Abstract
The atomic weight, nuclear charge, electron configuration of an atom and the total number of ielectrons in an atom belonging to iblock (i = s, or p, or d, or f) are considered as the fundamental characteristics of an element (atom). Only in the latter case, the true periodicity is achieved: the repetitions occur at regular intervals. The total number of ielectrons in an atom belonging to iblock is used as the new basis for the description of the periodicity. This made possible to propose a new formulation of the Periodic law and to describe the Periodic law by an equation. The equation provides opportunities for largescale prediction of the properties of elements and their compounds; it does not require special knowledge to work with it. Theoretical and applied aspects of the application of the new formulation and equation are outlined. Predictions are made for proton affinity and gas phase basicity of 20 elements, constants of inductive effects of 185 atoms and groups, electronic parameters of 222 neutral ligands. The suitability of the equation is exemplified by the description of the properties of atoms and elements, such as ionization energy, electron affinity, proton affinity, electronegativity, covalent atomic radii and the enthalpy of element formations in the gas phase. The equation makes possible to describe also the properties of compounds and their fragments: the acidic properties of the hydrogen compounds of the elements, the acidic properties of protonated atoms and molecules, gas phase basicity and proton affinity of compounds, inductive effects of ligands in coordination chemistry and substituents in organic chemistry, electronic parameters of neutral ligands, the electron effect constants of coordinating metals.
Introduction
The formulation of the Periodic law is improved along with the progress of science. This article will briefly review the evolution in the formulations of the Law and propose a new formulation.
Unlike other laws of the exact sciences, the Periodic law is given by a table. Let us refer, for instance, to the laws of universal gravitation, Coulomb and Ampère in physics or to the law of mass action in chemistry. All these laws are represented as equations. Attempts to describe the Periodic law by the equation had small success (see below). However, outstanding scientists and thinkers argued:
“There is no any certainty in the sciences where none of the mathematical sciences can apply.”
Leonardo da Vinci, XV century,
“In every department of natural science there is only so much real science as there is mathematics in it.”
Immanuel Kant, XVIII century,
“All sciences are divided into physics and the collecting of post stamps.”
Ernest Rutherford, XX century.
It will be shown in this article that the proposed new formulation of the Periodic law provides the possibility to describe the Periodic law by an equation. The applicability of this equation for the description of many properties of elements will be illustrated. The equation turns out to be effective not only for elements, but also for compounds.
New formulation of the Periodic law
In 1869, Dmitri Ivanovich Mendeleev formulated his famous law [1]:
The properties of the elements, and therefore the properties of the simple and complex substances they form, are periodically dependent on their atomic weight.
The state of science at that time did not allow understanding the reason for such a dependence: the electron was discovered by Thomson in 1897 [2], [3], [4], i.e. almost 30 years later. Furthermore, the presence of a nucleus in an atom was established by Rutherford even 14 years later, in 1911 [5].
Therefore, only in 1913, after 44 years, did the physical meaning of Mendeleev’s formulation become clear. This was done due to the theoretical works of van den Broek [6] and the experimental data of Mosley [7], [8]. As the main characteristic of the element in the formulation of the law, atomic weight was replaced by the charge of the atomic nucleus.
However, the cause of the periodicity remained unclear. Only 10 years later, Bohr explained the periodicity with repetitions of similar electron configurations of atoms [9], [10], [11]. In the modern textbook the following formulation is given [12]:
A successive increase of the charge of the atomic nucleus and the corresponding increase in the number of electrons results in a periodic formation of similar electron configurations of the atoms and, consequently, in a periodic formation of essentially similar chemical elements.
If one looks formally, then he/she can say that this is rather an explanation of the Law: the formulation of the law usually states it without explaining.
The formulation of the Periodic law is proposed, in which the azimuthal quantum number l [13] figures as a recurring property of chemical elements.
The considered formulations of the Periodic law turned out to be unsuitable for deriving the equation for this law. Until recently, this problem has not been resolved. A brief overview on the attempts to create the equation is given below in the section “Description of the Periodic law by Equation. History.”
