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Articles and Publication  Chemistry NEW FORMULATION OF MENDELEEV'S PERIODIC LAW.
NEW FORMULATION OF
MENDELEEV'S PERIODIC LAW.
© Naum S.
Imyanitov,
Dr. Sc.
Contact to the
author: naum@itcwin.com
VNIINeftekhim, St.Petersburg
Abstract.
A possibility of a
mathematical description and prediction of the periodically varying properties
of elements and their compounds is demonstrated, and the corresponding equations
are designed. Therewith a single equation is valid for a whole block (s + p, or
d, or f) of the periodic system; the number of all s + p-, or d-, or f-electrons
of the atom serve as independent variable. A corresponding new formulation of
the periodic law is suggested. Examples are presented of description with
equations developed for the ionization energies and the covalent atomic radii,
and also for the enthalpies of formation for elements in the gas phase. The wide
prospects for the description with these equations of the other properties of
elements and their numerous compounds are indicated.
___________________________________________________________________________
Background
In 100 years that have passed
since the discovery of the periodic law (1 - 4) about 700
of its versions has been published in the form of tables and diagrams.
Mazurs’s exhaustive study (5) analyzed them and reproduced
many of them. The number of versions published within the next 40 years has not
been counted, but even now the work of this kind is going on (2, 6,
7 - 9). Moore and Scerri grant that each of the many different
designements proposed has advantages and drawbacks, and we can use those to
improve pedagogy (2, 10). Bent believes that since the Periodic
Law is one of the central doctrines of the Central Science, tabular expressions
of it have many uses. There exist, consequently, many Periodic Tables (6,
11).
The attention should be drawn to
the fact that the presentation of Mendeleev’s law in a tabular form and
moreover, as a set of tables, is an exception from the general rule. Actually,
all the laws of exact sciences are formulated in the form of equations.
Let us mention as examples the laws of Newton, Coulomb, Faraday from physics,
the law of mass action from chemistry. However the eminent German philosopher Immanuel
Kant has advanced the aphorism that the study of nature is as
scientific as it contains mathematics.
It is difficult to presume that
all the details contained in hundreds of tables would be once possible to
describe by an equation and thus completely reject the tables. But no doubt, the
mathematization of the chemistry basis is necessary as is also the description
of its rules and laws with equations in conformity to the Kant’s science
criterion.
The search for an independent
variable
The problem of the periodic law
description by an equation was posed already by D.I.Mendeleev (12).
Himself, Thomsen (12,13), and Flavitskij (12, 14)
presumed a potential promise in trigonometric functions for this goal. While
Thomsen addressed to the functions of cotangent and sine square, Flavitskij
preferred cotangent. However we do not know any studies extending these purely
qualitative but undoubtedly historically interesting suggestions in the next
over hundred years.
The main success in the
mathematic description of the periodic system was attained in the theoretical
substantiation of the boundary values of a nucleus charge corresponding to the
beginning and the end of the formation of various electron subshells (5,
12, 15). Also a large number of rectilinear or monotone dependences of
properties of elements and their compounds on the nucleus charge or the number
of electrons
in their
outermost shell
(valence shell)
was established. However these dependences are valid only for small sets of
elements of the same type (12), for instance, within a group
of the periodic system. No report existed on a description with a single general
equation of any property of all elements of the periodic system or of its
essential part.
This problem was solved only
recently by designing equations involving several functions (16,17).
Let us present the logical sequence of the reasoning.
Take for example the ionization
energy, a very important property of an atom characterizing its ability to be
oxidized, to acquire a positive charge. The ionization energy of all elements as
a function of the nuclear charge is given in Figure 1. It is impossible to
describe the ionization energy with an equation (at least, by a relatively
simple equation).
Figure 1. Plot of ionization energies of
elements as a function of the nuclear charge.
Therefore let us exclude from
Figure 1 the data on the ionization energy of the transition metals, the
lanthanides, and the actinides. In the corresponding Figure 2 dependences of
similar kind remain, but the interval between them along the horizontal axis is
unequal: It is zero between Be and B, Mg and Al, 10 between Ca and Ga, Sr and In,
24 between Ba and Tl.
Figure 2. Plot of ionization energies of
elements as a function of the nuclear charge with the rejection of the data for
transition metals, lanthanides, and actinides.
To go further a very bold,
unordinary step should be done: These intervals should be omitted, and we should
go over from Figure 2 to Figure 3. This operation would provoke objections of a
chemist for it is quite inadmissible in the framework of the existing concept of
the governing role of the nucleus charge. Whereas on Figures 1 and 2 between Ca
and Ga, Sr and In, Ba and Tl occurs a smooth, gradual change of the nucleus
charge within periods, on Figure 3 these pairs occur near each other. In going
from one member of the pair to the other the charge grows with a jump, and as
sharply should change the properties.
