Language aspects of engineering students' view of entropy

Abstract

Entropy is a central concept in thermodynamics, but has been found to be challenging to students due to its abstract nature and the fact that it is not part of students' everyday language. Interviews with three pairs of engineering students (N = 6) were conducted and video recorded regarding their interpretation and use of the entropy concept, one year after a course on chemical thermodynamics. From a syntax perspective, students were asked to assess whether different sentences involving temperature, internal energy, and entropy make sense. With a focus on semantics, they were asked to rank a set of notions with regards to how closely they are related to entropy, how scientific they are, and how useful they are for explaining what entropy is. From a pragmatics point of view, students were asked to solve two qualitative problems, which involve entropy. The results show that these chemistry students regard internal energy, but not entropy, as a substance-like entity. The students' ranking of how closely related to entropy notions are and how useful they are for explaining entropy was found to be strongly negatively correlated to how scientific the notions were seen to be. For example, disorder was seen as highly unscientific, but very useful for explaining entropy. In the problem-solving tasks, Chemical Engineering students were comfortable relating entropy to enthalpy and Gibbs free energy, the three notions being seen to form a “trinity” in thermodynamics. However, the students had challenges grasping the unchanged entropy in reversible, adiabatic expansion of an ideal gas, in which they did not consider how entropy relates to the second law of thermodynamics. In final reflections on their learning processes, the students saw weak connections between their problem-solving skills and their conceptual understanding of entropy, although acknowledging that both aspects of learning are important.

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Introduction

Entropy is a word that students are likely to encounter first in their formal education. In contrast to many other terms in the thermal domain, such as heat, pressure, temperature, or energy, entropy does not play a prominent part in everyday language. In addition, entropy is abstract in the respect that it cannot be directly experienced through our senses, but is typically derived algebraically from other measurable physical quantities. Herein lies a fundamental challenge for teachers and students of thermodynamics: we cannot access entropy directly by perception or physical interaction, and we have limited help from everyday language. In this situation, we can approach the topic from a formal point of view where we consider how entropy is defined in terms of other physical quantities and what other quantitative relations have been established between them, or by use of microscopic models, for example the kinetic theory of gases. Another, more conceptual, approach is to make the best we can with limited, approximate meanings in everyday language, for instance by means of metaphorical expressions or anthropomorphic language. As recognised in chemistry education research on the matter (e.g.Taber and Watts, 2000; Lambert, 2002; Sözbilir and Bennett, 2007), however, different approaches to teaching of entropy will be characterised by their specific merits and shortcomings, argued on theoretical grounds or in conclusion from empirical analysis of students' understanding of the topic.

In our research, we investigate how one cohort of Swedish engineering students interpret the notion ‘entropy’, in connection with a course in chemical thermodynamics. Following a previous questionnaire study (Andersson et al., 2015; Haglund et al., 2015), which targeted the entire population of the course, in the present study three pairs of students (N = 6) were asked to interact collaboratively with four tasks. In line with the theme of the theme issue, ‘Language and the teaching and learning of chemistry’, the tasks were designed to shed light on a range of language aspects on students' ideas of entropy. These aspects involve primarily: syntax, in the study of the structure of language involving entropy; semantics, relating to the meaning of entropy, and; pragmatics, in terms of the communicative use of entropy in problem-solving dialogue.

Theoretical background

Language aspects of science learning

Language influences chemistry education – and science education at large – in many different ways, which is indicated by the diversity of potential topics suggested by Taber (2015) for the current theme issue. We would like to frame our study against the background of our interpretation of how different areas of linguistics have informed science education research.

Semantics can be characterised as the study of the meaning of entities in language, and is arguably the linguistic subfield that lies closest to the study of students' understanding of scientific concepts. Students' interpretation of what the words they meet in science class mean is closely connected to their understanding of the underlying concepts these words represent. A student's use of a word in a way that comes across as inadequate may reflect an underlying conception that is not in line with the scientific view, but not necessarily so. It could also be a matter of ‘labelling’ (Sutton, 1992), matching a word with a concept in an unorthodox way, such as ascribing force rather than linear momentum to an object in uniform motion, against the background of an otherwise sound conceptual understanding. Students' challenges relating to semantics include that certain words have several distinct, but related meanings in language, i.e. they are polysemous. As related to above, some words, such as ‘heat’, have clear definitions in science, and vaguer meanings in everyday language. This may lead to confusion in communication and learning. Other words, such as ‘entropy’ – the focus here, are abstract and can be expected to be unfamiliar to most students prior to their introduction of the term in science teaching. This leads to other challenges in education: What does this new word mean, and what does it refer to, either in the physical world or in the conceptual area of science? Furthermore, by means of instructional metaphors, students' interpretation of novel domains can be helped by comparison to other more known domains. However, also interpretation of metaphors involves vagueness. For instance, when a student is told that ‘entropy is disorder’ without further explication, it is not clear what sense of disorder is intended. Therefore, although metaphors are powerful, suggestive tools for students' learning, they are also, like analogies, “double-edged swords” (Glynn, 1989), since they may be interpreted in ways teachers do not intend, and will eventually break down at some point where they are no longer apt.

With the adoption of a socio-cultural perspective on science learning, there is an awareness of the situated nature of the educational practice (Sadler, 2009). We cannot pin down a student's understanding of a word or concept once and for all, but have to consider the social and material contexts in which it is expressed. This perspective has called for a more careful study of pragmatics, how language is used in practice, in educational research. Examples include Lemke's (1990) study of teachers' and students' discourse in the science classroom, in which he makes the parallel between science learning and learning a new language, coming to “talk science”, and Cameron's (2003) investigation of how teachers and students make use of metaphors in classroom communication.

Syntax, the structure of language, has long been a strong focus in linguistics, not least due to the effort of identifying a ‘universal grammar’ that is shared across and underlying all natural human languages, which was initiated by Chomsky (1965). Against this background, it is curious that the study of syntax seems to have received less attention than aspects of semantics and pragmatics in science education research. However, there are examples of research that has studied features of the structure of scientific language, adopting Halliday's (1985) functional grammar perspective. Fang (2005) points to the tendencies of written scientific language to incorporate dense language, with many content-related words in each sentence, nominalization, where processes and events, or entire clauses, are compressed into nouns, and the use of technical vocabulary (Halliday, 1993). Since students are not used to this kind of language from their everyday life, they are likely to have challenges when they are confronted with it in science class; this is part of developing scientific literacy. Other science education research has focused on syntactical aspects of particular words. With a focus on thermal phenomena, Amin (2001) shows that the nouns ‘heat’ and ‘temperature’ play quite distinct roles in sentences in everyday language, which points against the claim that students tend to conflate the concepts (e.g.Erickson, 1985). As another example, Brookes and Etkina (2007) employ functional grammar analysis (Halliday, 1985) in the study of students' language and underlying understanding of involved concepts in areas such as thermodynamics and quantum mechanics. An editorial with the title “Heat is not a noun” (Romer, 2001) also testifies to an interest in questions of grammar in science learning.

The field of cognitive linguistics adopts a view of language as being structured by our conceptualisation of largely shared embodied experiences. In opposition to Chomsky (1965), Lakoff and Johnson (1999) argue that syntax cannot be studied in isolation, but that we also have to take into account the meaning of the involved words, i.e. semantics, in a holistic fashion. Furthermore, they argue against the view of an innate universal grammar, and claim that we learn the structure of language through experiences of language in use, i.e. pragmatics. By means of conceptual metaphor (Lakoff and Johnson, 1980, 1999), the structure of concrete, familiar domains are projected onto abstract, less familiar domains. Such conceptual metaphors are found to be ubiquitous in everyday language in a systematic, yet subconscious manner. Conceptual metaphor analysis has received increasing attention in science education research as an approach to the study of how students can learn abstract concepts, as reflected in a recent special issue on the theme in the International Journal of Science Education (Amin et al., 2015). In a commentary on two of the contributions, Sherin (2015, p. 807) reflects on the potential merits of attending to subtleties of students' language by means of conceptual metaphor analysis, in comparison to previous approaches to educational research:

It is not that researchers such as myself ignored language; we certainly listened to what participants said. But to a great extent we looked through language rather than at language. We used the speech of our participants as a window into their thinking, but we did not focus as closely on the window itself.

Within semiotics, the study of signs and meaning-making, the focus is broadened from written and spoken language to other aspects of human communication. Drawing on Halliday's (1978) and Lemke's (1990) approaches to social semiotics, Airey and Linder (2015) argue that learning science can be seen in terms of coming to appropriately interpret and use disciplinary-specific semiotic resources. In this respect, semiotic resources are meaning-making tools, such as language, mathematical expressions, or diagrams, but also physical artefacts such as experimental apparatus. Against this background, Airey and Linder use the notion of ‘disciplinary affordance’ (Fredlund et al., 2012), in terms of the agreed meaning making functions that a semiotic resource fulfils for the disciplinary community. However, there is a challenge in that semiotic resources that skilful practitioners in a disciplinary community make use of in a fluent way may be largely inaccessible to the learner. As a practice develops, its semiotic resources become increasingly efficient and specialised, but therefore also opaque to uninitiated learners. The dense, precise character of scientific terminology, as mentioned above, or mathematical formalism are examples in case. In order for learners to be able to get gradually acquainted with a discipline, other semiotic resources, associated with ‘pedagogical affordance’ in terms of the aptness of a semiotic resource for teaching some particular educational content are required. Airey and Linder (2015) suggest an inverse relationship between pedagogical affordance and disciplinary affordance. In other words, the more powerful and succinct a resource appears to the trained practitioner, the less useful it is for learners, since it would require considerable unpacking before they can relate to it. Conversely, the greater the pedagogical affordance of a resource for the learner, the less useful it is to the practitioner, due to its cumbersome nature and lack of precision.

