Chapter 9 of Understanding Chemical Bonding: The development of A level students' understanding of the concept of chemical bonding
Learners' difficulties in understanding electron orbitals
I: Can you have an orbital with no electrons in it?
T: If you had that, then it wouldn't be an orbital.
T3.A336
§9.0: The orbital concept
As described in chapter 6 (§6.1.2), the learners in this study were following an examination syllabus which required them to learn about quantum numbers, atomic orbitals, energy levels, pairing of electrons, orbital overlap and delocalisation. All of these are concepts that are beyond the scope of a standard pre-A level course. The basic principle behind these concepts is that of quantization (of energy, angular momentum etc.), but in terms of the type of descriptions commonly used in A level chemistry the key concept is that of electron orbitals. The concept of an atomic orbital – "an allowed wave function of an electron in an atom obtained by a solution of Schrödinger's wave equation", according to one science dictionary (Pitt, 1977, p. 30) – is certainly one which learners may find abstract, as Tajinder's comment standing as the motto to this chapter hints.
In the present research it was found that some aspects of the orbital concept gave learners difficulty. The uncertainty about the meaning of electron spin, and the identification of orbital probability envelopes with 'boundaries' were not found to be be serious impediments to progress, but it was also found that
- learners commonly confused hybridized atomic orbitals with molecular orbitals;
- had difficulty remembering the designation of atomic orbitals;
- had difficulty understanding the relationship between orbitals, sub-shells and energy levels.
In this chapter aspects of the difficulties with the orbital concept elicited during this research will be illustrated with evidence selected from the database.
§9.1: The quantum hypothesis
"you're not going to be able to understand it … You see, my physics students don't understand it either. That is because I don't understand it. Nobody does."
Nobel laureate Richard Feynman (1985, p.9) embarking on a lecture on quantum theory.
A key principle in understanding some aspects of chemistry is that energy, as well as matter, is quantised. This principle may be used to explain the stability of atoms, the existence of discrete energy levels, and the consequent atomic and molecular spectra.
The notion of quantization was introduced as a heuristic device to 'save the phenomena', in that classical physics could not explain the distribution of energy in black body spectra, the sharp spectral lines in atomic spectra, nor the stability of atoms themselves (which as classical electrical oscillators should have radiated the orbital energy of the electron and collapsed to the density of neutron stars). It was accepted that the heuristic device was counter intuitive, so that even Bohr recognised that it required "a conscious resignation of our usual demands for visualisation and causality" (quoted in Petrucciolo, 1993, p.19. Cf. Feynman's warning in the motto above).
To understand the need for such a counter-intuitive hypothesis one needs to appreciate the problems faced by atomic theory before Planck and Bohr proposed quantization. In the present study it was clear some learners did not appreciate why the hypothesis was introduced.
One of the colearners, Edward, understood the consequence of the quantum hypothesis on the transitions between energy levels, as "you could put energy into it, … and the correct frequency, … which a particular electron would absorb, it would absorb a photon of energy and be promoted to another vacant orbital" (E2.A203). He also knew "the equation energy equals Planck's constant times the frequency of the radiation" (E2.A252). Despite this apparent understanding of how the quantum hypothesis explained the selective absorption of light, Edward did not appreciate how the quantization of atomic energies prevented the collapse of atoms. He knew that in an atom there would be forces "between the … negatively charged electrons and the positively charged … proton"s (E2.A289), however Edward was not very clear about why this did not result in the particles moving closer together. For Edward the problem was to explain why the electron was moving in the first place (he suggested that some initial energy was given "in creation", E2.289), but one month from his A level physics examination he did not appear to be aware that classically an atom would be expected to radiate energy as an electrical oscillator (see appendix 30, §A30.1.1).
Jagdish was another colearner who did not appear to understand the significance of quantization. In her third interview she stated that "the attractions from the nucleus, pulling in the electrons" were stronger than the repulsions (J3.A460). However she explained that the electrons did not fall into the nucleus, as although "they're being attracted, … the attraction isn't … that strong" (J3.A463). However, in the absence of quantization, a weaker net force would lead to a slower collapse of the atom rather than no collapse. Presumably Jagdish became aware of this fault in her logic as she comes to a stop part way through her argument, that "if you could actually physically make those electrons get closer to the nucleus then they would fall in because the attraction would be so strong that they'd …" (J3.A463). She is unable to produce an explanation here that she finds satisfactory.
An example of a learner becoming confused about electronic transitions between orbitals occurred when colearners Kabul and Tajinder were recorded discussing past examination questions, near the end of the first term of the second year of their course. When Kabul attempts to explain thermionic emission in metals he seems to confuse the emission of electrons with the emission of light photons during electronic transitions (see appendix 30, §A30.1.2).
§9.2: The relationship between orbitals, sub-shells, shells and energy levels
Prior to A level study learners are likely to have considered atomic structure in terms of electron shells. At A level they are expected to appreciate not only the notion of orbitals, but the related-yet-distinct concepts of sub-shells and energy levels. In the present study it was found that learners had some difficulty in making sense of these concepts, and when learners were first told about orbitals some of them seemed to take this as a synonym for shells, and for orbits.
§9.2.1: Learners confused electron orbitals and shells
So when students are first taught about orbitals, they do not seem to distinguish this new concept from their existing notion of an electron shell. For example, Annie in her first interview referred to "the quantum shell, on what the electrons sit" (A1.186), and in her second interview to electrons that "go round, like in orbitals, or in spherical things" (A2.378).
