- Email: [email protected]
A power law in the ordering of the elements of the periodic table
Journal Pre-proof A power law in the ordering of the elements of the periodic table Sergio Da Silva, Raul Matsushita, Murilo Silva
PII: DOI: Reference:
S0378-4371(19)31904-1 https://doi.org/10.1016/j.physa.2019.123408 PHYSA 123408
To appear in:
Physica A
Received date : 18 August 2019 Revised date : 29 October 2019 Please cite this article as: S. Da Silva, R. Matsushita and M. Silva, A power law in the ordering of the elements of the periodic table, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123408. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
Journal Pre-proof Highlights ⸰ We discover a power law in the periodic table between atomic number and atomic weight. ⸰ This power law overlaps Mendeleev’s periodic law.
Jo
urn
al
Pr e-
p ro
of
⸰ Its Pareto exponent is computed as 1.0909.
*Manuscript Click here to view linked References
Journal Pre-proof
A power law in the ordering of the elements of the periodic table Sergio Da Silva Department of Economics, Federal University of Santa Catarina, Brazil
Department of Statistics, University of Brasilia, Brazil
Murilo Silva
p ro
Department of Economics, Federal University of Santa Catarina, Brazil
of
Raul Matsushita
Corresponding author: Sergio Da Silva, Department of Economics, Federal University of Santa Catarina, Florianopolis SC 88049-970, Brazil. Email: [email protected]
Keywords Periodic table; periodic law; power law
1. Introduction
Pr e-
Abstract We discover a power law in the periodic table between atomic number and atomic weight that overlaps Mendeleev’s periodic law. Its Pareto exponent is computed as 1.0909. The power law can offer extra help in the quest for the next unknown element.
Jo
urn
al
The periodic table lists the elements that compose all earthly substances, arranged in a way that reveals patterns. Elements are fundamental substances from which all matter is made. Since the late 1600s, we know from Robert Boyle that such elements are not air, earth, fire and water. In 1789, Antoine Lovoisier established a list of 33 elements and classified them into groups – such as bromine, iodine, chlorine and lithium, sodium, potassium. Then, it was noted that certain groups had similar properties, in terms of melting points and types of crystals formed. In the early 1800s, it was realized that most atomic weights were multiples of the lightest element hydrogen. The atomic weight is the weight of an atom of each element. From 1860 on, deeper patterns in the arrangements of atomic weights were discovered, thanks to the insights from Stanislao Cannizzaro. Thus, the elements could be grouped by either their properties or atomic weights. Dmitri Mendeleev’s breakthrough 150 years ago was to combine both in his periodic table. He arranged the elements on one axis by atomic weight (in “periods”) and on the other axis by properties, in groups. His periodic law followed: as he ranked the elements by atomic weights, their chemical and physical properties recurred periodically. This framework allowed him to notice that three elements were missing, and these were discovered soon after and named as gallium, scandium and germanium [1]. Mendeleev’s generalization paved the way for the discovery of 55 more elements since 1869. In 1913, Henry Moseley proved that we should consider atomic number rather atomic weight to rank the elements [2]. Atomic number is the number of protons in each nucleus. However, the discovery of new elements still fits Mendeleev’s classification system. Why? The physical explanation is that
Journal Pre-proof
2. Materials and methods
Pr e-
p ro
of
the mass of an atom is concentrated on its nucleus, which is composed of protons and neutrons. However, because the number of neutrons in an element (isotopes) can vary, it is still not completely clear why replacing Mendeleev’s atomic weight with Moseley’s atomic number does not make much difference. This work offers such a missing justification. Any prediction faces the age-old problem of induction. Because it is logically possible for the prediction to be false while past evidence is true, the evidence does not conclusively establish the truth of the prediction [3]. The problem of induction cannot be removed, so why has Mendeleev’s periodic law succeed? [4][5] Because it offers a tunnel to hopefully track the next unknown elements. It converted unknown unknowns to known unknowns [6]. When we create a so-called tunnel, we focus on a few well-defined sources of uncertainty [7], thus leaving out others that fall outside a law domain. As stated by former U.S. Secretary Donald Rumsfeld, “There are known knowns. There are things we know that we know. There are known unknowns. That is to say, there are things that we now know we don’t know. But there are also unknown unknowns. There are things we do not know we don’t know.” The conversion of unknown unknowns to known unknowns has occurred repeatedly in chemistry through the formulation of hypotheses and experimental testing [8]. Here, we contribute to improving the roadmap given by the periodic table by showcasing a power law between atomic number and atomic weight. The discovery of this power law explains that atomic number could replace atomic weight with negligible prediction costs. The periodic law takes into account chemical relations between the elements, of course. In contrast, a power law is a regularity that is merely statistical. We conjecture that the power law describing the relationship between atomic number n and atomic weight (abundanceweighted average of isotopes) w in the periodic table is w a n , where w changes as a power of n. We then quantify this power law [9] from the data of the current periodic table and calculate its Pareto exponent [10] [11].