Students' understanding of entropy and teaching of entropy

In the light of the challenges inherent in the teaching and learning of entropy, it is hardly surprising that research on chemistry education (Johnstone et al., 1977; Carson and Watson, 2002; Sözbilir and Bennett, 2007) and physics education (Cochran and Heron, 2006; Christensen et al., 2009; Leinonen et al., 2015; Loverude, 2015; Smith et al., 2015) has found entropy to be difficult for students to grasp. For example, in a questionnaire study, Sözbilir and Bennett (2007) found that third-year chemistry students' problems with entropy included drawing erroneous conclusions from seeing entropy as disorder, and failure in disambiguation of a system and its surroundings. Similarly, Christensen et al. (2009) have found university physics students struggle with the fact that the sum of the entropy of a system and its surroundings increases when they exchange heat. Thomas and Schwenz (1998) and Loverude (2015) have concluded that students are reluctant to use the second law of thermodynamics and Clausius' expression dS ≥ đQ/T in problem solving.

Furthermore, Brosseau and Viard (1992) and Haglund and Jeppsson (2014) have found students conclude erroneously that the entropy increases when an ideal gas is allowed to expand adiabatically, in which they only consider the increased volume but not the decreased temperature and internal energy of the gas. It is very challenging to justify the unchanged entropy with a microscopic approach, even if they had considered the temperature, since it requires that the entropy increase due to increasing volume and entropy decrease due to decreasing internal energy cancel out. In this case, a macroscopic approach would be more useful, from which it can be argued that since no heat is exchanged, the entropy remains the same in this reversible process.

There is a debate in physics education research whether thermodynamics, and the entropy concept, should be introduced from a macroscopic or microscopic perspective. For example, Reif (1999) advocates a microscopic introduction of entropy in terms of the multiplicity of microstates. Loverude and colleagues (Loverude et al., 2002; Loverude, 2015) point to students' misinterpretations of microscopic models, and suggest a firm understanding of the macroscopic phenomena, such as temperature changes of bicycle pumps, before such models should be introduced. Baierlein (1994) argues that students need both the fundamental microscopic understanding and the ability to solve problems relating to phenomena at the macroscopic level, and that the main challenge lies in connecting the two levels. This relates to the issue of relating the macroscopic and submicroscopic levels to each other, as recognised in chemistry education research (Johnstone, 1991).

Geller et al. (2014) have identified that life-science students who take an interdisciplinary physics course involving thermodynamics are more used to relating to Gibbs free energy than to entropy from previous chemistry and biology courses, which provides another route to teaching the second law of thermodynamics.

Metaphors for entropy in science teaching

A theme in the educational research on the teaching and learning of entropy that touches directly on language issues is how metaphors for entropy can be useful in students' learning, and the relative merits and shortcomings of particular metaphors. The disorder metaphor of entropy is pervasive in science teaching. However, it has considerable shortcomings, in particular that it considers a single snap-shot view of a system rather than allowing students to understand the multiplicity of microstates, and that it entails a predominantly spatial interpretation in which energy distribution across energy levels is not considered (Styer, 2000). Lambert (2002) argues that the disorder metaphor is a ‘cracked crutch’, which we should avoid in teaching of entropy. As an alternative, other seemingly more apt metaphors have been suggested, such as interpreting entropy as spreading (Leff, 1996), or freedom (Styer, 2000).

Most educational research on metaphors for entropy has relied on theoretical arguments, but rarely involved empirical investigation of students' interpretations. One exception is Sato and Suganuma (2013), who asked university students who were assumed not to be familiar with the entropy concept to watch a set of ideal-gas animations pairwise, where the temperature and volume of the system varied. The students assessed which one of the animations best represented five different expressions, which are commonly used in Japanese teaching of entropy, including the equivalents of degree of randomness, activeness and diffusion. High ‘degree of randomness’ was found to be strongly correlated to temperature and entropy, but not related to the volume. In contrast, high ‘degree of diffusion’ was most strongly correlated to entropy of the five expressions, strongly correlated to volume, and somewhat correlated also to temperature. In line with Leff's (1996) view on entropy as spreading, Sato and Suganuma therefore conclude that ‘degree of diffusion’ “represents the essence of entropy” (p. 447). Gustavsson et al. (2013) conducted another empirical study of students' conceptions of entropy in conjunction with a course in engineering thermodynamics, through a questionnaire where students were asked to rate how strongly, on a 0–5 scale, they related entropy to a list of notions: probability, temperature, work, disorder, heat, and energy. The students associated entropy most strongly with disorder, and least strongly with work.

The syntax of “substance-like” physical quantities

In everyday language, we often use words that in science represent physical quantities as if they were some kind of substance, or “stuff” (Herrmann, 2000; Scherr et al., 2012). It has been argued that this substance metaphor can be used deliberately in science teaching of energy (Close and Scherr, 2015). In particular, it can be used to emphasise that energy is a conserved quantity, and in representation of energy transfer and transformation from one form to another. Within the Karlsruhe Physics Course, the substance metaphor is used systematically for a range of extensive physical quantities, i.e. quantities that depend on the size of a system, such as momentum, entropy, and amount of substance. In this way, by using systems dynamics, it is possible to model how these quantities are stored in, and flow into or out of systems (and in the case of entropy generated), and how different subfields of science are related to each other analogically (Herrmann, 2000).

However, as with all metaphors, the substance metaphor has limitations. Overall, it assumes a macroscopic perspective, where processes are seen as direct processes (Chi, 2005), like a water flow, where every constituent particle is representative for the whole phenomenon. In this way, the metaphor plays down microscopic aspects, which are integral to emergent processes (Chi, 2005), such as diffusion or heat conduction, where the pattern of the random behaviour of constituent particles emerges only at a collective, macroscopic level. Students have been found to confuse emergent processes for direct processes (Chi, 2005), presumably because of the influence of the substance metaphor.

There are also particular limitations of the substance metaphor in relation to energy. For instance, from this perspective it is hard to conceive of negative values of energy in relation to an arbitrary zero point (Dreyfus et al., 2014). Similarly, in the context of chemical bonds, it is tempting to see energy as a kind of glue or fuel that is stored in the bond, and thereby difficult to realise that energy is released when bonds form (Barker and Millar, 2000).

Apart from the use of the substance metaphor in relation to extensive state functions, such as energy and entropy, it also applies to a wider range of words in everyday language. For example, we tend to talk about heat as if it were an entity stored in warm objects, reminiscent of the obsolete caloric theory. This differs from the interpretation in thermodynamics, where heat is a process variable and refers to the process of energy transfer between two systems due to a temperature difference, or the amount of energy being transferred in that process. Brookes and Etkina (2015) show how individual students' use of the substance metaphor in relation to heat is strongly correlated to interpreting heat as a state function, thereby leading to a failure at solving problems involving isothermal compression where a system heats the surroundings at constant temperature.

As mentioned, Amin (2001) shows that the linguistic properties of these two words differ markedly, by use of examples from a text corpus of everyday language. Heat, as a noun in everyday language, is typically used to “designate a spatially localized causal entity that is independent of the object whose hotness is in question” (Amin, 2001, p. 7), sometimes by use of the substance metaphor:

The heat in the room was already intensely humid (Amin, 2001, p. 6).

As soon as the thigh meat is ready, remove it from the heat (Amin, 2001, p. 7).

In contrast, temperature is typically conceived of as a location along a vertical scale:

With almost clear skies and very little wind, the temperature will fall sharply this evening (Amin, 2001, p. 8).

We cannot readily talk about temperature as a substance. For instance, it does not make sense (denoted by the asterisk *) rewording the sentence above into substance-like language:

* The temperature will go away/disappear/leave this evening.

From the point of view of syntax, ‘heat’ and ‘temperature’ have different roles in how we structure sentences related to thermal phenomena in everyday language.

Talking about quantities in relation to a vertical scale is not limited to everyday language or to temperature. Dreyfus et al. (2014) show that negative energy or energy levels of a system are easily captured by the idea of energy along a vertical scale. Dreyfus et al. (2015) also provide examples of how teachers and students may use gestures referring to energy as a level on a scale on a graph, while at the same time talking of energy as if it were a substance, which shows the flexibility in how these construals can be used in conjunction.

In this study, we will use the words ‘more’ and ‘higher’ as alternative qualifiers of scientific terms, and argue that constructions involving ‘more’, such as ‘more heat’, indicate a substance-like interpretation. Following Karlsruhe Physics (Herrmann, 2000), it would be tempting to think that extensive quantities can generally be talked about as substances, whereas it is does not make sense for intensive quantities, such as temperature, which are not dependent on the size of a system. However, a closer look at other thermodynamics quantities gives a more complicated picture. Pressure, for example, is an intensive quantity. If two identical systems are considered together, the pressure of the combined system is the same as that of the original parts. Still, we can easily talk about pressure as if it were a substance, colloquially, but also in science discourse:

There is a lot of pressure on his shoulders.

As the reaction proceeds, the pressure builds up.

How much pressure should be applied to the system?

On the other hand, the substance metaphor does not apply to heat capacity, either in its extensive forms, C, or intensive forms, such as specific heat capacity, c, or molar heat capacity, C_mol. Compare, for example:

* There is a lot of heat capacity in the system.

The system has high heat capacity.

Only the second of these sentences makes sense.

Entropy is typically construed as a value along a vertical scale, just like temperature, and entropy change as rising and falling along this scale:

This strongly exothermic reaction results in an increase in the entropy of the surroundings as heat is released into them (Atkins, 1998, p. 102).

However, entropy may also take on some substance-like characteristics:

If hot spots do form, then the localized energy may subsequently disperse spontaneously and hence generate more entropy (Atkins, 1998, p. 102).

As we have seen, the Karlsruhe Physics Course (Herrmann, 2000), makes explicit use of the substance metaphor in the introduction of entropy:

We thus have the rule: The higher the temperature of an object, the more entropy is contained in it. The greater the mass of an object, the more entropy is contained in it (Herrmann, 2004, p. 38).

Even though talking of entropy as a substance-like entity might sound odd to the typical chemist or physicist, this approach is sometimes adopted also in engineering thermodynamics (e.g.Fuchs, 2010; Gaggioli, 2010), in modelling flows of matter, energy and entropy far from equilibrium, for instance in jet engines or turbines. Entropy, as in the case of amount of substance, is not conserved, but will be generated in irreversible processes, and this has to be brought up explicitly if the substance metaphor is to be adopted in the teaching of entropy.

