The logical and psychological structure of physical chemistry and its relevance to graduate students' opinions about the difficulties of the major areas of the subject

Abstract

In a previous publication, Jensen's scheme for the logical structure of chemistry was employed to identify a logical structure for physical chemistry, which was further used as a tool for analyzing the organization of twenty physical chemistry textbooks. In addition, science education research was considered for the study of the psychological structure of physical chemistry. In this companion paper, the findings are presented of a semi-structured interview study with seventeen chemistry graduates, which aimed to find out their opinion about the difficulties of the various areas of physical chemistry, their disposition towards the subject, and their explanations for the difficulties identified, and in this way to study further the psychological structure of physical chemistry. A mixture of an intensive inductive and a confirmatory data analysis was carried out that revealed ideas and trends and allowed for a reliable portrait of learners to emerge by identifying similarities and differences in the data. Students unanimously found the phenomenological subjects (classical thermodynamics, electrochemistry, chemical kinetics) easier than the submicroscopic subjects of quantum chemistry and statistical thermodynamics. The reasons invoked included that the latter subjects deal with more difficult and abstract concepts, and also their highly mathematical nature. Many students found classical thermodynamics simpler than quantum chemistry, because it “has logic”, includes “tangible examples”, and they had encountered related topics before (especially in high school). The findings for electrochemistry and chemical kinetics were more or less similar to those for classical thermodynamics. Implications, generalizability, and limitations of the findings and prospects for further research are discussed.

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Introduction

Physical chemistry originated as a distinct branch of chemistry toward the end of the 19th century with the publication in the year 1887 of the first physical chemistry journal, Zeitschrift für Physicalische Chemie. It started as a phenomenological discipline, with studies on solutions, chemical equilibria, reaction rates, catalysis, electrolytes, etc., leading to the development of the subjects of chemical thermodynamics, chemical kinetics, and electrochemistry. It is noteworthy that some physical chemists of the early period, such as Wilhelm Ostwald, doubted the very existence of atoms, arguing that, being not directly observable, they were hypothetical entities, for which there was no place in “the basic truths of science, which should be expressed in terms of energetics” (Sutton, 2003). Today, scientists agree that quantum mechanics is the cornerstone of chemistry, including physical chemistry, and so modern physical chemistry is dominated by topics that relate to the submicroscopic structure of matter, with the development of quantum chemistry, computational chemistry, and spectroscopy.

In a previous publication in this journal (Tsaparlis, 2014), Jensen's scheme for the logical structure of chemistry (Jensen, 1998) was used to lead to a logical structure for physical chemistry, which places for instance phenomenological/classical thermodynamics at the molar level of the energy dimension, and quantum chemistry at the electrical level of the structure dimension. Jensen's scheme distinguishes three dimensions (D1: Composition and structure; D2: Energy; D3: Time), with each dimension being treated at one of three levels (L1: Molar level; L2: Molecular level; L3: Electrical level). In this way, one can place the various physical chemistry subjects in the cells of the 3 × 3 Jensen grid (Tsaparlis, 2014). From the psychological perspective, the variation of dimensions and levels in Jensen's scheme should be considered on the basis of the conclusions of science education research on the molecular and electrical levels. Research has shown that the structural concepts of matter are ill understood and result in great conceptual difficulties among students at all levels of education (Tsaparlis, 1997a; Tsaparlis and Sevian, 2013). In addition, without implying that the phenomenological subjects are easy (they also are abstract, involving complicated concepts and mathematics), one has to admit that quantum chemistry constitutes the basis of the submicroscopic approach and therefore constitutes a serious learning impediment.

The logical and psychological structure of physical chemistry was coupled with J. W. Moore's two alternative approaches to teaching physical chemistry, the ‘analytical approach’ and the ‘synthetic approach’ (Moore, 1972). The former is close to the historical development of the subject, and starts with the molar/macro level, while the latter, pays emphasis to, and starts with the electric/submicro level. These approaches represent two opposite perspectives (two “outer limits”), that relate to the three principal areas of physical chemistry and their sequencing in teaching and learning: (1) equilibrium/classical thermodynamics, (2) structure/quantum chemistry and statistical thermodynamics, and (3) chemical kinetics or dynamics.

The division of physical chemistry into the traditional phenomenological and the modern structural subjects is reflected in the organization/sequencing of the major areas of physical chemistry in the physical chemistry textbooks. In the previous publication, an analysis of twenty physical chemistry textbooks showed that many authors favor the analytical approach, starting with the phenomenological subjects (usually with classical thermodynamics), while others focus on the molecular synthetic approach (starting with quantum chemistry), and some others take an intermediate route (Tsaparlis, 2014). Authors who favor the analytical approach consider the topics of quantum chemistry, spectroscopy and statistical thermodynamics more difficult. On the other hand, the fact that modern physical chemistry is based on quantum mechanics makes other authors consider that an early confrontation with quantum mechanics is advantageous to the student. However, many authors remain open to alternative approaches to their own.

The above division is confirmed by a recent survey of the undergraduate physical chemistry course in ACS certified institutions in the USA, with 331 physical chemistry instructors as participants (Fox and Roehrig, 2015), which revealed that: (i) almost all institutions in the survey divided physical chemistry into one semester of thermodynamics and one semester of quantum mechanics; (ii) 44% of the institutions required their students to take thermodynamics first, 20% of the institutions required students to take quantum mechanics first; and 36% gave the students the choice to take either thermodynamics or quantum mechanics first.

Considering the psychology of learning – research questions

As stated in the previous paper (Tsaparlis, 2014), the ultimate research question of this research program is:

Is there an optimum teaching-learning sequence for undergraduate instruction of the various areas of physical chemistry?

To this end, an effort needs to be made to ‘psychologize’ the subject matter of physical chemistry, by discussing ‘pedagogically sounder’ hierarchies/sequences for teaching and learning the various areas. To put psychology into the game, and using terminology borrowed from Dewey (1902), one needs to frame the problem as a dualism between the student and the curriculum. Dewey's advice to instructors for resolving such dualisms was to ‘psychologize’ the subject matter (or the curriculum). In our case, this can be translated into the design of a sequence for teaching and learning the various areas of physical chemistry, which on the one hand respects the logical structure of physical chemistry, and, on the other hand, is close to the abilities, the needs, and the interests of the students.

The present study takes a first pre-requisite step to the realization of the above target. It examines the opinions of students about the various areas of physical chemistry, their disposition towards the subject, and their explanations for the perceived difficulties. The focused research question of this study then is:

What were the experiences, the understanding, and the disposition which the students had developed with respect to the various areas of physical chemistry? In particular, how are the students disposed towards physical chemistry as a whole and its constituent areas, are there any recurring patterns in the difficulties that students are experiencing with the various areas of physical chemistry, and what explanations do they offer for these difficulties?

Qualitative research methodology is a tool that contributes to an understanding of “how people interpret their experiences, … and what meaning they attribute to their experiences” (Merriam, 2009, p. 5). Interviews are particularly useful for getting the story behind a participant's experiences, with the aim that the interviewer pursues in-depth information around the topic (McNamara, 1999). It was expected that the interviews would enable the researcher to answer the focused research question of this study. The semi-structured interview methodology was chosen because it allowed new ideas to emerge and be investigated as a result of what the interviewee says.

Method

The interviews

A qualitative research interview may seek to cover both a factual and a meaning level, though it is usually more difficult to interview at a meaning level (Kvale, 1996). This study focused on eliciting students’ factual knowledge and their opinions about and disposition towards physical chemistry and its various areas. The interviews were semi-structured, so a framework of themes to be explored had been prepared in advance as an interview guide. This helped to keep the focus on the topics of interest, without constraining them to a particular format, enabling questions to be tailored to the interview topic, and to the interviewed students. The interviews included open-ended questions about ‘what’, ‘which’, ‘where’, ‘for which reason/why’, and the issues were attempted to be probed in depth. Leading questions were restricted, but, even when leading questions were used, the students were allowed to express their own experiences and opinions. During the interviews, the interviewer emphatically asked students to try to give opinions that were as far as possible independent of their instructors and the textbooks. Appendix 1 gives an outline of the interview schedule, while Appendix 2 provides further details about the interviews.

Each interview consisted of two major parts. In the first part, the investigator asked the students to:

(a) Recall the titles and the contents of the various physical chemistry courses they had taken at university.

(b) State their liking of physical chemistry in general, as well as of the various areas/sub-disciplines of physical chemistry. [Often, the students were talkative on their own about the various areas of physical chemistry.]

The second part of the interview focused on:

(1) The difficulties the students had encountered with the various areas of physical chemistry.

(2) Students' explanations for the difficulties.