3. Results
urn
al
Few distributions follow a power law over their entire range – this emerges after a minimum threshold and sometimes vanishes after a maximum threshold [10]. This is the reason that a distribution is referred to have a power law tail. Considering that our conjectured power law becomes log w log a log n after taking base 10 logarithms (any base will work), a straightline power law form reveals itself only after a minimum threshold nmin . Pareto exponent essentially means that, relative to a given atomic number n, finding more elements in the periodic table is proportional to n [12][13]. Pareto exponent tracks the degree of uncertainty of repeating the discovery of a new element after a certain threshold atomic number n.
Jo
Table 1 displays the current periodic table of 118 elements, while Figure 1 showcases an example for n 6 . A power law reveals itself in a straight line of a log-log plot of the previous equation, where log a is the y-intercept = d, and is the slope. Thus, a 10d . From the fitting line in Figure 1, we find
log w 0.1994 1.1028 log n , where the estimates with their corresponding standard errors are = 1.1028 ± 0.0027 (slope), d = 0.1994 ± 0.0047 (intercept), and then a 100.1994 1.5826 . The implied power law is then
Journal Pre-proof
w 1.5826n1.1028 . To illustrate, the atomic weight of the next unknown element n = 119 is casually predicted as
Table 1. Current periodic table.
log n 0 0.30103 0.47712 0.60206 0.69897 0.77815 0.8451 0.90309 0.95424 1 1.04139 1.07918 1.11394 1.14613 1.17609 1.20412 1.23045 1.25527 1.27875 1.30103 1.32222 1.34242 1.36173 1.38021 1.39794 1.41497 1.43136 1.44716 1.4624 1.47712 1.49136 1.50515 1.51851 1.53148 1.54407 1.5563 1.5682 1.57978 1.59106 1.60206 1.61278 1.62325 1.63347 1.64345 1.65321 1.66276
Pr e-
p ro
Atomic weight, w 1.008 4.0026 6.94 9.0122 10.81 12.011 14.007 15.999 18.998 20.18 22.99 24.305 26.982 28.085 30.974 32.06 35.45 39.948 39.0983 40.078 44.956 47.867 50.942 51.996 54.938 55.845 58.933 58.693 63.546 65.38 69.723 72.63 74.922 78.971 79.904 83.798 85.468 87.62 88.906 91.224 92.906 95.95 98 101.07 102.906 106.42
al
urn
Rank, n 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 36 37 38 39 40 41 42 43 44 45 46
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Element Hydrogen Helium Lithium Beryllium Boron Carbon Nitrogen Oxygen Fluorine Neon Sodium Magnesium Aluminium Silicon Phosphorus Sulfur Chlorine Argon Potassium Calcium Scandium Titanium Vanadium Chromium Manganese Iron Cobalt Nickel Copper Zinc Gallium Germanium Arsenic Selenium Bromine Krypton Rubidium Strontium Yttrium Zirconium Niobium Molybdenum Technetium Ruthenium Rhodium Palladium
of
w 1.5826 1191.1028 307.8 .