There was a need in a new formulation the Periodic law, in other words, a new defining properties characteristic, a new independent variable. Why on the basis of the above formulations one could not derive the equation? From this point of view, let us consider in more detail the conventional formulation – the dependence of properties on the nuclear charge.
It turns out that, with a rigorous mathematical approach to the understanding of periodicity, the dependences of the properties of elements on the nuclear charge are not periodic! [14]. The periodic function is characterized by the repetitions at equal intervals of change of the independent variable.
Figure 1 shows a classic example of variations in the ionization energies of neutral atoms in the ground state with increasing nuclear charge.
The maximal values (He, Ne, Ar, Kr, Xe, Rn) are repeated. The minimal values (Li, Na, K, Rb, Cs, Fr) are also repeated. But repetitions do not occur at regular intervals. They are 8 for Li and Na, 8 for Na and K, 18 for K and Rb, 18 for Rb and Cs, and 32 for Cs and Fr (Fig. 1). There is no mathematically strict periodicity here. Therefore, there is no possibility to describe the dependence by an equation.
The situation does not change if the elements are divided into blocks. An increase in “autonomy” (selfsufficiency) of blocks was observed as the Periodic system developed [15]. The division into blocks occurs according to the configurations of outer electronic subshells. Outer subshell determines chemical properties. In accordance with the four configurations of the outer electronic subshells, the Periodic table is divided into four blocks: s, p, d and f.
If we take the pblock as an example, we get the dependence shown in Fig. 2. If we compare this figure with Fig. 1, the relationship becomes clearer. However, repetitions still occur at irregular intervals. They are 8 for B and Al, 18 for Al and Ga, 18 for Ga and In, and 32 for In and Tl.
The situation was significantly improved by replacing the independent variable. Instead of the charge of the nucleus, the sum of all the pelectrons in the atom was taken. This is done in Fig. 3, in which repetitions occur strictly periodically, the period is 6, where 6 is the number of electrons in each filled psubshell. The same approach with the same result was applied to s, d and fblocks.
A clear periodicity is achieved due to the fact that we divided the elements into blocks and changed the independent variable. In each block there can be outer subshells of only one type: s, or p, or d, or f. Wherein in each filled subshell of this type there can be only one and the same number of electrons: 2 – in the ssubshell, 6, 10 and 14 – in p, d and fsubshells. That is why now repetitions occur at regular intervals.
The dependencies obtained are so precise that they can be described by an equation. The ability to design an adequate equation is an essential argument in favor of a new formulation. The equations and their correctness will be discussed in detail below.
At this stage it is useful to recall that the refinement of the characteristic that determines the properties of the elements occurred in the sequence:
Atomic weight→Nuclear charge→Electronic configuration of an atom→Total number of ielectrons in an atom belonging to iblock, where i=s, p, d or f.
Thus, in formulating the Periodic law, the total number of ielectrons, which determine the assignment of an atom to an iblock, is a more adequate characteristic than the nuclear charge.
In other words, if we want to make the Periodic law mathematically exact and develop the equation for it, we need to take an independent variable that provides a mathematically strict periodicity. To do this, one must renounce the charge of the nucleus in favor of the total number of ielectrons, where i=s, p, d or f.
The stated results allow us to propose a new formulation of the Periodic law [14], [16], [17], [18], [19]:
Clear periodic regularities are observed when the properties of elements and their compounds are considered separately in blocks and as depending on the total number of the ielectrons in an atom belonging to the iblock (i=s, p, d or f).
Alternatively, this expression can be considered as an addition to the conventional formulation.
Description of the Periodic law by equation
Here, on the basis of a new formulation, an equation will be constructed.
History
The main advances in the mathematical description of the Periodic law were achieved in the theoretical substantiation of the boundary values of the nuclear charge corresponding to the beginning and the end of the formation of various types of electronic shells and subshells [20], [21], [22], [23], [24], [25], [26], [27], [28]. Also, a large number of dependencies of the properties of elements and their compounds on the nuclear charge or on the number of electrons in their outer (valence) shell, have been established. However, these dependencies are valid only for small sets of elements of the same type, for example, within the same group of the Periodic system. A complete review of these works would be immense, so we provide only references to several examples [22], [23], [24], [27], [29], [30], [31], [32], [33], [34], [35]. The authors of these works did not consider the description of the periodicity.