Similarly, in Debra's first interview she uses the term 'orbital', although her meaning seems closer to the notion of 'orbit'; so "the orbital closest to the nucleus … [is] the path the electron takes around the nucleus" as the electron "circles the nucleus in a sphere" (D1.32-6).
focal figure 1
In Edward's first interview (at the end of one year of A level study) he also seemed to confuse shells with orbitals, when he described focal figure 1 (which only showed shells) as a "representation of an atom, with er, its electron orbitals, erm in different shells, around the nucleus" (E1.2) and went on to explain that there were three orbitals containing two, eight and one electron respectively (E1.4). In his second interview (near the end of his course) he appeared to use the terms 'orbit', 'orbital' and 'shell' as synonyms. When questioned Edward explained that "an orbit's just a circular er thing, … a kind of neat way of describing an electron's motion" (E2.A050), but that the 'orbital' was "a better way of describing it, [i.e.] to say it occupies a volume – of space … and there's a probability that it will be found in that volume of space" (E2.A050). (See appendix 30, §A30.2.1.)
In Tajinder's first interview he used the term 'orbital', but also seemed to be using it as a synonym for shell, as "each orbital is like a sphere" (T1.B065). Later in his first term Tajinder undertook Kelly's construct repertory test, and one of constructs elicited by the test was 'shows rough placement of electrons in orbitals'. A number of triad elements were placed at the explicit pole of this construct, although in most cases what were shown were shells and not orbitals.
Shells and orbitals may be seen as components of distinct models used to discuss atoms. The idea of electron density introduces another variation.
focal figure 2
focal figure 7
Some learners found it difficult to relate their models of electrons in orbits with the notion of electron density, as Debra demonstrated when she compared focal figures 2 and 7, which,
"show different things, but the same sort of bonding. 'Cause [fig. 2] shows the outer shell electrons, erm, in the path, but [fig. 7] shows the electron cloud, where the electrons are most likely to be found … [which is] in the middle"
D1.306-8
Yet, Debra explains, the electrons in the cloud are the same ones in "the outer shell" (D1.314) "but they don't always stay there. They're not stationary. They're moving" (D1.318). Debra did not seem to find the apparent contradiction between her interpretations of the two diagrams problematic as "they're both correct … [but] show different things" (D1.322).
Tajinder's acceptance of alternative, and apparently incongruent, models in chemistry has been discussed in some detail in the previous chapter (§8.4.5). After his apparent confusion of orbitals and shells had been elicited in his first interview (see above) it transpired he had been told something of orbitals in his organic chemistry lesson, (by the other lecturer who taught the class). However when Tajinder attempted to explain the concept it seemed he was not sure what an orbital was, and that he found the idea "hard". (See appendix 30, §A30.2.2).
In his second interview, Tajinder continued to demonstrate some confusion over the orbital concept, and its relationship to the concept of electron shells, suggesting that "an orbital is just an area around … the nucleus of an atom, where electrons are likely to appear, or be held". He initially reported that there would be three orbitals in a sodium atom (T2.A085), before deciding there would be more than three, suggesting there would be "the s orbital, p orbital, d and the f" (T2.A085).
On reflection, Tajinder was aware that he was confusing material presented to him at A level, with his existing knowledge, "what we learnt in G.C.S.E." (T2.A112). Tajinder thought that the rings were "shells" and his G.C.S.E. level model "isn't wrong, but it's not totally correct". However he did not see how the new ideas fitted with the old, and at that time did not think they were related (T2.A123).
At this point in Tajinder's case his knowledge of electron shells seemed to be acting as an epistemological learning impediment (§1.5.5) to learning about orbitals, rather than as a suitable intermediate conception on an appropriate conceptual trajectory (§2.3.10).
§9.2.2: Learners confused shells, sub-shells and orbitals
Given that some learners had difficulty distinguishing the concepts of shell and orbital, it is not surprising that the additional concept of sub–shell added to the complications. For example, in her third interview, shortly before her final examination, Carol thought there could be eight electrons in the second shell of an atom (C3.387), but this would only require two orbitals (C3.389).
triad element 126
triad element 656
During the second term of Kabul's second year he undertook the repertory test exercise (January, 1994). One of the constructs that were elicited was "shows sub- shells", and Kabul construed triad elements 126 and 656 at the explicit pole, as showing sub-shells, although shells – but not sub-shells – were represented. In a subsequent interview Kabul confirmed that he thought the electrons were arranged "in sub-shells" (K5.A049), where "a sub-shell consists of orbitals. Like you know L is a sub-shell consisting of 2p and 2s orbitals" (K5.A077). So at this point Kabul described shells as sub-shells.
Tajinder made the complementary mistake of describing a sub–shell as a shell,
"if all the orbitals, say the p, p-orbitals are all full of electrons, if you work out where they are most likely to be, erm not sure what the word is, then it will show like a sphere shape of electrons smeared out, so then that is represented by the … shell, in the diagram that we learnt for G.C.S.E."
T2.145
§9.2.3: Learners’ confusion over energy levels
The notion of energy levels provided one further concept to be confused with shells, sub-shells and orbitals.
So, for example, in Annie's third interview (at the end of her course) her explanation of why electrons do not fall into the atomic nucleus invokes quantum shells, energy levels, orbitals and even hybrids, without suggesting that she has a clear idea of the distinction between these concepts,
"The electrons are held in, erm, in sort of levels, so, it's to do with sort of bonding, like you can only get two electrons in the first quantum shell. So that they are held in these shells. … Er, so, they're they're all held in quantum shells which are different energy levels, and you can sort of promote electrons should you need to in bonding, so, so if for example you need a bond to have, I don't know, an extra electron in a p orbital, you can donate an s, s electron across, to give you hybrids, things."
A3.10
focal figure 1
focal figure 7
focal figure 4
Carol's first interview took place after one term of A level work had been undertaken, and at this stage she described the lines in focal figure 1, which represented the shells of electrons, as the "energy levels around an atom" (C1. 18) of sodium. Focal figure 7 was intended to represent the electron density envelope around the hydrogen molecule, but Carol again refers to it as "an energy level" (C1.352) In the second interview Carol showed the same confusion between energy levels and shells, describing how in focal figure 4 the electrons "all look as though they're on the same energy level" (C2.102).