log w 0.00346 0.60234 0.84136 0.95483 1.03383 1.07958 1.14635 1.20409 1.27871 1.30492 1.36154 1.3857 1.43107 1.44847 1.491 1.50596 1.54962 1.6015 1.59216 1.60291 1.65279 1.68004 1.70708 1.71597 1.73987 1.74698 1.77036 1.76859 1.80309 1.81544 1.84338 1.86112 1.87461 1.89747 1.90257 1.92323 1.9318 1.9426 1.94893 1.96011 1.96804 1.98204 1.99123 2.00462 2.01244 2.02702
Journal Pre-proof 2.0329 2.05082 2.06001 2.07449 2.0855 2.10585 2.10348 2.11824 2.12354 2.13776 2.14272 2.14649 2.14894 2.15909 2.16137 2.17713 2.18174 2.19659 2.20119 2.21085 2.2173 2.22339 2.22772 2.23816 2.24296 2.25161 2.25755 2.26444 2.27 2.27928 2.28379 2.29022 2.29439 2.30231 2.31044 2.31639 2.3201 2.32015 2.32222 2.34635 2.3483 2.35411 2.35603 2.36556 2.36368 2.37663 2.37475 2.38739 2.38561 2.3927 2.3927 2.39967 2.4014 2.40993 2.41162 2.4133 2.4183 2.42651 2.42813 2.42975
p ro
of
1.6721 1.68124 1.6902 1.69897 1.70757 1.716 1.72428 1.73239 1.74036 1.74819 1.75587 1.76343 1.77085 1.77815 1.78533 1.79239 1.79934 1.80618 1.81291 1.81954 1.82607 1.83251 1.83885 1.8451 1.85126 1.85733 1.86332 1.86923 1.87506 1.88081 1.88649 1.89209 1.89763 1.90309 1.90849 1.91381 1.91908 1.92428 1.92942 1.9345 1.93952 1.94448 1.94939 1.95424 1.95904 1.96379 1.96848 1.97313 1.97772 1.98227 1.98677 1.99123 1.99564 2 2.00432 2.0086 2.01284 2.01703 2.02119 2.02531
Pr e-
107.87 112.414 114.818 118.71 121.76 127.6 126.904 131.293 132.905 137.327 138.905 140.116 140.908 144.242 145 150.36 151.964 157.25 158.925 162.5 164.93 167.259 168.934 173.045 174.967 178.49 180.948 183.84 186.207 190.23 192.217 195.084 196.967 200.592 204.38 207.2 208.98 209 210 222 223 226 227 232.038 231.036 238.029 237 244 243 247 247 251 252 257 258 259 262 267 268 269
al
urn
47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106
Jo
Silver Cadmium Indium Tin Antimony Tellurium Iodine Xenon Caesium Barium Lanthanum Cerium Praseodymium Neodymium Promethium Samarium Europium Gadolinium Terbium Dysprosium Holmium Erbium Thulium Ytterbium Lutetium Hafnium Tantalum Tungsten Rhenium Osmium Iridium Platinum Gold Mercury Thallium Lead Bismuth Polonium Astatine Radon Francium Radium Actinium Thorium Protactinium Uranium Neptunium Plutonium Americium Curium Berkelium Californium Einsteinium Fermium Mendelevium Nobelium Lawrencium Rutherfordium Dubnium Seaborgium
Journal Pre-proof 2.02938 2.03342 2.03743 2.04139 2.04532 2.04922 2.05308 2.0569 2.0607 2.06446 2.06819 2.07188
2.43136 2.42975 2.44404 2.44871 2.44716 2.45484 2.45637 2.4609 2.4609 2.46687 2.46835 2.46835
of
270 269 278 281 280 285 286 289 289 293 294 294
Pr e-
p ro
Bohrium 107 Hassium 108 Meitnerium 109 Darmstadtium 110 Roentgenium 111 Copernicium 112 Nihonium 113 Flerovium 114 Moscovium 115 Livermorium 116 Tennessine 117 Oganesson 118 Source: Royal Society of Chemistry
al
Figure 1. A power law in the periodic table.
