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Section 15. Chemistry
Simonyan Gevorg Sarkisovich, candidate of Chemiical Science, associatprofessor, the Faculty of Chemistry, Yerevan State University Head of the Departament Genetal Matematics and Natural Sciences of the Ijevan Branch of Yerevan State University, E-mail: [email protected], [email protected] Martiryan Davit Armenovich master student, the Faculty of Physics, Yerevan State University E-mail: [email protected]
A SERIES OF FIBONACCI NUMBERS AND CHEMICAL ELEMENTS AND MOLECULES
Abstract: The article discusses the structure of atoms of chemical elements in the light of the sequence of Fibonacci numbers. It is assumed that the first member of the series is the mass of the electron, and the second and third members of the series are, respectively, the mass of the proton and neutron. For example, the sixth member of the series is the lithium ion 5Li+ or the beryllium isotope 5Be. It is shown that the ratios of atoms in oxides of chromium of uranium and nitrogen are equal to the ratios of the Fibonacci numbers.It has been established that the potential number of fatty acids increases with the length of their chain in accordance with known Fibonacci numbers. It is shown that during the synthesis of oligo (3-hexylthiophene) the number of repeated units follows the Fibonacci numbers to length 21.
Keywords: Fibonacci series, atomic structure, chemistry, oxides of chromium and uranium, fatty acids, polymers.
Introduction numbers, in the series of which each number is the sum
In the beginning of the thirteenth century, in the fa- of the two preceding digits. The sequence of Fibonacci
mous book Liber abaci, the mathematician from Italy, numbers is as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,
Leonardo Fibonacci, first introduced the regularity of 144, 233, 377, etc.
Figure 1. Fibonacci's Rabbit Problem
Section 15. Chemistry
Fibonacci considers the development of population of rabbits, suggesting that: initially there is a newborn pair of rabbits (male and female); from the second month after their birth, rabbits begin to mate and every month produce a new pair of rabbits; rabbits never die (see figur 1). How many pairs of rabbits will be in a year?
More formally, the sequence of Fibonacci numbers Fn is given by a linear recurrence relation: F0 = 0, F1 = f2 = 1 F3 = 2, f4 = 3, f5 = 5, ... in general Fn+1 = F + F , n > 0 [1; 2].
n n-1;
It is known that uranium forms a series of oxides: U205, U308, U5013, U8021, U13034. The chromium oxides have the same composition: Cr205, Cr308, Cr5013, Cr8021. It turns out that the ratios of atoms are equal to the ratios of Fibonacci numbers [3].
The purpose of this paper is to review the structure of atoms of chemical elements and chemical compounds in the light of the laws of the Fibonacci number.
Results and discussion.
Consider the structure of atoms of chemical elements.
It is known that substances consist of atoms. The atom itself consists of a positively charged nucleus and a negatively charged electron cloud. In general, the atom is electrically neutral. The size of the atom is completely determined by the size of its electron
Table 1.- Elementary
cloud, since the size of the nucleus is negligible in compare to the size of the electron cloud. The nucleus consists of Z positively charged protons and N neutrons that do not carry a charge. Protons and neutrons are called nucleons, that is, core particles. p+ + e- = n0 + ve Thus, the nuclear charge is determined only by the number of protons and is equal to the ordinal number of the element in the periodic table. The positive charge of the nucleus is compensated by negatively charged electrons, which form an electronic cloud. The number of electrons is equal to the number of protons. The masses of protons and neutrons are equal (see Table 1.). The mass of the atom is mainly determined by the mass of its core, since the electron mass is approximately 1836 times smaller than the mass of the proton and neutron and is rarely taken into account in calculations.
A = Z + N
The appearance of the atoms of a chemical element with a nucleus consisting of a strictly defined number of protons and neutrons is called the nuclide. Different elements with the same total number of nucleons are called isobars, and with the same total number of neutrons are called isotones. Isotopes are atoms of the same element with the same number of protons, but with a different number of neutrons[4].
Particles of the Atom
Elementary particle Symbol Charge (C) Charge (conventional units) Actual Мass (g) Relative Мass (amu)
proton P 1.66-10 x 10-19 + 1 1.672 • 10 x 10-24 1
neutron n 0 0 1.674 ■ 10x10-24 1
electron e 1.66-10 x 10-19 -1 9.109 ■ 10 x 10-28 0
In our opinion, the first number in the Fibonacci series F = 0, is the electron mass m = 0.
0e
The second member of the series F = 1 is the proton mass
m = 1.
p
The third member of the series F = 1 is the neutron mass
2
m n = 1 or a hydrogen atom 1H,
F2 = F0 + F1, n0 = p + e- +. 1 = 0 +1. The fourth member of the series F = 2 is the sum of the masses of the proton and neutron, that is, the deuterium ion: 2H+ (D)+,
F = F + F = 1 + 1 = 2
3 1 2
The fifth member of the series F4 = 3 is a tritium ion: 3H+ (T)+ or helium isotope 3He,
F4 = F2 + F3 = 1 + 2 = 3
The sixth member F5 = 5, is a lithium ion 5Li+ or beryllium isotope 5Be.