In addition, their specific views on each of quantum chemistry, statistical thermodynamics, spectroscopy, and chemical thermodynamics were invited. Where appropriate, a comparison was sought between views on thermodynamics and those on quantum mechanics and statistical thermodynamics, in terms of concepts or in terms of the mathematics involved in each area. Finally, students' experiences regarding their exposure to concepts of quantum chemistry in introductory courses were also explored.

The second part of each interview is coupled directly to the main research question of the work (Is there an optimum teaching-learning sequence in undergraduate instruction for the various areas of physical chemistry?); that is, the evaluation by the students of the difficulties of the various areas of physical chemistry will help in formulating an answer to the main research question.

Before proceeding it is necessary to comment on a number of terms which have been used with regard to the affective domain: students' ‘experiences’, ‘opinions’, ‘disposition’, ‘liking’, about the different areas of physical chemistry. Other relevant terms, such as ‘interest’, ‘attitude’, and ‘relevance’, could also have been used. All these terms represent constructs that can be considered components of intrinsic motivation – motivation for learning coming from students themselves, and as such they are not necessarily equivalent (Rannikmäe et al., 2010). However, for the purposes of the present work, it is believed that no problems are likely to be caused.

Features of the interviewed students

The sample for the study was in part specially selected and in part it was a convenience sample (see footnotes to Table 1). It consisted of chemistry graduates who were beginning (post)graduate studies or had just started their graduate studies. Note that in Greece a considerable number of chemistry graduates go on to graduate studies. As a rule, these students are characterized by higher achievement in their undergraduate studies as compared to the average graduate. Our intention, however, was to cover a wide spectrum of achievement, and this was achieved with our sample (see below). In addition, it was desirable that the sample should vary in their disposition and attitude towards physical chemistry, so it included both students with positive disposition (two of them were even undertaking graduate studies in theoretical chemistry), as well as students who did not like and were not good at physical chemistry.

Table 1 Distribution and achievement of the interviewed students in three groups^a,b

Group A	Group B	Group C^c
a m: male; f: female; U1–U4 denote the institutions. In addition, for each student, year of graduation is given. b Students A1–A4 were specially selected by the investigator. Student A5 and all group B and group C students constituted a convenience sample. c Students C1–C5 were interviewed as a group – all other students were interviewed individually.
A1mU1 (2005)	B1fU1 (2005)	C1fU2 (2006)
A2mU1 (2005)	B2fU1 (2003)	C2fU2 (2006)
A3mU1 (2000)	B3fU1 (2004)	C3fU3 (2006)
A4mU1 (2001)	B4mU1 (2004)	C4fU4 (2004)
A5fU2 (2004)	B5mU1 (1999)	C5mU4 (2006)
		C6mU4 (1996)
		C7fU4 (2002)
Graduation grades range: 71.5–84.0%.	Graduation grades range: 50.0–70.0%.	Graduation grades range: 60.0–65.0%.
Mean grade: 77.7%	Mean grade: 64.2%	Mean grade: 62.1%

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In total, nineteen students were approached and all participated in the interviews. For one student (a male student from institution U1), the tape-recording was impossible to hear, so this student was removed from the study. Another student (also from institution U1) was a diligent undergraduate student in her final year, but she was not included in the analysis for methodological consistency. Her opinions were interesting and similar to other students' views. Consequently, this study is about the views of the remaining seventeen students.

Based on various features, such as the students' ranges of graduation grades, their liking or not for their undergraduate physical chemistry courses, and their field of graduate studies, the seventeen students were divided into three groups. Groups A and B contained five students each, while Group C contained seven. All Group A and Group B students, plus students C6 and C7 were interviewed individually. On the other hand, students C1–C5 participated (because of student availability) in a group interview. This sorting of participants into groups is consistent with the desired variation of the sample in ability and disposition towards physical chemistry. It must be stressed that the three groups are an artifact that was imposed on the data by the researcher and not one that derived from the analysis of the data; the purpose was just to facilitate the analysis of the findings and their interpretation.

With the exception of one student (A5, who got her first degree from institution U2), the other Group A students had got their first degree from institution U1. They included four males and one female. Student A1 was doing graduate studies in inorganic chemistry, A2 and A3 in theoretical chemistry, A4 in analytical chemistry, and A5 in organic chemistry. Group B had all five students from U1, while in Group C all students got their first degree from other than U1 institutions. In Groups B and C, the students were starting graduate studies in chemistry education. Group B included three females and two males, while Group C contained five females and two males. The uneven numbers from the four institutions were due to reasons of convenience. Note that the four institutions are the ones from which participants had got their first degrees; on the other hand, institutions where they were doing their graduate studies varied, with most participants continuing in the same university (this is common practice in Greece).

All Group A students had relatively high (71.5–84.0%) graduation grades. (Students with grades in the range of 85–100% are missing, but they are rare in Greece.) During the interviews, they all recalled their undergraduate physical chemistry courses well. In Group B, two students had good grades; two had moderate grades, and one a very low grade. According to their statements, all B group students were not comfortable with physical chemistry. All Group C students had moderate grades. Students C1–C5 stated that they did not like physical chemistry much. Student C7 had difficulties with everything from physical chemistry: “I did not understand gas laws etc. There was a problem for me with the physical chemistry courses”.

Table 1 shows the distribution of the seventeen students into the three groups. Each student is denoted with a code name that includes group (A, B, C), student number in the group, gender (m or f) and university (U1, U2, U3, U4). For each group, the graduation grades range, and the mean graduation grade are given. Also, for each student, year of graduation is shown. The interviews were conducted at the beginning of the academic year 2005–2006 for the participants of groups A and B plus students C6 and C7, and at the beginning of the academic year 2006–2007 for students C1–C5. Note that despite the fact that for some of the students there was a long delay between their undergraduate studies and graduate work, the undergraduate curricula experienced by the participants in each institution had remained the same until the times of data collection. Appendix 3 describes the content of the physical chemistry courses in the four institutions.

Teaching methodology

Apart from the content and its structure and organization, equally important for student disposition and learning is teaching methodology. As a rule, in Greek higher institutions, a didactic/instructor-centered teaching methodology is used. The instructor lectures and students attend, as a rule, silently. Often the instructor teaches or even reads the lecture from overhead transparencies or from a power-point presentation. Some students take notes, though few ask questions of the instructor during or at the end of the lecture. Students study using standard textbooks (as a rule, one textbook per course) or instructor-written notes. Passing a course is, as a rule, based on the final, end of semester examinations. All instruction is carried out in the Greek language, and the textbooks used and any instructor notes are also written in Greek.

Data analysis

The analysis of the interviews was carried out as follows: at the outset, the researcher attempted, as far as possible, to put any expectations and/or preferences aside, and reviewed the transcripts with the purpose of allowing trends from the learners' responses to emerge. First, a holistic examination of all responses was carried out. The interview transcripts were reviewed several times, looking for similarities and differences in sets of data, until ideas and trends emerged. As strong trends are noted throughout several interviews, occasions for which the tentative trend did not hold true were noted and probed further. This inductive approach to data analysis allows a reliable portrait of learners to emerge (Thomas, 2006). On the other hand, given the purposes, the questions, and the kinds of coded response categories, the analysis of the data could also be classified as confirmatory research, where analytical categories were largely determined in advance and a more deductive analytical approach was sensible. Actually, a mixture of an inductive and a confirmatory approach has been applied – with an intention to produce both a general account of student perceptions of course difficulties and preferences and the more idiosyncratic reasoning of the individuals involved.

Following the careful reading of each interview, a list of codes was drawn up as follows:

(i) Positive (+) or/and negative (−) disposition of the student towards each of the physical chemistry subjects (quantum chemistry, statistical thermodynamics, spectroscopy, chemical thermodynamics, electrochemistry, and chemical kinetics).

(ii) Encounter (+) of quantum chemical concepts in previous chemistry courses.

(iii) Encounter (+) of chemical thermodynamics concepts in previous courses.

(iv) Connection (+) of chemical thermodynamics with reality.

(v) Explicit comparison (+) of chemical thermodynamics with quantum chemistry in regard to subject difficulty.

