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Research: Science and Education edited by
Christopher F. Bauer University of New Hampshire Durham NH 03824-3598
Student Interpretations of Equations Related to the First Law of Thermodynamics Linda C. Hadfield* JILA, University of Colorado at Boulder, Boulder, Colorado 80309 *[email protected] Carl E. Wieman JILA, Department of Physics, University of Colorado at Boulder, Boulder, Colorado 80309 and Department of Physics, University of British Columbia, Vancouver, BC, V6T1Z3 Canada
The ability to understand the physical meaning of equations used in science and to manipulate those equations in conventional ways is an important scientific skill. Students in physics classes have been observed to interpret and use mathematical representations of physical ideas in a variety of ways (1). Qualitative diagrams of problems have been shown to help physics students logically construct mathematical representations (2). Employing mathematical equations to represent physical meaning is an integral aspect of thermodynamics classes at the undergraduate level and contributes to student difficulties with the course. Many researchers have studied student difficulties with conceptual aspects of thermodynamics (3, 4), including the relationship between students' conceptual ideas and graphical representations on pressure-volume plots (5). When reasoning about properties of gases such as pressure and volume that depend on multiple variables, students have been observed to treat those properties as if they depended on only one variable (6). Student interpretation or assignment of meaning to equations representing thermodynamic quantities has received less attention. In this study, we explore how undergraduate chemistry students completing their first semester of physical chemistry, a course devoted to thermodynamics and kinetics, interpret equations associated with the first law of thermodynamics. We investigated student ability to interpret the equations for the first law of thermodynamics, the mathematical definition of gas expansion work, and the mathematical definition of heat. These equations represent some of the most fundamental science concepts in thermodynamics, and in this course, many homework and exam questions involved manipulating these equations. The first law of thermodynamics is an important topic in many science disciplines including chemistry, physics, and engineering. The difficulties we observed among physical chemistry students are likely shared by students studying thermodynamics in other disciplines. Methods Chemical Thermodynamics and Kinetics (also known as firstsemester physical chemistry) is an upper-division undergraduate course taught in the chemistry department. Student interpretations of fundamental equations used in this course were investigated through a multiple-choice survey, a written-response survey, and interviews. Students in the fall 2005 and spring 750
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2006 semesters were given the two surveys in the final week of the course. The first survey contained multiple-choice questions, including true-false questions, that were based on course material and required 30-45 min to complete. Immediately after the multiple-choice survey, or in the following class period, students were asked to write a brief explanation of their answers to each of the multiple-choice questions. All students in the course took these surveys except students who were absent on the relevant days. Altogether, 55 students, or 83% of those enrolled in the fall 2005 and spring 2006 courses, completed both surveys. All the students involved consented to having their responses included in this study. The physical chemistry course was taught by different instructors in the two semesters, but the thermodynamics content of both courses was the same and homework and exam questions were similar. In both semesters, the first topic covered was the kinetic theory of gases. In the fall course, the second topic was chemical kinetics and then thermodynamics. In the spring course, thermodynamics was the second topic, taught immediately after the kinetic theory of gases. Multiple-choice questions were asked with “clickers” in every class period of the fall 2005 course, whereas students in the spring 2006 course were asked openresponse questions in class once a week and wrote out their answers. The research presented in this study focuses on three truefalse questions on the multiple-choice survey: The first law of thermodynamics may be stated as: energy can be converted from one form to another, but cannot be created or destroyed. For the following three mathematical statements, either choose true if the expression is a restatement of the first law or choose false if the expression does not restate the first law. R 1 w = - PdV, work is the integral of the pressure multiplied by theRchange in volume. 2 q= CvdT, heat is the integral of the heat capacity multiplied by the change in temperature. 3 ΔU = q þ w, a change in internal energy for a system can be the result of changes in heat and work.
Calculations involving these equations played a central role in the Chemical Thermodynamics and Kinetics course. In addition, these equations are often covered in the first-year general chemistry course that most of these students would have taken 2 to 3 years before this study.
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Research: Science and Education
Before the survey was given to the students, the validity of the questions was investigated. Five professors in the chemistry department reviewed these questions. All five professors answered false for the first two statements and true for the third statement, as expected by the authors. The professors thought these questions would be easy for their students to answer. In addition, 10 students who had completed the class in previous semesters were interviewed about these questions to confirm that students understood what the questions asked them to do. The survey was further validated by reviewing all students' written responses on the survey to determine if their responses reflected their understanding of the mathematical equations. Eight students wrote comments that indicated they had not thought about the meaning of the mathematical equations. Typical examples of these comments on the first question follow: • “Memorized eqn; Didn't see relation to 1st law” (Student chose true.) • “For this and the next question as well, I thought that since this is a part of the first law, it should be true.” (Student chose true.) • “Guessed” (Student chose true.)