Debra did not seem to relate the idea of energy levels to orbitals. In Debra's second interview she suggested that after a molecule had absorbed light "it's got more energy" (D2.63) and "it's excited the … molecule and … [the electrons are] vibrating, and moving more" (D2.69-73). Debra thought this would probably involve all of the electrons (D2.75), which normally "move round the, round the nucleus" (D2.77) and "they probably move faster" (D2.79). However Debra did not think the electrons would move into a different orbit (D2.83).
In her third interview Debra said it was not possible to excite a hydrogen atom (D3.82), and she did not know if it was possible to excite a sodium atom (D3.84). Only when asked to think about 'an experiment, in physics, not in chemistry, but in physics, where you have to work out spectral wavelengths' did she describe how "you promote an electron to a higher energy level. … And then it falls back and gives out the energy" (D3.102-4). It would seem that when Debra was taught about electronic transitions in physics she did not connect this to her work in chemistry (that is she suffered a fragmentation learning impediment, §1.5.2).
In Tajinder's fifth interview he discussed how an electron could be promoted to an excited state, but he was not sure whether he should refer to the electron moving between shells, or between orbitals. Only after he sketched out his recollection of the diagram he had seen on the board during a class did he conclude that the transitions are between energy levels (See appendix 30, §A30.2.3).
§9.2.4: Learners confused orbitals and probability envelopes
As some of the colearners in the study tended to think of electron motion in terms of orbits it is not surprising that the notion of representing orbitals of infinite extent with probability envelopes based on an arbitrary cut–off caused some confusion. It is not possible to draw a simple diagram of an orbital, so they are often represented by a probability envelope. By definition there is a small probability that the electron will be found outside the envelope, but the electron can not be outside of the orbital it occupies. This apparently self-evident statement depends upon a clear distinction between the two concepts (orbital, probability envelope) that the colearners did not always grasp.
Jagdish, in her second interview, referred to shells being "just something like arbitrary", before changing this to "orbitals were arbitrary" as "they were just regions of space that you had the highest probability of finding the electron" (J2.B218), whereas electron shells "show you the energy levels" (J2.B226). (Although imprecise, Jagdish's explanation shows some increase in sophistication over her first interview when she suggested that on cooling a gas would condense, because "you're stopping the electrons from moving around, because you're taking away their energy, so they form a solid", J1.B033).
In his third interview Tajinder also seems to confuse the orbital with the common diagrammatic representations in terms of a probability envelope. Tajinder seems to suggest that the electron is sometimes outside the orbital. He thought the orbital was "a probability of finding the electron in that certain area" (T3.A460). Tajinder demonstrated a similar confusion in his fifth interview. He thought that there was "just one" orbital in a hydrogen atom (T5.A433), but that "an orbital just distinguishes a sort of barrier around where you're most likely to find that electron, so it doesn't mean that it just sticks in that one place" (T5.A450). Although Tajinder was informed of his error, in a later interview he referred to how a 2p electron would "come out" of the p-orbital (T6.B090). (See appendix 30, §A30.2.4).
In Umar's final interview for the research, shortly before the end of his course, he described an orbital as the "space most likely to contain the electrons" (U4.B291) suggesting that he was also confusing the orbital with the envelopes drawn to represent them.
§9.2.5: Difficulties concerning the designation of atomic orbitals
For learners who find the orbital concept abstract, and confuse it with shells, sub– shells and energy levels, the designations given to orbitals may seem puzzling.
In Carol's final interview she suggested that an s-orbital was an "x, y, z, type of thing" (C3.309) and thought the next orbital to be occupied after 2s was "3p" (C3.365). Edward confused the labels for orbitals and configurations, referring to the atomic hydrogen orbital as "1s1" (E1.1050, E2.108).
In Tajinder's third interview he attempted to explain the electronic configuration of an oxygen atom in terms of orbital occupation, but was unsure how many p-orbitals there were in the atom's outer shell (T3.A336). In his fifth interview Tajinder labelled various orbitals with inappropriate designations (1s1, 1p, 1px, 1py, 1pz, 3s1 and 1d), and did not think that there could be 3s and 4s orbitals (T5.A288). (See appendix 30, §A30.2.5).
§9.2.6: Electron spin
Electron spin is a concept that is unhelpfully defined in one dictionary of chemistry by the statement that "properties of electrons can only be explained in terms of the electrons having spin, s = ± 1/2 " (Sharp, 1983, p.152).
However, another science dictionary provides a more helpful explanation, as
"the intrinsic angular momentum of a subatomic particle, nucleus, atom, or molecule, which continues to exist even when the particle comes to rest. A particle in a specific energy state has a particular spin, just as it has a particular electric charge and mass. According to quantum theory, this is restricted to discrete and indivisible values, specified by a spin quantum number. Because of its spin, a charged particle acts as a small magnet and is affected by magnetic fields."
Lafferty and Rowe, 1994, p.556
As angular momentum is not a concept that is itself referred to in the A level chemistry syllabus, the concept of electron spin might be expected to cause some difficulties.
In Edward's second interview he referred to electron spin direction, which he 'assumed' meant "that an electron moves about this this volume of space that's called an orbital in one particular direction, whereas the other moves in the opposite direction" (E2.A096), a description which does not relate to a property which continues to exist even when the particle comes to rest (see above). He had "also read it that they're spinning on their axes", but said that he did not "know what that means" (E2.A096).
In Quorat's first interview she had heard of electron spin, however, her understanding of this term was also based on the everyday meaning of spin as relating to motion, "because they're all going to be repelling each other and circling like that, always trying to get as far apart, 'cause that's why they're always spinning" (Q1.A344).