The coding of the data was carried out once by the researcher, and then it was repeated by him without reference to the first coding. Comparison of the two tables of coded data revealed very high (>90%) agreement. The few discrepancies were resolved by careful examination of the data. This procedure led to a self-consistent output, thus resulting in a valid and reliable approach to data analysis. A similar approach was applied to the selection of the interview excerpts that are used below to demonstrate the trends in the data: after a first selection of excerpts by the researcher, a second independent selection was made by him, and the two lists of excerpts were cross-checked, leading to the final selection, which is presented in the section with the findings. From the basic codes, well defined categories, themes and sub-themes were constructed, and these will be made evident in the presentation of the interview excerpts below. The themes and subthemes are as follows:

• On the difficulties students encountered:

– Views on quantum chemistry

– Views on statistical thermodynamics

– The role of mathematics in quantum chemistry and statistical thermodynamics

– Comparison of statistical thermodynamics with quantum mechanics

– Views on spectroscopy

• Views on (phenomenological) chemical thermodynamics

– Connection of thermodynamics with the real world

– Comparison of classical thermodynamics with quantum mechanics and with statistical thermodynamics

– Thermodynamics in previous courses

– Mathematics in chemical thermodynamics

• Views on electrochemistry and chemical kinetics

• Encountering quantum chemistry concepts in introductory courses.

Note that during the interviews the above themes and subthemes were not discussed independently of each other, in general, but were often interwoven together as a result of either the questioning or of the students' initiatives.

For the purpose of this report, and because the interviews were conducted in Greek, the interview transcripts were translated into English, and the accuracy of translation was checked by back translation of the English version. The two Greek versions (original and the back-translated one) were then compared and, after some changes to the English version, the agreement was judged satisfactory (Brislin, 1970, 1986).

Findings

Frequencies for the analyzed codes

Tables 2 and 3 give the results of code frequencies for the analyzed codes. In the tables, the positive signs represent ‘positive disposition’ or just occurrence of the corresponding situation in the data, and similarly for the negative signs. Table 2 represents the modern physical chemistry areas (quantum chemistry, statistical thermodynamics, and spectroscopy), while Table 3 represents the more traditional physical chemistry areas (chemical thermodynamics, electrochemistry, and chemical kinetics).

Table 2 Code^a frequencies for modern physical chemistry areas (quantum chemistry, statistical thermodynamics, spectroscopy)

Student	Quantum chemistry	Statistical thermodynamics	Spectroscopy	Encountering quantum chemistry concepts in introductory courses
a (+) indicates a positive disposition or occurrence of situation; (−) indicates a negative disposition or occurrence; (−) & (+) signifies the presence of both a negative and a positive disposition, with the negative disposition appearing before the positive one [and similarly for (+) & (−)].
A1mU1	(−)	(−)	(+)
A2mU1	(−)	(−)		(+)
A3mU1	(+) & (−)	(−)	(+) & (−)	(+)
A4mU1	(−)	(−)		(+)
A5fU2	(−) & (+)	(−)		(+)
B1fU1	(−)		(−)
B2fU1	(−)
B3fU1	(−)	(−)	(+)
B4mU1	(−)			(+)
B5mU1	(−)	(−)	(−)
C1fU2	(−) & (+)
C2fU2	(−) & (+)	(−)
C3fU3	(−)			(+)
C4fU4	(−)	(−)
C5mU4	(−) & (+)	(−)
C6mU4	(−)
C7fU4	(−)

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Table 3 Code^a frequencies for the more traditional physical chemistry areas (chemical thermodynamics, electrochemistry, and chemical kinetics)

Student	Chemical thermodynamics	Chemical thermodynamics in previous courses	Connection of chemical thermodynamics with reality	Chemical thermodynamics easier than quantum chemistry	Electro-chemistry	Chemical kinetics
a For the meaning of the various symbols, see footnote to Table 2. b Student C7 had difficulties with everything from physical chemistry.
A1mU1	(−)		(+)	(+)
A2mU1
A3mU1	(+)	(+)			(+)	(+)
A4mU1	(+)
A5fU2	(+) & (−)	(+)			(+)
B1fU1	(+) & (−)					(+)
B2fU1	+
B3fU1	(−) & (+)	(+)			(+)	(+)
B4mU1	(+)	(+)	(+)		(+)	(+)
B5mU1	(+)	(+)				(+)
C1fU2	(−) & (+)
C2fU2	(+)		(+)
C3fU3	(+)
C4fU4	(+)
C5mU4	(+)	(+)
C6mU4	(+)
C7fU4^b

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Inspection of Table 2 shows that all students had a ‘negative disposition’ towards quantum chemistry. This means that all found quantum chemistry difficult or very difficult for one or more reasons: complicated, abstract, with a different logic, with a lot of complicated mathematics. Most students expressed also a ‘negative disposition’ towards statistical thermodynamics. On the other hand, spectroscopy was not recognized as part of physical chemistry by many students, with only few students having positive or negative dispositions towards it. (Many students were more familiar with spectroscopy from other chemistry courses, especially organic chemistry.) Note that with regard to ‘Statistical thermodynamics’, ‘Spectroscopy’, and “Encountering quantum chemistry concepts in introductory courses’ many data are missing, especially with respect to the latter two entries. For statistical thermodynamics and spectroscopy, this was due mainly to the difficulty or even inability that many students demonstrated in describing these topics. For the quantum chemistry concepts in introductory courses, there was a lack of relevant discussion during the interviews.

A different picture emerges from Table 3 with regard to the traditional areas: most students had a positive disposition towards chemical thermodynamics for various reasons: because they have been taught it in previous courses (e.g. in physics) and/or because of its relevance to real life; and certainly they considered it easier than quantum chemistry and statistical thermodynamics. On the other hand, a number of students expressed negative views about chemical thermodynamics, admitting that it also has difficult concepts and involves complex mathematics – but for them both the concepts and the mathematics are less complex than those of quantum chemistry (and statistical thermodynamics). Finally, all students who expressed views about electrochemistry and chemical kinetics were positive towards them. However, many data are also missing in Table 3.

Detailed and specific student views will be presented below in the form of excerpts from the interviews. Representative quotes from all students are included.

On the difficulties students encountered in quantum chemistry, statistical thermodynamics, and spectroscopy

Group A students' views on the concepts and on the role of mathematics. The investigator (I) started the second part of the interviews with a question like the following, posed to student A1mU1:

I: Now, I want you to think irrespective of your instructors in your physical chemistry courses, as well as the textbooks you used – try to put these things aside – of course this is not easy. Did you find any of these physical chemistry or theoretical chemistry courses more difficult than other physical chemistry or theoretical chemistry courses? †

A1: Conceptually I had been confused mainly by “Quantum Chemistry I”, I had very great difficulties, especially with operators, Hermitian operators.

I: What about statistical thermodynamics, I want you to judge its difficulty – does it look more like classical thermodynamics or like quantum chemistry?

A1: Like quantum chemistry because it had a lot of mathematics.

I: What about spectroscopy?

A1: I like it.

I: But this is actually based on quantum chemistry.

A1: Yes it is, but you see it, you see the spectrum.

I: What about its theory?

A1: It is quantum chemistry.

I: So isn't it difficult again?

A1: May I tell you something? I learn something – I can learn something – much better when I need it, when I use it.

(This particular student had some training and practice with spectroscopy before the time of the interview as part of his graduate training, so, compared to the other participants, he had extra knowledge about spectroscopy.)

Further evidence was provided by various students about the difficulty of quantum chemistry and of statistical thermodynamics, as well as about the comparison of quantum chemistry with statistical thermodynamics:

A2mU1: More difficult was quantum chemistry because its concepts put you in a different logic. Particles move differently than what we know in the world we live in, and this is something peculiar. I found statistical thermodynamics difficult because of mathematics – probabilities and combinatorial analysis, topics I had not been taught about in high school.

A3mU1: Both quantum chemistry and statistical thermodynamics were mathematical, but I think that statistical thermodynamics was harder – I don't attribute this to the instructor, but to the subject itself, from what I can hardly recall.

A4mU1: I found more difficult the theoretical chemistry courses, and less so the other physical chemistry courses. There were some concepts we heard for the first time; they had a lot of mathematical operations and higher mathematics. The method of teaching certainly played a role. They involved many and difficult concepts, which we had to assimilate within a very short time interval.

I: Let us try to take out the method of teaching.

A4: Surely the concepts and the mathematical formulas are difficult. They do not make the relevant concepts understandable. That is, they are based more on mathematical reasoning than on (non-mathematical, physical) arguments.

A5fU2: Certainly, for us more difficult was quantum mechanics, it was the first time we met this concept, but with our professor (she mentions his name) we understood many things because he taught very well. Now both quantum mechanics and statistical thermodynamics were new concepts, and these caused difficulty to me – statistical thermodynamics was more (difficult) than quantum mechanics. … I think that statistical thermodynamics and quantum mechanics were more difficult.