A questionnaire study on views of entropy among chemical thermodynamics student

As mentioned, we performed a questionnaire study with the same cohort of chemical thermodynamics students as in the current study (Andersson et al., 2015; Haglund et al., 2015). Students were given questionnaires before and after the course in chemical thermodynamics, where they were asked to give a brief explanation of what entropy is to them, and list the most important scientific concepts that connect to entropy. The large majority of students engaged with microscopic explanations, involving molecular interaction (Haglund et al., 2015). Most of them made use of the disorder metaphor, however after the course often with a more problematising, nuanced view, where they related disorder to other explanations. Few of the students provided macroscopic accounts or brought up a connection between entropy and the second law of thermodynamics. Enthalpy and Gibbs free energy were the most common scientific concepts the students related to entropy, which was seen as an indication of their coming to adopt a chemistry disciplinary tradition (Christensen and Rump, 2008). Intriguingly, no correlations were found between the students' responses to the questionnaires and their results on the written exam of the course, which was dominated by quantitative problem-solving exercises. Some differences were found between students from different study programmes (Andersson et al., 2015). Students who had not taken a lot of chemistry before the course, for example those at the Environmental and Water Engineering programme, tended to adopt the disorder metaphor or use it in conjunction with other microscopic aspects of entropy. Students who had taken more chemistry courses before the course, at the Chemical Engineering programme, typically related to the disorder metaphor beforehand. These students tended to develop a more multifaceted view or abandon the disorder metaphor in the responses after the course. In all, it seems that seeing entropy as disorder, but also coming to problematise it, may be a useful resource at early stages in students' learning of the concept.

Purpose of the study

The aim of our overall research endeavour has been to contribute to the understanding of how engineering students interpret the entropy concept, in connection to a course in chemical thermodynamics. Building on the findings of our previous questionnaire study (Andersson et al., 2015; Haglund et al., 2015), the purpose of the present study was to probe deeper into how students talk about entropy and how their understanding of the concept has developed. In line with the overall intention of the theme ‘Language and the teaching and learning of chemistry’, we attended to language aspects of the students’ interpretations, in particular with the ambition to explore the syntax, semantics, and pragmatics of the use of entropy. Against this background, our research was guided by the following research questions:

• Do chemistry students treat entropy as a substance-like entity?

• How do chemistry students rank a set of given notions with regards to: how closely related to entropy the notions are; how scientific they are; how useful they are for explaining what entropy is?

• How do chemistry students approach qualitative problem-solving exercises, which involve entropy?

Method

Study design

We used pair interviews in order to enact a dialogue which closely resembles an authentic collaborative learning setting, in this case with one of the researchers moderating the dialogue. In this way, students were allowed to share ideas with each other in the thought process, work together to elaborate their thoughts, before coming to agree on how to express ideas.

The first task related to students' view of in what types of sentence structures entropy could be involved, i.e. the syntax of entropy. Students were presented with two thermodynamic systems which differed in that one had higher temperature than the other (see Appendix A). Using pictures of people and speech balloons, students were asked to assess whether it would make sense to say that system B has “higher temperature” than system A, or that it has “more temperature”. The same formulations were then used, but with internal energy and entropy replacing temperature. The task was designed to probe to what degree the students found it possible to consider the three physical quantities – temperature, internal energy, and entropy – as substance-like entities, as indicated by using ‘more’ as a qualifier of the terms.

The second task was designed to illustrate aspects of semantics: the students' view of the meaning of entropy. Through the technique of ranking tasks (O'Kuma et al., 2000), here applied to qualitative understanding (Larsson et al., 2011), students were asked to rank nine notions (see Appendix A), with respect to: how closely related to entropy they are; how scientific they are; and, how useful they are for explaining what entropy is. The set of notions were provided in the form of nine plastic cards to arrange.

The third task involved solving two qualitative problems (see Appendix A), which were introduced in order to study the pragmatics of language, in the sense of how entropy is used in dialogue in practice. The first problem was taken from the exam of the course, which they all had passed. It involved determining whether entropy and enthalpy increase, decrease or remain the same, as water leaves a hydrated salt in the gas state, and justifying the answer. The second problem, introduced in the theoretical background, involved determining what happens to the entropy of an ideal gas as it expands adiabatically (Brosseau and Viard, 1992).

A final fourth task involved students' reflection on how they have learned about entropy at different educational levels, and to what degree they see problem-solving skills as related to their conceptual understanding of entropy.

The students were selected and paired together against the background of their responses to the previous questionnaire (Haglund et al., 2015), and their exam results. We wanted all participants to have done well on the exam (all pass, above than average score) and represent a wide range of ideas about entropy as reflected in the questionnaire responses. In addition, they represent the two largest study programmes on the course, two pairs studying Chemical Engineering, and one pair studying Environmental and Water Engineering.

Students' prior experience of entropy

The involved students had first encountered entropy as part of the Swedish upper secondary chemistry curriculum, typically with a qualitative focus on entropy as disorder. The concept is mainly used in explaining spontaneous endothermic reactions. Prior to the course in chemical thermodynamics, they had taken general chemistry courses, although to different extent across the study programmes. With regards to entropy, the general chemistry courses focused on calculating changes of enthalpy, entropy and Gibbs free energy in given chemical reactions, still relying on the disorder metaphor for a qualitative understanding of entropy.

The chemical thermodynamics course used Atkins' physical chemistry, 9th edn (Atkins and De Paula, 2010) as course literature, and had a traditional structure with lectures, problem-solving sessions, and laboratory exercises. Apart from training quantitative problem-solving skills, students were encouraged to develop a conceptual understanding of the taught content, for instance by use of qualitative problems to be discussed in small-group settings. A microscopic understanding of entropy was emphasised, adopting Boltzmann's approach to entropy, as related to the number of microstates of a system. In the course, the lecturer explicitly pointed out limitations of the disorder metaphor, including that crystals may form into layers where molecules have high freedom to move in the layer (high ‘order’ going together with high entropy).

Data collection and data analysis

The pair interviews were conducted approximately a year after the students had finished the course. Informed consent was gathered from all participants and they volunteered to take part in the study. In order to guarantee anonymity, fictitious student names are used throughout the text. As a token of our appreciation, they were given two cinema tickets each for participating in the study. The interviews were conducted in a conference room by one of the researchers (JH), and data was gathered by video recording with a tripod-mounted video camera. All pair interviews were conducted during slightly less than one hour each.

All authors watched the videos separately and joined to agree on noteworthy sequences and themes, with a focus on how the three student pairs interacted with each task. One of the authors transcribed the selected sequences in the native Swedish, and translated them to English for presentation in the Results and Discussion section. Precise wordings of the translations have been discussed among the authors. Due to the language focus of the theme issue, we have taken care to make footnotes where we experienced problematic phrases, the meanings of which do not carry over easily into English. In our subsequent joint analysis, we attended to how the students' interactions with the tasks related to the responses to the preceding questionnaire study. In addition, we related their interaction to previous research on students' understanding of entropy.

Results and discussion

In this section, the student pairs' dialogue is presented and compared task by task. For transparency, we have chosen to share rather extensive transcript excerpts, with the ambition of allowing the reader to follow our interpretation and possibly come up with alternative readings of the dialogue.

Does it make sense? – “It is much more unreasonable to say more temperature than more entropy”

In the first task, the student pairs are asked to assess whether a set of statements involving thermodynamics terms make sense. The students' reasoning patterns and assessments of what sentences involving the terms temperature, internal energy and entropy sound reasonable are similar across the three pairs. They initially rely heavily on their intuition for the language used, in reporting what “sounds” or “feels” right, or otherwise wrong, “strange”, or “funny”. This is followed, to different extents across the pairs and terms, by some kind of rational justification, relating to what the three terms represent.

In relation to the first pictures, the student pairs all agree that it sounds odd to talk about “more temperature”, and spontaneously contrast with “more energy” or “more heat”, which makes more sense to them. In line with Amin (2001), the use of the word ‘temperature’ differs qualitatively from the use of ‘heat’ and ‘energy’ in the thermal domain, according to the students' intuition. This is exemplified by Anna and Nisse's reaction to “higher temperature” and “more temperature”:

Nisse: Well, the first statement [“higher temperature”] feels very reasonable… you talk about higher and lower temperatures, right…/…/ But you rarely… or hardly ever, really… talk about more temperature… You can think of more heat… but temperature is just… well, the unit, so to speak… for heat…

JH: [Turns to Anna] What would you say…? Do you have the same gut feeling?

Anna: Yes, precisely. It sounds really strange to say that something has more temperature.

Here Nisse seems to assume an everyday language view on heat as something that a system can contain, and temperature as a measure of its hotness.

This unease with “more temperature” is shared with Kajsa and Pelle, who also come further in their justification:

Pelle: “Higher” [temperature] certainly sounds more reasonable.

Kajsa: Yes.

Pelle: “More” feels really strange.

Kajsa: More temperature is maybe a strange thing to say, but is it wrong?

Pelle: But what would more temperature mean? More energy…?

Kajsa: Mm.

Pelle: Temperature is Celsius, right… the unit… and that is a measure of…/…/ some kind of motion…

Kajsa: So, if temperature is motion, then more temperature is more motion…/…/

Pelle: If we say that temperature is some kind of motion… let's go for that… then more temperature would mean that… [if] you have more water… then you would get more temperature… and that would be very irrational, in a way… if you think about how the temperature concept is used.

Kajsa and Pelle entertain the idea of what “more temperature” could mean, and by relating temperature to motion, as in the kinetic theory of gases, infer that more temperature might mean more motion. When probing further, however, Pelle realises that this would entail that more water would mean more temperature, which he deems “very irrational”. This irrationality seems to be grounded in an understanding of temperature being an intensive, size-independent quantity, which would not change by adding more water.

Next, the pairs are shown the statements involving “more internal energy” and “higher internal energy”. In contrast to the case of temperature, all pairs, to different degrees, are comfortable with both “more energy” and “higher energy”. The students in the third pair, Olle and Kalle, think that both “more internal energy” and “higher internal energy” make sense. With more energy in mind, the dialogue excerpts reveal a substance-like view of energy (Scherr et al., 2012), where it can be added to a system, in terms of more amount of substance, or quite viscerally as Olle says: “like something that you can put in a can”. Then again, in line with Dreyfus, et al. (2014), Olle sees an advantage in conceiving energy along a vertical scale, in that the zero level† can be positioned arbitrarily, whereas negative energies are difficult to imagine with a substance metaphor in mind:

Kalle: In one way, I think both work… quite well…

Olle: I think… more energy, I'd say… well, you can say higher energy, as well…

JH: In what way does internal energy differ from temperature, so that both these sound reasonable…?