One colearner who was able to provide an explanation of spin that was nearer the quantum-mechanical meaning was Umar, although he did not seemed satisfied with his understanding. In an interview near the end of the first year of his course he referred to electrons being "spin-paired", and was asked to explain this,
I: What do you mean by 'they're spin-paired'?
U: The the electrons are in same orbital.
I: Ah, what does 'spin-paired' mean exactly?
U: It's just to show that
• • [pause, c.2s]
U: they can be in the same orbital.
I: Ah, but what's all, sorry, what's the 'spin' business? What is the 'spin'?
U: It doesn't actually spin, but
••
U: (I dunno, I can't remember actually), it's not really spinning itself, but
••
U: it just means they, they're allowed to be together, I think and, they're in the same orbital, so they might be in opposite directions.
U3.B316
§9.3: Molecular orbitals
For a learner operating with the notions of shells and electron orbits (see above) bonding electrons may seem to be more confined than other valence electrons, as they must remain in the area of overlap of two shells.
So in Annie's first interview she suggested that electrons in a molecule "move around", (A1.198) except for the bonding electrons, "the ones that are involved in [bonding], they can't really move around, like all the way around the shell" (A1.202). Although by her final interview she was talking of 'orbitals', Annie did not appear to have a clear conception of a molecular orbital,
"each atom contributes, er an electron, well the electrons are shared equally between the atoms involved, so you haven't got dominance from one atom with the bonds, or of the electrons sorry … the electrons are sort of held in circuits, orbitals, because when they sort of combine together, they're sort of going around freely, so you've got all the forces, sort of just like they're being pulled in by the nucleus. Electrons are being pulled in, so you're, you've got sort of the nucleus pulling in, the electrons from the other, atom. So it helps them stay together"
A4.8-10
focal figure 2
In Lovesh's third interview, near the end of the first year of his course, he suggested that focal figure 2 (representing hydrogen) could be made more accurate if it was drawn with "orbitals instead, 'cause they show the probability of the electron being in that area" (L3.A282). At first he seemed to bring to mind atomic orbitals, and suggested the orbitals "would be sphere" shaped. But then he decided that "they form molecular orbitals", and then suggested that "it would be linear orbital" although he was "not sure" (L3.A282), perhaps confusing the idea of a 'linear combination' with the shape of the orbital.
§9.3.1: s (sigma-) and p (pi-) bonds
Where learners were able to discuss molecular orbitals appropriately, one area of difficulty was the categories of sigma and pi bonds.
Paminder used the construct 'pi-bonds' in Kelly's construct repertory test, and she was asked about this during the second interview. Paminder knew that a pi-bond differed from a sigma-bond, but was unable to offer any detail of what the pi-bond was, except that it was "like a hamburger" (apparently in terms of sandwiching the sigma bond),
"when you have something with a double bond, like say for example, … suppose you have carbon-carbon double bond, like an alkene, yeah? Like this is all to do with orbitals and things, like, suppose we have ethene, which is C double bond C, H H H H, and like, this, we're talking about orbitals now right, when the double bond is formed it's like an actual pi-bond is formed, it's not like a sigma bond, a sigma bond is like just simple overlap of like the orbitals. A pi-bond is slightly different, it's like, it's like a hamburger you could say. {Both laugh} You know like, this pi-bond like, if you look at the molecule like, three dimension, I think it's three dimensionally yeah? There's a pi-bond on top, pi-bond cloud there, pi-bond cloud there, that's the kind of thing."
P2.A327
She thought that a "sigma bond, … [is] just a simple overlap of like atomic orbitals" (P2.A327), and later added that "it's just linear overlap, like I think, if it's in the same plane, or something, something like that" (P2.A349). Although Paminder was thinking about bonding in terms of orbitals and orbital overlap, her understanding of molecular orbitals may be seen to have been tentative.
Another colearner to use the construct 'pi-bond' was Edward.
triad element 414
triad element 211
triad element 245
triad element 443
triad element 522
During the first term of the second year of his course he undertook the repertory construct test. Some of his assignments suggested that his understanding of pi- bonding was either confused, or idiosyncratic. He construed triad elements 414 (benzene molecule), 211 (an ethene molecule) and 245 (an oxygen molecule) at the emergent pole of either of his construct "pi-bond", although not triad element 443 (a molecule of sulphuric acid, with double bonds indicated as S=O), nor triad element 522 (a carbon dioxide molecule).
In Lovesh's final interview, shortly before the end of his course, he discussed the bonding in benzene where he thought there was "covalent bonding, and there's also some delocalisation". For him this meant "each carbon atom has got an unhybridized p-orbital with an electron in it and that form … two pi–bonds, and that's where the electrons can move around, in a pi bond" (L4.A424).
Lovesh thought the pi bonds were "above the ring and below the ring", and these were two separate bonds "one above and one below, the ring" (L4.A424). In other words, Lovesh was considering the two different volumes of electron density as being two separate bonds.
§9.3.2: Atomic orbitals, hybridization and molecular orbitals
In molecules the bonding electrons are considered to be in molecular orbitals which are formed by overlap of, and linear combination of, atomic orbitals. Often, although not always, the atomic orbitals involved are considered to be hybridized from the ground state atomic orbitals. (Electrons from inner atomic shells, and non-bonding electrons from the valence electron shells are usually considered to effectively remain in atomic orbitals.) In this research it was found that there were many examples of learners apparently confusing ground state atomic orbitals, hybridized atomic orbitals and molecular orbitals.