Figure 3. Plot of ionization
energies of sp-elements as a function of the number of all the s- and
p-electrons.
As will be seen further, the
departure from the commonly accepted interpretation proves to be justified.
However what is plotted on the horizontal axis of Figure 3 instead of the
nuclear charge of Figures 1 and 2? Note first that the number of electrons in an
atom equals the charge of the nucleus, therefore in the Figures 1 and 2 on the
horizontal axis might be plotted the overall number of electrons in the atom
instead of the charge. Inasmuch as in drawing Figure 3 we rejected the
transition metals, the lanthanides, and the actinides, therewith d- and
f-electrons were excluded. Only s-and p-electrons remained; consequently, on the
horizontal axis in Figure 3 is plotted the number of all the s- and p-electrons
in the atoms of the s- and p-blocks (of the sp-block).
Hence the jump-like change in the
nucleus charge in going from Figure 2 to Figure 3 is not of much importance. The
relationship is governed by the change in the number of the s + p-electrons that
is not affected by the rejection of d and f-elements. Actually (see above), in
going from Ca to Ga (from Ba to Tl) the nucleus charge grows by 10 (24) units,
but the number of the s + p-electrons, by 1 (1). In keeping with the small
difference in the number of these electrons the ionization energies of these
pairs are of close value (Fig. 2)
It seems that we gained the
clear periodic dependence in Figure 3 at a too great expense rejecting a
half of elements present in the periodic table. It is not however correct:
The rejected elements also form clear periodic relationships if the same
approach is applied to them. The properties of d-elements should be
considered as a function of the number of all the d-electrons (see below),
and of f-elements, as a function of the number of f –electrons,
disregarding the electrons of the other kinds (16 - 18).
Designing and application of
equations
Although in Figure 3 a clear
periodic dependence is observed it is not possible to restrict oneself to a
periodic function for its description: The obtained curves possess similar
“waves” (identical maxima and minima). In other words, the description does
not reflect the change in the elements’ properties in going from one period to
another, moving along the vertical in the periodic table.
The calculation studies showed
that the change in the height of the “waves” might be represented by
introducing an exponential function ebx. If the “waves” move up
or down, it is advisable to add the third function, linear one cx + d.
The most suitable for the role of the periodical
function are the fractional parts of functions whose magnitudes are obtained
from the initial function by rejecting the integer from the value of the
dependent variable.
Thus for the simplest version
ó = {x}
(the braces indicate the operation of rejection)
The plot
ó
= {x} is given in Figure 4.
Figure 4. Some examples of describing
periodical dependences based on the fractional parts of the dependent variable.
Sign { } means the rejection of the integer of the value, sign | | means going
over to the absolute value.
The ionization energy of atoms presented in
Figure 3 in keeping with the above reasoning is described by the following
equation
(1)
where y is the ionization energy
of an atom, x is the number of all the s + p – electrons, { } means the
operation of rejection of the integer of the value, 8 is the number of s- and
p-elements in the period, 3 is the number of s-
and p-electrons in Li, the first element in the dependence.
Interestingly, the performed
formal mathematical operation of rejecting the integer of the value of the
dependent variable has a clear physical sense: it
corresponds to a jump-like decrease in the ionization energy at the forced
arrival of an electron to an orbital more distant from the nucleus at the
beginning of each period.
Figure 5 demonstrates that the
curves obtained are well consistent with the experimental findings; the
correlation coefficient (R2) equals 0.92. A single equation described
the data of a large amount (42) of elements.
Figure 5. Plot of
ionization energies of sp-elements (y) as a function of the number of all the s-
and p-electrons (x). Eq 1
;
a = 2152, b = -0.031, c = 0, d = 412. R2 = 0.92. Number of
points (elements) 42.
In many cases the description
applying trigonometric functions also provided good results (17):
y = aebx
tn(ωx + φ) + cx + d (2)
y = aebx sin(ωx + φ) + cx
+ d (3)
Figures 6 and 7 show application
of trigonometric functions to sp-elements, and Figure 8, to d-elements.
Figure 6. Plot of covalent
atomic radii of sp-elements (y) as a function of the number of all the s- and
p-electrons (x). Eq 2, ω = π/8 (8 is the number of s- and p-elements
in the period), φ = 3π/16; a = -0.135, b = 0, c = 0.026, d = 0.67. R2
= 0.92. Number of points (elements) 33.
Figure 7 presents a sinusoid with
a decreasing amplitude, Figures 6 and 8, characteristic tangensoids. The
sinusoid of Figure 7 corresponds to the enthalpies of formation for elements in
the gas phase (number of elements is 43, R2 = 0.82). The tangensoid
on Figure 6 was employed for the description of the covalent atomic radii (number
of elements is 33, R2 = 0.92).
