Pedagogical Tools to Strengthen Engineering Student Comprehension of Entropy

Practice Paper Recommended citation: Al-Adili, A., & Scheicher, R. (2025). Pedagogical Tools to Strengthen Engineering Student Comprehension of Entropy. In Kangaslampi, R., Langie, G., Järvinen, H.-M., & Nagy, B. (Eds.), SEFI 53rd Annual Conference. European Society for Engineering Education (SEFI), Tampere, Finland. DOI: 10.5281/zenodo.17631877. This Conference Paper is brought to you for open access by the 53rd Annual Conference of the European Society for Engineering Education (SEFI) at Tampere University in Tampere, Finland. This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License. Pedagogical tools to strengthen engineering student comprehension of entropy Ali Al-Adili a,1 and Ralph Scheicher b a Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden, orcid.org/0000-0002-1233-0221 b Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden, orcid.org/0000-0001-5397-7753 Conference Key Areas: Fostering Engineering Education Research, Innovative Teaching and Learning Methods Keywords: entropy, thermodynamics, disorder, Boltzmann, peer-instruction ABSTRACT Entropy is a fundamental yet challenging concept for many students, requiring active engagement with its multiple definitions and their interconnections. This project seeks to enhance engineering students' understanding of the second law of thermodynamics through interactive exercises and the employment of peerinstruction. The initiative began with a literature review on entropy and workshops for thermodynamics instructors, promoting discussions on teaching strategies. Additionally, a large survey of STEM students who had completed a thermodynamics course revealed that the disorder metaphor was strongly prevalent in entropy definitions, despite instructors emphasizing that focusing exclusively on spatial disorder can result in misleading interpretations of entropy. Based on these insights, four exercises were developed to address different aspects of entropy, including the disorder metaphor, the duality of Clausius and Boltzmann’s definitions, and the engineering implications of Carnot’s ideal engine. These exercises were tested with a focus group of 11 engineering students, followed by interviews to assess their impact on students' conceptual understanding. Exam results from this group were also compared with a control group of 80 students to measure learning outcomes, indicating a slightly higher average score in the focus group. The developed instructional tools will be integrated into thermodynamics courses at the faculty to support student learning of entropy. 1 Corresponding Author A. Al-Adili Ali.Al-Adil[email protected]u.se 1 INTRODUCTION Thermodynamics is a fundamental area in physics which affects nearly every aspect of our universe. Its 4 fundamental theorems provide the foundation for advanced applications in both natural sciences and engineering. Among these laws, entropy stands out as a crucial yet conceptually challenging concept for students. While the first law of thermodynamics is relatively intuitive, stating that energy can neither be created nor destroyed, the second law is more complex. It introduces entropy, a quantity that is not directly measurable, not conserved, and can be defined in seemingly multiple ways. Ben-Naim, in his book on entropy, explores various interpretations such as "dispersed energy," "disorder," "information loss," and "irreversibility," noting that each can be misleading if not carefully contextualized (Ben-Naim, 2010). Indeed, few concepts in physics are as essential yet as difficult to grasp as entropy is. In teaching technical thermodynamics courses, conveying the meaning and importance of entropy to engineering students remains a challenge which instructors face. Clausius’s classical definition frames entropy for a reversible process, in terms of heat exchange across system boundaries at a given temperature: 𝑑𝑆 =𝛿𝑄 𝑇 ⁄ (1) where 𝛿𝑆 denotes a small change in entropy, 𝛿𝑄 is the heat transfer and 𝑇 is the temperature. His insights, developed through the study of Carnot’s idealized heat engine, introduced the notion of "equivalent value" as a conserved quantity. Clausius’s brilliance lay in challenging Carnot’s assumption that heat was fully conserved throughout the cycle, demonstrating instead the need for a new conceptual framework (Müller, 2007). Boltzmann expanded this understanding by linking entropy to the number of accessible microstates in a system, framing the second law as a fundamentally statistical law: 𝑆 = 𝑘!log*(Ω) (2) Where kB is Boltzmann’s constant, and Ω denotes the number of microstates (multiplicity). Given these two diverse viewpoints, the question arises: how can we effectively communicate this multifaceted concept to engineering students? The conceptual evolution of entropy mirrors broader developments in physics, marking a shift from the industrial era’s focus on heat engines and automobiles to the probabilistic nature of quantum physics. Understanding entropy is, in many ways, understanding the evolution of physics itself, an insight of great value to students. Numerous studies explore effective strategies for teaching thermodynamics, particularly entropy. For instance, Camacho et al. highlight how inconsistencies in textbook definitions can lead to confusion and suggest that a historical perspective may clarify its development (Camacho et al., 2015). The commonly used metaphor of "disorder" is often criticized, e.g. Haglund et al emphasizing the importance of addressing its limitations (Haglund, 2017). Carson argues that introducing these concepts early encourages a deeper understanding among chemistry students (Carson & Watson, 2002). Others focus on bridging the gap between Clausius’s macroscopic approach and Boltzmann’s statistical perspective, with Ben-Naim offering a thorough exploration of the latter (Ben-Naim, 2010). A key challenge lies in selecting relevant literature that not only provides a solid theoretical foundation but also inspires the development of innovative teaching strategies and exercises. By incorporating historical contexts, concept reasoning, and interactive learning tools, we can better equip students to grasp one of thermodynamics’ most intricate and fascinating concepts. It is worth noting that there is an interesting and extensive recent article reviewing various methods for teaching entropy (Natalis & Leyh, 2025). In the present work, we aim to create an active learning environment that incorporates peer instruction to enhance engagement and conceptual understanding. First, we would like to address the limitations of the disorder metaphor. Second, we work to resolve any apparent contradictions between the different approaches. Lastly, we try to shed more light on the role of Carnot cycles in understanding entropy, particularly in an engineering context. 