urn
This graphical exercise can be complemented by an alternative nonlinear least squares’ approach [9]. Here, we find the power law:
w 1.6683n1.0909 ,
Jo
where the standard errors for a and are, respectively, 0.036 and 0.0048. The Pareto exponent is thus statistically different from 1. Now, the atomic weight of the next unknown element n = 119 is predicted as w 1.6683(119)1.0909 306.54 .
Considering a Monte Carlo-based 95-percent prediction interval for this nonlinear regression, after 106 replications, we find [300.78; 312.20]. This is the tunnel for predicting what is offered by the power law. A final caution note is worthwhile. Consider the practical implication of rounding 1.0909 to either 1 (a so-called Zipf’s law) or 1.1. We warn this rounding is highly sensitive and the prediction error escalates as a result [14].
Journal Pre-proof 4. Conclusion
Acknowledgements
p ro
The authors thank the useful comments made by anonymous reviewers.
of
Mendeleev’s periodic law offers a tunnel to predict unknown elements from the periodic table. Replacing Mendeleev’s atomic weight with Moseley’s atomic number does not make much difference for such a prediction. We discover a power law between atomic number and atomic weight that justifies this, and compute its Pareto exponent as 1.0909 . The power law overlaps the periodic law and contributes to refine tunneling in the search for a new element by converting unknown unknowns to known unknowns.
Declaration of conflicting interests
The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.
Pr e-
Funding
The authors acknowledge the financial support from CNPq, CAPES and FAPDF.
References
[8] [9] [10] [11] [12] [13] [14]
al
[7]
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[4] [5] [6]
Scerri ER. The evolution of the periodic system. Sci Am 1998; 279(3): 78-83. Scerri E. The past and future of the periodic table. Am Sci 2008; 96: 52-58. Foster MR. Prediction and the problem of induction. Semantic Scholar 1998; 1-8. https://pdfs.semanticscholar.org/9a97/4d04be191df6779b60e54e2bf7cf3287a49a.pdf Armbruster P and Hessberger FP. Making new elements. Sci Am 1998; 279(3): 72-77. Scerri E. Cracks in the periodic table. Sci Am 2013; 308(6): 68-73. Matsushita R, Da Silva S, Da Fonseca R et al. Bypassing the truncation problem of truncated Lévy flights. Submitted. Taleb NN. The Black Swan: the impact of the highly improbable. New York: Random House, 2010. Loxdale HD, Davis BJ and Davis RA. Known knowns and unknowns in biology. Biol J Linn Soc 2016; 117(2): 386-398. Gabaix X. Power laws in economics: an introduction. J Econ Perspect 2016; 30(1): 185-206. Newman ME. Power laws, Pareto distributions and Zipf’s law. Contemp Phys 2005; 46(5): 323-351. Clauset A, Shalizi CR and Newman ME. Power-law distributions in empirical data. SIAM Rev 2009; 51(4): 661-703. Da Silva S, Perlin M, Matsushita R et al. Lotka’s law for the Brazilian scientific output published in journals. J Inform Sci 2018; doi: 10.1177/0165551518801813. Kawamura M, Thomas CD, Tsurumoto A et al. Lotka’s law and productivity index of authors in a scientific journal. J Oral Sci 2000; 42(2): 75-78. Da Silva S, Matsushita R, Giglio R et al. Granularity of the top 1,000 Brazilian companies. Physica A 2018; 512(C): 68-73.