Group B students' views on the concepts of quantum chemistry. Regarding the Group B students, it is reasonable to expect that when asked about the easiest and the most difficult subjects, their responses would not be significantly different from those of the very good students of Group A. In some instances they were, in fact, even more emphatic. B1fU1 had a very-well shaped view; she even declared that her view was shared by other fellow students:

B1: I do not know how objective I can be now, with the particular books I was taught from, and the particular instructors. What appeared very alien and very distant to me, and I considered difficult as a course – not so much difficult as distant – was quantum chemistry. This was also the conclusion any time I discussed this with many fellow students. Basically, it seemed to us to a large extent difficult to understand. On the other hand, I did not think that theoretical chemistry needed more effort, but it needs a way (a method), which, for some reason, we did not have. And I have heard this from many fellow students.

B2fU1 (asked to place in a teaching order the various physical chemistry subjects): I would place quantum chemistry later (in the course), not in the second year – it is more vague. Crystallography is also complicated and vague.

B3fU1: Theoretical chemistry was difficult. Basically, I do not know if the professors who taught us played a part in this. I consider theoretical chemistry difficult, and then the other (subjects) easier.

I: Is there, however, something that is beyond the instructors? That is, features of the subjects?

B3: Yes, the concepts of orbitals are, I think, more difficult, that is, personally I had a larger problem with them.

I: Why?

B3: I don't know, I couldn't understand them, there was a problem.

I: Some explanation for that? Was the difficulty due to the subject itself?

B3: Yes, but also the books contributed to that – we had a huge book (Atkins' Molecular Quantum Mechanics, translated into Greek).

B4mU1: … quantum chemistry is somehow more, the concepts are more abstract, in which case they appear more difficult to understand. …

B5mU1: Quantum chemistry was a course that I could not grasp/put in my mind. I got little out of it. There is nothing to understand (about it), nothing tangible.

Student B3 was further confused about statistical thermodynamics: “we did in that more maths, mathematical calculations, I could not see what chemistry there was there!” Also, student B5 did not understand many things from statistical thermodynamics, while for student B1, spectroscopy was of the upmost difficulty.

Group C students' views on quantum chemistry and statistical thermodynamics. All group C students had also found quantum chemistry more difficult:

C1fU2: I assumed that quantum chemistry was more difficult because it appeared that it had more exercises, which … a lecturer could ask in the exams. (On the other hand, quantum chemistry) scared me as a course – it shocked me a lot, I realized that the world is based on quantum mechanics etc., I looked at it mathematically, pure maths. It has many formulas, many calculations, a lot of maths.

C3fU3: Quantum mechanics because it has a lot of mathematics, just that.

C6mU4: For me it was quantum chemistry. It had, from what I can recall, very difficult concepts.

C4fU4: From all theories? (I found statistical) distributions (as the) more difficult (topic).

I: And where do you place quantum chemistry?

C4: Ah, yes, it is very difficult.

I: More difficult than distributions?

C4: Not more difficult than distributions.

I: Why do you consider quantum chemistry difficult?

C4: Because it involves a lot of mathematics.

Student C5mU4 did not recall statistical thermodynamics, while student C7fU4 had difficulties with everything in physical chemistry:

C7: I did not understand gas laws etc. There was a problem for me with the physical chemistry courses. [She attributed it to poor teaching in her time.] In quantum chemistry I did not understand anything. Quantum chemistry was very difficult, and this was also the view of the fellow students of my time.

On (phenomenological) chemical thermodynamics – comparison with quantum mechanics and statistical thermodynamics

It is evident from the above dialogues that theoretical chemistry/quantum mechanics/quantum chemistry and statistical thermodynamics were considered by the students complicated and difficult both on conceptual and manipulative grounds. Quantum particles “are peculiar”, “they move differently than what we know in the world we live in”. In addition, these subjects are more mathematical, they are “based more on mathematical reasoning than (non-mathematical, physical) arguments”.

Chemical thermodynamics and the real world. Do students have a different view about macroscopic/phenomenological topics such as classical thermodynamics, a subject that methodologically is independent of theories on the structure of matter, and yet is also conceptually and mathematically complicated? Although many students admitted that classical thermodynamics was also difficult, they thought that it was not as difficult as quantum chemistry, because it “has logic”, is “closer to reality”, it is “more interesting”, and “has “tangible examples”, its concepts are “perceptible”, “you understand them”:

A1mU1: Conceptually I had been confused mainly by “Quantum Chemistry I”/“Theoretical Chemistry I”. I had very great difficulties, especially with operators, Hermitian operators.

I: Classical thermodynamics wasn't also difficult?

A1: It seemed to me that it had logic.

I: Was it easier than quantum chemistry?

A1: Yes, it had logic. Also it was closer to reality, to what we were seeing. It dealt with heat and …

I: But it also had abstract things, such as the chemical potential and the Gibbs free energy. So quantum chemistry had wavefunctions, and classical thermodynamics had free energy.

A1: Yes it had, but in thermodynamics there were many tangible ‘examples’ (applications) – (for instance) I recall an experiment with molar volume.

B1fU1: For some reason, (classical thermodynamics) was just more familiar to me … not easier, not easier, more understandable, more familiar, and easier to work with. (I don't believe that organic chemistry is easier, I believe it is more difficult, but it is easier to understand.)

B2fU1: I remember thermodynamics better because I took the exam many times. Also it is more interesting.

C4fU4: It (classical thermodynamics) has (mathematics), but its concept is more perceptible, I don't know, maybe its concepts are more perceptible.

I: When you say perceptible?

C4: You understand them. You know what you are dealing with. With the others (the molecular ones), they are too mathematical to understand them.

One student, B3fU1, complained that in thermodynamics “you had to remember many – too many – formulas, they were not particularly difficult (but) the trouble was that you had to remember by rote all these for the exams. Two other students, C1fU2 and C2fU2, although admitting that thermodynamics was also demanding, found quantum mechanics more difficult but also interesting; they needed to try hard to grasp it:

C1: I thought that all (subjects) had difficult concepts, there is no doubt about this, that is, I had felt uncomfortable with thermodynamics and entropy, but I just thought that quantum mechanics was more difficult.

I: Regarding the nature of the quantum chemistry course?

C1: I thought that quantum mechanics was more interesting than (classical) thermodynamics. I liked quantum mechanics more, but we found that it was also more difficult. It has many (mathematical) equations, many (mathematical) manipulations, a lot of mathematics. That was tiring.

C2: We learnt more things in thermodynamics, it was more familiar, more understandable, but then, it took time for me to manage this. … Thermodynamics, yes, it was easier, but when I understood wave functions, I liked all these more. It took me time to understand them, after a lot of effort, when I read more books, solved exercises, dealt with them a lot.

I: Why did you like quantum mechanics more?

C2: It is difficult to say, I don't know.

Student C5mU4 had somehow a different and interesting view:

I: How do you compare quantum chemistry with the other subjects, with thermodynamics, more difficult, easier, more interesting?

C5: In principle different. Maybe more interesting than physical chemistry, such as electrochemistry and kinetics.

I: Why?

C5: Because it is a different world, you know. It is entirely different.

I: Regarding difficulty or easiness?

C5: Maybe it is little more difficult, you know, now it is more difficult, what to say?

I: For what reason (is it) a little more difficult?

C5: No, no, OK, it is different. But I can't say more difficult.

Thermodynamics in previous courses. The fact that students had encountered thermodynamic topics before (especially in high school) was a reason they gave for finding it easier and being more familiar with it:

A5fU2: Regarding thermodynamics, well, we did it in detail, chemical potentials entered also, that we hadn't met before.

I: Are there not difficult concepts here?

A5: There are, there are.

I: Concepts such as free energy, entropy.

A5: It had difficult concepts, but because I had done some of them in high school, they were closer to me. It had of course difficulties.

B4mU1: … we were taught thermodynamics in high school (in physics however), in which case it is easier when you had assimilated certain things. … (For instance) entropy is close to everyday reality.

B5mU1: I remembered thermodynamics from high school, together with chemical kinetics. Knowledge from high school helped. … ΔG as a concept was not clear to me, but I could imagine it somehow. The same about entropy. I want to be able to get a logical meaning (“a model”) in my mind.

C3fU3: I can't answer with certainty because there are many factors. One is that we did thermodynamics at school – at a very simple level – and that's why it appeared easier … I want to say that thermodynamics is clearly easier, but I don't know if generally it is easier. … It has (mathematics), but the formulas are shorter, more compact, things we were familiar with from high school.

I: Which subject did you find easier?

C5mU4: Thermodynamics.

I: Why?

C5: I liked thermodynamics, that's why.

I: Why did you like it?

C5: Because I had been familiarized with it.

I: At school?

C5: Yes.

