Article pubs.acs.org/jchemeduc
From Periodic Properties to a Periodic Table Arrangement Emili Besalú* Departament de Química i Institut de Química Computacional i Catàlisis, Universitat de Girona, C/Maria Aurèlia Capmany, 69, 17071 Girona, Catalonia, Spain S Supporting Information *
ABSTRACT: A periodic table is constructed from the consideration of periodic properties and the application of the principal components analysis technique. This procedure is useful for objects classification and data reduction and has been used in the field of chemistry for many applications, such as lanthanides, molecules, or conformers classification. From the information given, the whole procedure can be reproduced by any interested reader having a basic background in statistics and with the help of the supplementary material provided. Intermediate calculations are instructive because they quantify several concepts the students know only at a qualitative level. The final scores representation reveals an unexpected periodic table presenting some interesting features and points for discussion.
KEYWORDS: Upper-Division Undergraduate, Chemoinformatics, Physical Chemistry, Analogies/Transfer, Atomic Properties/Structure, Chemometrics, Periodicity/Periodic Table
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rincipal components analysis (PCA)1−4 and the periodic table (PT)5−7 are two topics that have been covered extensively but separately by this Journal. Here, both concepts are merged to obtain a PT representation according to PCA calculations made over a set of periodic properties. The procedure engages the students for several reasons. First, the analyzed properties are well-known to them. Second, the procedure is instructive because the relationships among the elements are also conceptually known by the students in advance. Finally, the graphical representation turns out to be graceful and, sometimes, unexpected for the audience.
experienced when introducing and teaching chemometric tools in a general framework.2 The needed procedural mathematical details are well stated in several sources, such as Cazar’s article in this Journal, among others, and also in the Supporting Information provided, which allows the calculations shown here to be reproduced by following the steps and using the online applications cited.
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SESSION One hour is enough to demonstrate the concept if all the needed material is prepared previously. The instructor can present the material in an interactive demonstration format, using an Internet connection to reproduce the calculations in a projection screen. However, a more interesting option is doing the session in a computer classroom where each student has access to an Internet connection. In this case, an activity guide is provided and the students reproduce the calculations with the cited Internet sites. The instructor’s activity guide is provided as a spreadsheet in the Supporting Information and contains the needed calculations: the initial data used, the list of complete and consecutive data manipulations, including screenshots of the Internet sites that have to be accessed to perform a matrix diagonalization, and the final graphical representation. The activity guide can be also formatted, hiding selected data, to constitute an in-class or a homework assignment. In the latter case, two classroom sessions are needed, the first one to give the instructions and the second one to discuss the results obtained.
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MOTIVE For many years, the author has been faced with the task of teaching the PCA chemometric technique to chemistry students. This is done in a second-year, one-semester course on mathematical methods in which a 1-h session is normally reserved for the subject. At our Faculty the course that incorporated this content is named “Mathematical Methods Applied to Chemistry”, which constitutes a revision of algebraic and numerical methods specifically related to chemistry and preparing for more advanced chemistry courses. Prior to this session, the needed statistical and linear algebra topics are presented. In the classroom, the strategy used to present the PCA procedure consists of developing a numerical example dealing with periodic properties. Periodic properties are familiar to the students and this allows them to engage in the classroom presentation. The particular data being manipulated not only allows for an effective presentation but also leads to an interesting final graphical display of object scores. This turns out to be exciting for the students, which is also what Cazar © XXXX American Chemical Society and Division of Chemical Education, Inc.
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Table 1. Data Used in This Worka Item No.
