In the Classroom
The Mole, the Periodic Table, and Quantum Numbers: An Introductory Trio Mali Yin Department of Chemistry, Sarah Lawrence College, Bronxville, NY 10708 Raymond S. Ochs* Department of Pharmaceutical Sciences, St. John’s University, Jamaica, NY 11439-0002;
[email protected] Few of us believe that our students absorb all of what we teach. The fundamental ideas of chemistry are known to be poorly grasped even after substantial exposure to the subject (1–3). The idea of providing a few great ideas in chemistry has been advanced by Gillespie (4). In this work, we propose three: the mole, the periodic table, and quantum numbers. How they should be presented, related, and developed is the substance of this contribution. Understanding the Mole The mole is a foundational concept of chemistry and yet is poorly understood by students (5–8). While some have advanced systems to improve problem-solving ability (7 ), or accounting-type systems (9), or favor a more historical approach (10–12), these solutions avoid teaching the underlying concept, which is of paramount importance (13). Rather than have precision of facts or a rigid order placed on presentation of material, we propose a radical shift: strip the concept to its essence. Eschew nonessential facts. Similar ideas have been proposed to explain the number–size relationships using jellybeans in a jar (14 ). One barrier to student understanding of the mole is the pedantic insistence on burdening the concept with needless detail. One egregious fact is that the atomic weights are based on assigning carbon the number exactly 12. Most texts also incur a historical regression, in which oxygen had previously been assigned as exactly 16, and before that, hydrogen had been assigned to be exactly 1. How is the beginning student supposed to assimilate this introduction to chemistry? The answer is evident from the resulting answers given by students: they don’t get it. Add to this the pedantic necessity of changing “atomic weight” to “atomic mass”. This makes an unfamiliar concept appear linked with another unfamiliar one. These quantities aren’t really weights, in any case; they are relative weights or masses.
A Slight Inaccuracy, a Greatly Improved Understanding The better way to teach the concept of the mole is to ignore the exact value or even any mention of values for C or O. The idea of the atomic weight itself is that the lightest element—hydrogen—should be set to one. The next question becomes, one what? We start with hydrogen, the lightest element, and consider a collection of hydrogen atoms that weighs 1 g as a standard. All other elements are then heavier than this by some multiple, which we determine by considering the same number of atoms in their collection. In this way, this number of atoms of carbon weighs 12 g because carbon is 12 times the weight of hydrogen; similarly, oxygen
is 16 times the weight of hydrogen. What we are really talking about is that some collection of atoms—defined as a mole— is obviously a very large collection, because all of us are aware that atoms are very small. It has a relative weight of 1 if it is a collection of hydrogen atoms, 16 if it is a collection of oxygen atoms, 12 if it is a collection of carbon atoms, and so forth. Now, this is a very powerful idea, because thinking about moles allows us to understand reactions. Atoms and molecules react in simple, whole-number ratios—as moles, not as grams. For example, 1 mole of C reacts with 1 mole of O2 to form 1 mole of CO2. Molar arithmetic is clearly different from ordinary arithmetic; abstracting the reaction in our example, we have 1+1=1 This is part of the unfamiliar territory that we must learn. Our ordinary senses return when we substitute the weights of the actual entities: C [12] + O2 [32] = CO2 [44]. The temptation to expand this point to issues such as isotopes should be resisted until the idea of moles and the fundamental notion of ratios of the atomic weights is entirely clear. In fact, isotopes as a topic have no part in our proposed introduction to chemistry. As for focusing on Avogadro’s number, there is little value in the beginning aside from calculation ability, and interest in such things is heightened only when students are clear as to what the number means in the first place (15). After all, even Avogadro didn’t know Avogadro’s number (16 ). To the Periodic Table It is a simple step from the idea of moles to the idea of the periodic table. One of the most imaginative is published by Atkins (17 ): an anthropomorphic romp through the various regions (e.g., western minerals), with clever asides. However, few students will sit still for this. On the other hand, textbook presentations (e.g. refs 18–20) do not join moles directly with the periodic table. Instead, these are placed in different sections, well removed from each other. Moreover, they have already lost the opportunity for a clear and direct connection by committing the sin of pedantry, asserting that the table is based on carbon-12! We propose the following description, which can be used directly after the mole has been explained. Suppose we arrange the elements in a simple order, from lightest to heaviest, all in a line. Our arrangement then becomes a simple onedimensional array, and that dimension we can identify as “weight”. Once again, the temptation to introduce isotopes
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In the Classroom
must be resisted. The listing, with the weights substituted for the element names is 1, 4, 7, 9, 11, 12, 14, 16, 19, 20, 23, 24, 27, 28, 31, 32, 35, 40, … Now this is clearly one possible representation of the elements, but we will soon run out of room on the paper. In the 19th century, it was noticed that a second dimension could be introduced into the display. It was observed, for example, that elements with weights 1, 7, and 23 shared similar reactivities. Also, those with weights 4, 20, and 40 were very similar; they had essentially no reactivity! We can rearrange the list to produce the familiar periodic table. The periodic part refers to the recurrence of properties such as reactivity when the elements occur in the same column, reflecting this second dimension. Any two-dimensional table describes two things about the entries in that table. This table describes: 1. The relative weights (which come from the original mole idea), shared by rows; 2. The reactivities (which come from experimental observations), shared by columns.