These students seemed to choose true or false based on the accuracy of the mathematical expressions or their recognition of the equations instead of their understanding of the meaning of the equations or their understanding of the true-false questions. These students did not appear to interpret the physical ideas represented in the equations. Therefore, the responses of these eight students are presented separately. This left 47 students, or 71% of the class, who completed both forms of the survey and appeared to answer these questions based on their interpretation of the equations. Only one student indicated on the long-answerresponse form of the survey that she would change the answer she gave on the multiple-choice version. Ten students were interviewed in depth on the three questions, between 1 and 8 months after they completed the course. Two of the interviewed students had completed the written surveys in class but the other eight took the class in a semester when the surveys were not given and therefore they had not seen the questions prior to the interview. Eight of the interviewed students had the same instructor, who was also the instructor of the spring 2006 course. In the interviews, students were asked to talk about their thinking as they answered the survey questions. The interviewer asked many follow-up questions to clarify student thinking. This work was carried out with IRB approval and all of the students gave written informed consent. In selecting students for interviews, we asked students with a range of grades who had successfully completed the course to be interviewed about the course. One of the interviewed students did not complete the course in the chemistry department, but completed a similar thermodynamics course in the engineering department. We interviewed all the students who agreed to participate and paid them for their time. However, we did not have access to specific student grades. As we are unable to characterize all of the students by their final grades, we describe their experiences and approach to problems. Survey Data Results Of the 47 surveyed students who appeared to answer these questions based on their interpretation of the mathematical
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Table 1. Student Responses on the Multiple-choice Survey Correct Number of Students Number of Students Equation Answer Answering True Answering False R 1. w = - PdV False 23 (49%) 24 (51%) R False 15 (32%) 32 (68%) 2. q = CvdT 3. ΔU = q þ w True
44 (94%)
3 (6%)
Table 2. Surveyed Students' Combined Responses to Questions 1 and 2 Number of Students Answering Question 2 Number of Students Answering Question 1
True
True
13 (28%)
False
3 (6%)
False 9 (19%) 22 (47%)
equations, nearly half chose true for the equation for work and a third chose true for the equation for heat on the multiple-choice survey (Table 1). These are incorrect responses. None of the surveyed professors chose these responses. Although 94% of the students in our sample answered the third question correctly, our interviews suggest that many students also held inaccurate interpretations of this equation. The 8 students who were considered separately, as discussed in the Methods section, had a low success rate in answering the first two questions. All 8 answered the first question incorrectly and half answered the second question incorrectly. All 8 answered the third question correctly. For the 47 students who appeared to answer based on their interpretation of the equations, we reviewed students' combined responses to questions 1 and 2 (Table 2). Every combination of responses to questions 1 and 2 was represented among the 47 students in our sample. However, even when combined with interview data, we found nothing conclusive in these student response patterns. In all cases, interviewed students' responses to questions 1 and 2 were consistent with their interpretation of the equations for work and heat. We do not present a combinatorial analysis involving question 3 because the overwhelming majority of students answered this question correctly. Only three survey students answered question 3 incorrectly. However, these three students did not share a common response pattern to questions 1 and 2. Therefore, student responses to question 3 did not appear correlated with responses to question 1 or 2. In addition, we can offer no insight about why students incorrectly chose false for question 3. The surveyed students who chose false did not provide explicit reasoning and none of the interview subjects chose false for question 3. A deeper understanding of student thinking about these equations was obtained from the 10 student interviews. The interviewed students were randomly selected from a group who had received a range of grades in the class. Also, the interviewed students' responses to the three true-false questions (Table 3) were not significantly different from students who completed surveys in class. (This is based on the t test, performed for these three questions. For work, t = 1.21, for heat t = 1.08, and for the first law t=0.811. In all cases, df=55, and so the differences were not statistically significant.) On the basis of their survey responses and their random selection, we believe the interview
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Research: Science and Education Table 3. Interviewed Students' Responses to the True-False Questions Correct Number of Students Number of Students Equation Answer Answering True Answering False R 1. w = - PdV False 7 (70%) 3 (30%) R False 5 (50%) 5 (50%) 2. q = CvdT 3. ΔU = q þ w True
10 (100%)
0 (0%)
subjects well represented the opinions of all students who completed the class. Interview Data Results Several themes emerged from the interviews. Students who answered that the equations defining work and heat represented the first law of thermodynamics believed that these equations represented conversion and conservation of energy. In addition, even though all interviewed students answered the third true-false question correctly, none gave a correct explanation for how the equation ΔU = q þ w represents conversion and conservation of energy. We will review student responses to each question individually. First, we will review student comments about the equation for work, then the equation for heat, and finally the equation for the first law of thermodynamics. Throughout, “S” represents a student and “I” represents the interviewer. The Equation for Work R The equation for work, w = - PdV, indicates how to calculate the quantity of work associated with specific changes in pressure and volume. Students who provided this explanation were thought to have a good interpretation of the work equation. In contrast, all seven students who incorrectly chose “true” for question 1 interpreted the mathematical definition of work to represent energy conversion and conservation. There was a diversity of thought about how energy conversion and conservation were represented in this equation. One student said that pressure and volume were different forms of energy. He described the different types of energy as follows: S 2: “Work is essentially changing energy into different forms, whether it be, you know, heat energy being converted to mechanical energy or pressure energy being converted to a volume. So, since this (referring to the equation for work) is a pressure energy being converted into a volume, I'm going to say ... true ...” I: “... So where that energy is going from and going to, I guess is the question.” S 2: “Energy in the form of pressure ... that's allowed to translate from pressure energy into volume, so it causes the molecules to expand and form the volume that they want to form.”
This student erroneously thought pressure and volume were different forms of energy and that the work equation showed energy being converted between these different forms. This student also described how the equation for work represented energy conservation. I: “... What if we think also about the conservation of energy, the idea that energy is not being created.” S 2: “Doesn't go away?” I: “Yeah.” S 2: “Well that goes along with what I said before. We're not
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creating it. We're just changing its form.” I: “Ok” S 2: “So work is just a way of changing forms of energy.”
This student attributed conversion and conservation of energy, ideas of the first law of thermodynamics, to the definition of work. This student began a thermodynamics course in the chemistry department but withdrew without completing the course. He then completed a thermodynamics course in the engineering department that focused on these topics. He used these equations repeatedly to solve homework and exam questions. Another student believed that, in the equation for work, moving from the variables on the right, PdV, to the variable on the left side, w, represented conversion of energy between different forms. After the student briefly recounted his answer, the interviewer asked further questions to explore his thinking. I: “So, in work = -PdV, do you see that as energy being converted from one form to another, or did you just say this looks like something in the first law, so therefore this is the first law?” S 10: “Well, I mean, it is energy being, (pause), well I mean, that is, work is energy. Heat is energy. I: “Ok. But the question's saying, `Does the statement express the first law?' Was that clear, that that's what we were asking?” S 10: “It seems like it is. I mean, because yeah, I mean, work is a form of energy, and converting, is moving from here to there. (The student pointed to one side and then the other side of the equation for the definition of work.) It's still being conserved. It doesn't breach the first law. I guess it's consistent with the first law, but is it actually the first law, no, I guess not. ... Well that's a form of energy, and that's a form of energy. (again referring to the two sides of the equation) They're both in the law, must be all part and parcel. ... I didn't actually make the discrepancy between what was actually the first law and what was implied by the first law. It all seems the same.”
Although the definition of work is meant to show that w R and - PdV are the R same thing, this student erroneously interpreted w and - PdV to represent different forms of energy and he believed the definition for work represented the conversion of energy between these forms. This student asked questions in class and attended problem-solving sessions and exam-review sessions outside of class. In our third example, a student interpreted the equation for work to represent the motion of energy between a system and its surroundings. I: “... Another idea of the first law is the idea of conservation of energy. Do you see that being represented in this equation?” S 9: “Yes, um, the work relates to something, someone, somebody. I guess it's sort of related to conversion too, but putting out energy and some other system taking up that energy, so the energy is just conserved, it's just from one system to another.” I: “Ok, so in the equation it sounds like you started thinking about two systems, or a system and a surrounding maybe interacting.” S 9: “Yeah, right, and this would be the surroundings doing work on the system, and this is the system's energy that they are getting from the surroundings.” (Student referred to the two sides of the equation for work).