Mathematics in thermodynamics – (phenomenological) chemical thermodynamics vs. statistical thermodynamics. Many students, who identified mathematics as a factor leading to difficulty with quantum chemistry and statistical thermodynamics, admitted that phenomenological thermodynamics also involves mathematics. But they still found both the concepts and the mathematics of quantum chemistry more difficult:

I: Classical thermodynamics was also mathematical.

A3mU1: As far as I can recall, it had first and second derivatives, but in what we did, the concepts were easy.

I: Is enthalpy, for instance, easier than the momentum operator?

A3: I would say, yes.

For student C6mU4 thermodynamics has less mathematics:

C6: (Quantum chemistry) had very difficult concepts.

I: Does this imply that the other subjects didn't have difficult concepts – for instance in thermodynamics the concept of free energy?

C6: I understood these better as they had less mathematics.

Placing statistical thermodynamics in the frame, leads to the conclusion that, in terms of concepts and structure, it is closer to quantum mechanics than to classical thermodynamics:

I: What about statistical thermodynamics, I want you to judge its difficulty – does it look more like classical thermodynamics or like quantum chemistry?

A1mU1: Like quantum chemistry because it had a lot of mathematics. Thermodynamics had also mathematics, but statistical thermodynamics …

I: So is it mathematics that makes the differences or something else? That is, doesn't statistical thermodynamics deal with tangible things or does it also mix things up?

A1: It mixes things up. I recall the (case of the) drunken man.

I: So here did we also have non-real things?

A1: Yes, I could not make the connections.

I: But classical thermodynamics also involves mathematical reasoning, doesn't it?

A4mU1: It does, but it also has examples from chemistry, it is more tangible.

I: What about statistical thermodynamics?

A4: Statistical thermodynamics is of more mathematical interest.

I: So, is it getting closer to theoretical chemistry/quantum chemistry?

A4: It does not get so close in terms of difficulty and in the concepts – its nature is just more difficult.

Views on electrochemistry and chemical kinetics

Only few students expressed views about electrochemistry and phenomenological chemical kinetics. One reason was that not much attention was paid to these subjects during the interviews. Another was that in some institutions one or both of these subjects were only covered in the physical chemistry laboratory course (see Appendix 3). In any case, the findings were more or less similar to those for classical thermodynamics:

A2mU1: Chemical kinetics was easy – I knew it from high school, but also I liked it. Electrochemistry is also an area that we did in other courses.

A3mU1: I find thermodynamics, electrochemistry, and kinetics easier. Quantum chemistry has concepts with which we did not come in contact in earlier courses. It had new things and, generally speaking, it was more difficult to understand.

A5fU2: Electrochemistry was easier, it had more exercises.

B4mU1: Kinetics, I think, and electrochemistry were among the most approachable.

Encountering quantum chemistry concepts in introductory courses

It is apparent from the above dialogues that almost all students found classical thermodynamics (as well as electrochemistry and chemical kinetics) as easier because they were familiar with them from previous high school physics and chemistry courses. To this argument, the interviewer responded with the fact that they also had encountered basic quantum chemistry concepts in other introductory courses (for instance, orbitals in inorganic chemistry). The students based their responses on the fact that in the introductory courses quantum chemistry had been qualitative, without mathematics:

A2mU1: In that course we did them in a simple way, while in quantum chemistry we did the tools, and the mathematics was complicated.

A3mU1: We did very few things about orbitals in inorganic chemistry – the concepts were simple and easy there, even if we did them there for the first time. For instance, there were no operators there as in quantum chemistry.

A5fU2: Yes, but all these, the Schrödinger equation, the Hamiltonian were given qualitatively.

Asked if she had understood them qualitatively, A5 responded:

A5: I had difficulty with these in inorganic chemistry as well as (including group theory), they were more theoretical.

Student C3fU3 found the concepts simple and easy to understand in the inorganic chemistry course, but was not very happy with her encounter with the concepts in the quantum chemistry course:

C3: (In high school, we did orbitals) but not at such a theoretical level and (at a) low level.

I: When you say theoretical and low level?

C3: Without mathematics, with no equations, there were just five things about the orbitals.

I: The concepts you received in high school about the orbitals, the qualitative concepts, without mathematics. Had you understood them?

C3: … I had understood them. They were a drop in the ocean compared with the mass of information and concepts I had to learn.

I: Having done quantum chemistry with mathematics, etc., did it help you to understand better these qualitative (things) you had met before?

C3: Certainly (yes), but not special things. There was a lot of mathematics, the mathematics was I think very difficult.

Conclusions and discussion

The analysis of the physical chemistry textbooks that was presented in the previous relevant paper concluded that many authors favor the traditional analytical treatment, while other authors focus on the molecular synthetic approach (Tsaparlis, 2014). The former authors consider the topics of quantum chemistry, spectroscopy and statistical thermodynamics more difficult; the latter authors accept that an early confrontation with quantum mechanics is advantageous to the student. As a consequence, a dichotomy between the macroscopic/phenomenological and the submicroscopic molecular/atomic/electronic approaches to quantum chemistry emerged.

The findings of the interview study confirm the dichotomy. Asked if there were features of quantum chemistry that are intrinsically difficult regardless of the instructor, one of the students (B3fU1) responded: “The concepts of orbitals are, I think, more difficult, that is, personally I had a larger problem with them/I don't know, I couldn't understand them”. Student A2mU1 asked in the interview about the difficulty of the various parts of the undergraduate physical chemistry course commented: “More difficult is quantum chemistry because its concepts put you in a different logic: particles move differently than what we know in the world we live in, and this is something peculiar”. For another student (C5mU4) quantum mechanics represents “a different world/an entirely different world”. It is noteworthy, however, that there were students (C1fU2 and C2fU2), who, while admitting that they needed to try hard to grasp quantum mechanics, found it interesting: “When I understood wave functions, I liked all these more” (C2).

As already commented (Tsaparlis, 2014), the physics of quantum chemistry is complicated and different from classical physics, requiring thinking abilities beyond Piagetian formal operations (Castro and Fernandez, 1987) or quantum logic (Birkoff and von Newmann, 1936). Very relevant here is the opinion of a recent (2014–2015) final (fourth-year) chemistry student, who liked mathematics (especially “mathematics in chemistry”) and physics, with very good grades in her courses (including physical chemistry). She liked both theoretical chemistry and classical thermodynamics, but found theoretical chemistry more difficult, more complicated, with much more difficult mathematics. For her, theoretical chemistry is more difficult “because it is theoretical; it is chaotic indeed; we talk about concepts. When I have theoretical chemistry in my mind, it is (like) a room where concepts just float in the air. I don't know, there is no structure … Thermodynamics is more structured”.

Doctoral student A1mU1 liked physical chemistry but did not like abstract things, while concrete things attracted him (“if I see an abstract painting, I don't like it”). The same student, asked about the difficulty of spectroscopy (a subject which he liked), admitted that its theory was based on quantum chemistry, and yet for him was concrete because “you see it, you see the spectrum.” This is reflected in Vemulapalli's experience that: “an early introduction to spectroscopy (including an elementary coverage of vibrational and rotational spectra), before bonding theory … makes quantum theory more interesting and less abstract” (Vemulapalli, 2010, p. xiii). It is also interesting to compare this student's view with the disposition of scientist John Clauser, who conducted the first experimental tests on Bell's theorem (Clauser and Shimony, 1978). Clauser has stated that he always disliked abstract reasoning:

“I am not really a very good abstract mathematician or abstract thinker. Yes, I can conceptualize a Hilbert's Space, etc. I can work with it, I can sort of know what it is. But I can't really get intimate with it. I am really very much of a concrete thinker, and I really kind of need a model, or some way of visualizing something in physics” (Clauser, 2002 – see also Greca and Freire Jr., 2014).

The mathematical character of quantum chemistry (such as the use of operators) is another source of difficulty, which many interviewed students invoked. Student A4mU1 commented: “Surely the concepts and the mathematical formulas [of quantum chemistry] are difficult. They do not make the relevant concepts understandable. That is, they are based more on mathematical reasoning than (non-mathematical, physical) arguments”. It is remarkable that this mathematical complexity must have led even many practicing chemistry researchers to have adopted a quasi-quantum character to the quantum chemistry tools they employ in their practice (Sánchez Gómez and Martín, 2003). A unique feature of the mathematics of quantum chemistry is that it is more esoteric. For Pauling and Wilson Jr. (1935), “Quantum mechanics is essentially mathematical in character, and an understanding of the subject without a thorough knowledge of the mathematical methods involved and the results of their application cannot be obtained” (p. iii). This view has been re-enforced by Coulson (1974) who stated: “Mathematics is now so central, so much ‘inside’, that without it we cannot hope to understand our chemistry. … These (quantum-chemical) concepts have their origin in the bringing together of mathematics and chemistry”.