The hybridization process is considered to provide combinations of atomic orbitals with a more suitable geometry for overlap, so as Carol explained about p orbitals, "you hybridize them because you've got half of the, the p-orbital out the other side where you don't need it" (C3.455), otherwise there might be a "bit of a waste" (C3.507) because "you've got half of it not being used" (C3.509). hybridization is often accompanied by unpairing of electrons to give a greater number of half-occupied orbitals for overlap, an aspect that Carol seemed less clear about,
"something about, [when there is a] load of electrons say all together, and you, split them all up. So … say in, group five you've got five different electrons … in their separate, little bit. So … other electrons can come and bond [sic] with them, 'cause they've got a space …, 'cause they're not spin paired."
C3.515
In the study there were examples of colearners assigning ground state atomic orbitals to molecular orbitals. In his first term Tajinder undertook Kelly's construct repertory test (November, 1992), and one of the constructs elicited was 'shows s and p orbitals'. However, most of the triad elements construed at the emergent pole actually represented molecular orbitals.
In Paminder's third interview, near the end of her first year of A level, she suggested that in a hydrogen molecule the electrons would be in an "s orbital" (P3.A379) and in a tetrachloromethane molecule the electrons in a bond were in a chlorine "3p" orbital, and a carbon "2p" orbital (P3.A387). Paminder thought that the four carbon bonding electrons were in the 2px, 2py and 2s orbitals, and indeed that two of the bonding electrons, i.e. in two different bonds, were in "the 2s" orbital (P3.A403).
Although in real chemical systems promotion of electrons, rehybridization of atomic orbitals, and formation of molecular orbitals are not discrete processes, they are usually presented as if discrete when formulated as a conceptual scheme. Despite this learners may confuse the ideas of rehybridized atomic and molecular orbitals, as when Brian referred to "the bonding sp3 hybrids" (B3.690).
At the end of his first year Lovesh referred to molecular orbitals in tetrachloromethane and methane, as hybridized atomic orbitals, and during his second year he suggested the presence of atomic orbitals in molecular systems (graphite, benzene) where those orbitals would have been 'used' in the bonding (see appendix 30, §A30.3.1).
This type of error could occur even when a learner had demonstrated a grasp of the principles involved. So although at the end of her course Debra demonstrated that she understood the concept of molecular bonding in the simple case of the hydrogen molecule (see appendix 30, §A30.3.2), she thought that the bonding electrons in benzene were "in molecular orbitals" (D3.518) which were "hybrids" (D3.520).
Similar data was obtained from Edward who was able to explain the electronic configuration of carbon, and the process of hybridization ("where you put energy into the system, in the hope that, you'll get a more stable resultant, structure", E1.884), but did not seem to appreciate the formation of molecular orbitals from overlap of atomic orbitals (see appendix 30, §A30.3.3).
A different error was to assume atomic orbitals always need to be rehybridized to form bonds. So near the end of her course Carol described a π bond as "overlapping of p-orbitals" (C3.297), and in the specific example of the oxygen atom in forming an oxygen molecule, she suggested the hybridization would be "π [pi], hybridization, … or something like that" (C3.295). Carol thought that "you have to hybridize them otherwise they don't overlap fully" (C3.299), not realising that this particular bond was formed by the overlap of unhybridized atomic orbitals.
Some of the difficulties experienced by learners may be illustrated though the example of colearner Kabul (see appendix 30, §A30.3.4). At the end of his first year he demonstrated that he understood the basic notion of hybridization. However at the end of the first term of his second year Kabul undertook a College test on multiple bonding and explained the bonding in carbon dioxide in a way that was not only incorrect from the curriculum science perspective (with the wrong hybridization on the carbon) but which contained internal contradictions – the same orbital being described as both p-orbital and hybrid – that make it difficult to produce a consistent interpretation of Kabul's thinking. It seems likely he was himself unsure of a coherent scheme.
When he undertook the repertory test during the second term of his second year Kabul construed molecules as having "sp3 hybridization", assignments he confirmed during a subsequent interview (K5.A331). When he was asked about hybridization he explained that different combinations of orbitals could be hybridized, although he put this down to "what they want to form" (K5.A356). (Learners' use of anthropomorphic language is considered in chapter 11, §11.3). When he was asked about forming the ammonia molecule Kabul was confused about the hybridization required on nitrogen, the number of hybrids formed by the sp2 and sp3 hybridization, and suggested that in each case one hybrid would be "not that similar" to the others. Kabul appeared to be confusing the nitrogen atomic hybrid containing the lone pair, with the molecular orbitals formed from the overlap of the other hybrids with the hydrogen atomic orbitals. Although Kabul seemed to have some grasp of the key points of the hybridization concept, particularly the requirement "to get good overlap", he also showed some confusion. It would seem he had still not completely separated the concepts of the hybridized atomic orbitals from the molecular orbitals that resulted from overlap: thus the suggestion that one of a set of hybrids would be dissimilar because it contained a lone pair.
This issue was further explored later in the fifth interview (see appendix 30, §A30.3.5). During an extended exchange Kabul's belief that the atomic orbitals still exist in molecular species was elicited, and then challenged. Kabul thought that the orbitals present in the hydrogen molecule were "s orbitals, 1s orbitals", none others, "just 1s orbitals" (K5.A420). In methane he thought there would be "1s and p orbitals … like 2p on carbon, and 1s on hydrogen" (K4.A424). Indeed Kabul at first suggested that on carbon there would be not one p orbital, but "four, 2px, y and z, … four [sic]" 2p orbitals (K5.A428). As well as these 2p orbitals, he thought that "there are [other orbitals on carbon] but they're, they don't take part in bonding" (K5.A428).
When Kabul was asked about methane he seemed to confuse two aspects of the hybridization process (the geometry and orientation of four similar sp3 hybrids that makes them suitable for overlap with orbitals on other species, and the similarity of the energy levels of two overlapping atomic orbitals that leads to a molecular orbital of significantly lower energy). Although Kabul had talked of the overlap involved in forming bonds, he then went on to describe the orbitals present as atomic orbitals.