The tangensoid on Figure 8
describes the ionization energies of the d-elements (number of elements is 30, R2
= 0.83). In the latter case evidently the number of all the d-electrons in an
element served as the independent variable.
Figure 7. Plot of enthalpies
of formation for sp-elements in the gas phase (y) as a function of the number of
all the s- and p-electrons (x). Eq 3, ω = 2π/8 (8 is the number of s-
and p-elements in the period), φ = 7π/8; a = 331, b = -0.035, c =
-5.38, d = 332. R2 = 0.82. Number of points (elements) 43.
Figure 8. Plot of ionization
energies of d-elements (y) as a function of the number of all the d-electrons
(x). Eq 2, ω = π/10 (10 is the number of d-elements in the period),
φ = 9π/20; a = 14.8, b = 0.032, c = 3.69, d = 681. R2 =
0.83. Number of points (elements) 30.
The problem of the best-founded
application of a certain periodic function in each definite case is yet to be
solved. It is however clear that the use of the fractional part of the dependent
varialble is more general: this method can transform any function into a
periodic one; some simple examples are given in Figure 4.
The designing of the above
equations and the fitting of their coefficients are carried out using standard
computer software (for instance, Origin®).
Besides the mentioned
properties the following characteristics were described by the equations:
the electron affinity of sp-elements (17), the electron
affinity, the electronegativity, and other data of d-elements and also of
their coordination compounds (18).
It is exceptionally important to
note that the above equations are suitable not only for the characterization of
elements’ properties but also those of their compounds whose quantity is by
several orders of magnitude greater than the number of elements. The equations
were designed for the quantitative characteristics of the following properties:
• acidities of hydrides ElHn:
CH4, SiH4, GeH4, NH3, PH3,
AsH3, H2O, H2S, H2Se, HF, HCl, HBr,
and HJ from the enthalpy change for the process ElHn = ElH-n-1
+ H+ (17),
• acidities of protonated
molecules AH+: NH4+, PH4+,
AsH4+, H3O+, H3S+,
H3Se+, H2F+, H2Cl+,
H2Br+, H2J+, NeH+, ArH+,
KrH+, and XeH+ from the change in the free energy of the
process AH+ = A + H+ (17),
• gas-phase basicities and
proton affinities of compounds ElRn (19),
• inductive effects of atoms
and groups of the general formula Rn-1El- (ligands in inorganic
chemistry, substituents in organic chemistry) (20),
• electronic parameters of
neutral ligands ElRn in the coordination compounds (21).
Therewith in every of the cited
cases a single equation was sufficient.
The developed equations made it
possible in the same way as had done Mendeleev but at a new level to predict the
properties of elements and their compounds (18), in particular, of
superheavy elements synthesized in an amount of several atoms or not yet
synthesized.
New formulation of the
Periodic Law
D.I. Mendeleev formulated the
periodic law as follows:
“The properties of elements and
also of simple and complex substances formed from them are in periodical
dependence on their atomic weight”.
In the first quarter of the 20th
century it was discovered that a more precise formulation is as follows:
“The properties of
elements and also of simple and complex substances formed from them are in
periodical dependence on the charge of atomic nuclei”.
The law is illustrated by Figure
1.
Based on the content of this
paper it is possible to suggest the following new formulation of the periodic
law:
“Clear periodic
regularities are observed when considering the properties of elements and their
compounds, separately in blocks and as depending on the total number of
the s-electrons in the atom of s-block, p-electrons in p-block, s + p- electrons
in s p-block,
d-electrons in d-block
or f-electrons in f-block. The values of cycles are, respectively, 2, 6, 8, 10
and 14.”
The new formulation of the
periodic law is illustrated by Figures 5 – 8.
It is interesting also to
consider the problem from a wider standpoint. The periodicity is
characteristic not only of the periodic law but nearly for every more or less
complex development. The known periodic cycles exist in the nature and the
ecology, in the demography, the technology, the economy and social and
political life, in the science, the culture, and the education. The equations
treated in this paper might be useful also in these events.
Conclusions
Based on the content of this
paper it is possible to suggest the following new formulation of the periodic
law:
“Clear periodic
regularities are observed when considering the properties of elements and their
compounds, separately in blocks and as depending on the total number of
the s-electrons in the atom of s-block, p-electrons in p-block, s + p- electrons
in s p-block,
d-electrons in d-block
or f-electrons in f-block. The values of cycles are, respectively, 2, 6, 8, 10
and 14.”
Proposed above can be considered
as further clarifying the formulation of the Periodic Law (for characteristics
determining the properties) in the sequence: the atomic weight - the nuclear
charge - the total number of electrons in an atom defining belonging to a
particular element of the block.
Literature Cited
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Publishing date: March 3, 2010
Source: SciTecLibrary.ru
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