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[1] [2] [3]
Journal Pre-proof *Declaration of Interest Statement
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
of
☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
Jo
urn
al
Pr e-
p ro
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
PII: DOI: Reference:
S0378-4371(19)31904-1 https://doi.org/10.1016/j.physa.2019.123408 PHYSA 123408
To appear in:
Physica A
Received date : 18 August 2019 Revised date : 29 October 2019 Please cite this article as: S. Da Silva, R. Matsushita and M. Silva, A power law in the ordering of the elements of the periodic table, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123408. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
Journal Pre-proof Highlights ⸰ We discover a power law in the periodic table between atomic number and atomic weight. ⸰ This power law overlaps Mendeleev’s periodic law.
Jo
urn
al
Pr e-
p ro
of
⸰ Its Pareto exponent is computed as 1.0909.
*Manuscript Click here to view linked References
Journal Pre-proof
A power law in the ordering of the elements of the periodic table Sergio Da Silva Department of Economics, Federal University of Santa Catarina, Brazil
Department of Statistics, University of Brasilia, Brazil
Murilo Silva
p ro
Department of Economics, Federal University of Santa Catarina, Brazil
of
Raul Matsushita
Corresponding author: Sergio Da Silva, Department of Economics, Federal University of Santa Catarina, Florianopolis SC 88049-970, Brazil. Email: [email protected]
Keywords Periodic table; periodic law; power law
1. Introduction
Pr e-
Abstract We discover a power law in the periodic table between atomic number and atomic weight that overlaps Mendeleev’s periodic law. Its Pareto exponent is computed as 1.0909. The power law can offer extra help in the quest for the next unknown element.
Jo
urn
al
The periodic table lists the elements that compose all earthly substances, arranged in a way that reveals patterns. Elements are fundamental substances from which all matter is made. Since the late 1600s, we know from Robert Boyle that such elements are not air, earth, fire and water. In 1789, Antoine Lovoisier established a list of 33 elements and classified them into groups – such as bromine, iodine, chlorine and lithium, sodium, potassium. Then, it was noted that certain groups had similar properties, in terms of melting points and types of crystals formed. In the early 1800s, it was realized that most atomic weights were multiples of the lightest element hydrogen. The atomic weight is the weight of an atom of each element. From 1860 on, deeper patterns in the arrangements of atomic weights were discovered, thanks to the insights from Stanislao Cannizzaro. Thus, the elements could be grouped by either their properties or atomic weights. Dmitri Mendeleev’s breakthrough 150 years ago was to combine both in his periodic table. He arranged the elements on one axis by atomic weight (in “periods”) and on the other axis by properties, in groups. His periodic law followed: as he ranked the elements by atomic weights, their chemical and physical properties recurred periodically. This framework allowed him to notice that three elements were missing, and these were discovered soon after and named as gallium, scandium and germanium [1]. Mendeleev’s generalization paved the way for the discovery of 55 more elements since 1869. In 1913, Henry Moseley proved that we should consider atomic number rather atomic weight to rank the elements [2]. Atomic number is the number of protons in each nucleus. However, the discovery of new elements still fits Mendeleev’s classification system. Why? The physical explanation is that
Journal Pre-proof
2. Materials and methods
Pr e-
p ro
of
the mass of an atom is concentrated on its nucleus, which is composed of protons and neutrons. However, because the number of neutrons in an element (isotopes) can vary, it is still not completely clear why replacing Mendeleev’s atomic weight with Moseley’s atomic number does not make much difference. This work offers such a missing justification. Any prediction faces the age-old problem of induction. Because it is logically possible for the prediction to be false while past evidence is true, the evidence does not conclusively establish the truth of the prediction [3]. The problem of induction cannot be removed, so why has Mendeleev’s periodic law succeed? [4][5] Because it offers a tunnel to hopefully track the next unknown elements. It converted unknown unknowns to known unknowns [6]. When we create a so-called tunnel, we focus on a few well-defined sources of uncertainty [7], thus leaving out others that fall outside a law domain. As stated by former U.S. Secretary Donald Rumsfeld, “There are known knowns. There are things we know that we know. There are known unknowns. That is to say, there are things that we now know we don’t know. But there are also unknown unknowns. There are things we do not know we don’t know.” The conversion of unknown unknowns to known unknowns has occurred repeatedly in chemistry through the formulation of hypotheses and experimental testing [8]. Here, we contribute to improving the roadmap given by the periodic table by showcasing a power law between atomic number and atomic weight. The discovery of this power law explains that atomic number could replace atomic weight with negligible prediction costs. The periodic law takes into account chemical relations between the elements, of course. In contrast, a power law is a regularity that is merely statistical. We conjecture that the power law describing the relationship between atomic number n and atomic weight (abundanceweighted average of isotopes) w in the periodic table is w a n , where w changes as a power of n. We then quantify this power law [9] from the data of the current periodic table and calculate its Pareto exponent [10] [11].