When he was asked about diamond Kabul first suggested there were only ground state atomic orbitals present, but he later changed this to include hybridized atomic orbitals.
When it was put to Kabul that there were no 1s orbitals in the hydrogen molecule, and no sp3 hybrids in the methane molecule, he did not agree. Only after agreeing that there was a molecular orbital present (which he recognised was "made up of, … two atomic orbitals, [which] combine together to form a molecular orbital" , K5.A487) did Kabul then agree that there were no 1s orbitals in the hydrogen molecule.
However, when asked about the methane molecule again, it took Kabul a short time to transfer the same argument to this context, and agree that there were no (valence shell) atomic orbitals in the methane molecule. After this though he did immediately recognise that he was wrong about the hybridized orbitals in graphite: that in the structure there were actually molecular orbitals. In this interview Kabul had demonstrated that he had both the necessary knowledge, and the competence, to discuss the orbitals present in hydrogen, diamond and methane from a molecular orbital perspective – one might say such a description was within his zone of proximal development (§2.2.2) – but that he spontaneously tended to think of the molecules in terms of the atomic orbitals present before bonding. His knowledge of atomic orbitals and hybridization seemed to act as an epistemological learning impediment (§1.5.5) to thinking about molecular orbitals.
§9.4: Resonance
Some structures of interest to chemists cannot be readily represented in terms of drawing single, double and triple bonds between specific atoms. These structures are often better represented in terms of a molecular orbital description. However, it is usually also possible to consider the actual structure to be 'in between' a number of valence-bond structures, called canonical forms. The actual structure is said to be a resonance of these canonical forms.
In the present study there was evidence of learners having difficulty conceptualising resonance. In particular there was evidence of students considering the resonance to mean an alternation between the canonical forms, rather than something intermediate to them.
§9.4.1: Resonance in benzene
focal figure 12
In the interview study a range of interpretations of the bonding in benzene were elicited from colearners, and – as would be expected – individual colearners' understandings changed over time. Three themes may be identified from the comments made. The first theme is that of the interpretation of the circle used to represent aromaticity, which was seen by some colearners as indicating some type of electron reservoir inside the ring. A second theme concerns the use of the term delocalisation, but in the absence of a molecular orbital interpretation that makes the notion vague, and even unrelated to the bonding. Thirdly, although learners may refer to resonance, they may mean an alternation between single and double bonds, although perhaps one which occurred very rapidly.
These three themes may all be illustrated in the case of Annie. In Annie's first interview she thought that the circle "shows where the electrons are, because it's electron rich" (A1.464) and "they're denser in the circle" (A1.466) – not all the electrons – "just the ones from the carbon" (A1.472). In her second interview Annie showed an awareness of delocalisation, although this did not appear to be related to any concept of molecular orbitals. Her interpretation was that the electrons "go around in the ring, so they sort of charge around and, … they're not fixed anyway, they don't belong to anything in particular, so they're, they're free-flowing" (A2.295).
In the third interview Annie demonstrated some notion of the resonance, in terms of single and double bonds that move around the ring,
"If you've got the benzene ring, erm, with the double bond and the single bond and then, I don't know somehow, a simplistic way of looking at it, and the bond moves, …"
A3.233
Each of these three interpretations were reflected in data from other learners.
The electron reservoir. Where Annie referred to a circle "where the electrons are, because it's electron rich" (A1.464), in Brian's second interview he demonstrated a more technical rationale for a similar interpretation. He took the circle inside the hexagon as a literal representation of where the delocalisation occurred (B2.58-60). Brian reported that the p-orbitals "of the ring overlap, and the electrons can pass from orbital to orbital, to become delocalised" (B2.76), but he though that the p- orbital used to overlap to give the delocalised system was in the plane of the ring, so "in carbon there's four valence electrons, three of which are used in bonding, and the one in the other p-orbital" (B2.76) which was in "the plane of the" hexagon (B2.82).
Carol's interpretation in her early interviews was less technical, in terms of spare electrons. So in her first interview she explained that "there's six spare electrons in the middle" (C1.579), which were "just spinning around" (C1.581). There were spare electrons as "carbon, it's got a valency of four, and, because it's only got three … bonds … , it's got to have one [electron] still whizzing round itself" (C1.585-587). She thought that these 'spare' electrons were "attracted to their own nucleus" (C1.623) and were localised (C1.625). In her second interview Carol maintained that in benzene there were "spare electrons from the carbon" (C2.452) "because carbon's got a valency of four so it can form four bonds and yet it's only formed three, so they're, like, left in the middle" (C2.454) She thought that the 'spare' electrons were not involved in bonding, and "you show that by the circle" (C2.456).
In Debra's first interview she had a similar interpretation, that the circle was "where the extra electrons are from the outer, shell. 'Cause it usually has a valency of four, [so] there's one electron from each carbon, in there" (D1.504). She thought that these extra electrons were "sort of free to move … in between carbons" (D1.510-512). Even in her second interview, when she was considering the resonance as between single and double bonds, Debra thought that the circle "shows the electrons" (D2.423) which are "sort of, within the carbons, within the ring … Or in the middle" (D2.425-7). She was able to explain the hybridization in the molecule, and the overlapping of the unhybridized p-orbitals, but despite this, Debra still thought that the ring drawn in diagrams of benzene represented the "spare electrons" (D2.605).
Kabul demonstrated a sophisticated interpretation of the circle, that it represented "the electron density, of carbon atoms" (K6.A557), which was shown for benzene "because … the outermost electrons [from "just the carbon atoms" (K6.B004)] are equally attracted by the whole ring of atoms, not just one atom" (K6.A562). However, even here the circle symbol appears to be taken too literally, as the electron density represented is actually above and beneath, not inside, the hexagon.