Olle: Energy, I've seen it more like something that you can put in a can…/…/ You can put more and more… [gestures moving things horizontally] whereas temperature has been more a thermometer [gestures two levels horizontally] a scale you can fill up… or, I mean, it increases to a higher value… [gestures upwards] and then, more energy feels more reasonable to use… than more temperature… but in both… you still use the notion of higher energy…

Kalle: Yes, I think… I feel comfortable with both… [points to both speech bubbles]/…/

Olle: On the other hand… with energy, you can put the zero wherever you want…

Kalle: Yes, that's true…

Olle: But I… I think I'll go with the left alternative here too [points to “higher internal energy”] that it's higher energy, still…/…/

Overall, Kalle's conclusion that both “more energy” and “higher energy” make sense, and Pelle's fine distinctions between what they would mean, adheres to the view of Dreyfus, et al. (2014), in that these two ways of conceiving energy capture different aspects of the physical quantity, and may both be useful in different contexts.

As for Kajsa and Pelle, they also start by considering what feels or sounds good. Kajsa reads aloud, “savours” the words and wonders: “can you say higher energy?” Similar to Kalle and Olle, they conclude that both ‘higher energy’ and ‘more energy’ make sense, albeit with subtle differences. Higher energy has an intensive character to Pelle, as he sees it as a case of more energy per particle or for a given amount of substance, while more energy entails adding more substance:

Pelle: Well, then suddenly ‘more’ [internal energy] feels better, I think!

Kajsa: Yes, but spontaneously, I'd say that both work fine. [Reads aloud:] “System B has higher internal energy”, higher energy… can you say higher energy…?

Pelle: You can, but…

Kajsa: …but it's more correct to say more energy…

Pelle: Then you would say, like, more potential energy… [gestures a change in vertical position] more… it feels like a quantity you have…/…/ But if someone said higher energy to me, I probably wouldn't…

Kajsa: …have noticed that it was strange… like the other one, with more temperature…/…/

JH: Would they mean roughly the same thing [points to the statements of the speech bubbles] or would there be any differences?

Pelle: I mean, I think it has to do with… in this case, how much… well, more energy… [gestures holding something with both hands] you can fix more energy by having more particles… [gestures adding more particles] at the same temperature. So, if we had two moles, then we'd have had more energy… in, like, that part of the room… but higher energy… I'd say that you increase the energy [gestures along vertical scale] of this mole./…/

Kajsa: Yes, that seems quite reasonable. So, you mean that higher energy is per particle?

Pelle: Yes, I look at per particle. More energy is the sum of all… So if I took another mole of water, I wouldn't have got higher energy… it would have been more energy.

Finally, the three pairs are given the speech balloons involving “higher entropy” and “more entropy”. Although the student pairs intuitively prefer “higher entropy” to “more entropy”, they are less assertive in their judgements than in the case of temperature. According to Kajsa and Pelle, “more entropy” sounds “strange”, and Pelle explicitly gives the contrasting case that with more energy, the number of ways to arrange the system will increase, and thereby also the entropy will increase. Still, they have difficulties providing justifications of why “more entropy” sounds strange:

Pelle: Here, I want “higher” [entropy], actually…! [they laugh] Rather than “more”.

Kajsa: Yes, quite spontaneously.

Pelle: But it's hard to say why./…/ But with more energy, we should get more ways to arrange ‡ the particles… we get more energy levels… [gestures movement between levels vertically] one can move…

Kajsa: Well, evidently, warmer water has more… higher entropy…/…/ and I'd say that it is higher entropy…

Pelle: But could we say that more is wrong…?

Kajsa: No, I couldn't say so, spontaneously…

JH: Does it sound… [puts images of temperature and entropy next to each other] in comparison to “system B has more temperature”… and “system B has more entropy”… does this [“more temperature”] sound even more strange?

Kajsa/Pelle: Yes.

JH: But this [“more entropy”] also sounds a bit strange.

Kajsa: Yes, if I choose between these two… if I were writing a lab report, I'd write higher…

Pelle: But it's hard to say if it is wrong or not…!

JH: But this [“more temperature”] is more clearly…

Kajsa: It's more wrong! [laughter]

Olle and Kalle also prefer “higher entropy” to “more entropy”. However, they come a bit further in their justification, by returning to the idea of energy as something that you can put in a can or bowl, which in their view is not the case with temperature or entropy. In addition, Olle describes entropy in terms of a location along a path or vertically, which follows a pattern that has been identified in textbooks (Amin et al., 2012) and in student problem-solving dialogue (Jeppsson et al., 2013):

Olle: Well, language wise, I think higher entropy… that's what one uses most often…

Kalle: Yes, I agree… higher entropy… I would also say that… more entropy feels…/…/

Olle: On the other hand, if you connect to… well, in the way I think… connect to a degree of disorder… there is more… or a higher degree of disorder, or… than, that… [points to the speech balloon with “higher entropy”] how shall I put it… it isn't disorder… but something along those lines…

Kalle: A higher degree of freedom to move…/…/ Well, I think this sounds [points to “higher entropy”] the best… but I actually cannot explain why I think so… [they laugh] because actually, it shouldn't be wrong to say both.

JH: So, hypothetically, why is it that also that one [points to “more entropy”] could make sense, in some way…?

Kalle: With the same line of reasoning we had before… that I think energy can be something… more energy in a system… but, well… that sounds funny… [laughs]

JH: [brings forward the temperature picture] If you compare with temperature… which you thought… you thought that it was completely unreasonable to say more temperature… is it just as unreasonable to say that a system has more entropy…?

Kalle: No, I think it is much more unreasonable to say more temperature than more entropy… on a scale… it is closer to OK to say more entropy, than it is to say more temperature…/…/ I think, to me, it is a matter of… with entropy, I think degree of disorder... and when it is a matter of degrees, then it is like temperature… higher degree… and not more degree… but it is really just a language thing.

Olle: But what I think, spontaneously… those on the right side [points to the statements involving “more temperature”, “more internal energy” and “more entropy”] that use the notion “more”… you have this bowl… you can put things in it [they both gesture adding stuff into a bowl] you can put energy in there…/…/ or put marbles in the bowl… whereas for the other [statements involving “higher”] it feels more like… [gestures movement vertically] something you measure…

Kalle: How something is, really… You cannot put in entropy [gestures adding something to a bowl] but it is how something is…

Olle: Or how far it has travelled along a path [gestures movement forward] or up a mountain [gestures upwards]/…/ And then, entropy feels more where you are located along the path…

Kalle: So these two [images involving temperature and entropy] are more states… and this [“internal energy”] is something that you can actually, “physically” [gestures quotation marks] put in and take out [gestures adding and taking away something]

In our reading Olle and Kalle end up with the position that temperature and entropy are abstract state functions, while energy can be given a more concrete substance-like interpretation, something that can be put into a system. It is also worth noting that Pelle engaged the intensive character of temperature in deeming “more temperature” unreasonable, and in interpreting “higher energy” as related to a given amount of particles or amount of particles. In contrast, he does not consider the extensive character of entropy in his line of reasoning here. The fact that the students see “more temperature” as “more wrong” than “more entropy” may stem from a difference in exposure to temperature and entropy, in everyday life, as well as in science discourse. They may not yet have appropriated entropy in a way to be able to say what sounds stranger than something else with confidence. Another interpretation might be that they appreciate the extensive character of entropy implicitly, in the light of which “more entropy” is not inconceivable, as argued by e.g.Fuchs (2010).

In addition, in this excerpt, Olle and Kalle involve “degree of disorder” in talking about entropy. The other pairs also relate entropy to disorder in this task. In particular, Nisse points out that disorder is a “dirty word”§ and that entropy really has more to do with molecular motion than disorder. This metaphorical use of entropy as disorder relates closely to how students on the course explained entropy in our preceding questionnaire study (Haglund et al., 2015), and, as we will see in the subsequent ranking task, is an important tool for them in interpreting what entropy is.

Ranking tasks – “It is important to know it, but I wouldn't be able to justify why”

In the second task, the student pairs are asked to rank nine notions with regards to: how closely related to entropy they are; how scientific they are, and; how useful they are for explaining what entropy is. For a summary of the overall result among the interviewed students, the averages of the student pairs' rankings are shown in Table 1 (the individual student pairs' rankings are detailed in Appendix B).

Table 1 Averages across the student pairs' rankings of: how strongly notions are related to entropy; how scientific they are; and how useful they are for explaining what entropy is

	Related to entropy	Scientific	Explaining entropy
Disorder	2.17	7.67	1.50
Freedom	2.17	7.33	4.50
Microstates	4.67	4.00	4.50
Motion	5.67	6.17	4.83
Heat	5.67	6.83	5.50
Second law of thermodynamics	5.67	2.33	5.83
Spreading	6.00	7.00	6.00
Gibbs free energy	6.33	1.67	6.17
Enthalpy	6.67	2.00	6.17

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The first thing we would like to point out is the striking similarity in ranking between the relatedness and explanation categories. The average orders in which the notions are ranked are identical, and the correlation between the variables (measured with Spearman's rho) is ρ = 0.966, with a 2-tailed significance of 0.000024. For the relatedness rankings, ‘disorder’ and ‘freedom’ stand out as clearly more related to entropy than the others, which are grouped quite closely together. In the ranking of how useful the notions are for explaining entropy, ‘disorder’ is seen as exceptionally useful, and the others are grouped together. These patterns emerge clearly at this average level, against the background of some variability across the pairs' individual rankings (as seen in Appendix B). We come back to some of the notions with the largest differences across the pairs in the detailed analysis below.

In contrast to the similarity between how related to and useful for explaining entropy the notions are, the ranking of how scientific the notions are is very different. In fact, this variable shows a negative correlation with the relatedness (ρ = −0.724, significance: 0.028) and the usefulness (ρ = −0.731, significance: 0.025) variables. In other words, the more scientific a notion is seen to be, the less related to and useful for explaining entropy it is seen to be among these students.