3. Results
urn
al
Few distributions follow a power law over their entire range – this emerges after a minimum threshold and sometimes vanishes after a maximum threshold [10]. This is the reason that a distribution is referred to have a power law tail. Considering that our conjectured power law becomes log w log a log n after taking base 10 logarithms (any base will work), a straightline power law form reveals itself only after a minimum threshold nmin . Pareto exponent essentially means that, relative to a given atomic number n, finding more elements in the periodic table is proportional to n [12][13]. Pareto exponent tracks the degree of uncertainty of repeating the discovery of a new element after a certain threshold atomic number n.
Jo
Table 1 displays the current periodic table of 118 elements, while Figure 1 showcases an example for n 6 . A power law reveals itself in a straight line of a log-log plot of the previous equation, where log a is the y-intercept = d, and is the slope. Thus, a 10d . From the fitting line in Figure 1, we find
log w 0.1994 1.1028 log n , where the estimates with their corresponding standard errors are = 1.1028 ± 0.0027 (slope), d = 0.1994 ± 0.0047 (intercept), and then a 100.1994 1.5826 . The implied power law is then
Journal Pre-proof
w 1.5826n1.1028 . To illustrate, the atomic weight of the next unknown element n = 119 is casually predicted as
Table 1. Current periodic table.
log n 0 0.30103 0.47712 0.60206 0.69897 0.77815 0.8451 0.90309 0.95424 1 1.04139 1.07918 1.11394 1.14613 1.17609 1.20412 1.23045 1.25527 1.27875 1.30103 1.32222 1.34242 1.36173 1.38021 1.39794 1.41497 1.43136 1.44716 1.4624 1.47712 1.49136 1.50515 1.51851 1.53148 1.54407 1.5563 1.5682 1.57978 1.59106 1.60206 1.61278 1.62325 1.63347 1.64345 1.65321 1.66276
Pr e-
p ro
Atomic weight, w 1.008 4.0026 6.94 9.0122 10.81 12.011 14.007 15.999 18.998 20.18 22.99 24.305 26.982 28.085 30.974 32.06 35.45 39.948 39.0983 40.078 44.956 47.867 50.942 51.996 54.938 55.845 58.933 58.693 63.546 65.38 69.723 72.63 74.922 78.971 79.904 83.798 85.468 87.62 88.906 91.224 92.906 95.95 98 101.07 102.906 106.42
al
urn
Rank, n 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 36 37 38 39 40 41 42 43 44 45 46
Jo
Element Hydrogen Helium Lithium Beryllium Boron Carbon Nitrogen Oxygen Fluorine Neon Sodium Magnesium Aluminium Silicon Phosphorus Sulfur Chlorine Argon Potassium Calcium Scandium Titanium Vanadium Chromium Manganese Iron Cobalt Nickel Copper Zinc Gallium Germanium Arsenic Selenium Bromine Krypton Rubidium Strontium Yttrium Zirconium Niobium Molybdenum Technetium Ruthenium Rhodium Palladium
of
w 1.5826 1191.1028 307.8 .