The fact that many ideas can be derived intuitively from the periodic table explains why it is so popular among chemists; but among students, it often appears to be just one more thing to memorize! Others have argued that at least nonchemistry majors should be spared the details of the periodic tables, and just given the essence (21). Various formats have been suggested (22, 23). We suggest that only the simplest concepts should be presented at first, as depicted here, for all students. The Next Step: The Quantum Numbers For the next step, we propose a simple introduction to quantum numbers. There is some debate about whether quantum numbers belong in a beginning chemistry course, let alone at the very beginning. However, we have found from experience that this topic can be introduced early in courses for both liberal arts students and chemistry majors. Relating the empirical foundations of chemistry represented by the periodic table to simple aspects of the more modern quantum theory can, we believe, lead to a clear understanding of basic aspects of chemistry. The quantum numbers can be described as below, without recourse to the Schrödinger equation, except to mention that the numbers are the solutions of this equation. There are four quantum numbers: n, l, mᐉ, and ms. While they can be associated with their names as a first introduction, students find the symbols themselves less daunting than wondering about what concepts such as “spin” and “angular momentum” mean.
Quantum Number n The quantum number n is the row number for the main group elements in the periodic table. It has values of 1, 2, 3, … and is also the “energy shell” of the atom. Quantum Number l The quantum number l is the block of the periodic table. Its values are 0, 1, 2, …, (n – 1), but these are also named s, p, d, f, … . These symbols have more historic than educational value, as they represent associations with lines in the atomic 1346
emission spectra (sharp, etc.). We believe they, like the quantum numbers themselves, should remain as symbols for this introductory treatment. Linking them to the periodic table, the first subshell, s, is the small block to the left; the second, p, is the larger block to the right; the third, d, is an even larger block interspersed between the s and p (in standard presentations). The f is the wide block placed below the others. The quantum number l is also the “subshell” of the atom, and it determines the shape of the orbitals within the subshell. The letters s, p, d, f, … are the subshell labels. An orbital is associated with a region in space where an electron is likely to be found. Every orbital in a subshell has a characteristic shape, which relates to the probability of finding an electron in a specific region.
Quantum Number mᐉ The quantum number mᐉ almost dictates how wide the blocks will be. It is also the orientation of the subshells in the atom. To remove the “almost” from the above, we need a fourth quantum number—— Quantum Number ms The quantum number ms, when combined with mᐉ, completely specifies how wide the blocks will be. From this, one can derive the exact widths of the blocks: s, p, d, and f are respectively 2, 6, 10, and 14 elements wide. Thus, the combination of mᐉ and ms specifies both the width of the blocks in the periodic table and the number of electrons held in each subshell in the atom. From the quantum numbers and the periodic table, students can readily write the electron configurations of atoms. Traditionally, students must first encounter energy levels of atomic orbitals. However, the electron configurations for the atomic ground states can be taken from the periodic table immediately after our introduction to the quantum numbers. We have students work with partners on combinations of quantum numbers, writing the electron configurations directly from the periodic table. They begin with H, going through each block, putting electrons into s, p, and d subshells (in our introduction, we don’t actually include atoms with electrons in the f subshell). The students have an early introduction to and an immediate grasp of the relations of the periodic table to the quantum numbers, and an easy and successful exercise. As indicated by class discussions and homework problems, this is far more successful than methods that rely on memory or mnemonic devices. There are arguments about whether quantum mechanics is even necessary (24–26 ). However, at a simple level, it is a foundational concept; avoiding complex details, it can provide chemical insight even at the start of an introductory course. Obviously, as students are not presently learning the basics of chemistry, they tend not to enjoy it or attempt to apply it to their lives. In this work, we suggest that a key to reversing that situation is to identify the most important ideas and be rigorous about sticking with them to the exclusion of higher order extraneous information. After the presentation of this material, more information can be introduced, taking the opportunity to repeat the basic information whenever possible. Paraphrasing Strunk and White (27 ) slightly, repetition of the basics is never enough; repetition of the basics is never enough.
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In the Classroom
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