We discussed quantum chemistry at length because, in Jensen's scheme, it lies at the electrical level of the first dimension (the molar one). It follows that subjects such as statistical thermodynamics, which are at the demanding electrical level and are based on quantum chemistry, take us one step further into the energy dimension, making statistical thermodynamics more complicated and difficult, as many of the interviewed students expressed. Also statistical thermodynamics was considered as being closer to quantum mechanics than to classical thermodynamics: “(It is difficult) like quantum chemistry because it had a lot of mathematics” (student A1mU1). The mathematical calculations involved in statistical thermodynamics made student B3fU1 not able to see “what chemistry there was there!” In regard to the concepts, statistical thermodynamics “mixes things up” (A1). Similar considerations apply to theoretical spectroscopy: “It is of the outmost difficulty” (B1fU1).

In any case, referring to classical thermodynamics, many interviewed students admitted that it was difficult too, but they were definite that it is simpler than quantum chemistry, because classical thermodynamics “has logic”, is “closer to reality”, and has “tangible examples”, its concepts are “perceptible”, “you understand them”. A similar argument applies to electrochemistry and chemical kinetics. Another reason the students invoked was that they had encountered topics in these areas before (especially in high school).

Finally, students' encounter with basic quantum chemistry concepts in other introductory courses was considered simpler by them because the treatment was qualitative, without mathematics. It must be commented here, however, that this is a false/unfounded perception. In point of fact, there is ample research evidence that many students hold a lot of misconceptions, thinking for instance in terms of old quantum theory, assuming that the term “orbital” is another word for an “orbit,” considering that orbitals are unique and represent a well-bound fixed space. In this way, students exhibit deterministic views about the orbitals, failing to understand the probabilistic nature of quantum mechanics. In addition, they assume that hydrogen-like orbitals are as exact for many-electron atoms as they are for the one-electron case (Tsaparlis, 1997b, 2013; Taber, 2002a, 2002b, 2005; Tsaparlis and Papaphotis, 2002, 2009; Papaphotis and Tsaparlis, 2008; Stefani and Tsaparlis, 2009).

In conclusion, irrespective of their fields of graduate studies and irrespective of group A, B, or C, all students found quantum chemistry and statistical thermodynamics the most difficult both from the conceptual and the mathematical perspectives. On the other hand, for many students classical thermodynamics “has logic” and “tangible examples”, and they had encountered relevant topics before (especially in high school). The findings for electrochemistry and chemical kinetics were more or less similar to those for classical thermodynamics. Of course, these findings were expected, but it is remarkable that even the group A students, who had the highest range of graduation grades among the participants, liked physical chemistry and were more knowledgeable about their undergraduate physical chemistry courses, did not differ in their opinions from the other two groups.

Implications for programs of study, textbook writing, and instruction

Considering that the present study is a companion one to that on the organization of the physical chemistry textbooks (Tsaparlis, 2014), it is appropriate that the following discussion pertains to the findings of both studies. The objective is to contribute to answering the central research question about an optimum teaching-learning sequence for undergraduate instruction of the various areas of physical chemistry. In addition, suggestions for relevant improved pedagogy will be made with respect to more general issues of teaching and learning that derive from the findings of science education research.

First, let us return to the logical structure of physical chemistry as that follows from Jensen's scheme for the logical structure for chemistry (Jensen, 1998). Various hierarchies of presenting the different areas of physical chemistry result from both the structure of physical chemistry as well as from the analysis of the physical chemistry textbooks (Tsaparlis, 2014). The structure of physical chemistry was further analyzed in relation to the psychology of learning, with special attention paid to the concepts of quantum mechanics and of statistical thermodynamics. It was pointed out that the structural concepts of matter result in poor understanding and cause great conceptual difficulties among students at all educational levels (Tsaparlis, 1997a; Tsaparlis and Sevian, 2013). The logical and psychological structure of physical chemistry was coupled with J. W. Moore's two alternative approaches to teaching physical chemistry, the ‘analytical approach’ (which starts with the molar/macro level), and the ‘synthetic approach’ (which starts at the electric/submicro level) (Moore, 1972). As was stated at the introduction to this paper, the central question is concerned with whether either of these approaches is preferable from the psychological perspective.

The psychological structure of physical chemistry leads to the conclusion that the sequence in which its various areas and topics are presented can be crucial for student understanding. To Vemulapalli (2010), “the uneasiness of many students with the subject is not related to its abstract nature”, but to the fact that many important topics like orbitals, entropy, Gibbs free energy, and magnetic resonance are introduced ahead of the proper sequence and before the student has acquired the necessary background” (p. xv). The psychology of learning and the findings from the students' interviews favor the analytical approach, which is also adopted by many textbook authors. J. W. Moore (1972) for instance, starts his book with thermodynamics, “which is based on concepts common to the everyday world” (p. 3).

For Atkins (2008), “the first question that comes to mind when considering a course in physical chemistry is its order… there are the camps inhabited by thermodynamics first and those inhabited by quantum first” (p. 45). The author discusses the advantages of each approach, and concludes that “one serious disadvantage of the quantum-first approach is the unfamiliarity and depth of the mathematics needed to do anything in quantum mechanics … especially when heavy mathematics is in alliance with bizarre concepts” (p. 47). For Van Hecke (2008), “it probably really does not matter which approach is used”, although after having used both, he has “settled on the traditional macro to micro approach” (p. 15). The sequence of topics has also been discussed by Mortimer (2008), with emphasis on the need for a careful treatment of thermodynamics and quantum mechanics before the presentation of statistical thermodynamics. Of interest is the proposal by LoBue and Koehler (2008) for teaching kinetics first, a topic that is “mathematically more accessible” and “highly relevant to modern physical chemistry”.

In the opinion of the author of the present work, and based on the findings of the student interview study, it should be repeated that a sequence that is close to the analytical approach may be preferable, but this should be dependent upon the particular instructor, the teaching methodology, the resources available, and last but not least the students themselves. (See also below the limitations of the current study.) On the other hand, and as commented on in the previous paper (Tsaparlis, 2014), the analytical approach is fulfilled in a way by precursor courses such as high school chemistry and general chemistry, but the problem with such courses is that they may not be scientifically rigorous, thus promoting many misconceptions. In addition, in a study of the impact of general chemistry on student achievement and progression to subsequent chemistry courses (Shultz et al., 2015), it was estimated that general chemistry had some impact on performance in Physical Chemical Principles, without, however, providing a statistically significant estimate of passing the course. Also, taking general chemistry did not impact student progression to Physical Chemical Principles.

For chemistry students, a spiral, three-cycle approach to core physical chemistry, which is logically sound but also is psychological acceptable, would be preferable. In cycle 1 (first year) an elementary physical chemistry course would substitute the physical chemistry part of the general chemistry course. The elementary course should differ from its general chemistry counterpart by paying more respect to the rigor of the subject. In cycle 2 (second year), a mainly phenomenological course would follow, including phenomenological theory of gases, thermodynamics, electrochemistry and chemical kinetics. Some topics at the molecular level might also be included (e.g. reaction mechanisms). The final cycle 3 (partly in second year and partly in third year) would be the molecular approach (molecular and electric levels in Jensen's scheme): kinetic theory, quantum chemistry, statistical thermodynamics, spectroscopy, and special topics (e.g. statistical interpretation of thermodynamic functions, thermodynamic functions from spectroscopy, gas-phase reaction dynamics, solids, polymers).

A major issue with the physical chemistry course is mathematical depth and coverage. Engel and Reid (2012) consider that mathematics is central to physical chemistry, but also “can distract students from ‘seeing’ the underlying concepts” (p. v). Textbooks, lecturers, and students are often lost in a mass of mathematical formulas, which hide the conceptual background, and thus make the subject cumbersome, boring, and to a great extent useless. Manipulating numerical equations and algorithmic problem solving is not equivalent, hence does not presuppose conceptual understanding. Sözbilir (2004) suggests that qualitative understanding of the concepts and laws of physical chemistry should precede their mathematical derivations and numerical calculations. Berry et al. (2000) state as the aim for their book “to keep the reader's mind on the science rather than on the mathematics, especially at the beginning”.

Of course, without mathematics, it is not possible to arrive at even an elementary understanding of the concepts and avoid or cure misconceptions in physical chemistry. Addressing the student in his book, Winn (1995) states: “Pay more attention to the physical reason for an integration (or other mathematical operation) than to its mathematical evaluation … but the fine points – the mathematical techniques – take you toward fluency in physical chemistry” (p. xx). In addition, according to Levine (2009), “to use a (mathematical) equation properly, one must understand it. Understanding involves not only knowing what the symbols stand for but also knowing when the equation applies and when it does not apply, for instance, many student errors in thermodynamics result from the use of equations in situations where they do not apply” (pp. xv, xvi).