Atomic Number
Element Symbol
Atomic Weight/ amu
Atomic Radius/ pm
First IP/ (kJ mol−1)
First EA/ (kJ mol−1)
Electronegativity (Pauling scale)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
1 3 4 5 6 7 8 9 11 12 13 14 15 16 17 19 20 31 32 33 34 35 37 38 49 50 51 52 53 55 56 81 82 83 84
H Li Be B C N O F Na Mg Al Si P S Cl K Ca Ga Ge As Se Br Rb Sr In Sn Sb Te I Cs Ba Tl Pb Bi Po
1.008 6.94 9.01 10.81 12.01 14.01 16 19 22.99 24.31 26.98 28.09 30.97 32.06 35.45 39.10 40.08 69.72 72.64 74.92 78.96 79.90 85.47 87.62 114.82 118.71 121.76 127.60 126.90 132.91 137.33 204.38 207.20 208.98 209
25 145 105 85 70 65 60 50 180 150 125 110 100 100 100 220 180 130 125 115 115 115 235 200 155 145 145 140 140 260 215 190 180 160 190
1310 519 900 799 1090 1400 1310 1680 494 736 577 786 1011 1000 1255 418 590 577 784 947 941 1140 402 548 556 707 834 870 1008 376 502 590 716 703 812
73 60 0 27 122 −7 141 328 53 0 43 134 72 200 349 48 2 29 116 78 195 325 47 5 29 116 103 190 295 46 14 19 35 91 174
2.2 1 1.6 2 2.6 3 3.4 4 0.93 1.3 1.6 1.9 2.2 2.6 3.2 0.82 1.3 1.6 2 2.2 2.6 3 0.82 0.95 1.8 2 2.1 2.1 2.7 0.79 0.89 2 2.3 2 2
a
The data have been collected from Atkins’ textbook8 and the atomic radii have been taken from Wikipedia, which in turn come from Slater’s work.9,10
Table 2. Symmetric Diagonalized Correlations Matrix R among the Variables
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Property
Atomic Weight
Atomic Radius
First IP
First EA
Electronegativity
Atomic Weight Atomic Radius First IP First EA Electronegativity
1.000 0.580 −0.351 0.005 −0.111
1.000 −0.848 −0.355 −0.753
1.000 0.597 0.911
1.000 0.699
1.000
DATA Table 1 lists five atomic properties for a set of 35 elements (the representative ones excluding noble gases): atomic weight, experimental covalent atomic radius, first ionization potential (IP), first electron affinity (EA), and electronegativity. Precise numbers are not necessary because the goal is to provide the general structure of the collected data. Thus, the numbers available in many textbooks are appropriate.
dimension reduction, the proposal is to present and teach the PCA method by applying it to the data in Table 1. This can be presented in a lecture format or even interactively, using standard statistical programs or online applications.11 A practical guide is available in the Supporting Information. The first treatment is to standardize the last five columns of Table 1 (z transformation obtaining matrix Z), making them dimensionless. Next, the correlations matrix R of the five variables is obtained (Table 2). The author always felt comfortable demonstrating the PCA method by means of this example. The students show immediate engagement because they “know” the data. The implication is reinforced because moving through the numerical method progressively
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PROCEDURE After the introduction of general information and chemometric concepts such as data simplification, data interpretation, and B
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although some of them can feel more comfortable picking three. In the present case, the standard criterion suggests selecting the first two PCs because each one condenses information equivalent to more than one of the original variables.12 After a decision is made regarding the number of PCs to keep, it is now prescriptive to look at the structure of these new variables and discuss the loadings or eigenvector coordinates (see Table 4, which presents the two main columns of matrix L, the ones attached to the biggest eigenvalues). From the composition of the new PCs, it can be inferred that the first one is mainly related to pure periodic properties (all the original variables contribute appreciably to the eigenvector except the atomic weight). The first component is a linear combination dominated by the ionization potential, the electronegativity value, and, with an opposed sign, by the atomic radius (in accordance with the correlations in Table 2). The first PC can be interpreted as defining a variable related to the PT periods, despite the small influence of the atomic weight. On the other hand, the second PC is clearly dominated by the atomic weight, and this new variable can be mainly associated with the variation of the elements along the PT groups. All in all, the two first components are related to the periods and groups of the PT. The scores (matrix S) are the new coordinates of the described objects (the elements) in the new reduced set of variables or PCs. The elements are depicted in Figure 1, after
reveals quantitative evidence of qualitative knowledge the students acquired in previous courses. For instance, the data in Table 2 show the notable correlations existing between some of the periodic properties. The first ionization potential and the Pauling electronegativity scale are positively correlated (r = 0.911), and the latter is also slightly correlated with the electron affinity (r value of almost 0.7). As is well-known by the students, an alternative to the Pauling electronegativity scale is Mulliken’s scale, which is the arithmetic mean of electron affinity and ionization potential. The students are normally aware that both electronegativity scales are positively correlated, but it is important that at this stage they realize the level of correlation being dealt with. Another illustrative datum worth mentioning is the negative correlation between the ionization potential and the atomic radius (−0.848), a qualitative concept also well-known by first-year students that is quantified here. The next procedural step consists in diagonalizing the correlations matrix and obtaining the eigenvalues and loadings matrices. Once the symmetric matrix R is given, the diagonalization consists in solving the equation RL = LD, being D a diagonal matrix (the eigenvalues matrix codified in Table 3) and L a unitary matrix (the eigenvectors or loadings Table 3. Eigenvalues of the Correlations Matrix PC
eigenvalue
Retained Variance (%)a
Cumulated Variance (%)a
1 2 3 4 5
3.24 1.19 0.43 0.09 0.05
64.9 23.7 8.6 1.8 1.0
64.9 88.6 97.2 99.0 100.0
a
The percentages stand for the portion of retained variance of the data.