However, the task of describing the Periodic law by the equation was already articulated by D.I. Mendeleev [22]. It seemed obvious that the use of periodic functions was necessary to describe the Periodic law, and for this reason D.I. Mendeleev considered that the use of trigonometric functions could be promising.
In his pioneering works, Flavitskij [22], [36], [37], [38], [39] refused to use sine and cosine and preferred tangent and cotangent. As an independent variable, he used the function of atomic weight in accordance with the formulation of Mendeleev. Flavitskij proposed equation
where “e” is “electropositivity” or “electronegativity”, P is atomic weight, ϕ(P)=(P–1.4)/35.6 from Li to Cl, this is ϕ(P)=(P–37)/46.8 from K to Br and from Rb to I.
With the Flavitskij values of constants (1.4, 35.6; 37 and 46.8), the value of the function cot [2π·ϕ(P)] decreases monotonically (starting with positive and ending with negative values) along the period (from Li to F, from Na to Cl, and, etc.). In the transition to the next period (from F to Na, from Cl to K, and so on), this function abruptly changes the sign to the opposite. In exactly the same way, “electropositivity” monotonously turns into “electronegativity”, and then abruptly – back, into “electropositivity”.
Later Thomsen [22], [40], [41] on the basis of similar reasoning came to a similar equation
where “a” is the atomic weight; in square brackets is the term for the elements from Li to Cl; from K to I, it is (a–36) π/48, and then (a–132) π/80.
On the basis of the established relationship, Thomsen suggested that at the points of discontinuity of the function cotangent with a change of sign (−∞ to +∞), between F to Na, Cl to K, and so on) there should be “inactive” elements. These elements have neither electropositivity, nor electronegativity, their valence is equal to 0. At that time, only Ar was discovered, and there was no place for it in the Periodic table. Based on an analysis of eq. (2), Thomsen justified the existence of an inert gas group and indicated its place in the Periodic table.
Thomsen also composed an equation to describe changes in valence
Here the terms in square brackets are the same as for eq. (2).
Returning to the similarity between eqs. (1) and (2), it should be mentioned that Flavitskij [38], [39] defended his priority in relation to Thomsen’s work. It is also necessary to pay attention to the purely qualitative, illustrative nature of the dependence on the atomic weight in the Flavitskij and Thomsen equations.
As noted above, after the work of van den Broek and Moseley (1913), the nuclear charge (atomic number) began to be considered as a characteristic defining properties. The report [42] notes that complicated character of chemical periodicity can be modeled using a set of periodical functions with distinct frequencies. The representations of any property of the chemical elements or substances as a sum of sine and cosine functions of atomic number allows the use of digital filtration for the analysis of chemical periodicity. On this basis, a method for quantitative prediction of the properties of chemical elements has been elaborated. Predictions of the properties of superheavy elements using autoregression models reveal that these elements might be volatile [42].
There are studies of interest, in which the Gauss function was used to describe the physical properties of elements in a threedimensional space depending on the period number and the number of electrons on the outer subshell [43], [44], [45], [46]. It is important, that in these works the elements of each block were described separately, which already partially corresponds to the novel formulation proposed by us above.
The properties of the elements were represented by the equation [46]:
where n is the principal quantum number,
x is the value determined by the number of outer s, p, d and s+delectrons,
c is the principal quantum number of period that served as a reference period.
First term of function (4) gives the curve of reference period.
Thanks to these works [43], [44], [45], [46], significant progress has been achieved in the formal mathematical description of periodic dependencies by a set of similar nonperiodic functions. However, this approach is not based on periodicity, which is the essence of the Periodic law.
As for the progress in the quantitative description of the Law by the periodic equation, this progress was hampered, firstly, by considering all the elements together, as in Fig. 1, and, secondly, by the idea that the nuclear charge as a universal and ideal characteristic of the element. The first was due to the fact that until the last quarter of the 20^{th} century, the conventional version of the Periodic did not contain the division of elements into blocks: s and pelements formed a single structure, in which delements were scattered as side subgroups [15]. Overcoming these stereotypes allowed us to propose the above new formulation of the Periodic law. On the basis of a new formulation, the equation of the Periodic law is obtained.