It was commented earlier that mathematics is central to the understanding of quantum chemistry, which has an essentially mathematical character (Pauling and Wilson Jr., 1935, p. iii), and that quantum-chemical concepts have their origin in the bringing together of mathematics and chemistry” (Coulson, 1974). However, one should be careful about the extent of coverage and functioning of mathematics because “an excessively mathematical treatment (of quantum chemistry) would obscure the physical ideas for most undergraduates”, while “a purely qualitative treatment does little beyond repeat what students have learned in previous courses” (Levine, 2009; p. xvi). In any case, it could be argued that physical chemistry concepts can be understood at an acceptable level with only a minimal mathematical treatment. In quantum chemistry, for instance, one needs mathematical equations and functions but there is no need to solve differential equations or perform other complicated mathematical operations. The underlying physical picture and its connections with mathematics should be emphasized (Tsaparlis, 2008, 2013; Tsaparlis and Papaphotis, 2009).

Generalizability of the findings, limitations, and prospects for further research

As stated in the previous article (Tsaparlis, 2014) as well as at the beginning of this paper, the ultimate research question of this research program is:

Is there an optimum teaching-learning sequence for undergraduate instruction of the various areas of physical chemistry?

Textbook organization (which was treated in the previous paper) guided the above programmatic research question, while the interview study which is reported here was designed to contribute to answering the question about which teaching-learning sequence reflects the students' views about the difficulty of the various areas of physical chemistry and their explanations for the difficulties. The extensive set of the physical chemistry textbooks and their international character provided rich insights into the central question, but one has to admit that textbook authors vary in their approaches and philosophies (Tsaparlis, 2014). On the other hand, the interview study had the limitation of the restricted national context; however, the variation in the student sample in terms of coverage of four (out of the five) Greek chemistry departments, coupled with the strong similarities of their views, provides sufficient support to the generalizability and hence to the global usefulness of the findings. There exist, however, a number of issues, which have been pointed out by an anonymous reviewer of this article. These might limit the generalization of the findings, and should be considered carefully.

Students' beliefs are likely influenced by a variety of factors that may bias their reasoning. Results from psychological and education research have shown that, in general, people are poor judges of when we are learning well and when we are not. Students might create an “illusion of understanding” about thermodynamic concepts due to repeated exposure to them through high school and university. Consequently, one cannot be certain that students' learning difficulties with physical chemistry would be much alleviated by changing course sequence, without a more in-depth reconceptualization of how students are actually introduced and asked to think about core concepts and ideas in each of the courses. Also students' beliefs and recollections might be difficult to separate from the ways in which they are traditionally exposed to the subject matter. Historically, classical thermodynamics has been taught in an inductive way (starting from experimental results to build the theory), while quantum chemistry has been taught in a deductive way (starting from the theory to explain experimental results). This difference in approach might be more responsible for students' views than the actual nature of the two fields. If quantum chemistry were to be taught following an inductive approach, students might judge it differently.

At this point it is imperative to emphasize that there are strong supporters of the synthetic approach, which has the advantage that it brings students into the realm of modern science (quantum chemistry, spectroscopy, etc.) early in the course, and thus it might catch their interest and motivate them. According to R. J. Moore and Schwenz (1992), the physical chemistry instructor should present the material “in a manner that excites students, illustrates the usefulness of the material, and generates an understanding of the chemistry, rather than as a series of dull mathematical abstractions upon which the foundations of chemistry are laid” (p. 1001). The solutions suggested by these authors included reorganizing the curriculum to focus on the study of quantum mechanics first. In this way, students' interests in topics such as chemical bonding, intermolecular interactions, and spectroscopy would better be addressed. In the same spirit, and talking to the student, McQuarrie and Simon (1997) comment in their book, “Armed with the tools of quantum mechanics, you shall learn that thermodynamics can be formulated in terms of the properties of the atoms and molecules that make up macroscopic chemical systems” (p. xviii). Also, Kuhn et al. (2009) state in their book: “We feel that an early confrontation with quantum mechanics is advantageous to the student” (p. xxv). A call for the inclusion of new physical chemistry research and applications in undergraduate curricula has been made by various researchers and physical chemistry educators (Moore and Schwenz, 1992; Schwenz and Moore, 1993; Zielinski and Schwenz, 2004). In any case, it should be pointed out that an approach such as the molecular approach to physical chemistry by McQuarrie and Simon (1997) is excellent as a meta-cognitive tool for mature and graduate students and scientists.

Consideration of a wider range of information is still required in order to come closer to a more secure answer to the ultimate research question. Thus, in addition to the students' views on the difficulty of the various areas of physical chemistry and their explanations for the difficulties, the following very important aspects should be considered:

• Input from teachers of the courses would provide additional insights (Mack and Towns, 2016).

• An analysis of the conceptual relationships within the content of physical chemistry – a massive job in itself.

• Data to show where students actually go wrong or cannot answer questions during the undergraduate course – as opposed to what they may remember some years later.

• Data that present the level of success, as well as the depth of understanding of physical chemistry among students who will be taught in different teaching sequences.

Also, apart from the content and its structure and organization (which is a teacher-centered and curriculum-oriented focus), teaching methodology that takes into account the psychology of learning, that is, how people learn (Åkerlind, 2008; Tsaparlis, 2008) is equally important. Depth of coverage should take precedence over length of content, with content and emphasis being addressed in accordance with students' needs and interests (Zielinski and Schwenz, 2001). In addition, the literature of known student alternative conceptions and frameworks should be taken into account and further be coupled with constructivist methods of teaching physical chemistry, such as the Process Oriented Guided Inquiry Learning (POGIL) (Spencer and Moog, 2008). Very conducive to students interest and the learning of the modern topics of physical chemistry is also the introduction of a computational laboratory into the undergraduate physical chemistry program, such as the one implemented at the University of Michigan, which covers four principal computational methods: (1) Monte Carlo methods, (2) Molecular mechanics methods, (3) Molecular dynamics simulations, and (4) Quantum chemical calculations (Sension, 2008). Levine's book also includes “modern ab initio, density functional, semiempirical, and molecular mechanics methods, … so that students can appreciate the value of such calculations to nontheoretical chemists” (Levine, 2009, p. xvi). For further information on various methods of teaching physical chemistry, see Tsaparlis (2008). Very relevant to the issue of establishing a research-based sound hierarchy/sequencing of the various areas of physical chemistry are two major topics in the science education literature, that of learning progressions (Stevens et al., 2010; Duschl et al., 2011; Merritt and Krajcik, 2013; Sevian and Stains, 2013), and that of teaching-learning sequences (Méheut and Psillos, 2004).

Finally, one may take a critical stand with respect to the ultimate research question of this program, as two other anonymous reviewers of this manuscript commented. One of them posed the question: “Is it possible to find an (one) optimum teaching-learning sequence for low-high achievers with different attitudes towards physical chemistry and interest?” For the reviewer, “if all instructors followed a similar teaching-learning sequence (the optimum one), the aspect of students' interest, attitudes and understanding will be automatically (so to speak) addressed” – or, in the opinion of the author of this manuscript, at least minimized. This, however, needs to be proved. In any case, “proper, validated ways to teach are, therefore, important for all teachers”. On the other hand, according to the other reviewer, it must be accepted that “even if a well designed and comprehensive research will come up with a particular optimum sequence” it is doubtful “if all teachers of the subject matter will agree with it. Furthermore, each specific sequence means teaching the various topics of physical chemistry at a different level because the students may not have the proper background at earlier stages of their undergraduate studies. Therefore the learning outcomes may be very different and the comparison between different sequences is far from being obvious”. So our above extensive discussion and the proposed lines for further research enforce a re-phrasing of the leading research question into the following more appropriate one: “What are the considerations for choosing, and implications of applying, various teaching-learning sequences for undergraduate instruction of physical chemistry?”