Vague notions of delocalisation. Annie's notion of delocalised electrons not belonging [sic] to anything, not fixed to anything and free-flowing (A2.295) does not make sense from a valence-bond perspective, and was not supported by any kind of molecular orbital interpretation, but rather seemed to have been learnt as an isolated piece of information (c.f. fragmentation learning impediments, §1.5.2). Although Annie had also learnt that the circle could indicate "an unsaturated, aromatic or something" (A2.287), she thought aromatic simply meant "that it smells" (A2.289).
During Brian's first interview he demonstrated that he had acquired the notion that benzene had "a delocalised system" (B1.434), but he thought this just meant meant that "the double bonds aren't in any set place … they're not in specific places on every benzene … molecule" (B1.436-440). In Carol's final interview she described the benzene molecule as having both rings of electrons, and delocalised electrons, that is "kind of like a ring [with] like electron thing underneath it, and electron thing on the top, isn't it, because they're π-bonds … and then you've got delocalised electrons in the middle, but I don't know what they look like" (C3.1019 -1021). She later she referred to the 'electron thing' as "the electron density below and above it, kind of thing" (C3.1037), however she did not realise that this 'electron density' was the delocalised electrons. Whereas the electron density was 'underneath' and 'on top', the delocalised electrons were inside the ring, and were not seen as relevant to the bonding.
In Kabul's last interview he referred to how "the electron density … [of the] carbon atoms, overlap with one another, and they form a delocalised structure" (K6.B031). However his explanation of this delocalisation was simply that the three electrons from each carbon atom involved in carbon-carbon bonding were "attracted equally between carbon atoms" (K6.B041).
Resonance as alternation. In Annie's second interview she was unsure about the existence of double bonds in benzene, first suggesting that benzene had "single" (A2.277) "covalent bonds" (A2.275), although "one carbon to carbon bond would be a double bond" (A2.279), and then that "they're all single bonds" (A2.281).
Although Annie referred in her third interview to "the double bond and the single bond and then … the bond moves" she recognised that this was "a simplistic way of looking at it" (A3.233). Annie was aware that the canonical forms were not accurate representations of the structure of benzene – as "they don't really exist, it's sort of something that scientist has in their minds to show, to explain something", and "in nature, they don't actually perform that way". However, Annie did not seem to consider the role of molecular orbitals, and was limited to discussing benzene in valence bond terms ("you haven't got single bonds all the way around, you've got to have three double bonds"). (See appendix 30, §A30.4.1).
In Brian's first interview he seemed to imply that single and double carbon-carbon bonds were fixed in particular molecules (although "the double bonds aren't … in specific places on every benzene … molecule" (B1.436-440). Carol however clearly entertained the notion of alternation of the bonds within the molecule. In her final interview, as well as discussing the electron density above and below the ring, and the electrons inside the ring, she also talked of how,
"it will be double bond, single bond, double bond, single bond, double bond, single, … and, to make the resonance, you draw a little two way arrow, and where there was a double bond in one diagram there would be a single bond in the other one."
C3.1047-9
Carol seemed confused over these three different models (the spare electrons, the π clouds and the alternating single/double bonds), and immediately after referring to the double bonds suggested there were "just single bond all the way around, and the delocalised electrons in the middle, in a ring" (C3.1051). She also thought that the bond order "could be any, anything you wanted to" (C3.1051). She went on to suggest an alternative interpretation of the circle in the figure: that it "shows that, you can either have a double bond, or a single bond, and it happens so quickly that you might as well just have a single bond" (C3.1057), but although the alternation was rapid, she thought it was literal, so that the bonds were actually "both … sometimes single, sometimes double" (C3.1060-1061).
By Debra's second interview she was referring to resonance in benzene in terms of an alternation, so that – of the carbon-carbon bonds – "some of them are double and some of them single, … there's three that are doubly bonded, but the electrons are free to move between, between the carbons so you don't actually write on the diagram as a double bond" (D2.391-393).
However, she contradicted this view of alternating bonds when she went on to suggest that the bonds were "in between the length of a single and a double" (D2.403) and that the bonds were not single or double but "midway between them" (D2.411) – although Debra could only suggest this might mean "three" electrons per bond (D2.417). By this interview Debra had some understanding of hybrid orbitals. So whereas in preparing methane "you hybridize" the carbon orbitals (D2.464) to give "four" (D2.470) "sp3" hybrids (D2.466), which point in a "tetrahedral" arrangement (D2.474), in benzene each carbon atom has to bond with "three" other atoms (D2.493), so the hybridization required would be "sp2" (D2.497), which leaves "a p orbital" (D2.501), where Debra supposed "the electrons are [in an] overlapping p-orbital so they're paired" (D2.679). However, Debra did not seem to be able to integrate her consideration of the single and double bonds with this discussion in term of orbitals.
Kabul's explanation of the bonding in his final interview also demonstrated some ambiguity. He knew that each carbon atom would "use one of its electrons to form bond with the hydrogen, uses one of the bonds [sic] to form bond with the carbon, another carbon, and uses two of its electrons to form bond with another carbon" (K6.B018.) Kabul also seemed to recognise that delocalisation gave rise to a symmetrical distribution of electron density, but he did not seem able to explain how this related to the valency requirements of the atom,
"the actual structure of benzene, you know, where you've got a single bond, double bond, single bond, double bond, alternating …But, but when you see it overall, … the electron density … [of the] carbon atoms, overlap with one another, and they form a delocalised structure."
K6.B031
During Quorat's second year she prepared a concept map for 'multiple bonding' as a revision exercise. This map included a reference to "resonance structures" and "canonical forms" (see appendix 30, §A30.4.2). However her explanatory notes suggested that Quorat construed 'resonance' as a device for overcoming ignorance about which bonds were double and which were single ("since the actual positions are not known, it is better shown as a delocalised system"), rather than as a means of representing bonds with non-integral bond order, e.g. those which were between single and double bonds.