This negative correlation between the degree to which a notion is scientific and how useful it is for explanations may be surprising at first sight. Would not scientific terms, which are clearly defined in relation to other terms in a coherent system, such as the International System of Quantities (ISQ) or the International Systems of Units (SI), be expected to have higher explanatory value than a set of vague ambiguous everyday words? However, this result fits well with the tension between disciplinary affordance and pedagogical affordance, as proposed by Airey and Linder (2015). Semiotic resources that have high disciplinary affordance, such as mathematical relations between physical quantities, may be hard to decipher for learners. Other semiotic resources may have high pedagogical affordance, in helping students to understand what entropy is, but be practically useless to the expert scientist due to their cumbersome and imprecise nature. Various metaphors for entropy may belong to this latter category.

If we look at the rankings of the individual notions, ‘enthalpy’, ‘Gibbs free energy’, and ‘the second law of thermodynamics’ were typically seen as highly scientific, but not strongly related to entropy or useful for explaining what it is. Conversely, ‘disorder’ was seen as strongly related to entropy and useful for explaining it, but very unscientific. To the expert scientist, Gibbs free energy provides an efficient way to operationalise the second law of thermodynamics, by condensing enthalpic and entropic aspects of the interaction between a thermal system and its surroundings into one single formula. Gibbs free energy has high disciplinary affordance, in the sense of being a valuable resource for conducting and communicating research in thermodynamics. In contrast, for a novice who is unfamiliar with thermodynamics, the relation between Gibbs free energy and entropy is opaque. In order to become useful for the novice, this relationship has to be unpacked in a carefully designed teaching effort. Conversely, the idea of entropy as disorder has, as expressed by Lambert (2002), very limited value for the scientist and in more advanced levels of teaching. It has little disciplinary affordance. However, as seen in the evidence here, and in our preceding questionnaire study with the same student group (Haglund et al., 2015), undergraduate students find the disorder metaphor useful for explaining entropy, even though they are aware that disorder is not a scientific term and that the metaphor has shortcomings. Thus it appears to have great pedagogical affordance.

In their reasoning of whether a notion is scientific of not, the students typically contrast scientific notions with words that are used in everyday language, or, as Nisse expresses it, “at home in your kitchen”. In this way, ‘microstates’ and ‘Gibbs free energy’ are seen as scientific, while ‘disorder’ and ‘freedom’ are not. This is a kind of negative definition, which describes what characterises a non-scientific term, but not scientific terms themselves. The students may have been led to this one-dimensional scale by the nature of the ranking task. It does not cater for ambiguous, polysemic terms, such as ‘heat’, which has a precise, scientific meaning, but also other, related meanings in everyday language.

While the students do not comment on subtleties across the scientific−everyday divide, they bring up that ‘disorder’, ‘freedom’, and ‘spreading’ can have different everyday meanings, such as whether it is a matter of freedom to move or freedom in terms of many alternative locations. For instance, Kalle comments on the ranking of how strongly related ‘spreading’ is to entropy: “Well, that depends on how you define ‘spreading’!” When probed about the criteria for assessing how scientific the terms are, Olle says: “Well, most of these [notions] can be interpreted in many different ways. But it feels as if the ones that are more precise [the more scientific terms] are at the top, and then it gets more and more diffuse.” They rank ‘microstates’ as 4 and ‘spreading’ as 9, which Olle explains: “It is clearer what a microstate is than a spreading.”¶

The students' ways of interacting with ‘disorder’ stand out in the ranking tasks. The pairs rate it as strongly related to entropy (ranking: split 1, 2, and 3, among the nine notions), useful for explaining entropy (ranking: 1, 1, and split 2), but unscientific (ranking: 9, 9, and 5). There is an emotional charge that is expressed most clearly in the ranking of how scientific the terms are. Following directly after the assessment of the degree to which they are related to entropy, in which entropy was at the top, they quickly remove the entropy card when they hear ‘scientific’. During the lectures of the course, Pelle was made aware that entropy as disorder is not a very apt metaphor. They were given the example that crystals may form in layers, characterised by high entropy but a seemingly ordered pattern. In analogy, books may have freedom to move about if they are placed in layers in a box, as opposed to when they are placed in a disorderly way and interlock. However, now in the exercise, Pelle considers this as an exceptional case, and still finds entropy strongly related to disorder: “In 2000 cases against one, it deserves to be there. And then, on some occasion, it does not.” In other words, even though disorder is not a comprehensive description of entropy, Pelle sees disorder as strongly related to entropy, and inadequate in only one out of 2000 cases.

In our view, the second law of thermodynamics is a central aspect of entropy. The students do not immediately see this connection. For instance, Kalle and Olle do not consider the second law of thermodynamics as strongly related to entropy, in spite of being aware that it entails that entropy is an increasing variable:

Kalle: I guess it [the second law of thermodynamics card] should be somewhere up here [ranked as strongly connected to entropy] but I think it is so messy, so it's not the first thing I get to think about…

Olle: Isn't it?

Kalle: No. I use it very… I don't really know it by heart…

Olle: Well, it says… Should I say what it says [looks at Kalle]?/…/ …that for every spontaneous reaction in an unlimited system… the entropy increases…

Kalle: But that is so abstract… so that it's hard for me to grasp it… I know that it exists, and kind of what it says, but I don't know how to, like, use it. Well, it increases… So…? Good for that one! [laughter]

In addition, they come back to considering the second law of thermodynamics in relation to how useful it is for explaining what entropy is, where they find it to have little practical value. Kalle expresses that he is aware that the second law of thermodynamics is supposed to be important, but that he cannot really relate to it himself. He sees it as too abstract, and does not know what to do with it. If Kalle were to explain entropy to junior students at supplemental instruction (Blanc et al., 1983), he would only be able to connect it to the second law of thermodynamics in the instrumental sense that it is likely to appear in the exam:

Olle: I think the second law of thermodynamics is a very… [places the card in the middle of the ranking] or, I don't know… maybe, I don't quite know what it means… but it is still a funny law to write down… [laughter]

Kalle: I have reservations about that, because I think it is really… explanation wise, I think it is terrible… it doesn't say anything…

Olle: That's true. To explain the entropy concept…

Kalle: I think… if I were to explain it at an SI [supplemental instruction] meeting… discuss entropy… I would maybe mention that it exists, and that it is important to know it, but I wouldn't be able to justify why it is important, or why one should know it… apart from that it will be in the exam…

The perceived weak connection between entropy and the second law of thermodynamics reinforces the findings from our previous questionnaire study with the student group, where only few of them related entropy to the second law of thermodynamics, and those who did, in terms of entropy ‘striving’ towards a maximum (Haglund et al., 2015). The finding also reflects students' unwillingness to apply the second law of thermodynamics in problem solving exercises, for instance by use of the formula ΔS ≥ Q/T, as reported in previous research (Thomas and Schwenz, 1998; Haglund and Jeppsson, 2014; Loverude, 2015).

While Kalle found it difficult to apply the second law of thermodynamics in problem solving, this is not the case for Gibbs free energy. Olle gives examples of calculations of changes in Gibbs free energy, once enthalpy and entropy changes are known, from which the spontaneity of reactions can be assessed. Kalle indeed sees these three quantities as a “trinity” in chemistry, which reflects the high prevalence of enthalpy and Gibbs free energy as scientific concepts the students identified as closely related to entropy in the preceding questionnaire study (Haglund et al., 2015):

Kalle: I think Gibbs free energy is a… comes pretty high up [in the ranking] because if you are going to explain something, it is good to have an application… and then Gibbs is good… you use it in this way…/…/

JH: In what kinds of applications…?

Kalle: I mean, if you want to look at equilibrium… whether a reaction will happen… and how the entropy relates to this… I think Gibbs is good to have…

Olle: In our exams… there were often tasks based on, largely… this system is getting colder… or a fizzling bicycle tyre… one could decide whether the entropy increased and the enthalpy increased… and thereby, whether it was spontaneous, if [the change in] Gibbs free energy was negative… and in that way, one could connect the entropy to Gibbs free energy. And there, the enthalpy is just as important, I think… [places enthalpy at the same ranking as Gibbs free energy] if you connect it to a system… You need these two together… [points to enthalpy and Gibbs free energy] in order to give an explanation of the entropy…

Kalle: They are like a trinity… when you talk about them… in chemistry, at least… Gibbs, the enthalpy and the entropy.

The connection to Gibbs free energy reinforces the view of the students having come to adopt a chemistry disciplinary tradition (Christensen and Rump, 2008). As reported by Geller et al. (2014), Gibbs free energy may be a productive, familiar starting point in discussions of entropy and the second law of thermodynamics for life science students, and as we see here is useful also for chemistry students.

Microstates are one of the notions with the largest differences across the three student pairs' rankings. Olle and Kalle adopt a view from statistical mechanics, when assessing how strongly related microstates are to entropy:

Olle: Microstates… I have the image of entropy from a chess board… where you can put different pieces with different colours [gestures placing the pieces on the board] and that will be the actual microstate…

Kalle: Exactly. What it looks like at that moment… and if you change it… [gestures shuffling the pieces] you will get another microstate.

Olle: Precisely… so, in how many ways you can arrange || it. And then I think that this one [picks up the microstates card] ends up very high. I think it is above disorder, because perfect order is also a microstate, which yields an entropy.

In contrast, Anna and Nisse associate microstates to the microscopic level in a more vague fashion. When they have placed out some cards as strongly related to entropy, JH asks if any of the remaining notions are less related to entropy:

Nisse: Well, yes and no… I mean, microstates are maybe not the first thing you get to think of... but still, it deals with atoms and molecules…

Anna: Uhum… [nods]

Nisse: And that is on a very… well, on a micro scale… even smaller… it's small…

Consequently, Anna and Nisse find microstates to be the notion with the weakest connection to entropy out of the nine ones. In our view, there is a difference between the pairs in the degree of sophistication with which they have adopted a microscopic interpretation of entropy, which is reflected in the previous questionnaire study (Haglund et al., 2015). Students were seen to incorporate more microscopic aspects into their explanations of entropy after the course than before, but with large individual variation. In addition, this variation between students was found to depend on their study programmes (Andersson et al., 2015). This is the case also here, where Olle and Kalle study the Chemical Engineering programme, and Anna and Nisse study Environmental and Water Engineering, where the chemistry content is less pronounced.