Concluding comments

In physical chemistry there are two factors that appear to make the subject difficult for the students: (i) a high degree of abstraction in the concepts; (ii) a high degree of complexity in the mathematical relationships between the concepts. This can be coupled to Maton's Legitimation Code Theory (LCT) (Maton, 2014), which distinguishes between semantic gravity and semantic density. According to Maton, semantic gravity (SG) indicates ‘the degree to which meaning relates to context’, that is, to the degree of abstraction; on the other hand, semantic density (SD) indicates ‘the degree to which meaning is condensed within symbols (terms, concepts, phrases, expressions, gestures, etc.)’, that is, to the degree of complexity. These two factors can be stronger (+) or weaker (−); thus high abstraction relates to weak semantic gravity, while high complexity relates to strong semantic density. The two factors are assumed to be independent of one another, and are conveniently represented by orthogonal axes, leading to four possible combinations: SG(−)SD(+); SG(−)SD(−); SG(+)SD(+); SG(+)SD(−). Both quantum chemistry and statistical thermodynamics are highly abstract, that is, have weak semantic gravity, SG(−); on the other hand, both are mathematically complex, that is, have strong semantic density, SD(+). Classical thermodynamics, despite being a macroscopic/phenomenological subject, which is not based and is above and beyond the theories on the structure of matter, is also conceptually and mathematically complicated. One has to admit, however, that, compared to classical thermodynamic, quantum chemistry and statistical thermodynamics have a weaker semantic gravity and a stronger semantic density.

Last but not least, and despite the need for shifting the emphasis from the complicated mathematical operations and derivations to deep conceptual understanding (for which, however, mathematics is also essential), we must always remember Swante Arrhenius' (1912) comment that physical chemistry is an exact science, which has a “profound quantitative character from the science of physics. … The theoretical side of physical chemistry is and will probably remain the dominant one … Physical chemistry may be regarded as an excellent school of exact reasoning for all students of the natural sciences” (pp. xix, xx). And as such, it should always occupy a central and integral part in the education of chemists, with great emphasis placed on the underlying physical picture and its connections with mathematics.

Appendix 1: an outline of the interview schedule

(1) Do you remember the physical chemistry lecture courses (not the practical ones), which you were taught during your undergraduate studies? In your account, include, if you remember, the content of each course.

(2) How do you compare your relevant achievement in your BSc chemistry course, with your achievement in the lecture courses of the physical chemistry sector?

(3) Among the lecture courses of the physical chemistry sector are there one or more courses in which you did well? Or you didn't do well?

(4) Do you remember anything from these courses?

(5) (If the student does not remember anything): For which reason?

(6) I want you to think irrespective of your instructors in your physical chemistry courses, as well as the textbooks you used – try to put these things aside. Did you find any of the physical chemistry courses more difficult than other physical chemistry courses? Or, did you find any one of these courses easier than others?

(7) What about quantum chemistry/theoretical chemistry? For which reason?

(8) What about statistical thermodynamics? For which reason?

(9) What about spectroscopy? For which reason?

(10) (Compare the mathematics of quantum chemistry with the mathematics of statistical thermodynamics.)

(11) What about classical chemical thermodynamics? For which reason?

(12) What about electrochemistry? For which reason?

(13) What about chemical kinetics? For which reason?

(14) (Can you compare quantum chemistry with statistical thermodynamics in terms of difficulty? Justify your answer.)

(15) (Can you compare chemical thermodynamics with quantum chemistry? Justify your answer.)

(16) (Compare the concepts of quantum chemistry with the concepts of chemical thermodynamics.)

(17) (Compare the mathematics of quantum chemistry with the mathematics of chemical thermodynamics.)

(18) Have you encountered any quantum chemistry concepts in introductory/previous chemistry courses? Had you understood the concepts in these courses? Were the concepts in these courses easy to understand? Justify your answer.

Note: Depending on the context of an interview, questions within parentheses were asked only if they were appropriate.

Appendix 2: further information about the interviews

The interviews were semi-structured, including open-ended questions about ‘what’, ‘which’ ‘where’, for which reason/why’, in an attempt to probe the issues in depth:

Student: I found statistical distributions (as the) more difficult (topic).

Investigator: And where do you place quantum chemistry?

S: Ah, yes, it is very difficult.

I: More difficult than distributions?

S: Not more difficult than distributions.

I: For what reason do you consider quantum chemistry difficult?

S: Because it has a lot of mathematics

A different student: In high school, we did orbitals, not at such a theoretical level but at a low level.

I: When you say theoretical and low level?

S: Without mathematics, with no equations, there were just five things about the orbitals.

Leading questions were restricted:

I: What about statistical thermodynamics, I want you to judge its difficulty – does it look more like classical thermodynamics or like quantum chemistry? (a non-leading question)

But even when leading questions were used, the students were encouraged to express their own views:

S: Conceptually I had been confused mainly by quantum chemistry. I had very great difficulties, especially with operators, Hermitian operators.

I: Do you think that classical thermodynamics wasn't difficult?

S: It seemed to me that it had logic.

I: Was it easier than quantum chemistry?

S: Yes, it had logic. Also it was closer to reality, to what we were seeing. It dealt with heat and …,

I: But it also had abstract things, such as the chemical potential and the Gibbs free energy. So quantum chemistry had wavefunctions, and thermodynamics had free energy.

S: Yes it had, but in thermodynamics there were many tangible ‘examples’ (applications) – (for instance) I recall an experiment with molar volume.

In total, nineteen students participated in the interviews. For one student (a male student from institution U1), the tape-recording was impossible to hear, so this student was removed from the study. Another student (also from institution U1) was a diligent undergraduate student (UGS) in her final year, but she was not included in the analysis for methodological consistency. Her opinions were interesting and similar to those expressed by the other students:

UGS: Quantum chemistry is always difficult.

I: What is the meaning of always?

UGS: It is somehow more … you have to spend several hours to understand. In terms of written exams, you have to solve many equations.

I: But classical thermodynamics had equations, hadn't it? Is there some difference?

UGS: It has, it has. But we are more familiar with thermodynamics from high school, we did certain things (about it), so I found it easier. Quantum chemistry is more theoretical; we have not been familiarized with it.

I: What about statistical thermodynamics?

UGS: It is difficult, very difficult because it has mathematics in it.

I: Is the problem purely a mathematical one, or is it conceptual too?

UGS: It is difficult to understand.

I: What about entropy and chemical potential? Aren't they difficult to understand?

UGS: Actually, they probably are more difficult, but we encounter them more frequently.

Appendix 3: the content of the physical chemistry courses in the four Greek institutions

The following information about the content of the physical chemistry courses in the four Greek institutions is based on various sources: the personal experience of the author; the programs and contents of study that are published on the websites of the corresponding departments; and participant students’ recollections from their undergraduate studies.

Institution U1 had four physical chemistry (PC) plus two theoretical chemistry (TC) courses. PC1 was about classical thermodynamics; PC2 had two sections, one on electrochemistry and one on phenomenological chemical kinetics (reaction rates); PC3 was on kinetic theory and statistical thermodynamics; and PC4 was a theoretical course about spectroscopy. TC1 had the fundamentals of quantum mechanics, with emphasis on the mathematical manipulation of simple systems; and TC2 was about molecular quantum mechanics.

Institution U2 had four physical chemistry courses. PC1 was mainly classical thermodynamics; PC2 was about statistical thermodynamics. PC3 was on quantum chemistry; and PC4 included applications of quantum chemistry. Note that statistical thermodynamics was taught before quantum chemistry. Electrochemistry and phenomenological chemical kinetics were only covered in the physical chemistry laboratory course.

Institution U3 had two physical chemistry courses; PC1 was about the basics of quantum mechanics, electronic structure and bonding, and spectroscopy; PC2 was about classical and statistical thermodynamics, chemical kinetics (including phenomenological and theoretical treatments). Electrochemistry was covered only in the physical chemistry laboratory course.

Institution U4 had the traditional subjects (classical thermodynamics, states of matter, phase and reaction equilibrium, kinetics, and electrochemistry) in its physical chemistry courses. Quantum chemistry was taught as a standard course by the section of inorganic chemistry. Students from this department appeared not to be familiar with statistical thermodynamics (they stated that they had not been taught statistical thermodynamics). Also they did not have a theoretical course on spectroscopy from a physical chemistry perspective. But they had been taught two laboratory courses on applied spectroscopy, one in inorganic chemistry, the other in organic chemistry.

Tuition in all four institutions also included physical chemistry laboratory courses.

Acknowledgements

The author wishes to thank all graduate students who participated in the interview study and made possible the realization of this work. I also thank Professor Chryssa Tzougraki of the Department of Chemistry of the University of Athens who arranged for students from the graduate program “Chemistry Education and New Educational Technologies” (DiCheNET) to participate in the interviews. Thanks are also due to the journal editor and the anonymous reviewers of this manuscript who made numerous very reasonable and useful comments that have contributed greatly to its amelioration and refinement. Finally, the author is grateful to Dr Bill Byers who read the manuscript and made suggestions for a better presentation.

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Footnote

† In institution U1, there were two theoretical chemistry courses, covering the fundamentals of quantum mechanics/quantum chemistry and molecular quantum mechanics/quantum chemistry respectively. In addition there were four physical chemistry courses. See Appendix 3.

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