log w 0.00346 0.60234 0.84136 0.95483 1.03383 1.07958 1.14635 1.20409 1.27871 1.30492 1.36154 1.3857 1.43107 1.44847 1.491 1.50596 1.54962 1.6015 1.59216 1.60291 1.65279 1.68004 1.70708 1.71597 1.73987 1.74698 1.77036 1.76859 1.80309 1.81544 1.84338 1.86112 1.87461 1.89747 1.90257 1.92323 1.9318 1.9426 1.94893 1.96011 1.96804 1.98204 1.99123 2.00462 2.01244 2.02702
Journal Pre-proof 2.0329 2.05082 2.06001 2.07449 2.0855 2.10585 2.10348 2.11824 2.12354 2.13776 2.14272 2.14649 2.14894 2.15909 2.16137 2.17713 2.18174 2.19659 2.20119 2.21085 2.2173 2.22339 2.22772 2.23816 2.24296 2.25161 2.25755 2.26444 2.27 2.27928 2.28379 2.29022 2.29439 2.30231 2.31044 2.31639 2.3201 2.32015 2.32222 2.34635 2.3483 2.35411 2.35603 2.36556 2.36368 2.37663 2.37475 2.38739 2.38561 2.3927 2.3927 2.39967 2.4014 2.40993 2.41162 2.4133 2.4183 2.42651 2.42813 2.42975
p ro
of
1.6721 1.68124 1.6902 1.69897 1.70757 1.716 1.72428 1.73239 1.74036 1.74819 1.75587 1.76343 1.77085 1.77815 1.78533 1.79239 1.79934 1.80618 1.81291 1.81954 1.82607 1.83251 1.83885 1.8451 1.85126 1.85733 1.86332 1.86923 1.87506 1.88081 1.88649 1.89209 1.89763 1.90309 1.90849 1.91381 1.91908 1.92428 1.92942 1.9345 1.93952 1.94448 1.94939 1.95424 1.95904 1.96379 1.96848 1.97313 1.97772 1.98227 1.98677 1.99123 1.99564 2 2.00432 2.0086 2.01284 2.01703 2.02119 2.02531
Pr e-
107.87 112.414 114.818 118.71 121.76 127.6 126.904 131.293 132.905 137.327 138.905 140.116 140.908 144.242 145 150.36 151.964 157.25 158.925 162.5 164.93 167.259 168.934 173.045 174.967 178.49 180.948 183.84 186.207 190.23 192.217 195.084 196.967 200.592 204.38 207.2 208.98 209 210 222 223 226 227 232.038 231.036 238.029 237 244 243 247 247 251 252 257 258 259 262 267 268 269
al
urn
47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106
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Silver Cadmium Indium Tin Antimony Tellurium Iodine Xenon Caesium Barium Lanthanum Cerium Praseodymium Neodymium Promethium Samarium Europium Gadolinium Terbium Dysprosium Holmium Erbium Thulium Ytterbium Lutetium Hafnium Tantalum Tungsten Rhenium Osmium Iridium Platinum Gold Mercury Thallium Lead Bismuth Polonium Astatine Radon Francium Radium Actinium Thorium Protactinium Uranium Neptunium Plutonium Americium Curium Berkelium Californium Einsteinium Fermium Mendelevium Nobelium Lawrencium Rutherfordium Dubnium Seaborgium
Journal Pre-proof 2.02938 2.03342 2.03743 2.04139 2.04532 2.04922 2.05308 2.0569 2.0607 2.06446 2.06819 2.07188
2.43136 2.42975 2.44404 2.44871 2.44716 2.45484 2.45637 2.4609 2.4609 2.46687 2.46835 2.46835
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270 269 278 281 280 285 286 289 289 293 294 294
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Bohrium 107 Hassium 108 Meitnerium 109 Darmstadtium 110 Roentgenium 111 Copernicium 112 Nihonium 113 Flerovium 114 Moscovium 115 Livermorium 116 Tennessine 117 Oganesson 118 Source: Royal Society of Chemistry
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Figure 1. A power law in the periodic table.