Problem solving – “If the temperature decreases, even though the particles get more space to move about, these two cancel out”

In the third task, the student pairs are given two problem-solving exercises. In relation to the first problem, all three student pairs correctly say that the entropy will increase as a hydrated salt separates into the pure salt and water in the gas state. All three pairs make use of what Olle calls a “simple rule” in justifying their answer, in that “the entropy increases in the direction where there are more moles of gas”. Kajsa simply says: “There are more particles”, in reference to the products of the reaction. Although not a water-tight principle, it is a rule of thumb that helps the students get an initial feeling for the situation. The rule is also reflected in chemistry textbooks on the matter, e.g.: “Fewer molecules means fewer possible configurations./…/ In general, when a reaction involves gaseous molecules, the change in positional entropy is dominated by the relative numbers of molecules of gaseous reactants and products” (Zumdahl, 1998, p. 416, italics in the original). In addition, Anna points to the increased disorder in going from solid to gas states, while Olle and Kalle turn more fundamentally to the multiplicity of ways the water molecules can be arranged in the system.

Assessment of whether the enthalpy increases or decreases turns out to be more challenging for the pairs. Anna and Nisse justify their answer with the idea that gases are generally warmer than solids, and that you need to add heat to “melt” solids; hence the enthalpy increases. The other two pairs, however, relate the enthalpy change to the entropy change by considering that spontaneous changes in a system are associated with a decrease in Gibbs free energy, which makes it more complicated:

Kalle: Whether it's spontaneous or not doesn't have to do with delta H… because it has to do with both delta H and delta S… if the entropy increases, the enthalpy can also increase… it's just a matter of the sum between them…

Olle: What did you say?

Kalle: I mean... we have concluded that the entropy increases… [writes with his finger on the table] then all of it… all is negative…

Olle: You mean delta G is negative…?

Kalle: Delta G is negative, if the enthalpy is the same… so the enthalpy could actually increase a bit… that wouldn't matter…

Olle: No. But can we say anything about the enthalpy from this? Could it be zero, positive, negative…?

Kalle: We don't know if we heat it or not…

By considering the formula ΔG = ΔH − TΔS, Kalle points out that the increasing entropy gives a negative contribution to the change in Gibbs free energy. They realise that this does not help them in concluding whether the enthalpy increases, decreases or remains the same. The enthalpy may increase and thereby give a contribution in the unfavourable direction, so long as it is counteracted by a greater entropic contribution. This reinforces their view of a “trinity” between entropy, enthalpy and Gibbs free energy within the chemistry disciplinary tradition (Christensen and Rump, 2008), and the use of teaching entropy by reference to Gibbs free energy (Geller et al., 2014). By taking into account the molecular interaction, however, Olle and Kalle eventually settle correctly for the answer that the enthalpy increases for the given reaction, since you need to heat the hydrated salt to break the bonds between the salt and the water.

Two of the pairs engage the notion of activation energy, which may contribute to making the assessment of what happens to the enthalpy more challenging than expected. Nisse relates the activation energy to the issue of whether a reaction will happen spontaneously or not, whereas Kajsa considers the activation energy in relation to the energy needed to break chemical bonds, and the resulting enthalpy increase. This is similar to the findings of Cakmakci (2010), according to which students tend to confuse notions in thermodynamics and chemical kinetics, in particular believing that activation energy is identical to the amount of energy released by (or taken up in) a reaction.

As for the second problem, all three pairs initially intuit that the entropy will increase as an ideal gas undergoes reversible, adiabatic expansion. They reason that as the volume increases, the involved particles will have more available spatial configurations, and hence, the entropy increases. Two of the pairs engage disorder in their explanations. Kajsa says: “More space for the same number of particles, which gives the entropy… that is the disorder, possibility for disorder increases, so the entropy increases”. Similarly Anna says: “If there are x number of molecules in this [gas], they have more space to arrange** themselves. And, anyway, that is what I think entropy is.” These findings follow the results of previous research on students' challenges with reversible, adiabatic expansion, with an exclusively spatial interpretation (Brosseau and Viard, 1992; Haglund and Jeppsson, 2014).

Kalle and Olle engage the chessboard model of the system, and consider how it can be adapted to the situation of expanding volume:

Kalle: If you make, like, a square grid [gestures squares] with a gas in a volume… like a square grid… and then, you double the volume [gestures increasing squares] so you get squares of the “double” size… Can you think in that way…? [looks at Olle] Or is it wrong…? Do you get twice as many squares…?

Olle: You get the same number of pieces, but double the number of squares./…/

Kalle: It's like that, right…? You get more squares, but more squares are empty…

Here, Kalle starts off agnostic as to whether the squares increase in size or in number. The idea of increasing size of the squares holds the potential to visualise the constant phase space (representing positions and momenta of involved particles) or the unchanged Boltzmann distribution (Cartier, 2011), but the students leave the line of reasoning at this stage. They settle for that the number of squares increases, so that the configurational entropy increases.

In spite of this predominately spatial focus, the two Chemical Engineering pairs consider the temperature change that the system undergoes. Olle recognises a temperature decrease, but deems it irrelevant for determining the entropy change:

Olle: Let's see… the volume doubles… the pressure decreases, and then, also the temperature decreases, right…? It expands… So, we made it a bit too simple… But, I don't think that matters, really… spontaneously, I think the entropy increases.

Pelle, in contrast, thinks that the temperature influences the entropy of the system:

Pelle: [Reads:] “Without heat exchange” What happens to the temperature… just for laughs…? What should happen…? Something has to happen… It feels as if, from what I remember of this… it feels like the temperature of the gas must decrease…

Kajsa: Your male intuition?

Pelle: Precisely! Ouch! [laughs]

JH: So, according to your male intuition, what happens to the temperature…?

Pelle: I'd like the temperature to decrease… but I really cannot say why…/…/ doesn't it get warmer if you compress a gas…?/…/ so, logically, if we take the gas and pull it outwards, it gets colder… I mean, a bicycle pump gets warm… when you pump…

JH: But, if we imagine that the temperature decreases, would that have anything to do with the entropy…?

Pelle: Yes, because if the temperature decreases, then the number of ways… every particle gets less energy…

Halfway through the problem, JH reveals that the entropy of the system is constant through the reversible, adiabatic expansion, and asks the pairs to reconcile this with their ways of thinking. Since Pelle has already started to consider that the entropy may be related to the temperature, it is not inconceivable to him that the entropy might be constant during the expansion:

Pelle: OK, this could be the case. If the temperature decreases… even though they [the particles] get more space to move about… they also get less temperature††… and therefore, these two cancel out./…/ For think about… you set out… [gestures moving pieces] you have this chess grid… a square grid… but then, every particle has different rotation, vibration… and that means that you can distribute balls on top of each other… [gestures levels vertically] Do you remember that?/…/

Kajsa: Balls on top of each other…? [laughter]

Pelle: But I'm just thinking of the lecture here!

JH: So, what do these balls on top of each other mean?

Pelle: It's kind of a poor… but when you have different rotation and vibration states, you can see them as different energy levels… then you can place… one ball would be a particle… that you can have at different levels…

Kajsa: So, there is a lower temperature…?

Pelle: Yes.

In Pelle's view, there is a positive contribution to the entropy due to volume increase, but also a negative contribution due to decreased temperature. These contributions may well cancel out. Just like Kalle and Olle, Pelle also justifies his argument by comparing to a chess board, where balls can be positioned at different vertical levels on each square, corresponding to energy levels of the involved particles. Lower temperature means less energy and fewer ways to distribute the energy across energy levels; hence a negative contribution to the entropy.

Similarly, when Kalle is told about the constant entropy, he comes back to his model of expanding squares on a chess board:

Kalle: Well, I think about what I… if you have a chess board… with a certain number of pieces… and they have to be set out in a certain way… positions in a square grid… and then you increase the size of the chess board… [gestures expansion] then you have to increase the number… the size of the squares, as well… because the pieces have to be at an optimal distance from each other… so basically, there are just as many positions left…

It should be noted that all pairs use microscopic approaches to the problem. None of them makes use of the second law of thermodynamics, and the macroscopic equality ΔS = Q/T = 0 J K⁻¹, which holds for this reversible process. This reinforces the unease with the second law that was expressed in the ranking task, mirroring previous research (Thomas and Schwenz, 1998; Loverude, 2015) and the fact that few students connected entropy to the second law in the preceding questionnaire study (Haglund et al., 2015). From our experience, only graduate students, or undergraduate students who have encountered the problem before, have used a macroscopic approach to solving it (Jeppsson et al., 2015).

After Pelle has provided his microscopic account of why the entropy can be constant, JH offers the alternative macroscopic take on the problem, and comments that both ways work. However, Pelle realises that with the microscopic approach, you only know that the contributions counteract each other, not that they cancel out:

Pelle: The other way [the microscopic approach] we cannot say anything for sure from that one… one of them [decreased temperature] makes it decrease… and the other [increased volume] makes it increase…

JH: So, it's not unreasonable…

Pelle: …right, not unreasonable that it cancels… [gestures balancing]

JH: But, not evident, either… far from evident that they cancel out exactly.

Pelle: The other one [the macroscopic approach] is better. If we had remembered the equations …

JH: Which have to do with the second law of thermodynamics! [points to this card from the ranking tasks, and they laugh]

Here, Pelle refers to the relevance of remembering the relevant formulae. In fact, all three pairs comment that they could have made use of a collection of formulae and tables. In relation to the first problem, tables of the changes in enthalpy and Gibbs free energy would be useful, whereas they searched for appropriate thermodynamics formulae in relation to the second problem. Nisse comments that the second problem would have been much easier with a collection of formulae:

Nisse: Because then, with not too much effort, you can recall roughly what it says…/…/ but it's a bit more tricky for us to recall the course a year afterwards, because we don't use it much in our everyday life or our education…

Anna: Plus, we are engineers and engage in surface learning! [laughter]

Nisse says that the thermodynamics content is a bit distant now, since they have not dealt with these matters much since the course a year ago. Finally, Anna alludes – tongue in cheek – to the expectation of engineering students to adopt surface approaches to learning (Marton and Säljö, 1976), which incidentally at a meta level shows insight into educational theory.