§9.4.2: Resonance in the ethanoate ion
focal figure 13
Focal figure 13 was intended to represent the two canonical forms for the ethanoate ion. In Debra's first interview, she thought the two canonical forms were the same molecule seen from a different angle (D1.574), as she did not think the same molecule could change between the two configurations (D1.578). However in her second interview she had changed to a more common interpretation in terms of alternation between the two structures. Debra suggested that the double headed arrow in focal figure 13 "represents that it can change from one to the other" (D2.689) when "the electrons that are in, that double bond, move over to the other carbon and the oxygen, and form a double bond there" (D2.693).
In Brian's second interview he identified that the arrow meant "resonance between the two forms" (B2.106). He recognised that in the two different forms shown a "different oxygen" centre was charged (B2.114), although he thought resonance meant "it alters between the two states" (B2.110) so that at any one moment an oxygen "could be minus, or it could have no charge" (B2.145), but the change over was "instantaneous" (B2.156). By his third interview Brian's thinking had developed further to match the intended meaning of resonance, so he thought that the arrow represented "resonance" (B3.286) which meant that "it exists sort of between the two forms" (B3.288), the bond order was "one and a half" (B3.298), and the molecule would exist in a "form in between [the canonical forms] with, single bonds to both oxygens and a delocalised system between the two oxygens" (B3.322) comprising of "overlap of p-orbitals" (B3.324-332) on the carbon and both oxygens.
Kabul also developed beyond seeing the resonance as alternation, although his thinking did not shift as far as Brian's. In Kabul's fourth interview he thought that "the negative charge is being shared you know by both the oxygen atoms" (K4.B330). This was according to "something which [Kabul had] read", but he could not "recall now" (K4.B330). However, he thought the arrow symbol was meant to represent "resonance" which meant "sometimes it could be this thing, sometimes could be the bottom thing" (K4.B346), although he could not suggest how this could be. However, in his final interview Kabul explained that the diagram showed "a resonance between … two forms of a compound" which meant that "it can either exist in this form or other form, you know, the actual structure is in between both the forms" (K6.B056). Kabul agreed that he was saying that the structure swaps round between the two, so at any one moment it could be one, then it would flip over to the other one (K6.B063). As he seemed to be implying an alternation of the structure Kabul was asked what he meant by saying that the actual structure is between the two. He explained that "the electron cloud will be shared equally between … two oxygen atoms, so that's the actual structure" (K6.B066). So Kabul appeared to hold two distinct models of the structure in his head, one with a stable smeared electron density, but the other involving alternation between valence-bond structures such as those in the figure he had been shown.
§9.4.3: Resonance in boron trifluoride
focal figure 14
A third context for discussing resonance was that of boron trifluoride. The focal diagram (number 14) showed three canonical forms, each with two covalent bonds and one ionic. This is one mode of representing the bonds as polar.
In the first interview Brian describes the diagram as depicting "an alternation between the states of boron fluoride" (B1.539). He identifies the bonds present as "ionic and covalent" (B1.541): so "the majority of the time [a particular bond] is covalent, but occasionally it is ionic" (B1.560). In his second interview the polar bonding in boron trifluoride was thought to consist of "some covalent, and some ionic" bonding (B2.122), and Brian thought the electron density distributions would be different in the two types of bond (B2.137-139). In the third interview Brian describes the bonding as "sort of fifty-fifty … co-ionic … [or] polar" (B3.434-8). Although Brian reports that "all three of" the bonds will be polar (B3.442), as "they'll all be the same" (B3.444), he describes the electron density in terms inconsistent with such a belief,
"polar towards the fluorine … on the two … fluorines that are not charged. There will be a greater electron density around the fluorine. And on the one that is charged it will be completely around the fluorine."
B3.452
Brian's explanations are contradictory, as he subsequently maintains that all the bonds are equally polar (B3.470). In this less familiar example of resonance Brian does not seem to fully appreciate the way canonical forms showing ionic and covalent bonds are able to stand for a resonance that does not alternate between them, but rather has a form 'in between' them. He apparently interprets the bond type differently depending upon whether he is labelling the bonds verbally, or construing them in terms of electron density patterns, an interesting finding which mirrors Kabul's distinct conceptualisations of the ethanoate ion structure above (§9.4.2).
In Debra's first interview she did not recognise the resonance represented in focal figure 14, but instead interpreted one of the canonical forms literally: that there were "two covalent bonds, and one ionic bond" (D1.624).
In Kabul's fourth interview, he was asked to consider the diagram, which he thought showed that "sometimes, one of, this fluoride ion is negative, so sometimes this fluoride, another fluoride ion can be negative. It shows like resonance" (K4.B381). The curriculum science meaning of the term resonance would be that all three of the bonds were something in between ionic and covalent, but further questioning suggested that this was not what Kabul meant (see appendix 30, §A30.4.3). Although Kabul referred to resonance, his spontaneous expectation was that the bonds would be covalent (K4.B377), and his interpretation of the figure was that individual bonds would at any one time be covalent or ionic.
In his final interview Kabul thought that the figure represented "resonance" which meant "it can exist in either [sic] form. It can be either of them, at any time" (K6.B084). Kabul thought that the diagram probably meant that "it's bonded covalently with two and ionically with one" (K6.B096). Changing between the different forms required the "atom" to "just flick around" (K6.B102). So even near the end of his course Kabul did not consider the curriculum science meaning of the resonance.
This particular example, representing polar bonds as a resonance, may demonstrate something more than a difficulty with a molecular orbital description, as it will be suggested in chapter 11 that learners in this study commonly showed some reluctance to label bonding as intermediate to ionic and covalent (§11.6.2).