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This graphical exercise can be complemented by an alternative nonlinear least squares’ approach [9]. Here, we find the power law:
w 1.6683n1.0909 ,
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where the standard errors for a and are, respectively, 0.036 and 0.0048. The Pareto exponent is thus statistically different from 1. Now, the atomic weight of the next unknown element n = 119 is predicted as w 1.6683(119)1.0909 306.54 .
Considering a Monte Carlo-based 95-percent prediction interval for this nonlinear regression, after 106 replications, we find [300.78; 312.20]. This is the tunnel for predicting what is offered by the power law. A final caution note is worthwhile. Consider the practical implication of rounding 1.0909 to either 1 (a so-called Zipf’s law) or 1.1. We warn this rounding is highly sensitive and the prediction error escalates as a result [14].
Journal Pre-proof 4. Conclusion
Acknowledgements
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The authors thank the useful comments made by anonymous reviewers.
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Mendeleev’s periodic law offers a tunnel to predict unknown elements from the periodic table. Replacing Mendeleev’s atomic weight with Moseley’s atomic number does not make much difference for such a prediction. We discover a power law between atomic number and atomic weight that justifies this, and compute its Pareto exponent as 1.0909 . The power law overlaps the periodic law and contributes to refine tunneling in the search for a new element by converting unknown unknowns to known unknowns.
Declaration of conflicting interests
The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.
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Funding
The authors acknowledge the financial support from CNPq, CAPES and FAPDF.
References
[8] [9] [10] [11] [12] [13] [14]
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[7]
urn
[4] [5] [6]
Scerri ER. The evolution of the periodic system. Sci Am 1998; 279(3): 78-83. Scerri E. The past and future of the periodic table. Am Sci 2008; 96: 52-58. Foster MR. Prediction and the problem of induction. Semantic Scholar 1998; 1-8. https://pdfs.semanticscholar.org/9a97/4d04be191df6779b60e54e2bf7cf3287a49a.pdf Armbruster P and Hessberger FP. Making new elements. Sci Am 1998; 279(3): 72-77. Scerri E. Cracks in the periodic table. Sci Am 2013; 308(6): 68-73. Matsushita R, Da Silva S, Da Fonseca R et al. Bypassing the truncation problem of truncated Lévy flights. Submitted. Taleb NN. The Black Swan: the impact of the highly improbable. New York: Random House, 2010. Loxdale HD, Davis BJ and Davis RA. Known knowns and unknowns in biology. Biol J Linn Soc 2016; 117(2): 386-398. Gabaix X. Power laws in economics: an introduction. J Econ Perspect 2016; 30(1): 185-206. Newman ME. Power laws, Pareto distributions and Zipf’s law. Contemp Phys 2005; 46(5): 323-351. Clauset A, Shalizi CR and Newman ME. Power-law distributions in empirical data. SIAM Rev 2009; 51(4): 661-703. Da Silva S, Perlin M, Matsushita R et al. Lotka’s law for the Brazilian scientific output published in journals. J Inform Sci 2018; doi: 10.1177/0165551518801813. Kawamura M, Thomas CD, Tsurumoto A et al. Lotka’s law and productivity index of authors in a scientific journal. J Oral Sci 2000; 42(2): 75-78. Da Silva S, Matsushita R, Giglio R et al. Granularity of the top 1,000 Brazilian companies. Physica A 2018; 512(C): 68-73.
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[1] [2] [3]
Journal Pre-proof *Declaration of Interest Statement
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.