The method of rectangular hyperbolas is developed for the first time, by which a means for estimating the upper bound of the Periodic Table is established in calculating that its last element has an atom mass of 411.663243 and an atomic number (the nuclear charge) of 155. The formulating law is given.
1 Introduction. The mathematical basis
The periodic dependence of the properties of the elements on their atomic mass, as discovered by D. I. Mendeleev in 1869, predicted the existence of new elements in appropriate locations in the Periodic Table.
Progress in synthesis and in the study of the properties of the far transuranium elements has increased interest in the question of the upper limits of the Periodic Table. G. T. Seaborg, J. L. Bloom and V. I. Goldanskii emphasized that the charge of the atomic nucleus and the position occupied by the element "define unambiguously the structure of electron jackets of its atoms and characterize the whole set of its chemical properties". They suggested the existence of nuclei containing 114, 126 and 164 protons, 184, and 258 neutrons and the Table arrangement of the relevant elements [1, 2].
The objective of this study is to determine the possible number of chemical elements, along with atomic masses and atomic numbers up to the final entry in the Periodic Table.
The calculations were performed on the basis of IUPAC [3] table data for all known elements. The basic principle resides in the idea that the proportion of the defined element Y in any chemical compound of molecular mass X should be related to its single gram-atom. In this case, if K is the atomic mass, the equation Y = K/X would represent a rectangular hyperbola in the first quadrant (K > 0). Its asymptotes conform to the axis coordinates, and semi-axis a=b= =[square root of 2 [absolute value of K]]. The peak of the curve should occur on the virtual axis inclined at an angle of 45[degrees] to the positive direction of the abscissa axis. The necessary conditions associated with this chemical conception are: Y [less than or equal to] 1 and K [less than or equal to] X.
The foregoing equation differs only in the atomic mass for each element of the Periodic Table and allows calculation of the proportion of the element in any compound. Accuracy plotting the curve and the associated straight line in the logarithmic coordinates depends on the size of the steps in the denominator values, which can be entirely random but must be on the relevant hyperbola in terms of X. Consequently, it can be computed without difficulty by prescribing any value of the numerator and denominator. In Table 1a are given both known oxygen containing compounds and random data on X arranged in the order of increasing molecular mass. Fig. 1 depicts the hyperbola (the value of the approximation certainty [R.sup.2] = 1), calculated for 1 gram-atom of oxygen.
[FIGURE 1 OMITTED]
Estimation of the unobserved content in the chemical compound as determined by the formula is expressed on the plot by the polygonal line (Table 1b, Fig. 1). It is obvious from the Fig. 2a that the hyperbolic function of the elemental proportion in chemical compounds plotted against molecular mass, by the example of the second Group, is true ([R.sup.2] = 1). In the logarithmic coordinates (Fig. 2b) it is represented as the straight lines arranged in the fourth quadrant (to the right of hydrogen) all with slope 1. With the view to expansion of the basis of the arguments, this example is given for the first Group including "roentgenium" No. 111, a more recently identified element, and the predicted No. 119 and No. 155. The real axis is shown here, on which the peaks of all hyperbolas of the Periodic Table are arranged (see below).
[FIGURE 2 OMITTED]
2 Using the theorem of Lagrange
It is clear from the Fig. 2a that with the rise of the atomic mass the curvature of the hyperbola decreases (the radius of curvature increases), and the possibility to define its peak, for example, by means of graphical differentiation, becomes a problem due to errors of both subjective and objective character (instrument, vision and so on). Therefore, to estimate the curve peak of the hyperbola the mathematical method of the theorem of Lagrange was used [4].
For example, the coordinates of the peak for beryllium are as follows: X = 60.9097, Y = 0.14796, the normal equation is Y = 0.0024292X. Taking into consideration that the semiaxis of the rectangular hyperbola a=b= [square root of 2 [absolute value of K]], the coordinates of the point [X.sub.0] = [Y.sub.0] = [square root of K].
Let us examine this fact in relation to elements with the following atomic masses (K): beryllium Be (9.0122), random Z (20), chromium Cr (51.9961), mercury Hg (200.59), No. 126 (310), random ZZ (380), No. 164 (422), random ZZZ (484). In this case [X.sub.0] = [Y.sub.0] = [square root of K], and correspondingly, 3.00203, 4.472136, 7.210825, 14.16298, 17.606817, 19.493589, 20.54264, 22.
The obtained values are the coordinates of the rectangular hyperbola peaks ([X.sub.0] = [Y.sub.0]), arranged along the virtual axis, the equation of which is Y = X (because tan [alpha] = 1).
3 The point of crossing and the scaling coefficient
Our attention is focused on the point of crossing of the virtual axis with the line Y = 1 in Fig. 3 when the atomic mass and the molecular mass are equal, i.e. K = X. It is possible only in the case when the origin of the hyperbola and its peak coincide in the point with the maximum content Y according to the equation Y = K/X.