Reflections on the learning of entropy – “to reason is good, because then you understand why you do the calculations”

As a final and fourth task, the student pairs are asked to reflect on their learning process regarding entropy, and how their conceptual understanding relates to their ability to solve problems. The students share the experience that in the introductory chemistry courses, taken prior to the course in chemical thermodynamics, entropy was primarily treated as a variable whose change was looked up in a table, in order to calculate Gibbs free energy and assess whether or not a particular reaction was spontaneous. This could be done without reflecting on what entropy really is. They were told that entropy could be seen as disorder, but it remained unclear in what respect. In the course in chemical thermodynamics, they were introduced to a microscopic view of entropy as related to the number of available configurations for the involved particles, and to tasks of a more conceptual character, such as the one involving water leaving a hydrated salt.

When asked about how problem-solving skills relate to conceptual understanding, Kalle does not see a strong connection in the structure of the courses he has experienced:

Kalle: I don't think they are connected… extremely well… [laughter] in all cases…/…/ I have been able to calculate the entropy really well… but the fundamental understanding… you need both… one of them was to calculate to pass the exam… and the other to reason and understand what it was actually all about… and they were not always connected, I think…

JH: What do you need both for?

Kalle: Well, learning to calculate is good, because that is what appears in the exam… to reason is good, because then you understand why you do the calculations…

In spite of the weak connection between problem-solving skills and understanding, Kalle still thinks that they are both important aspects in learning. This is interesting, not least in the light of our previous finding that students' results on the exam of the course were not correlated with their responses on the questionnaire with a focus on their understanding of what entropy is (Haglund et al., 2015).

Kajsa and Pelle, in contrast, see some connections. Kajsa appreciates that if you have in-depth understanding of involved concepts, you can assess whether the results of your calculations are reasonable. However, in previous courses, as Pelle expresses, it has been largely a matter of looking up values in tables in SI chemical data (Blackman et al., 2014), for which conceptual understanding was not required:

JH: Does this conceptual understanding help you in problem solving…?/…/

Kajsa: Yes, it is another way, and you can double-check… if you didn't know… shoot, I didn't find this in SI data then you could think in that direction, instead… you can go many ways to reach the same goal…

JH: OK, so two examples of ways… what could they be…?

Kajsa: Well, either pure calculation… or see if it is plausible… make a plausibility analysis… by use of your knowledge of entropy…

Pelle: You are more confident… whether you have written right or wrong… it feels better if you're on top of what you're doing… and not just plug things into formulae…/…/ but you can get by without it, also… understanding…

Kajsa: In particular, since most exam questions do not presume that you understand what entropy is…

Pelle: Typically, you open up SI data, and get a large table… it says enthalpy… entropy… [points to imagined columns] you get the values… add the temperature… next temperature… and then you calculate…/…/ In [the general chemistry course], if I understand the concept or do not understand the concept… I would probably have been able to solve the tasks equally well…

Kajsa: But in [chemical] thermodynamics, it was more advanced…

Pelle: They wanted to check that you knew it…

Kajsa: Discuss and find out why… explain in that way…

In addition, we would like to point to another potential connection between problem solving and conceptual understanding that the students do not touch upon. Students might solve problems with the intention to reach conceptual understanding through a procedural route (Case and Marshall, 2004). For instance, in the problem-solving dialogues, we see how Pelle realises that a positive and a negative contribution to the entropy change could cancel in adiabatic expansion, and how Olle and Kalle draw conclusions from the signs and values of the terms that add up to the change in Gibbs free energy when considering the expression ΔG = ΔH – TΔS. Now, Kalle indirectly draws on his understanding of the same formula when reflecting on his initial understanding of entropy: “some funny sort of energy… that you didn't really know what it was… and it didn't really matter at low temperatures, so you didn't have to consider it…” In other words, the lower the value of T, the lower the value of the term TΔS and the less significant it gets. In our view these are examples of how the consideration of mathematical formulae may contribute to qualitative, conceptual understanding of the thermodynamic processes at hand. In this regard, as recognised by Sherin (2001) in the context of physics learning, equations have a rich conceptual content, in the form of quantities that are opposing, competing, balancing, cancelling one another, etc. Learning to handle such equations is part of understanding what the involved quantities mean.

Finally, Kajsa and Pelle, and Anna and Nisse agree that they have not thought much of entropy since the thermodynamics course, and have forgotten much of the details. Olle and Kalle, in contrast, think that entropy is touched upon in many courses:

Olle: It comes back often… the entropy concept… in many courses, really… if you take chemistry… and there are slightly different descriptions of what it is, every time… often, you have some picture of it… and that picture, in my experience… changes quite often… since you take another course, get some more understanding… so, now, we're working with crystals, you think in one way… when you come to thermodynamics, you think in another way… if you turn to spectroscopy, a third way, maybe… you talk about the same thing, but from different perspectives… and then, you stress…

Kalle: …the most important parts…

JH: Do you feel that these parts… these dimensions… are they connected… or is it like talking about something new every time…?

Kalle: I think they are connected… quite well…

Olle: Yes, they are connected, at most times… but sometimes, different things are stressed as the most important ones… and then, they may get a bit distanced from each other…/…/ but, I think they are connected quite well… if you understand what entropy is… that it is possibilities to arrange things or whatever way you think about it… you will be helped in most courses… it's roughly the same thing you talk about…

Here, Olle expresses that as chemistry students they meet entropy in many courses, all with slightly different interpretations. However, against the background of a foundational, microscopic understanding of the concept, he feels that these interpretations are not too distant from each other, but rather relate to one another. In our view, these different perspectives contribute to developing a richer understanding of the complexity of the entropy concept. Still, as we have seen, the students in the present study embrace a primarily microscopic view of entropy, while macroscopic approaches seem to be limited to looking up values in tables to calculate the change in Gibbs free energy of different reactions. As pointed out by Baierlein (1994) there is a challenge in seeing how these different approaches to entropy relate to one another. It is worth noting that Olle and Kalle's experience of having encountered entropy in many courses differs from that of Kajsa and Pelle, even though they have taken the same compulsory courses at the Chemical Engineering programme. One significant difference might be that Olle and Kalle, but not the others, have engaged as mentors in supplementary instruction (Blanc et al., 1983), where they have had the opportunity to reflect more on the meaning of entropy.

Conclusions

We now turn to drawing conclusions from the results by revisiting the research questions of the study.

Do chemistry students treat entropy as a substance-like entity?

The brief answer to this question is: no, the student pairs do not treat entropy as a substance-like entity. From their experience of approaching entropy in a chemistry tradition, they are not comfortable with the expression “System B has more entropy than system A”, which would entail a substance-like conception of entropy. Instead, they prefer “System B has higher entropy than system A”, which more closely follows the conventionalised language in relation to entropy in chemistry.

In contrast, “more internal energy” makes sense to them, as something concrete that “you can put in a can”, as if it were an object or some kind of substance. “Higher internal energy” also sounds reasonable, but with slightly different connotations, involving intensity, such as energy per mole.

Still, the students find the notion of “more temperature” less reasonable than “more entropy”, and are able to articulate and justify their stance in relation to temperature better than they do with regards to entropy. This difference in security may be due to their larger exposure to temperature in everyday language, in school and at university, in comparison to entropy. Alternatively, it may reveal sensitivity to differences between extensive and intensive properties.

How do chemistry students rank a set of given notions with regards to: how closely related to entropy the notions are; how scientific they are; how useful they are for explaining what entropy is?

There are strong positive correlations between the student pairs' rankings of how closely related to entropy the notions are and how useful they are for explaining entropy. In contrast, these rankings are strongly negatively correlated to how scientific the notions are seen to be. As an example, disorder is seen as strongly related to entropy and useful in explaining what it is, but very unscientific. Conversely, Gibbs free energy and the second law of thermodynamics are highly scientific, but less strongly connected to entropy and less useful in explanations.

In our interpretation, this lends support to the suggestion by Airey and Linder (2015) of an inverse relationship between disciplinary affordance and pedagogical affordance. As pointed out by Lambert (2002) and others, the disorder metaphor for entropy has considerable shortcomings, and is far from a watertight principle in the consideration of many physical phenomena. There is no doubt that it has low disciplinary affordance, in the sense that is not very useful to practicing chemists. Then again, as expressed by the students in the current research, seeing entropy as disorder is useful for their understanding of what entropy is. To them, it may fail in one out of 2000 cases, but is still good enough to have explanatory value. From this perspective, talking about entropy as disorder is a semiotic resource with high pedagogical affordance. In contrast, the second law of thermodynamics is regarded as central to science, which entails high disciplinary affordance, but deemed to have limited value in the students' understanding of entropy and in practical problem solving, hence with low pedagogical affordance.

Bearing in mind the language focus of the current theme issue, it is worth pointing out that these Swedish students' interpretations of notions in scientific and everyday Swedish seemingly follow previous studies of the corresponding notions in English. The students' readiness to use the verb ‘ordna’, corresponding literally to the verb ‘order’ in English, in the context of combinatorics, how many ways a system can be arranged or configured, is an exception. This formal interpretation of order in mathematics and science in Swedish may have had an influence on the related disorder metaphor for entropy being more acceptable in Swedish than in English, where order comes across as vague and subjective (Lambert, 2002).

How do chemistry students approach qualitative problem-solving exercises, which involve entropy?

The student pairs adopt predominately microscopic approaches to both problems, where they consider the molecular interactions involved in the studied processes. Together with their comments that they would like to have had access to collections of formulae and tables, this microscopic perspective reveals identification with a chemistry disciplinary tradition (Christensen and Rump, 2008).

In their problem solving, the students make use of a range of models or imagery. For instance, they use the example of interlocked books in a box as an exceptional case where what looks like a disorderly state may come together with low entropy. As another example, they use a model of a chess board to discuss the multiplicity of possible configurations of chess pieces. The chess-board model is very useful and productive, in that the student pairs come back to it in different situations across the tasks. They also modify it in order to formulate their understanding of the problem-solving exercises. Pelle relates the chess-board model to stacking pieces on top of each other, which you do not do in a chess game, and Kalle considers whether gas expansion should be seen as a matter of squares increasing in size or getting more squares. Such models can be seen as another kind of semiotic resource, which complements the spoken language relating to thermodynamics, which the students have only partially come to master.