2 CONTEXT AND PRACTICAL WORK 2.1 Study stages The methodology used in this project followed several stages: 1. Conduct a literature review and distribute an online survey to hundreds of students from various academic years: The literature review aimed to explore how educators across various disciplines approach the teaching of entropy. Several relevant articles and books were identified, specifically addressing instructional strategies for this complex concept. Additionally, a quantitative survey with 157 participants examined how students' understanding of entropy evolves over time, highlighting which definitions and interpretations are most commonly retained. 2. Develop an initial test version of the exercises: To facilitate group discussions within a problem-based learning (PBL) framework, four exercises were developed. PBL has been recognized as an effective teaching method (Nilson, 2010). In our problem-solving sessions, groups of three student were used to structure discussions, with two students defending a position and one acting as an opponent. Each student (designated A, B, or C) alternated between the roles of defender and opponent. 3. Collect feedback from alumni students to validate the exercises and revise them accordingly: The exercises were tested by three alumni students to incorporate a student perspective that is more relevant to their experiences. The feedback from the alumni students was valuable in refining the exercises. 4. Carry out a pilot study with a focus group: A group of 12 students were chosen randomly from the 2023 technical thermodynamics course, to participate in a pilot study. The selection process ensured equal gender balance through statistical sampling. However, several students declined to participate, necessitating the recruitment of new students to reach the required number of focus group participants. One student was absent due to illness. The other 11 were divided into four groups (one group with two students) and exposed to the four interactive exercises. The instructors observed and took notes during the two-hour session. Afterward, follow-up interviews and evaluations were conducted to assess the participants’ understanding and engagement. 5. Analysis of exam results to measure the effectiveness of the exercises and disseminate the educational material: Exam results from the focus group were analysed and compared with those of a control group, while evaluation forms provided additional insights to further refine the exercises. 2.2 The developed exercises on entropy Exercise 1: The first exercise presents various scenarios requiring students to assess entropy changes. The first scenario examines heat transfer, the second focuses on ink diffusion in water, and the third investigates phase transitions. The primary pedagogical goal is to encourage students to critically analyse entropy in different states and relate these concepts to everyday phenomena. Each of the three students is responsible for explaining one of the scenarios, thus ensuring active participation from all. The first scenario is relatively straightforward, as both heat and temperature play a prominent role and can be easily understood in terms of Equation 1. The diffusion example, on the other hand, is particularly interesting because it is not directly linked to conventional thermodynamic observables. Analysing it from a statistical perspective can be rather challenging. To introduce additional complexity, we deliberately reversed the commonly depicted time order (i.e., from left to right) in this scenario, choosing to present it instead from right to left. This inversion may have disrupted students' conventional reasoning patterns, making the conceptual challenge even greater. The third scenario, which involves randomly placed ice cubes, is particularly effective in illustrating the complex relationship between entropy and spatial disorder. Many students may mistakenly believe that the random placement of ice cubes appears more disordered and "chaotic," thereby corresponding to higher entropy than the uniform distribution of water molecules in liquid form. Exercise 2: The second exercise aims to connect the concept of spatial molecular disorder with momentum space, inspired by (Bhattacharyya and Dawlaty, 2019). In this exercise, a noble gas is heated, and students are asked to determine which of two cases exhibits higher entropy and which displays greater disorder. While participants correctly identified the case with higher entropy, the concept of disorder proved more challenging. When examining a snapshot of the molecular positions, it is not immediately apparent how spatial disorder increases within a heated sealed box, making this a true test of the disorder metaphor. Only when students also consider momentum space and the velocity distribution of the atoms does it become clear that entropy has indeed increased. Exercise 3: The third exercise highlights a common misunderstanding about entropy definitions, potentially creating the impression of a contradiction between the versions of Boltzmann and Clausius. In this problem (Bhattacharyya & Dawlaty, 2019), an isolated piston-cylinder device undergoes an adiabatic compression, i.e. no heat exchange with the surroundings. One student is asked to explain the entropy change in the system using Boltzmann's statistical approach (Eq. 2), while