[FIGURE 3 OMITTED]
The atomic mass of this element was calculated with application of the scaling coefficient and the value of the slope of the virtual axis (the most precise mean is 0.00242917): tan [alpha] = y/x = 0.00242917, from which x = y/tan [alpha]. Due to the fact that at this point k = x we have: y/tan [alpha] = 1/tan [alpha] = =411.663243. This value is equal to the square of the scaling coefficient too: [20.2895.sup.2] =411.6638, [DELTA] A=0.0006.
The coefficient was calculated from matching of the coordinates of the peak hyperbola for beryllium: [X.sub.0] = [Y.sub.0] = [square root of K] and X = 60.9097, Y = 0.14796. Using this data to construct two triangles (Fig. 4), one easily sees an inversely proportional relationship: X/[X.sub.0] = [Y.sub.0]/Y, whence X/[X.sub.0] = 60.9097/3.00203 = 20.2895041 and [Y.sub.0]/Y = 3.00203/0.14976 = 20.28947013, [DELTA] = 0.000034.
[FIGURE 4 OMITTED]
The calculated value M = 20.2895 is the scaling coefficient. With its help the scale of system coordinates can be reorganised.
Now if one rectangular hyperbola peak is known, [X.sub.0] = = [Y.sub.0] = [square root of K], then the new coordinates will be: X = [X.sub.0] M or X = M [square root of K], Y = [square root of K]/M. Furthermore, tan [[alpha].sub.0] = [Y.sub.0]/[X.sub.0] = 1, so tan [alpha] = Y/X = 1/[M.sub.2]. At the same time at Y = 1 and K = X, X = Y/tan [alpha] or K = Y/tan [alpha] = 1/tan [alpha] = [M.sup.2].
The results obtained are plotted in Fig. 5 in comparison with the hyperbolas of such elements as Be, Cr, Hg and the hypothetical No.126 (atomic mass = 310), No.164 (atomic mass = 422), ZZZZ (atomic mass = 411.66). It is obvious that it is practically impossible to choose and calculate precisely the curve peak for an atomic mass exceeding the value 250 without the application of the mathematical method adduced herein.
[FIGURE 5 OMITTED]
The rated element ZZZZ is the last in the Periodic Table because the hyperbola No.164 after it crosses the virtual axis at the point which coordinates are: [X.sub.0] = [Y.sub.0]= [square root of 422]= =20.5426386.
After scaling we have X =20.2895x20.5426386=416.8 and Y =20.5426386/20.2895=1.0125, but this makes no sense because Y cannot exceed the value 1. In addition, the hypothetical atomic mass 422 occurred higher than the molecular mass 416.8, i.e. X < K, but that is absurd. Similarly, it is obvious from Fig. 2b how the virtual axis (the equation Y =X -6.0202 where Y = ln y, X = ln x) crossing all the logarithmic straight lines at the points corresponding to the hyperbola peaks, takes the value ln x=6.0202 at ln y = 0, or after taking logarithms, X = 411.66, Y = 1.
4 The atomic (ordinal) number
To determine important characteristics of the atomic number some variants of graphical functions of the atomic mass versus the nucleus of all the elements were studied, including No.126. One of them is exponential, with the equation Y = =1.6091 [e.sup.1.0992x] (where y is the atomic mass, x is ln No) at [R.sup.2] = 0.9967. After taking the logarithm of the both sides and inserting the atomic mass of 411.66 we have No.155. The calculations also demonstrated that the ordinal No.126 should have the atomic mass 327.2 but not 310.
Finally, the following atomic masses were obtained: No.116--298.7, No.118--304.4, No.119--307.2, No.120--310, No.126--327.3, No.155--411.66.
5 The new law
Based on the foregoing, the heretofore unknown Hyperbolic law of the Periodic Table of Elements is established. This law is due to the fact that the element content Y when estimated in relation to 1 gram-atom, in any chemical combination with molecular mass X, may be described by the adduced equations for the positive branches of the rectangular hyperbolas of the type Y = K/X (where Y [less than or equal to] 1, K [less than or equal to] X), arranged in the order of increasing nuclear charge, and having the common virtual axis with their peaks tending to the state Y =1 or K =X as they become further removed from the origin of coordinates, reaching a maximum atomic mass designating the last element.
References
[1.] Seaborg G. T. and Bloom J. L. The synthetic elements. Scientific American, 1969, v.220(4), 56.
[2.] Goldanskii V. I. About connections nuclear and chemical physics. Progress in Physical Sciences, 1976, v. 118, issue 2.
[3.] IUPAC Commission on Atomic Weights and Isotopic Abundances/ Atomic Weights of the Elements, 2001.
[4.] Weisstein E.W. Mean-value theorem. CRC Concise Encyclopaedia of Mathematics, 2nd edition, CRC Press, Boca Raton (FL), 2003, 1881.