Table 4. The First Two Eigenvectors Showing the New Variable (PCs) Loadings
a
Original Property
PC 1
PC 2
Atomic Weight Atomic Radius First IP First EA Electronegativity
−0.237 -0.497 0.536 0.381 0.515
-0.771a −0.306 −0.021 −0.493 −0.262
The most relevant loadings are shown in bold type.
matrix partially reproduced in Table 4). Diagonalization can be performed using statistical packages. There are also online tools11 that allow matrix R (Table 2) to be input by means of a manual copy and paste action. Then, the application automatically returns the eigenvalues (Table 3) and the corresponding eigenvectors (Table 4). The eigenvalues read in Table 3 are sorted. The percentages tell us about the portion of information or variance retained by the new generated variables L, also called principal components (PCs). This information is crucial in PCA techniques, and it is a good idea to deal with the concept of dimensionality reduction by asking the audience for the appropriate number of new variables needed to describe our original data in less than the original five dimensions. At this stage, students normally agree that two dimensions are enough (more than 88% of information is kept in our example),
Figure 1. The new scores of the 35 elements reproducing the typical arrangement in the PT. Original groups and periods are connected by segments. Note the particular position of H and the apparent displacement of C and N.
the change of basis, according to these new coordinates computed from the matrix product S = ZL. Surprisingly to many students, the scores map reveals a well-known object arrangement: the periodic table. Of course, the classical tabular and perfect rectangular layout has been lost, but the picture reveals that the information contained in the collected numerical data is related to the familiar bidimensional C
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arrangement.13 It reproduces the main groups and periods and demonstrates how the original numerical information that was entered incorporates information related to the periodic arrangement.14
visualized in Figure 1 (see the following alignments: Li−Na−K, Be−Mg−Ca, K−Rb−Cs, Ca−Sr−Ba, Al−Ga−In, Si−Ge−Sn, P−As−Sb, S−Se−Te, and Cl−Br−I). This is a consequence of the fact that the second PC is dominated by the atomic weight (see PC 2 in Table 4). So, the progressions of three elements according to the respective atomic weight are preserved in the pictures. As a consequence, the middle element of each triad is equidistant from the other two, and the three elements are almost perfectly aligned.
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RESULTS In Figure 1, groups and periods are clearly distinguished, especially if they are manually connected by segments. Every period goes down from left to right mainly due to the collateral effect of the atomic weight (it has negative loadings in PCs, partially responsible for the tendency to place the heaviest elements low in the figure, configuring the groups structure). Regarding the ionization potentials and electron affinity values, it is well-known that the variation irregularities found across the periods are mainly due to effects of the filled s shells and halffilled p subshells. These effects can be also seen in the depicted periods as a sort of zigzag series (see for example the element sequences Li−Be−B, Na−Mg−Al, K−Ca−Ga, Rb−Sr−In, Cs− Ba−Tl, and also the C−N−O, Si−P−S, Ge−As−Se, Sn−Sb− Te, and Pb−Bi−Po sequences).
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ASSOCIATED CONTENT
S Supporting Information *
Instructor’s activity guide. This material is available via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
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The authors declare no competing financial interest.
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DISCUSSION Figure 1 is a very informative mesh. Students will find that the position of the hydrogen atom in the arrangement is irregular, although some of them will be aware that its position is still undecided. With regard to its electronic structure alone, it can be claimed that hydrogen is not an alkali but that it can be also considered a particular case of halogen (i.e., viewing it as having an unfilled 1s2 noble gas configuration). From an intermediate perspective, and taking into account chemical, thermodynamic, and physical properties, it has been argued that the most plausible place for hydrogen in the PT is above the carbon element.15 This interpretation agrees with the results of the present study (see Figure 1) because in this PCA treatment three relevant properties have been taken into account that justify the positioning near the carbon element: ionization potential, electron affinity, and electronegativity. Figure 1 is consistent with the PT depicted by Cronyn.15 Additionally, according to this PCA study, H could plausibly be placed at the head of many groups (note that all the groups seem to converge to H, except the O and F groups). In the bidimensional picture, H also approaches N.16 Interestingly, a more careful inspection of the grid in Figure 1 reveals how the carbon seems to be displaced from its “natural” or “expected” position, and N looks displaced at least as much as C. Figure 1 shows N to be seriously out of line with the other elements of its period, whereas C appears to be moderately out of line with the other elements of its group. The displacement of N is part of the zigzag feature described in the text due to half-filled p orbitals, but to a considerably larger degree than the other groups. Hence, in the previous paragraph, the arguments relative to the hydrogen placement also depend on the fuzzy position of the C and N elements. So, the PCA results corroborate that H can be placed near C, but also near N. It is also worth mentioning the effect of a slight squeezing in the upper-left part of the mesh depicted in Figure 1. Some elements are grouped by pairs: Li−Mg, Na−Ca. Perhaps the whole set of distortions in this zone constitutes a reminiscent feature related to the diagonal relationships present in the PT. Döbereiner’s rule17,18 states that the middle element of a triad of similar elements bears an atomic weight equal to the arithmetic mean of the other two. Döbereiner’s triads are also
ACKNOWLEDGMENTS The author acknowledges the University of Girona for funding. Four anonymous referees are also acknowledged for their valuable comments and suggestions, which improved the article contents. Also, one of the referees provided valuable language supervision after a careful reading.