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As the first two students did, this student thought the equation for work represented conversion and conservation of energy. Slightly different from the other two, this student believed the work equation represented conversion of energy as motion of energy in or out of a system. This student received an A in the course and tutored less-advanced students. A common student misinterpretation was that the equation for work represented conversion of energy between potential and kinetic forms. Three of the seven students who chose true for question 1 explicitly said the equation for work expressed changes between potential and kinetic energy. For example, S 6: “So does that have to do with energy being converted from one form to another. So work, being converted into pressure is volume. So I would say that that is true, because work can be converted to, if you put work on the system you can change the pressure and the volume, so you would be changing the, I guess, the energy, yeah, because it would go from kinetic energy to potential energy.”
These students erroneously interpreted the equation for work to represent conversion between kinetic and potential energy. These students also seemed to use the words potential or potential energy to refer to internal energy, which may indicate that students have difficulty separating the idea of internal energy from potential energy. In summary, students who responded “true” to question 1 believed the work equation represented conversion and conservation of energy in a variety of ways. It is likely that the 23 students (49% of those surveyed) who answered true for question 1 on the written survey had difficulties similar to the interviewed students who chose that answer. Even the three interviewed students who answered question 1 correctly demonstrated difficulties understanding the physical meaning of the work equation. Each had a unique interpretation. For example, S 3: “I think of the integral sort of like a huge sum of things, of changes here, small changes in this property, the negative sort of sum of all the little changes in pressure times change in volume.” I: “Ok, and so in this question there are sort of two ideas of the first law, one idea being that energy is converted between different forms, another idea of the first law being that energy is conserved, not being created or destroyed. When you look at this equation, do you see it expressing one of those ideas or both of those ideas or neither of those ideas? S 3: “... It's the sum of all the little changes, so it seems like conservation is being stated there.” This student summarized his thinking by saying: S 3: “... I mean this is kind of weird, I can't really bring this math out into words, so if the change in internal energy is just heat, then mathematically you have to add the work to account for the total amount of energy then it seems like, that statement, the change, would apply to this, maybe if you add work then you're adding energy to the system and it could be created, you know. You'd be creating energy, which is illegal under the first law or something like that.”
This student had difficulty expressing his ideas about the meaning of the work equation, and was confused about how energy due to work should be considered and mathematically represented. This student thought about each question for a long time and initially tried to derive the equation for work
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from the ideal gas law. Another student expounded on her own answer: S 8: “I'm going to say that, without doing any math or anything, that this (the work equation) really isn't a first statement, or a restatement of the first law, because, um, well first, first comes to mind is that it's not even dealing with energy in like a flat out sense. ... So I guess I'm really not seeing the relationship between work and energy in that sense. ...” I: “So do you think of work as energy?” S 8: “Well eventually, but like first, right away, not right away, because I mean you do have to have energy to do work, but just purely for, again, just what the equation's, just how it's simply written out, no, I wouldn't say that until I had some other variable to relate that to energy. So right away I wouldn't.”
This student seemed to understand that work is a form of energy, but at the same time considered work to be distinct or removed from energy. In general, this student seemed confident, was eager to answer questions, and quickly wrote down equations and manipulated them as part of her answers. She described her approach by saying, “I really think of things in a mathematical sense, that's because it makes sense to me.” The third student who answered question 1 correctly then explained his answer as follows: S 1: “Individually each of these are like mathematical equations of the first law, but separately they cannot be a restatement of the first law. They cannot, both the heat and the pressure.”
This student spoke at length about his thinking. However, he was unable to discuss these equations in a manner that we understood. Before the physical chemistry course, this student had completed a competitive honors general chemistry course. He frequently wrote down and manipulated mathematical equations during the interview. On the basis of comments from the interviewed students who answered question 1 correctly, we believe that many of the students who responded correctly to question 1 on the written survey would similarly have difficulty interpreting the physical meaning of the work equation. The Equation for Heat R The equation for heat, q = CvdT, indicates how to calculate the quantity of thermal energy associated with specific changes in temperature for a substance having the heat capacity, Cv. Students who provided this explanation were thought to have a good interpretation of the heat equation. For question 2, five of the R 10 interviewed students incorrectly chose true, that is, q = CvdT represents the first law of thermodynamics. These students expressed a variety of ideas about how energy conservation and conversion were represented in the equation. Two students expressed the erroneous idea that the equation represented the transfer of thermal energy between a system and its surroundings. An example is S 2: “... So in this one we're going to be taking, thermal energy and converting it into, I guess thermal energy also ...Taking thermal energy from outside and converting it into thermal energy inside...”
This student also mistakenly referred to the transfer of thermal energy as energy conversion. Another student who chose
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true for question 2 reflected on energy conservation in all three equations: S 7: “... I see all of these as an expression of the conservation of energy. You wouldn't be able to make these equations if not, if you weren't conserving energy in some sense, you know, because you're conserving the energy produced by heat and by work and turning into a measure of internal energy. ... So I think all these equations are only, they only happen or only can be stated because of the first law through conservation of energy.”
This student erroneously believed that the work and heat equations express energy conservation and are a result of the first law. The fourth student tried to remember the physical meaning of variables in these equations but did not get far beyond that. For example, when asked to explain the equation for heat this student said, S 4: “Oh, I really don't know. ... when I think of Cv its like heat capacity, and um, but this is a specific, we have Cv and we have Cp as well, and there were equations relating these, they all, I don't, it's equal to R or something. Manipulating them is equal to R in some way. (quietly reads question) So heat capacity is basically when a certain, um, thing, has, it's like a constant basically. (sighs)”
This student, who received a C in the class, seemed unable to provide an interpretation of the heat equation. Considering these interviews of students who answered question 2 incorrectly, it is likely that the 15 students (32%) who incorrectly answered question 2 on the written survey had similar difficulties interpreting the heat equation. Even among the five interviewed students who chose the correct response to question 2, two students did not have a good understanding of the physical meaning represented in the heat equation. One of these students described the equation for heat as S 3: “Um, a false statement, mathematically. Or, it could be true in some scenarios, but it's not always true. Like, if work is zero, I guess, the change in internal energy could be the enthalpy of this.”