Albert Khazan
E-mail: albkhazan@list.ru
Table 1: Content of oxygen Y in compounds X per gram-atom (Table 1a)
left and summarized O (Table 1b) on the right.
K X Y = K/X ln X ln Y Compound
15.9994 15.999 1 2.77255 0 O
15.9994 17.007 0.9408 2.83363 -0.0611 1/2[H.sub.2][O.sub.2]
15.9994 18.015 0.8881 2.8912 -0.1187 [H.sub.2]O
15.9994 20 0.8 2.99573 -0.2232 --
15.9994 22 0.7272 3.09104 -0.3185 --
15.9994 23.206 0.6895 3.14441 -0.3719 1/3[B.sub.2][O.sub.3]
15.9994 25.01 0.6397 3.21928 -0.4467 BeO
15.9994 28.01 0.5712 3.33256 -0.56 CO
15.9994 30.006 0.5332 3.4014 -0.6288 NO
15.9994 33.987 0.4708 3.52598 -0.7534 1/3 [Al.sub.2]
[O.sub.3]
15.9994 37 0.4324 3.61092 -0.8384 --
15.9994 40.304 0.397 3.69645 -0.9239 MgO
15.9994 44.012 0.3635 3.78446 -1.0119 [N.sub.2]O
15.9994 50.663 0.3158 3.9252 -1.1526 1/3 [Cr.sub.2]
[O.sub.3]
15.9994 53.229 0.3006 3.9746 -1.2021 1/3 [Fe.sub.2]
[O.sub.3]
15.9994 56.077 0.2853 4.02673 -1.2542 CaO
15.9994 60.075 0.2663 4.09559 -1.323 COS
15.9994 71.844 0.2227 4.2745 -1.5019 FeO
15.9994 79.545 0.2011 4.37632 -1.6038 CuO
15.9994 86.905 0.1841 4.46482 -1.6923 [Cl.sub.2]O
15.9994 108.6 0.1473 4.6877 -1.9151 1/3 [La.sub.2]
[O.sub.3]
15.9994 128.41 0.1246 4.85523 -2.0827 CdO
15.9994 143.09 0.1118 4.96348 -2.1909 [Cu.sub.2]O
15.9994 153.33 0.1043 5.03257 -2.26 BaO
15.9994 216.59 0.0739 5.37801 -2.6055 HgO
15.9994 231.74 0.069 5.44562 -2.6731 [Ag.sub.2]O
15.9994 260 0.0615 5.56068 -2.7881 --
15.9994 300 0.0533 5.70378 -2.9312 --
15.9994 350 0.0457 5.85793 -3.0854 --
15.9994 400 0.04 5.99146 -3.2189 --
15.9994 450 0.0356 6.10925 -3.3367 --
15.9994 500 0.032 6.21461 -3.4421 --
15.9994 600 0.0267 6.39693 -3.6244 --
Compound X Y = n K/X
O 15.9994 1
[H.sub.2]O 18.015 0.88811546
BeO 25.01 0.63972011
CO 28.01 0.57120314
NO 30.006 0.53320669
[H.sub.2][O.sub.2] 34.01 0.94089974
MgO 40.304 0.39698293
[N.sub.2]O 44.012 0.36353722
CaO 56.077 0.28532197
COS 60.075 0.26633375
[B.sub.2][O.sub.3] 69.618 0.68947686
[N.sub.2][O.sub.3] 76.01 0.63149586
CuO 79.545 0.20114401
[Cl.sub.2]O 86.905 0.18410908
Cr[O.sub.3] 99.993 0.4800336
[Al.sub.2][O.sub.3] 101.96 0.47077285
[N.sub.2][O.sub.5] 108.008 0.74068588
CdO 128.41 0.12460089
[Cr.sub.2][O.sub.3] 151.99 0.31581025
[Fe.sub.2][O.sub.3] 159.687 0.30058803
C[O.sub.2][O.sub.3] 165.86 0.2894007
[V.sub.2][O.sub.5] 181.88 0.43985045
W[O.sub.2] 215.84 0.14825797
[Fe.sub.3][O.sub.4] 231.53 0.27642206
U[O.sub.2] 270.027 0.11850667
[Ag.sub.2]C[O.sub.3] 275.75 0.174064
U[O.sub.2][Cl.sub.2] 340.94 0.0938546
[Gd.sub.2][O.sub.3] 362.5 0.132409
[Tl.sub.2][O.sub.3] 456.764 0.10508709
[Bi.sub.2][O.sub.3] 465.96 0.103009
[Re.sub.2][O.sub.7] 484.4 0.231205
[Tl.sub.2]S[O.sub.4] 504.8 0.1267781
[Ce.sub.2][(S[O.sub.4]).sub.3] 568.43 0.33776