another employs Clausius' thermodynamic definition (Eq. 1). The third student takes on the role of an opponent, critically examining the arguments of the other two participants, with some guiding questions provided by the tutors. The challenge lies in the seemingly conflicting conclusions that arise depending on which definition of entropy is applied, prompting students to reconcile the two perspectives. Student A may argue that, since no heat is exchanged with the surroundings, the entropy change must be zero according to Clausius' formula (Eq. 1). Student B, however, can counter that the significant reduction in volume restricts the space available for gas molecules to move, leading to fewer accessible microstates and, consequently, a decrease in entropy. A third perspective could arise from the observation that the temperature increases during compression, which might increase entropy. It is therefore possible for students to argue that entropy can either increase, decrease or even remain constant in the presented scenario. The resolution lies in reconsidering the situation in terms of momentum space. The multiplicity (Ω) remains unchanged due to a compensating effect: while the reduction in volume limits the spatial distribution of the gas molecules, the increase in temperature expands the range of possible velocities in momentum space. The decrease in spatial multiplicity is precisely offset by the increase in momentum space. This realization can be challenging for many students, especially since it requires a mathematical treatment to be fully understood. However, conceptually it underscored an important insight, namely that there is no contradiction between Clausius’ and Boltzmann’s definitions of entropy. Rather, they represent two complementary perspectives on the same fundamental concept. Exercise 4: Lastly, a fourth exercise was created to investigate a practical application: the Carnot cycle and its relation to entropy. In this exercise, students were assessed on their understanding of entropy changes in a steam turbine under both adiabatic and non-adiabatic conditions, applying both Clausius' and Boltzmann's equations (Equations 1 and 2). The goal of this problem was to deepen students' understanding of how entropy operates in real-world thermodynamic systems, particularly in the context of energy conversion in steam turbines. 3 RESULTS AND INSIGHTS 3.1 Online survey The online survey was conducted to assess students' understanding of the second law of thermodynamics. All participants had completed a thermodynamics course and represented various academic years. There were 67 students from the second and third academic years and 90 students from year 4 and 5. The survey included questions such as: 1) Rank the following concepts in terms of their relation to entropy, from most to least related: disorder, heat, temperature, and probability. 2) Which mathematical definition of entropy do you recall? 3) Yes or No answers to the following questions: i) Is entropy conserved? ii) Is entropy a state function? iii) Can entropy decrease in a closed system? iv) Is a Carnot engine both internally and externally reversible? Figure 1 shows key findings from the survey, highlighting a strong correlation in the students’ mind between entropy and disorder. Around 80 % of the students though disorder was mostly related to entropy! Other concepts (heat, temperature, and probability) were perceived as equally less linked to entropy. Notably, 77.3% of students were unable to recall any formula for entropy. Boltzmann’s definition was remembered by 11.3%, Clausius’ by 10%, and only 1.3% recalled both definitions. There was no significant difference in recall between students in the 2nd/3rd and 4th/5th academic years. Additionally, many students were uncertain whether or not entropy is a state function, and most students confused a closed system with an isolated system, believing that entropy cannot decrease in a closed system. Furthermore, students struggled with the concepts of external and internal reversibility, finding it particularly difficult to understand what makes a thermodynamic cycle reversible. Collectively, these findings suggest that entropy certainly remains an elusive concept, necessitating dedicated discussion sessions and practical examples to facilitate a clearer understanding. We would like to point out that the article by Bhattacharyya & Dawlaty (2019) contains an alternative questionnaire that can be regarded as complementary to the format chosen by us. 3.2 Focus group observations and reflections During the problem-based session, almost all students were actively engaged, and the group size seemed well-matched to the exercises. As instructors, we chose to observe rather than participate in the discussions, allowing students to navigate the problems independently. Due to time constraints, the scientific feedback was limited to the end of the session. Some students found this approach frustrating, as they felt the need to consult a tutor for clarification after each exercise. One student specifically commented on this: “Interesting questions and good discussion. I wished I had received the correct answer or had the opportunity to discuss it with the teacher. In the future, increased teacher interaction may be beneficial to clarify potential misconceptions. After completing the exercises, the focus group filled out an evaluation form and expressed overall satisfaction with all four problems. The first problem received an average rating of 4.2/5.0, followed closely by the fourth problem at 4.0/5.0, while problems two and three both were rated at 3.8/5.0. Students were also asked where they felt more emphasis should be placed, on Boltzmann's probabilistic description, Clausius's interpretation of entropy, the disorder metaphor, or engineering relevance. The feedback indicated general satisfaction with the current approach, with a preference for a stronger focus on engineering relevance and the statistical perspective. One student reflected on the discussions with the following comment: “It was interesting to see a conflict between entropy and