This student erroneously suspected the heat equation was true only under special circumstances. As this interviewed student did, it is likely that some of the students who correctly answered question 2 on the written survey were unable to interpret the heat equation accurately. The First Law Equation The first law equation, ΔU = q þ w, mathematically represents both conversion and conservation of energy. This equation represents energy conversion as both heat and work, two different forms of energy, are written as contributions to internal energy. The change in internal energy is exactly equal to heat plus work for a system, as symbolized by the equal sign (instead of g or e). This indicates conservation of energy, the idea that energy is converted between different forms of energy without being created or destroyed. Even though all 10 interviewed students answered question 3 correctly, most were not able to accurately relate the ideas of energy conservation and conversion to the symbols in the equation for the first law of thermodynamics. In interviews, students were asked how the first law equation represented energy conversion and conservation. Four students justified their answer 754
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by referring to the scenario in which ΔU = 0 and q = -w: two of the interviewed students wrote ΔU=0 and q=-w and the other two described this result in words. This was the most common response. However, this reasoning is not consistent with standard use of these symbols. Two examples follow: S 8: “...basically what I wrote was delta U equals zero, q = w, and I think, what to me that's saying that there is no, um, there is nothing created in the universe, therefore q=delta, or q = -w, and that's saying that they can be converted to one another but it's not created overall. So delta U is not changing. So I'm going to put true for that one.” ... I: “Ok. So was there anything else that you thought about in looking at this question?” S 8: “I guess, ... the whole change in the universe, q þ w, q = -w, ... I just kind of associated that one right away with the first law of thermodynamics, I really don't know why, just because I've seen it so many times, and seen it rewritten. ...” I: “... and there's another idea of the first law, which is the idea of conservation of energy. Do you see that in these equations?” S 8: “I see it more in the last one that I do the other ones, just because ... well, q=-w, these ones, I can't really go right away, I can't go the q = the negative of something right there. ... I can't see that right away.”
The previous student thought that the condition when ΔU = 0 demonstrated conservation of energy. However, the equation, ΔU = 0, only shows that there was no change in the internal energy of a system. Furthermore, changes in internal energy of a system are often nonzero, a condition that is consistent with energy conservation. The condition q = -w is not a requirement or proof of conversion or conservation of energy and there are many cases in which q 6¼ -w. Another student said: S 6: “And the next one, delta U equals q plus w, the change in the internal energy can be the result of changes in heat and work. So yeah, that one I definitely would because U, your internal energy, or the energy within the system is not going to change, but it can change from heat to work or work to heat. So therefore that definitely would be the first law.” I: “... maybe you could just go back and look at them again and talk about, do you see the idea of conservation of energy?” S 6: “... this one would definitely be conservation, because the internal energy delta U, wouldn't change, or would be, this would be conserved, whereas these two would vary, the heat and the work would vary, whereas delta U would stay consistent.”
Again, this student seemed to think the first law is satisfied only when the internal energy of a system does not change. Many students do not appear to have reconciled the idea that nonzero changes in the internal energy of a system can be consistent with conservation of energy even though numerous homework problems and exam questions in this course dealt with changes in the internal energy of a system. Three of these students stated that a ΔU of zero represented energy conservation, rather than recognizing that conservation of energy is represented by the equal sign in the first law equation. When considering the energy of the universe, the expression ΔU = 0 could indicate that the total energy does not change. However, in that case, both heat and work are zero, which is different from heat and work canceling each other. None of the interviewed students provided the correct explanation that the
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equation ΔU = q þ w shows that without gain or loss, heat and work can be converted to internal energy of a system. The other six students gave a range of incomplete justifications for their answers. For example, two of these students said that this equation represented the first law because it included both heat and work. On the basis of these interview results, it is likely that few students completing this class are able to properly interpret the physical meaning of the fundamental equation expressing the first law of thermodynamics. Discussion and Conclusion The students in this study had successfully completed the Chemical Thermodynamics and Kinetics course but most of them remained unable to properly interpret the physical meaning of some of the most fundamental mathematical expressions that were frequently used in the course. The majority of lectures, homework, and exam questions in this course focused on mathematical problem solving and instructors for this course thought students would easily answer these survey questions about the meaning of the equations for work, heat, and the first law of thermodynamics. Through interviews it became apparent that most students were unable to correctly relate the physical ideas of the first law, energy conservation and conversion, to the mathematical representation, ΔU = q þ w. Approximately half of the students seemed unable to differentiate the physical meaning of the first law equation from equations that simply define how to calculate work and heat. A common student misinterpretation was that, in the equation forR work, w represented a different form of energy than - PdV. Similarly, some students interpreted the equation for heat to represent conversion of thermal energy between a
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system and its surroundings. This study suggests that students need explicit instruction to accurately interpret the physical meaning of equations commonly used in thermodynamics. Previous studies have shown that, even after instruction, many undergraduate students have difficulty understanding and distinguishing among concepts of work, heat, and internal energy, the basic components of the first law of thermodynamics (3, 5). Similarly, students in this study appear to have difficulty understanding how work, heat, and the first law of thermodynamics are represented mathematically. A number of students in the present study generated similar, partially formed or incorrect interpretations of fundamental mathematical expressions indicating that there are intuitive but unconventional ways for students to assign meaning to these equations. Instruction that addresses the most common incorrect interpretations would probably be helpful. Acknowledgment We would like to thank the instructors who allowed us to interact with their students. This work was funded by NSF. Literature Cited 1. Sherin, B. L. Cognit. Instruct. 2001, 19, 479–541. 2. Van Heuvelen, A.; Zou, X. Am. J. Phys. 2001, 69, 184–194. 3. Loverude, M. E.; Kautz, C. H.; Heron, P. R. L. Am. J. Phys. 2002, 70, 137–148. 4. Thomas, P. L.; Schwenz, R. W. J. Res. Sci. Teach. 1998, 35, 1151– 1160. 5. Meltzer, D. E. Am. J. Phys. 2004, 72, 1432–1446. 6. Rozier, S.; Viennot, L. Int. J. Sci. Educ. 1991, 13, 159–170.
