. Phys. 71, 1188–1195 共2003兲. 18 Reference 3, pp. 227–230. 19 We have found that students who incorrectly answer questions based on the ideal gas law are often misled by incorrect or incomplete microscopic models. Since the second post-test question starts from a microscopic perspective, we regard it as more difficult than the first. 20 Some physicists take a different instructional approach. They argue that introducing a microscopic model makes it easier for students to think about both macroscopic and microscopic phenomena. See, for example, R. W. Chabay and B. A. Sherwood, “Bringing atoms into first-year physics,” Am. J. Phys. 67, 1045–1050 共2001兲; F. Reif, “Thermal physics in the introductory physics course: Why and how to teach it from a unified atomic perspective,” ibid. 67, 1051–1062 共2001兲. 1071 Am. J. Phys., Vol. 73, No. 11, November 2005 Kautz et al. 1071 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 68.232.23.162 On: Fri, 12 Feb 2016 17:14:09 
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