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REFERENCES
(1) Ribone, M. É.; Pagani, A. P.; Olivieri, A. C; Goicoechea, H. C. Determination of the Active Principle in a Syrup by Spectrophotometry and Principal Component Regression Analysis. An Advanced Undergraduate Experiment Involving Chemometrics. J. Chem. Educ. 2000, 77 (10), 1330−1333. (2) Cazar, R. A. An Exercise on Chemometrics for a Quantitative Analysis Course. J. Chem. Educ. 2003, 80 (9), 1026−1029. (3) Rusak, D. A.; Brown, L. M.; Martin, S. D. Classification of Vegetable Oils by Principal Component Analysis of FTIR Spectra. J. Chem. Educ. 2003, 80 (5), 541−543. (4) Horovitz, O.; Sârbu, C. Characterization and Classification of Lanthanides by Multivariate-Analysis Methods. J. Chem. Educ. 2005, 82 (3), 473−483. (5) Diener, L. News from Online: The Periodic Table of the Elements. J. Chem. Educ. 2009, 86 (10), 1163−1166. (6) Winter, M. J. Diffusion Cartograms for the Display of Periodic Table Data. J. Chem. Educ. 2011, 88 (11), 1507−1510. (7) Bonifácio, V. D. B. QR-Coded Audio Periodic Table of the Elements: A Mobile-Learning Tool. J. Chem. Educ. 2012, 89 (4), 552− 554. (8) Atkins, P.; Jones, L. Chemical Principles. The Quest for Insight, 5th ed.; W. H. Freeman and Company: New York, 2010. For the first electron affinity, some negative values were set to 0 in this bibliographic reference because their absolute value is quite small. (9) Wikipedia. http://en.wikipedia.org/wiki/Atomic_radius (accessed on May 2013). (10) Slater, J. C. Atomic Radii in Crystals. J. Chem. Phys. 1964, 41 (10), 3199−3205. (11) See, for instance, the Colby University chemistry department Web page http://www.colby.edu/chemistry/PChem/eigen.html or the Bluebit software Web page http://www.bluebit.gr/matrixcalculator/ (both accessed on May 2013). The first application allows the lower triangular part of the matrix to be entered (as written in Table 2), whereas the second application requires the full square matrix to be entered. D
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(12) If the correlations matrix R has been diagonalized, Kaiser’s rule says to retain the eigenvectors having an eigenvalue greater than one. The rule arises from the fact that the sum of the eigenvalues (or equivalently, the trace of matrix R in Table 2) adds to the total number of variables. Hence, PCs attached to eigenvalues greater than 1 can be viewed as enriched factors condensing more than one of the original variables. (13) Eigenvectors are undetermined by a sign, an undetermined phase. As a consequence, depending on the data sorting and the diagonalization algorithm considered by the software, the sign of the PCs may fluctuate from application to application (as it can occur in the online applications of reference 11). For a better visual inspection and impact over the audience, it is convenient to force the appropriate PCs signs to avoid obtaining a visually “inverted” or “reversed” periodic table. (14) The author performed a factor analysis (varimax rotation) in order to rotate the original PCs. The aim was to get new variable loadings more directly related to the row and column arrangement of the classical rectangular periodic table, but no qualitative (and even quantitative and pictorial) gain was obtained. (15) Cronyn, M. W. The Proper Place for Hydrogen in the Periodic Table. J. Chem. Educ. 2003, 80 (8), 947−951. (16) The first three element coordinates arising from matrix S, once conveniently scaled, can be arranged in a .pdb file, making it very easy to be viewed in 3D by many chemistry-related programs. In the 3D representation obtained when considering the three first PCs, H is seen to be placed in an almost equidistant position from N and C. (17) Ibrahim, S. A. Predicting the Atomic Weights of the TransLawrencium Elements: A Novel Application of Dobereiner’s Triads. J. Chem. Educ. 2005, 82 (11), 1658−1659. (18) Scerri, E. R. The Periodic Table: Its Story and Its Significance; Oxford University Press: New York, 2007.
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