The students make little use of macroscopic lines of reasoning. This mirrors a reluctance of students to use the second law of thermodynamics as found in previous research (e.g.Thomas and Schwenz, 1998), a scarcity of macroscopic considerations in the students' questionnaire responses (Haglund et al., 2015), and the conceived limited use of the second law of thermodynamics as expressed in the ranking tasks. In the case of reversible, adiabatic expansion, this was a clear obstacle in the students' problem-solving process, where a macroscopic approach is more suitable than microscopic considerations. The two Chemical Engineering student pairs did consider Gibbs free energy, however, which involves interaction between macroscopic state functions, lending support to Gibbs free energy as a useful notion for students in entropy considerations (Geller et al., 2014).

Implications for the chemistry education research and practice

This qualitative study shows how three pairs of students think and talk about entropy across four tasks of different character. The strength of the study lies in the richness of the students' dialogue, how they make use of spoken language in closing in on what entropy means to them. Whether the expressed ideas and types of language use can be generalised to other groups of students and contexts is an important issue for further exploration in chemistry education research and practice.

The chemistry students we have studied adopt microscopic perspectives on entropy and thermodynamics at large. The fact that they consider particle interactions, in terms of molecules' disorder, or freedom to move about, is hardly surprising, given the chemistry focus of their education. However, as pointed out by Baierlein (1994), there lies an educational challenge in showing how microscopic approaches to entropy, involving the multiplicity or probabilities of microstates, relate to macroscopic thermodynamics, with a focus on the second law of thermodynamics and Clausius' inequality, dS ≥ đQ/T. It is apparent from our study that the students have difficulties seeing this connection.

From another point of view, however, in line with the findings of Geller et al. (2014), the students from the Chemical Engineering programme are comfortable seeing the influence of entropy changes on Gibbs free energy, by considering the expression ΔG = ΔH − TΔS, and drawing conclusions from tabulated values of changes of these quantities in chemical reactions. This is a macroscopic perspective, which involves changes in state functions of a system, but somehow its connection to the second law of thermodynamics has been lost to the students. As teachers, we have to help students make this connection, both as a fundamental aspect of what entropy is, and in practical problem solving.

We are intrigued and concerned about the fact that the students experience weak connections between problem-solving skills and conceptual understanding of thermodynamics. In this regard, the student pairs' reflections on the matter in the present study match the result in the preceding questionnaire study (Haglund et al., 2015), where their interpretations of what entropy is were uncorrelated with their exam results. If students' interactions with thermodynamics are characterised by a ‘shut up and calculate’ (Kaiser, 2002) attitude, there is little wonder they found the topic unengaging (Ugursal and Cruickshank, 2015). Then again, in their reflections the students acknowledge the value of conceptual understanding, and in our analysis we see that dealing with mathematical expressions is helpful to the students in their reasoning about the involved physical quantities. Once again, they will need help from teachers and in the learning environment at large to see such connections. Our impression is that lectures often incorporate conceptual approaches to taught topics, but that there is a scarcity of collaborative student exercises where they can put words on how they interpret the involved terms. In addition, exams are typically dominated by quantitative problem-solving, which at the end of the day determines what students prepare for.

Finally, we draw some implications from our study with direct bearing on the language focus of the current theme issue. Our analysis of the students' interaction with the tasks lends support to the view that matters of semantics, syntax, and pragmatics are highly interrelated, in line with Lemke's (1990) study of authentic language in the science classroom, Halliday's (1985) approach to functional grammar, and Lakoff and Johnson's (1999) overall thoughts on cognitive linguistics. For example, the reluctant reaction to the grammatical structure in ‘more entropy’ has bearing on the students' interpretations of what entropy means. A conceived lack of a connection between entropy and the second law of thermodynamics may well have prevented macroscopic approaches to the problem-solving tasks. Then again, relating disorder to the configuration of the pieces on a checkboard as a representation of possible energy states provided a productive context for the interpretation of this notoriously vague word.

This study was conducted in the native Swedish of the participating students and researchers. What would have been different in the context of another language, such as English? We experienced few problematic instances in the translation of dialogue transcripts from Swedish to English. As mentioned, one example of a difference between the languages is the adoption of the verb ‘ordna’ in formal scientific Swedish, which may have paved the way for a more positive view of the disorder metaphor than in English. In all, however, we have no reason to believe that inherent differences between the languages would have led to radically different results, if the study had been conducted in an English-speaking context. As for less closely related, non-Germanic languages, the matter might be quite different. Scholars such as Whorf (1936) have argued for linguistic relativity in the sense that differences among languages cause differences in the underlying patterns of thought. In his study of the native American language Hopi, Whorf (1936, p. 131) noted that the use of different verb forms was particularly sensitive to characteristics of vibratory motion, and claimed: “The Hopi actually have a language better equipped to deal with such vibratile phenomena than is our latest scientific terminology”. This was particularly striking at a time when novel wave-like aspects of nature were revealed in modern physics. One may speculate whether other non-Indo-European languages are particularly apt for expressing certain aspects of entropy, such as its probabilistic nature (although cf.Pinker, 1995, for a summary of critique against linguistic relativism in general).

The debate on metaphors in relation to entropy has largely dealt with identifying the most apt metaphor or combination of metaphors to be introduced to students in teaching (Styer, 2000; Lambert, 2002), on theoretical grounds. In the light of our findings, we rather suggest turning the focus to providing students with a broad range of complementary descriptions and explanations of what the complex concept entropy might be. In addition, we should invite students to analyse the merits and shortcomings of each description, because at some point every model, analogy or metaphor breaks down, where it is no longer a useful tool for understanding the topic at hand (Glynn, 1989). This is true for the disorder metaphor, but also for the spreading metaphor or entropy as diffusion which may entail interpreting entropy as a process rather than a state function, the kinetic theory of gases with unrealistic assumptions of particle interaction, etc.

More generally, for teaching purposes, we should strive for developing semiotic resources that have high pedagogical affordance, but are not necessarily efficient, succinct, and generally applicable, properties of resources with high disciplinary affordance. Ideally, some semiotic resources may have both high pedagogical affordance and disciplinary affordance. In this study, Gibbs free energy is probably the best candidate for such a combination. In addition, even though the chemistry students in this study do not treat entropy as a substance-like entity, it can be useful for teachers to be aware of this teaching approach, which has been adopted in engineering thermodynamics.

Appendix A. Interview questions

Task 1. Does this sentence make sense (cartoon speech bubbles pair-wise)?

Consider two systems. System A consists of 1 mole of water at 10 °C. System B consists of 1 mole of water at 30 °C.

Discuss to what degree it would make sense to say:

– System B has higher temperature than system A.

– System B has more temperature than system A.

– System B has higher internal energy than system A.

– System B has more internal energy than system A.

– System B has higher entropy than system A.

– System B has more entropy than system A.

Task 2. How does it relate to entropy?

(a) Rank how strongly these notions (provided on laminated plastic cards) relate to entropy. Justify the ranking.

– Enthalpy

– Gibbs free energy

– Microstates

– Heat

– The second law of thermodynamics

– Disorder

– Freedom

– Movement

– Spreading

(b) Rank how scientific the notions are.

(c) Rank how good they are for explaining entropy.

Task 3. Problem solving

(a) Specify for the following reaction, with justification, whether ΔH^θ and ΔS^θ, respectively, should be negative or positive.

Na₂SO₄·10H₂O (s) → Na₂SO₄ (s) + 10H₂O (g)

(b) An ideal gas is allowed to expand adiabatically (no heat exchanged with the surroundings), and reversibly to the double volume. What happens to the entropy of the gas in the process?

Task 4. Reflect upon what you have learnt about entropy

How has your understanding of entropy developed at different stages in your education? How does your understanding of entropy relate to your ability to solve problems in text books and exams?

Appendix B. The student pairs' ranking of notions in relation to entropy

	Olle and Kalle			Anna and Nisse			Kajsa and Pelle
	Related	Scientific	Explain	Related	Scientific	Explain	Related	Scientific	Explain
Disorder	2	5	2.5	1.5	9	1	3	9	1
Freedom	4	6	2.5	1.5	8	9	1	8	2
Spreading	8	9	9	8	7	2	2	5	7
Gibbs free energy	5	2.5	4.5	6	1	6	8	1.5	8
Enthalpy	7	2.5	4.5	4	2	5	9	1.5	9
Heat	6	8	8	5	6	4	6	6.5	4.5
2nd law of thermodynamics	3	1	6	7	3	7	7	3	4.5
Movement	9	7	7	3	5	3	5	6.5	4.5
Microstates	1	4	1	9	4	8	4	4	4.5

Acknowledgements

We would like to thank the interviewed students for their participation in the study. In addition, we are grateful towards John Airey for valuable discussions on the notion of affordance and comments on the text. The study was partly funded by the Centre for Discipline-Based Education Research in Mathematics, Engineering, Science, and Technology at Uppsala University.

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Footnotes

† Note that in contrast to energy, temperature and entropy have determinate zero levels. According to the third law of thermodynamics, at the absolute temperature of 0 K, the entropy of a perfect crystal is 0 J K⁻¹, which corresponds to one unique microstate.

‡ As a note on the translation from Swedish, Pelle uses the verb ‘ordna’, which corresponds literally to ‘to order’ in English. Whereas ‘ordna’ is an adequate term in combinatorics and probability theory in Swedish, other words would be used in English, such as ‘arrange’ or ‘configure’. This adequacy of ‘ordna’ in Swedish mathematics language may have had an influence on the view of the relative merits of the related disorder metaphor for entropy in Swedish, compared to in English.

§ Nisse uses the Swedish expression “med fula ord” and rolls his eyes to indicate that he knows that ‘disorder’ is not appropriate in relation to entropy in the science setting. The expression encompasses traditional “bad language”, but also sloppy, careless language of low prestige in general. It conveys the negative emotional charge of the disorder metaphor from the course experience.

¶ In Swedish, Olle says “en spridning”, which corresponds literally to “a spreading” in English. Both expressions come across as slightly awkward to us in this context, which corresponds to a low ranking of spreading across all three tasks from Olle and Kalle (8, 9, and 9, respectively).

|| Once again, Olle says ‘ordna’ in Swedish. See footnote.‡

** In Swedish, she says ‘ordna’. See footnote.‡

†† Possibly by a slip of the tongue, he makes use of the substance metaphor in relation to temperature.

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