Equation design
As shown above, the new formulation allows to obtain clear periodic dependencies, a sample of which is shown in Fig. 3. For their description, equations are constructed to include several functions, one of which is necessarily periodic [14]. From periodic functions, those functions in which periodicity is achieved by elimination of the integer part of the dependent variable (Fig. 4), seem to be particularly suitable. This elimination ensures periodicity: dependent variable grows until the integer part appears, then it is eliminated, and the growth begins again (Fig. 4, the three lower graphs).
At first glance, the operation of elimination of the integer part seems formalmathematical. However, this operation has a clear physicochemical meaning. In the case of ionization energies, this elimination simulates a jumplike drop in the energy when the electron is forced to place into the more remote subshell. This happens every time during the transition from one period to another. This can be seen in Figs. 1–3, for example, when moving from Ne to Al in Fig. 3.
The constructed equation of the Periodic law has the form [14]:
where {} means elimination of the integer part of the number in these braces,
y is the value of the property under consideration, for example, the ionization energy in Fig. 3,
x is the sum of all i (s, or p, or d, or f) electrons in an atom,
ϕ is the number of these electrons in the atom, starting the first of the considered rows in the block (ϕ=3 for Li in sblock and 1 for B in p, Sc in d, and La in fblocks),
δ is the number of electrons in each filled isubshell, equal to 2, 6, 10 and 14 for i=s, p, d and fblocks, respectively.
For example, for Mg and Cl fragments of the calculation are in the following form:
One equation quantitatively describes the considered property of all atoms (elements) of a given block of the Periodic table.
Let’s consider the physicochemical meaning of the parameters of eq. (5) in the example of dependencies in Fig. 3. The value of “ϕ” is chosen, so that the elements that start the curves (B, Al, Ga, In, Tl) lie on a single straight line. For this, “ϕ” in the pblock should be equal to 1, then (x–1)/6=0, 1, 2, 3, 4. These are integers, after elimination of them one gets 0, {(x–ϕ)/δ}=0, and ae^{bx}{(x–ϕ)/δ}=0. As a result, elements that start curves actually lie on the line y=cx+d, and the parameter “c” reflects the change in the values of “y” for these elements when moving from one curve to another. If “c” is close to zero (Fig. 3) y=d, and the parameter “d” is the average value of “y” for elements that start curves.
The “b” parameter characterizes the deviation from a straight line dependence and a change in the range of values of the property under consideration when going from the first curve to the next. When b<0, the interval decreases (Fig. 3), and when b>0, it increases.
The parameter “a”, apparently, does not have a physicochemical meaning: it brings the values exp(bx){(x–ϕ)/δ} in accordance with “y”. In particular, when b=0, and considering that {(x–ϕ)/δ} varies from 0 to 1, the need for “scaling” (“normalization”) is manifested clearly.
It should be noted that the above physicochemical meanings of the parameters are an example rather than a universal explanation.
Working with the considered equation does not require special knowledge. Calculations are carried out on a personal computer using a routine program for scientific and engineering calculations, for example, Origin^{®}. The duration of the calculation is measured in fractions of a second.
It is not difficult to calculate the total number of s, p, d, or felectrons in an atom. Nevertheless, we present Table 1, in which these total numbers are given.