Systems of inorganic and organic chemistry are fundamental systems for building the macrocosm in which we exist.
They are created according to certain rules and are of decisive importance in the subsequent construction of matter.
At the end of the 19th century the French scientist Proust came to the conclusion that the compounds have a strictly constant composition, independent of the conditions of their formation. Most compounds are stoichiometric. This means that the ratio between the number of atoms or ions in a compound is expressed by simple integers. And another French scientist Berthollet argued that chemical compounds have a variable composition, which depends on the conditions for their production and the mass of the reactants. It was denied that their compounds can have a strict, permanent composition. The non-stoichiometry of these compounds is due to the presence of defects in their crystal lattices. In the report of 1914, "The Connection and the Chemical Individual" N. S. Kurnakov suggested calling daltonids, and compounds of variable composition - berthollids[5]. The difference between daltonides and bertholids is that in the composition-property diagram for
daltonides there is a narrow interval of composition variation with unchanged properties, and for bertholids, a wide range. As already shown, there are series of oxides whose composition corresponds to the Fibonacci numbers. Thus, for chromium, the following compounds are known: Cr02 Cr03, Cr205, Cr308, Cr5013, Crg021. A number of intermediate compounds are formed between the uranium oxides UO2 and UO3: U205, U308, U5013, U8021, U13034. As we see, the ratios of atoms are equal to the ratios of Fibonacci numbers [3]. For nitrogen oxides, the oxygen coeficients also correspond to N2O, NO, NO2, N2O3, N2O5 Fibonacci numbers [6].
The authors of the work [7] have noted that the potential number of fatty acids increases with the length of their chain according to the famous Fibonacci numbers, when cis / trans isomerism is neglected. It is shown that it grows according to the famous Fibonacci numbers when cis/trans isomerism is neglected. Since the ratio of two consecutive Fibonacci numbers tends to the Golden section, 1.618, organisms can increase fatty acid variability approximately by that factor for per carbon atom invested. Moreover, it is shown that, under consideration of cis/ trans isomerism and/or of modification by hydroxyl- and/or oxo- groups, diversity can be described by generalized Fibonacci numbers (e.g. Pell numbers). The recursive definition of Fibonacci numbers paves the way to construct all structural formulas of fatty acids in an automated way.
The paper[8] had been described a new synthetic approach to regioregular monodisperse oligo(3-alkylthiophene) s allowing for simple separation of regioregular material in gram quantities. The number of repeat units follows the Fibonacci numbers up to a length of 21. In a small adaption of this approach, introduction of a protecting group was used to synthesize an oligo(3-hexylthiophene) with 36 repeating units, the longest regioregular 3-hexylthiophene oligomer synthesized to date.
Conclusion
1. The structure of atoms of chemical elements in the light of the sequence of Fibonacci numbers is considered. It is assumed that the first member of the series is the electron mass, and the second and third members of the series are, respectively, the mass of the proton and neutron.
2. It was shown that the ratios of atoms in chromium oxides: Cr02 Cr03, Cr205, Cr308, Cr5013, Cr8021, uranium:
U20^ U30^ ^ UA^ U13°34 and nitrogen: ^
N2O2, N2O3, N2O5 are equal of ratio of Fibonacci numbers.
3. It has been established that the potential number of fatty acids increases with the length of their chain in accordance with known Fibonacci numbers.
4. It is shown that in the synthesis of oligo (3-hexylthio-phene) the number of repeated units follows the Fibonacci numbers up to length 21.
References:
1. Vorobyov N. N. Fibonacci numbers.- M.: Nauka, 1978.-144p. (in Rusian).
2. Plaza A., Falcon S. Identities for generalized Fibonacci numbers: a combinatorial approach.//International Journal of Mathematical Education in Science and Technology.- 2008.- V 39.- Issue 4.- P. 563-566.
3. Vasyutinskij N. A. Fibonacci numbers in stoichiometry of chemical compounds //Izvestiya Akademii Nauk SSSR, Neor-ganicheskie Materialy.- 1989.- V. 25.- No. 5.- P. 799-803.
4. Ishkhanov B. S., Kabin E. I. Physics of the nucleus and particles. The twentieth century. - M.: Publishing House of Moscow University, 2000.- 200p. (in Rusian).
5. Kurnakov N. S. A compound and a chemical individual.- St. Petersburg. F type. Imp. AN, 1914.- 18 p.(in Rusian).
6. Adamyan R. Kh. Inorganic chemistry (in two volumes).- Vol. 1. Book 2. Chemistry of intransitive elements. - Yerevan: YSU, 2018.-282 p. (in Armenian).
7. Schuster S., Fichtner M., Sasso S. Use of Fibonacci numbers in lipidomics - Enumerating various classes of fatty acids // Scientific Reports. - 2017.- V. 7: 39821. doi: 10.1038/srep39821
8. Koch F. P. V Smith P., Heeney M. Fibonacci's Route to Regioregular Oligo(3-hexylthiophene)s // J. Am. Chem. Soc.-2013.- V. 135.- No. 37.- P. 13695-13698. DOI: 10.1021/ja4057932