disorder because one often thinks they are synonymous. There were many discussions with different opinions!” Finally, to assess any changes in perception, participants were asked to define entropy in three words both before and after the active session. Figure 2 presents a word cloud plot, where the size of each word reflects its frequency of use by the students. It is evident that "disorder" was the most frequently used term before the session. However, the exercises appeared to enhance students' understanding of Boltzmann’s definition, as "probability" was used more often. Additionally, the range of definitions became more varied and diverse following the active session. The importance of heat was also increased, pointing towards a better understanding of Clausius definition. Despite all this, the "disorder" definition of entropy remained prevalent among the students. This metaphor seems deeply rooted and challenging to fully move beyond. Nevertheless, some students began to recognize that relying solely on the disorder metaphor could be misleading. An interesting observation was the mention of the word “contradiction” in a student’s feedback after the active session, suggesting that at least one participant may have left the discussion feeling more confused. In the free-text responses, two students expressed: “I got the impression entropy is not easy to define.” “This exercise was also good, but the problem, I feel, is that entropy has two definitions that sometimes contradict. Therefore, it is hard to know what is correct or wrong and what should be used.” These comments highlight that, for some students, two hours of discussion were not sufficient to clarify the complexities of entropy. This suggests that additional efforts are needed to further support their understanding of the concept. Fig. 1. Results from an online survey assessing the ranking of items associated with entropy. The survey was completed by 157 students from the science and technology faculty, covering years 2 to 5, all of whom have completed an introductory thermodynamics course. Students were asked to rank four entities based on their relevance to entropy. The disorder metaphor emerged as the predominant theme. Exam outcomes Upon completion of the course, we analysed the final exam results for both the focus group and the control group. The exam is structured around four main learning goals, and students must pass all four sections to succeed. One of these sections specifically assesses knowledge of the second law of thermodynamics. Figure 3 presents the performance of the focus group (11 students) and the control group comprised of the rest of the class (89 students) in both the entropy part and the remaining three parts of the exam. To account for overall performance differences, we compare student results in the other three sections as a reference. The focus group obtained a higher average score than the control group, achieving an average grade of 0.67 (out of 1) in the three non-entropy related exam parts, compared to 0.56 in the control group. However, specifically in the entropy section, the focus group achieved an average score of 0.64, while the control group scored 0.48. This corresponds to a 20% overall increase in the average score for the focus group and a 33% increase in the average score in the entropy section. This amounts to a 13% higher average score for the focus group in the exam performance in the entropy section. Despite the higher average score, the statistical sample was very small and not normally distributed, thus we cannot assert a statistically significant improvement. It is important to acknowledge a potential bias in our study, as the focus group was not drawn from the initial random samples. Instead, new samples had to be selected after some students declined participation, which may have led to a selection bias favouring students who were more engaged in the course and had stronger academic performance. Nevertheless, our findings suggest that the activation session may have contributed to an improvement in students' understanding. 0 20 40 60 80 Frequency (%) Disorder Which is mostly related to entropy? Heat Temperature Probability 0 10 20 30 40 Frequency (%) Which is least related to entropy? Temperature Probability Disorder Heat Fig. 2. Word cloud plot with the font size of each word reflecting how often it was given by the students. The left panel depicts the results before the session, while the right panel shows the outcome after the session. 4 CONCLUSIONS AND IMPLICATIONS In this study, we have shown that active learning approaches, including peer instruction, can potentially enhance students' conceptual understanding of complex physical phenomena. The focus group obtained a higher average score in the entropy part of the exam as compared to the control group on the exam, which we interpret as evidence for the potential usefulness of these methods. Additionally, our findings indicate that the disorder metaphor is deeply ingrained in students' perceptions of entropy, while the mathematical definitions of entropy are seldom recalled. It appears that students tend to associate entropy primarily with its chaotic and random nature, even when instructors emphasise clearly that entropy is more than just disorder. The problem-based session will be incorporated into thermodynamics teaching when introducing entropy, facilitating a clearer connection between Clausius' and Boltzmann’s definitions. Fig. 3. Results from the final exam in the course comparing the average points obtained by the focus and the control group. 5 ACKNOWLEDGEMENTS The authors would like to express their gratitude to the Faculty Pedagogical Development Fund (TUFF) for their generous grant, which made this study possible. We also wish to acknowledge TUR for their continuous support and guidance throughout the project. Furthermore, we are grateful to the alumni students who tested the first version of the exercises and provided valuable feedback, as well as the focus group for their active participation and engagement in discussions, despite the challenging nature of the subject.