Group →  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  

Period ↓  
1  1 ^{s=1}H 
2 ^{s=2}He 

2  3 ^{s=3}Li 
4 ^{s=4}Be 
5 ^{p=1}B 
6 ^{p=2}C 
7 ^{p=3}N 
8 ^{p=4}O 
9 ^{p=5}F 
10 ^{p=6}Ne 

3  11 ^{s=5}Na 
12 ^{s=6}Mg 
13 ^{p=7}Al 
14 ^{p=8}Si 
15 ^{p=9}P 
16 ^{p=10}S 
17 ^{p=11}Cl 
18 ^{p=12}Ar 

4  19 ^{s=7}K 
20 ^{s=8}Ca 
21 ^{d=1}Sc 
22 ^{d=2}Ti 
23 ^{d=3}V 
24 ^{d=4}Cr 
25 ^{d=5}Mn 
26 ^{d=6}Fe 
27 ^{d=7}Co 
28 ^{d=8}Ni 
29 ^{d=9}Cu 
30 ^{d=10}Zn 
31 ^{p=13}Ga 
32 ^{p=14}Ge 
33 ^{p=15}As 
34 ^{p=16}Se 
35 ^{p=17}Br 
36 ^{p=18}Kr 

5  37 ^{s=9}Rb 
38 ^{s=10}Sr 
39 ^{d=11}Y 
40 ^{d=12}Zr 
41 ^{d=13}Nb 
42 ^{d=14}Mo 
43 ^{d=15}Tc 
44 ^{d=16}Ru 
45 ^{d=17}Rh 
46 ^{d=18}Pd 
47 ^{d=19}Ag 
48 ^{d=20}Cd 
49 ^{p=19}In 
50 ^{p=20}Sn 
51 ^{p=21}Sb 
52 ^{p=22}Te 
53 ^{p=23}I 
54 ^{p=24}Xe 

6  55 ^{s=11}Cs 
56 ^{s=12}Ba 
*  71 ^{d=21}Lu 
72 ^{d=22}Hf 
73 ^{d=23}Ta 
74 ^{d=24}W 
75 ^{d=25}Re 
76 ^{d=26}Os 
77 ^{d=27}Ir 
78 ^{d=28}Pt 
79 ^{d=29}Au 
80 ^{d=30}Hg 
81 ^{p=25}Tl 
82 ^{p=26}Pb 
83 ^{p=27}Bi 
84 ^{p=28}Po 
85 ^{p=29}At 
86 ^{p=30}Rn 
7  87 ^{s=13}Fr 
88 ^{s=14}Ra 
**  103 ^{d=31}Lr 
104 ^{d=32}Rf 
105 ^{d=33}Db 
106 ^{d=34}Sg 
107 ^{d=35}Bh 
108 ^{d=36}Hs 
109 ^{d=37}Mt 
110 ^{d=38}Ds 
111 ^{d=39}Rg 
112 ^{d=40}Cn 
113 ^{p=31}Nh 
114 ^{p=32}Fl 
115 ^{p=33}Mc 
116 ^{p=34}Lv 
117 ^{p=35}Ts 
118 ^{p=36}Og 
*Lanthanoids  57 ^{f=1}La 
58 ^{f=2}Ce 
59 ^{f=3}Pr 
60 ^{f=4}Nd 
61 ^{f=5}Pm 
62 ^{f=6}Sm 
63 ^{f=7}Eu 
64 ^{f=8}Gd 
65 ^{f=9}Tb 
66 ^{f=10}Dy 
67 ^{f=11}Ho 
68 ^{f=12}Er 
69 ^{f=13}Tm 
70 ^{f=14}Yb 

**Actinoids  89 ^{f=15}Ac 
90 ^{f=16}Th 
91 ^{f=17}Pa 
92 ^{f=18}U 
93 ^{f=19}Np 
94 ^{f=20}Pu 
95 ^{f=21}Am 
96 ^{f=22}Cm 
97 ^{f=23}Bk 
98 ^{f=24}Cf 
99 ^{f=25}Es 
100 ^{f=26}Fm 
101 ^{f=27}Md 
102 ^{f=28}No 
Note: For example, ^{p=10}S means that S has 10 pelectrons [15]. When i=d, the total number of delectrons is given for the electron configuration of an atom with two selectrons in the outer subshell.