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Christopher F. Bauer University of New Hampshire Durham NH 03824-3598
Student Interpretations of Equations Related to the First Law of Thermodynamics Linda C. Hadfield* JILA, University of Colorado at Boulder, Boulder, Colorado 80309 *[email protected] Carl E. Wieman JILA, Department of Physics, University of Colorado at Boulder, Boulder, Colorado 80309 and Department of Physics, University of British Columbia, Vancouver, BC, V6T1Z3 Canada
The ability to understand the physical meaning of equations used in science and to manipulate those equations in conventional ways is an important scientific skill. Students in physics classes have been observed to interpret and use mathematical representations of physical ideas in a variety of ways (1). Qualitative diagrams of problems have been shown to help physics students logically construct mathematical representations (2). Employing mathematical equations to represent physical meaning is an integral aspect of thermodynamics classes at the undergraduate level and contributes to student difficulties with the course. Many researchers have studied student difficulties with conceptual aspects of thermodynamics (3, 4), including the relationship between students' conceptual ideas and graphical representations on pressure-volume plots (5). When reasoning about properties of gases such as pressure and volume that depend on multiple variables, students have been observed to treat those properties as if they depended on only one variable (6). Student interpretation or assignment of meaning to equations representing thermodynamic quantities has received less attention. In this study, we explore how undergraduate chemistry students completing their first semester of physical chemistry, a course devoted to thermodynamics and kinetics, interpret equations associated with the first law of thermodynamics. We investigated student ability to interpret the equations for the first law of thermodynamics, the mathematical definition of gas expansion work, and the mathematical definition of heat. These equations represent some of the most fundamental science concepts in thermodynamics, and in this course, many homework and exam questions involved manipulating these equations. The first law of thermodynamics is an important topic in many science disciplines including chemistry, physics, and engineering. The difficulties we observed among physical chemistry students are likely shared by students studying thermodynamics in other disciplines. Methods Chemical Thermodynamics and Kinetics (also known as firstsemester physical chemistry) is an upper-division undergraduate course taught in the chemistry department. Student interpretations of fundamental equations used in this course were investigated through a multiple-choice survey, a written-response survey, and interviews. Students in the fall 2005 and spring 750
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2006 semesters were given the two surveys in the final week of the course. The first survey contained multiple-choice questions, including true-false questions, that were based on course material and required 30-45 min to complete. Immediately after the multiple-choice survey, or in the following class period, students were asked to write a brief explanation of their answers to each of the multiple-choice questions. All students in the course took these surveys except students who were absent on the relevant days. Altogether, 55 students, or 83% of those enrolled in the fall 2005 and spring 2006 courses, completed both surveys. All the students involved consented to having their responses included in this study. The physical chemistry course was taught by different instructors in the two semesters, but the thermodynamics content of both courses was the same and homework and exam questions were similar. In both semesters, the first topic covered was the kinetic theory of gases. In the fall course, the second topic was chemical kinetics and then thermodynamics. In the spring course, thermodynamics was the second topic, taught immediately after the kinetic theory of gases. Multiple-choice questions were asked with “clickers” in every class period of the fall 2005 course, whereas students in the spring 2006 course were asked openresponse questions in class once a week and wrote out their answers. The research presented in this study focuses on three truefalse questions on the multiple-choice survey: The first law of thermodynamics may be stated as: energy can be converted from one form to another, but cannot be created or destroyed. For the following three mathematical statements, either choose true if the expression is a restatement of the first law or choose false if the expression does not restate the first law. R 1 w = - PdV, work is the integral of the pressure multiplied by theRchange in volume. 2 q= CvdT, heat is the integral of the heat capacity multiplied by the change in temperature. 3 ΔU = q þ w, a change in internal energy for a system can be the result of changes in heat and work.
Calculations involving these equations played a central role in the Chemical Thermodynamics and Kinetics course. In addition, these equations are often covered in the first-year general chemistry course that most of these students would have taken 2 to 3 years before this study.
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Research: Science and Education
Before the survey was given to the students, the validity of the questions was investigated. Five professors in the chemistry department reviewed these questions. All five professors answered false for the first two statements and true for the third statement, as expected by the authors. The professors thought these questions would be easy for their students to answer. In addition, 10 students who had completed the class in previous semesters were interviewed about these questions to confirm that students understood what the questions asked them to do. The survey was further validated by reviewing all students' written responses on the survey to determine if their responses reflected their understanding of the mathematical equations. Eight students wrote comments that indicated they had not thought about the meaning of the mathematical equations. Typical examples of these comments on the first question follow: • “Memorized eqn; Didn't see relation to 1st law” (Student chose true.) • “For this and the next question as well, I thought that since this is a part of the first law, it should be true.” (Student chose true.) • “Guessed” (Student chose true.)