The new formulation can also be described by other periodic functions, for example, trigonometric [16], [48], [49]:
Description by the equation of a wide range of properties
The proposed eqs. (5–7) are effective not only to describe the ionization energies (Fig. 3). These equations were able to describe other properties of atoms and elements:
Electron affinity [48]
Proton affinity and gas phase basicity [17]
Electronegativity [48]
Covalent atomic radii [48]
and others [47]
The equations are correct for describing the properties of not only the elements, but also the compounds, as well as their fragments:
Acidic properties of the hydrogen compounds of the elements ElH_{n}: CH_{4}, SiH_{4}, GeH_{4}, NH_{3}, PH_{3}, AsH_{3}, H_{2}O, H_{2}S, H_{2}Se, HF, HCl, HBr, and HJ by enthalpy changes in the reaction ElH_{n}=ElH^{−}_{n−1}+H^{+} [48]
Acidic properties of protonated atoms and molecules: AH^{+}: NeH^{+}, ArH^{+}, KrH^{+}, XeH^{+}, NH_{4}^{+}, PH_{4}^{+}, AsH_{4}^{+}, H_{3}O^{+}, H_{3}S^{+}, H_{3}Se^{+}, H_{2}F^{+}, H_{2}Cl^{+}, H_{2}Br^{+}, and H_{2}I^{+} by changes in the free energy in the reaction AH^{+}=A+H^{+} [48]
Gas phase basicity and proton affinity of the compounds ElR_{n}, where El are elements of groups 2, 13, 14, 15, 16 and 17 [49]
Inductive effects of atoms and groups R_{n−1}El– (ligands in coordination chemistry and substituents in organic chemistry), where El are elements of groups 1–17 [50]
Electronic parameters of neutral ligands, compounds of elements of groups 14, 15, 16 and 17 [51]
Areas of the application of the new formulation and equation
Interesting theoretical results can be obtained by comparing and explaining the differences for the equations obtained for different blocks of elements and different sets of elements inside blocks. This applies, in particular, to kainosymmetry (absence of “primogenic repulsion”) as well as to internal and secondary periodicity [17], [18], [53].
It would be useful to see within the general picture how all the underlying ielectrons influence on the outer ielectrons.
The proposed equation opened up the possibility of largescale prediction of the properties. For example, were the following properties were predicted:
Proton affinity and gas phase basicity of 20 elements [17]
Constants of inductive effects of 185 atoms and groups [50]
Electronic parameters of 222 neutral ligands [51]
The most promising for forecasting are the following areas:
Prediction of the properties of element compounds. The number of elements is relatively small and they are studied in detail, but the number of their compounds is hundreds and thousands of times more
Forecast of the reaction rates of atoms in various interactions. There are no such comprehensive and accurate data on kinetics for an atom of each element, such as, for example, ionization energies. Here is a completely new, very extensive field for the predictive application of equations. The next stage is the transition from reactions of atoms to reactions of compounds.
The challenge for the future seems to be the spread of periodic equations to the description of the bottom of the Periodic table. Here relativistic effects [53], [54], [55] operate, which are not taken into account by eqs. (5–7). Starting from period 6, these effects significantly affect the properties. Two solutions appear to be possible: (1) a description of the upper and lower parts of the periodic table by two different modifications of the mentioned equations or (2) the introduction of an additional term in the equation reflecting relativistic effects.
Specifics of the equation
It should be noted that the above eq. 5 is not a simple (statistical) correlation: it describes a causeandeffect relationship. The fact that the number of ielectrons in the outer subshell of an iblock atom, their principal and orbital quantum numbers determines the properties of an element is generally recognized in chemistry. These characteristics strongly correspond to the total number of ielectrons. If it is necessary to take into account other electrons, it is essential to acknowledge that their characteristics are also strongly related to the total number of ielectrons. When considering the properties of elements (atoms) separately in blocks, it seems natural to view the total number of ielectrons as the prime cause.
One equation quantitatively describes the considered property of all atoms (elements) of a given block of the Periodic table. It is necessary to remind, that there are 28–40 atoms in the block, with the exception of the sblock. For another property or another block, the same equation is applied, but with different parameters.
If we assume that a new equation arises upon the change of parameters, then to describe the Periodic law, we need four times (by the number of blocks) more equations than there are properties. There is a need for a huge number of equations, which again looks ugly compared to the laws of exact sciences mentioned at the beginning of the article.
Although only very distant analogies are relevant, let us consider this problem in more detail. Each of the laws of the exact sciences describes only one dependence, e.g.: the law of universal gravitation – the force of attraction between objects (masses), the law of Coulomb – the force of attraction (repulsion) between charges. For the force with which a magnetic field acts on a conductor with current, a third law is derived – the Ampère law. For each force there is a law. There is a multitude of laws, which corresponds to the multitude of forces. In the Periodic system, in a similar way, there are many dependencies, only these dependencies are not called laws. Therefore, it seems natural to have a large number of equations in the case of the Periodic law. In addition, the formula of the equation remains the same, only the values of the parameters change.
It was mentioned earlier that not only eq. 5 can be used to describe a Periodic law, but also eqs. 6 and 7. This unusual multiplicity has analogies in a huge number of suggested periodic tables (see below, “Adequacy of the equation….”), as well as in a large number of mathematically equivalent descriptions (formulations) of quantum mechanics.
Adequacy of the new formulation and the equation when fundamental variations occur in blocks and periods
While the number of formulations of the Periodic law can be counted on the fingers of one hand, the number of tables and other graphic images has exceeded one thousand [16], [56], [57], [58]. Such a huge amount of variants often causes perplexity. Back in 1911, when the number of versions was measured in dozens, Nernst wrote: “… this region, which especially needs scientific tact for its development, has become the playground of dilettante speculations, and has fallen into much discredit” (quoted from [59]).
Of course, part of the published graphic representations can be attributed to the “art for the sake of art” or to the development of the territory between art and science. However, the above opinion seems to be an unjustified extreme. The developed versions focus attention on different essential aspects of the Periodic law. Only by taking into account many of the options, one can get a fairly complete picture of the Law [15], [60].
An understanding of the entire set of tables and other graphical representations of the Periodic law as a continuum [57], [61] seems to be very adequate. Wherein chemically accurate systems are at one extreme, the more abstract or more philosophical tables are lying at the other end of the spectrum. If one seeks for maximal chemical utility one should opt for the more “unruly” tables. If one seeks maximal elegance and orderliness above all, one should favor the more regular representations.
Everything discussed earlier in this article is based on a conventional table. It seemed natural to find out how much the new formulation and the equation derived from it are applicable to systems that differ fundamentally. For this, a table of doublecharged cations [18] was designed.
The boron atom, losing one p and one selectrons, turns into a doublecharged scation
The carbon atom, losing two pelectrons, also turns into a doublecharged scation:
The scations of boron and carbon from the pblock are shifted to the left into the sblock (Table 2).
Similarly, the satoms of sodium and magnesium are converted into pcations:
They are moved to the pblock, at the end of the previous period (Table 2).
The resulting Table 2 of doublecharged cations differs substantially from the original conventional table of neutral atoms:
s and pblocks of the Periodic table of cations in composition differ appreciably from these blocks in the Periodic table of atoms (elements)
Thus, Table 2 is a good model for testing the applicability of a new formulation far beyond the conventional table. This analysis is particularly significant in connection with the change in the composition of the blocks, since the basis of the new formulation is the consideration of the properties separately by blocks.
The ionization energies of the cations were analyzed as described above for atoms (Figs. 1–3).
Figure 5 shows the dependence on the nuclear charge. As described earlier for atoms (Fig. 1), the location of points for cations cannot be described by the equation, mainly due to the fact that repetitions occur at unequal intervals. Only consideration separately by blocks and using as an independent variable the total number of electrons that determine the belonging to the block, led to a clear periodicity in Fig. 6.
This made it possible to describe the dependence by equation [18].
Thus, the new formulation and the equation derived from it retains applicability with significant changes in the composition and mutual arrangement of blocks and periods.
Conclusions
Unlike other laws of the exact sciences, the Periodic law is not represented by an equation, but by a table. At the same time, the fundamental characteristic of an element (atom) is the charge of the nucleus. However, the use of a nuclear charge as an independent variable in the description of the element (atom) properties leads to repetitions at different intervals. There is no periodicity in the strict mathematical sense, and, as a result, there is no possibility of describing dependencies by a periodical equation.