These students seemed to choose true or false based on the accuracy of the mathematical expressions or their recognition of the equations instead of their understanding of the meaning of the equations or their understanding of the true-false questions. These students did not appear to interpret the physical ideas represented in the equations. Therefore, the responses of these eight students are presented separately. This left 47 students, or 71% of the class, who completed both forms of the survey and appeared to answer these questions based on their interpretation of the equations. Only one student indicated on the long-answerresponse form of the survey that she would change the answer she gave on the multiple-choice version. Ten students were interviewed in depth on the three questions, between 1 and 8 months after they completed the course. Two of the interviewed students had completed the written surveys in class but the other eight took the class in a semester when the surveys were not given and therefore they had not seen the questions prior to the interview. Eight of the interviewed students had the same instructor, who was also the instructor of the spring 2006 course. In the interviews, students were asked to talk about their thinking as they answered the survey questions. The interviewer asked many follow-up questions to clarify student thinking. This work was carried out with IRB approval and all of the students gave written informed consent. In selecting students for interviews, we asked students with a range of grades who had successfully completed the course to be interviewed about the course. One of the interviewed students did not complete the course in the chemistry department, but completed a similar thermodynamics course in the engineering department. We interviewed all the students who agreed to participate and paid them for their time. However, we did not have access to specific student grades. As we are unable to characterize all of the students by their final grades, we describe their experiences and approach to problems. Survey Data Results Of the 47 surveyed students who appeared to answer these questions based on their interpretation of the mathematical
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Table 1. Student Responses on the Multiple-choice Survey Correct Number of Students Number of Students Equation Answer Answering True Answering False R 1. w = - PdV False 23 (49%) 24 (51%) R False 15 (32%) 32 (68%) 2. q = CvdT 3. ΔU = q þ w True
44 (94%)
3 (6%)
Table 2. Surveyed Students' Combined Responses to Questions 1 and 2 Number of Students Answering Question 2 Number of Students Answering Question 1
True
True
13 (28%)
False
3 (6%)
False 9 (19%) 22 (47%)
equations, nearly half chose true for the equation for work and a third chose true for the equation for heat on the multiple-choice survey (Table 1). These are incorrect responses. None of the surveyed professors chose these responses. Although 94% of the students in our sample answered the third question correctly, our interviews suggest that many students also held inaccurate interpretations of this equation. The 8 students who were considered separately, as discussed in the Methods section, had a low success rate in answering the first two questions. All 8 answered the first question incorrectly and half answered the second question incorrectly. All 8 answered the third question correctly. For the 47 students who appeared to answer based on their interpretation of the equations, we reviewed students' combined responses to questions 1 and 2 (Table 2). Every combination of responses to questions 1 and 2 was represented among the 47 students in our sample. However, even when combined with interview data, we found nothing conclusive in these student response patterns. In all cases, interviewed students' responses to questions 1 and 2 were consistent with their interpretation of the equations for work and heat. We do not present a combinatorial analysis involving question 3 because the overwhelming majority of students answered this question correctly. Only three survey students answered question 3 incorrectly. However, these three students did not share a common response pattern to questions 1 and 2. Therefore, student responses to question 3 did not appear correlated with responses to question 1 or 2. In addition, we can offer no insight about why students incorrectly chose false for question 3. The surveyed students who chose false did not provide explicit reasoning and none of the interview subjects chose false for question 3. A deeper understanding of student thinking about these equations was obtained from the 10 student interviews. The interviewed students were randomly selected from a group who had received a range of grades in the class. Also, the interviewed students' responses to the three true-false questions (Table 3) were not significantly different from students who completed surveys in class. (This is based on the t test, performed for these three questions. For work, t = 1.21, for heat t = 1.08, and for the first law t=0.811. In all cases, df=55, and so the differences were not statistically significant.) On the basis of their survey responses and their random selection, we believe the interview
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Research: Science and Education Table 3. Interviewed Students' Responses to the True-False Questions Correct Number of Students Number of Students Equation Answer Answering True Answering False R 1. w = - PdV False 7 (70%) 3 (30%) R False 5 (50%) 5 (50%) 2. q = CvdT 3. ΔU = q þ w True
10 (100%)
0 (0%)
subjects well represented the opinions of all students who completed the class. Interview Data Results Several themes emerged from the interviews. Students who answered that the equations defining work and heat represented the first law of thermodynamics believed that these equations represented conversion and conservation of energy. In addition, even though all interviewed students answered the third true-false question correctly, none gave a correct explanation for how the equation ΔU = q þ w represents conversion and conservation of energy. We will review student responses to each question individually. First, we will review student comments about the equation for work, then the equation for heat, and finally the equation for the first law of thermodynamics. Throughout, “S” represents a student and “I” represents the interviewer. The Equation for Work R The equation for work, w = - PdV, indicates how to calculate the quantity of work associated with specific changes in pressure and volume. Students who provided this explanation were thought to have a good interpretation of the work equation. In contrast, all seven students who incorrectly chose “true” for question 1 interpreted the mathematical definition of work to represent energy conversion and conservation. There was a diversity of thought about how energy conversion and conservation were represented in this equation. One student said that pressure and volume were different forms of energy. He described the different types of energy as follows: S 2: “Work is essentially changing energy into different forms, whether it be, you know, heat energy being converted to mechanical energy or pressure energy being converted to a volume. So, since this (referring to the equation for work) is a pressure energy being converted into a volume, I'm going to say ... true ...” I: “... So where that energy is going from and going to, I guess is the question.” S 2: “Energy in the form of pressure ... that's allowed to translate from pressure energy into volume, so it causes the molecules to expand and form the volume that they want to form.”
This student erroneously thought pressure and volume were different forms of energy and that the work equation showed energy being converted between these different forms. This student also described how the equation for work represented energy conservation. I: “... What if we think also about the conservation of energy, the idea that energy is not being created.” S 2: “Doesn't go away?” I: “Yeah.” S 2: “Well that goes along with what I said before. We're not
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creating it. We're just changing its form.” I: “Ok” S 2: “So work is just a way of changing forms of energy.”
This student attributed conversion and conservation of energy, ideas of the first law of thermodynamics, to the definition of work. This student began a thermodynamics course in the chemistry department but withdrew without completing the course. He then completed a thermodynamics course in the engineering department that focused on these topics. He used these equations repeatedly to solve homework and exam questions. Another student believed that, in the equation for work, moving from the variables on the right, PdV, to the variable on the left side, w, represented conversion of energy between different forms. After the student briefly recounted his answer, the interviewer asked further questions to explore his thinking. I: “So, in work = -PdV, do you see that as energy being converted from one form to another, or did you just say this looks like something in the first law, so therefore this is the first law?” S 10: “Well, I mean, it is energy being, (pause), well I mean, that is, work is energy. Heat is energy. I: “Ok. But the question's saying, `Does the statement express the first law?' Was that clear, that that's what we were asking?” S 10: “It seems like it is. I mean, because yeah, I mean, work is a form of energy, and converting, is moving from here to there. (The student pointed to one side and then the other side of the equation for the definition of work.) It's still being conserved. It doesn't breach the first law. I guess it's consistent with the first law, but is it actually the first law, no, I guess not. ... Well that's a form of energy, and that's a form of energy. (again referring to the two sides of the equation) They're both in the law, must be all part and parcel. ... I didn't actually make the discrepancy between what was actually the first law and what was implied by the first law. It all seems the same.”
Although the definition of work is meant to show that w R and - PdV are the R same thing, this student erroneously interpreted w and - PdV to represent different forms of energy and he believed the definition for work represented the conversion of energy between these forms. This student asked questions in class and attended problem-solving sessions and exam-review sessions outside of class. In our third example, a student interpreted the equation for work to represent the motion of energy between a system and its surroundings. I: “... Another idea of the first law is the idea of conservation of energy. Do you see that being represented in this equation?” S 9: “Yes, um, the work relates to something, someone, somebody. I guess it's sort of related to conversion too, but putting out energy and some other system taking up that energy, so the energy is just conserved, it's just from one system to another.” I: “Ok, so in the equation it sounds like you started thinking about two systems, or a system and a surrounding maybe interacting.” S 9: “Yeah, right, and this would be the surroundings doing work on the system, and this is the system's energy that they are getting from the surroundings.” (Student referred to the two sides of the equation for work).
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Research: Science and Education
As the first two students did, this student thought the equation for work represented conversion and conservation of energy. Slightly different from the other two, this student believed the work equation represented conversion of energy as motion of energy in or out of a system. This student received an A in the course and tutored less-advanced students. A common student misinterpretation was that the equation for work represented conversion of energy between potential and kinetic forms. Three of the seven students who chose true for question 1 explicitly said the equation for work expressed changes between potential and kinetic energy. For example, S 6: “So does that have to do with energy being converted from one form to another. So work, being converted into pressure is volume. So I would say that that is true, because work can be converted to, if you put work on the system you can change the pressure and the volume, so you would be changing the, I guess, the energy, yeah, because it would go from kinetic energy to potential energy.”
These students erroneously interpreted the equation for work to represent conversion between kinetic and potential energy. These students also seemed to use the words potential or potential energy to refer to internal energy, which may indicate that students have difficulty separating the idea of internal energy from potential energy. In summary, students who responded “true” to question 1 believed the work equation represented conversion and conservation of energy in a variety of ways. It is likely that the 23 students (49% of those surveyed) who answered true for question 1 on the written survey had difficulties similar to the interviewed students who chose that answer. Even the three interviewed students who answered question 1 correctly demonstrated difficulties understanding the physical meaning of the work equation. Each had a unique interpretation. For example, S 3: “I think of the integral sort of like a huge sum of things, of changes here, small changes in this property, the negative sort of sum of all the little changes in pressure times change in volume.” I: “Ok, and so in this question there are sort of two ideas of the first law, one idea being that energy is converted between different forms, another idea of the first law being that energy is conserved, not being created or destroyed. When you look at this equation, do you see it expressing one of those ideas or both of those ideas or neither of those ideas? S 3: “... It's the sum of all the little changes, so it seems like conservation is being stated there.” This student summarized his thinking by saying: S 3: “... I mean this is kind of weird, I can't really bring this math out into words, so if the change in internal energy is just heat, then mathematically you have to add the work to account for the total amount of energy then it seems like, that statement, the change, would apply to this, maybe if you add work then you're adding energy to the system and it could be created, you know. You'd be creating energy, which is illegal under the first law or something like that.”
This student had difficulty expressing his ideas about the meaning of the work equation, and was confused about how energy due to work should be considered and mathematically represented. This student thought about each question for a long time and initially tried to derive the equation for work
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from the ideal gas law. Another student expounded on her own answer: S 8: “I'm going to say that, without doing any math or anything, that this (the work equation) really isn't a first statement, or a restatement of the first law, because, um, well first, first comes to mind is that it's not even dealing with energy in like a flat out sense. ... So I guess I'm really not seeing the relationship between work and energy in that sense. ...” I: “So do you think of work as energy?” S 8: “Well eventually, but like first, right away, not right away, because I mean you do have to have energy to do work, but just purely for, again, just what the equation's, just how it's simply written out, no, I wouldn't say that until I had some other variable to relate that to energy. So right away I wouldn't.”
This student seemed to understand that work is a form of energy, but at the same time considered work to be distinct or removed from energy. In general, this student seemed confident, was eager to answer questions, and quickly wrote down equations and manipulated them as part of her answers. She described her approach by saying, “I really think of things in a mathematical sense, that's because it makes sense to me.” The third student who answered question 1 correctly then explained his answer as follows: S 1: “Individually each of these are like mathematical equations of the first law, but separately they cannot be a restatement of the first law. They cannot, both the heat and the pressure.”
This student spoke at length about his thinking. However, he was unable to discuss these equations in a manner that we understood. Before the physical chemistry course, this student had completed a competitive honors general chemistry course. He frequently wrote down and manipulated mathematical equations during the interview. On the basis of comments from the interviewed students who answered question 1 correctly, we believe that many of the students who responded correctly to question 1 on the written survey would similarly have difficulty interpreting the physical meaning of the work equation. The Equation for Heat R The equation for heat, q = CvdT, indicates how to calculate the quantity of thermal energy associated with specific changes in temperature for a substance having the heat capacity, Cv. Students who provided this explanation were thought to have a good interpretation of the heat equation. For question 2, five of the R 10 interviewed students incorrectly chose true, that is, q = CvdT represents the first law of thermodynamics. These students expressed a variety of ideas about how energy conservation and conversion were represented in the equation. Two students expressed the erroneous idea that the equation represented the transfer of thermal energy between a system and its surroundings. An example is S 2: “... So in this one we're going to be taking, thermal energy and converting it into, I guess thermal energy also ...Taking thermal energy from outside and converting it into thermal energy inside...”
This student also mistakenly referred to the transfer of thermal energy as energy conversion. Another student who chose
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true for question 2 reflected on energy conservation in all three equations: S 7: “... I see all of these as an expression of the conservation of energy. You wouldn't be able to make these equations if not, if you weren't conserving energy in some sense, you know, because you're conserving the energy produced by heat and by work and turning into a measure of internal energy. ... So I think all these equations are only, they only happen or only can be stated because of the first law through conservation of energy.”
This student erroneously believed that the work and heat equations express energy conservation and are a result of the first law. The fourth student tried to remember the physical meaning of variables in these equations but did not get far beyond that. For example, when asked to explain the equation for heat this student said, S 4: “Oh, I really don't know. ... when I think of Cv its like heat capacity, and um, but this is a specific, we have Cv and we have Cp as well, and there were equations relating these, they all, I don't, it's equal to R or something. Manipulating them is equal to R in some way. (quietly reads question) So heat capacity is basically when a certain, um, thing, has, it's like a constant basically. (sighs)”
This student, who received a C in the class, seemed unable to provide an interpretation of the heat equation. Considering these interviews of students who answered question 2 incorrectly, it is likely that the 15 students (32%) who incorrectly answered question 2 on the written survey had similar difficulties interpreting the heat equation. Even among the five interviewed students who chose the correct response to question 2, two students did not have a good understanding of the physical meaning represented in the heat equation. One of these students described the equation for heat as S 3: “Um, a false statement, mathematically. Or, it could be true in some scenarios, but it's not always true. Like, if work is zero, I guess, the change in internal energy could be the enthalpy of this.”