To construct the equation of the Periodic law, a new formulation was required, in other words, a new basis for describing periodicity, a new independent variable. Such an independent variable was found when considering the elements separately by blocks. In each block there can be outer subshells of only one type: s, or p, or d, or f. At the same time, in each filled subshell of given type there can be only one and the same number of electrons: 2 – in the ssubshell, 6, 10 and 14 – in p, d and fsubshells. That is why repetitions here occur at regular intervals.
In accordance with the above, a new basis has been proposed for describing the periodicity: the total number of ielectrons in an atom belonging to iblock (i=s, p, d, or f).
Proceeding from this basis, a new formulation of the Periodic law is given:
Clear periodic regularities are observed when the properties of elements and their compounds are considered separately in blocks and as depending on the total number of the ielectrons in an atom belonging to the iblock (i=s, p, d, or f).
Alternatively, this expression can be considered as an addition to the conventional formulation.
Thanks to the application of the proposed new basis as an independent variable, it became possible to construct adequate equation of the Periodic law:
It is emphasized that the proposed equation is not a simple correlation: it represents a causeandeffect relation.
One equation characterizes the property of the atoms (elements) or their compounds of the whole block. For another property or another block, the same equation is applied, but with different parameters. The diversity of descriptions of the properties (dependencies) given by this equation is analogous to the diversity of equations in the exact science laws.
The effectiveness of equation for describing many properties of a wide range of elements and their compounds is shown. The proposed equation opened up the possibility of largescale prediction of the properties. Predictions are made for proton affinity and gas phase basicity of 20 elements, constants of inductive effects of 185 atoms and groups, electronic parameters of 222 neutral ligands.
Interesting theoretical results can be obtained by comparing and explaining the differences for the equations obtained for different blocks of elements and different sets of elements inside blocks. This applies, for example, to kainosymmetry (absence of “primogenic repulsion”) as well as to internal and secondary periodicity.
It would be useful to see within the general picture how all the underlying ielectrons influence on the outer ielectrons.
A promising applied field of the use of the designed equation is the prediction of the compound properties. The number of elements is relatively small and they are studied in detail, but the number of their compounds is hundreds and thousands of times more. A completely new, very extensive field for predictive application of equation is the prediction of the reaction kinetics of atoms in various interactions.
Working with the considered equation does not require special knowledge. Calculations are carried out on a personal computer using a routine program for scientific and engineering calculations, for example, Origin^{®}. The duration of the calculation is measured in fractions of a second.
In connection with the huge number of periodic tables proposed in the literature, it seemed natural to find out how the new formulation and the equation derived from it are applicable for systems, that principally differ from the conventional one. For this, a table of doublecharged cations was constructed. On this model, it was shown that the new formulation and the equation retain their correctness upon fundamental changes in the composition and mutual arrangement of the blocks and periods.
n  n (s^{1}, s^{2})  n (p^{1}, …p^{6})  (n–1) (f^{1}, …f^{14})  n (d^{1}, …d^{10})  Number of cations in period 

1  Li^{+2} Be^{2+}  2  
2  B^{+2} C^{+2}  N^{+2}…Na^{+2}, Mg^{+2}  8  
3  Al^{+2} Si^{+2}  P^{+2}…K^{+2}, Ca^{+2}  Sc^{+2}…Zn^{+2}  18  
4  Ga^{+2} Ge^{+2}  As^{+2}…Rb^{+2}, Sr^{+2}  Y^{+2}…Cd^{+2}  18  
5  In^{+2} Sn^{+2}  Sb^{+2}…Cs^{+2}, Ba^{+2}  La^{+2}…Yb^{+2}  Lu^{+2}…Hg^{+2}  32 
6  Tl^{2+} Pb^{2+}  Bi^{2}^{+}…Fr^{2+}, Ra^{2+}  Ac^{2+}…No^{2+}  Lr^{2+}…Cn^{2+}  32 
Article note
A collection of invited papers based on presentations at Mendeleev 150: 4^{th} International Conference on the Periodic Table (Mendeleev 150), held at ITMO University in Saint Petersburg, Russian Federation, 26–28 July 2019.
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