This student erroneously suspected the heat equation was true only under special circumstances. As this interviewed student did, it is likely that some of the students who correctly answered question 2 on the written survey were unable to interpret the heat equation accurately. The First Law Equation The first law equation, ΔU = q þ w, mathematically represents both conversion and conservation of energy. This equation represents energy conversion as both heat and work, two different forms of energy, are written as contributions to internal energy. The change in internal energy is exactly equal to heat plus work for a system, as symbolized by the equal sign (instead of g or e). This indicates conservation of energy, the idea that energy is converted between different forms of energy without being created or destroyed. Even though all 10 interviewed students answered question 3 correctly, most were not able to accurately relate the ideas of energy conservation and conversion to the symbols in the equation for the first law of thermodynamics. In interviews, students were asked how the first law equation represented energy conversion and conservation. Four students justified their answer 754
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by referring to the scenario in which ΔU = 0 and q = -w: two of the interviewed students wrote ΔU=0 and q=-w and the other two described this result in words. This was the most common response. However, this reasoning is not consistent with standard use of these symbols. Two examples follow: S 8: “...basically what I wrote was delta U equals zero, q = w, and I think, what to me that's saying that there is no, um, there is nothing created in the universe, therefore q=delta, or q = -w, and that's saying that they can be converted to one another but it's not created overall. So delta U is not changing. So I'm going to put true for that one.” ... I: “Ok. So was there anything else that you thought about in looking at this question?” S 8: “I guess, ... the whole change in the universe, q þ w, q = -w, ... I just kind of associated that one right away with the first law of thermodynamics, I really don't know why, just because I've seen it so many times, and seen it rewritten. ...” I: “... and there's another idea of the first law, which is the idea of conservation of energy. Do you see that in these equations?” S 8: “I see it more in the last one that I do the other ones, just because ... well, q=-w, these ones, I can't really go right away, I can't go the q = the negative of something right there. ... I can't see that right away.”
The previous student thought that the condition when ΔU = 0 demonstrated conservation of energy. However, the equation, ΔU = 0, only shows that there was no change in the internal energy of a system. Furthermore, changes in internal energy of a system are often nonzero, a condition that is consistent with energy conservation. The condition q = -w is not a requirement or proof of conversion or conservation of energy and there are many cases in which q 6¼ -w. Another student said: S 6: “And the next one, delta U equals q plus w, the change in the internal energy can be the result of changes in heat and work. So yeah, that one I definitely would because U, your internal energy, or the energy within the system is not going to change, but it can change from heat to work or work to heat. So therefore that definitely would be the first law.” I: “... maybe you could just go back and look at them again and talk about, do you see the idea of conservation of energy?” S 6: “... this one would definitely be conservation, because the internal energy delta U, wouldn't change, or would be, this would be conserved, whereas these two would vary, the heat and the work would vary, whereas delta U would stay consistent.”
Again, this student seemed to think the first law is satisfied only when the internal energy of a system does not change. Many students do not appear to have reconciled the idea that nonzero changes in the internal energy of a system can be consistent with conservation of energy even though numerous homework problems and exam questions in this course dealt with changes in the internal energy of a system. Three of these students stated that a ΔU of zero represented energy conservation, rather than recognizing that conservation of energy is represented by the equal sign in the first law equation. When considering the energy of the universe, the expression ΔU = 0 could indicate that the total energy does not change. However, in that case, both heat and work are zero, which is different from heat and work canceling each other. None of the interviewed students provided the correct explanation that the
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Research: Science and Education
equation ΔU = q þ w shows that without gain or loss, heat and work can be converted to internal energy of a system. The other six students gave a range of incomplete justifications for their answers. For example, two of these students said that this equation represented the first law because it included both heat and work. On the basis of these interview results, it is likely that few students completing this class are able to properly interpret the physical meaning of the fundamental equation expressing the first law of thermodynamics. Discussion and Conclusion The students in this study had successfully completed the Chemical Thermodynamics and Kinetics course but most of them remained unable to properly interpret the physical meaning of some of the most fundamental mathematical expressions that were frequently used in the course. The majority of lectures, homework, and exam questions in this course focused on mathematical problem solving and instructors for this course thought students would easily answer these survey questions about the meaning of the equations for work, heat, and the first law of thermodynamics. Through interviews it became apparent that most students were unable to correctly relate the physical ideas of the first law, energy conservation and conversion, to the mathematical representation, ΔU = q þ w. Approximately half of the students seemed unable to differentiate the physical meaning of the first law equation from equations that simply define how to calculate work and heat. A common student misinterpretation was that, in the equation forR work, w represented a different form of energy than - PdV. Similarly, some students interpreted the equation for heat to represent conversion of thermal energy between a
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system and its surroundings. This study suggests that students need explicit instruction to accurately interpret the physical meaning of equations commonly used in thermodynamics. Previous studies have shown that, even after instruction, many undergraduate students have difficulty understanding and distinguishing among concepts of work, heat, and internal energy, the basic components of the first law of thermodynamics (3, 5). Similarly, students in this study appear to have difficulty understanding how work, heat, and the first law of thermodynamics are represented mathematically. A number of students in the present study generated similar, partially formed or incorrect interpretations of fundamental mathematical expressions indicating that there are intuitive but unconventional ways for students to assign meaning to these equations. Instruction that addresses the most common incorrect interpretations would probably be helpful. Acknowledgment We would like to thank the instructors who allowed us to interact with their students. This work was funded by NSF. Literature Cited 1. Sherin, B. L. Cognit. Instruct. 2001, 19, 479–541. 2. Van Heuvelen, A.; Zou, X. Am. J. Phys. 2001, 69, 184–194. 3. Loverude, M. E.; Kautz, C. H.; Heron, P. R. L. Am. J. Phys. 2002, 70, 137–148. 4. Thomas, P. L.; Schwenz, R. W. J. Res. Sci. Teach. 1998, 35, 1151– 1160. 5. Meltzer, D. E. Am. J. Phys. 2004, 72, 1432–1446. 6. Rozier, S.; Viennot, L. Int. J. Sci. Educ. 1991, 13, 159–170.
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