Size of the Periodic Table Answering a Philosophical Question about Possibilities and Limitations John R. Huizenga University of Rochester, Rochester, NY 14627-0216 In the late 1860's, Dmitri Mendeleev and Lothar Meyer, working independently, first arranged the elements in the order of their atomic weights and showed that there are regular recurrences of a great many chemical and physical properties. The modem periodic table is organized by increasing atomic number Z rather than atomic weight. I t places elements into groups (arranged vertically) and periods (arranged horizontally). As an example of periodicity, the elements in group VIII (helium, neon, argon, krypton, xenon, and radon) are called noble gases because of their relative inertness toward chemical compound formation. Several scientists have made contributions to our present periodic table, some of which are major. For example, Henry Moseley put the atomic number on a firm basis. Glenn T. Seaborg participated in the discovery of many of the transactinium elements and also arranged their placement in the periodic table. Explaining the Size of the Table Because the chemical elements cannot be decomposed into simpler substances by chemical processes, they represent fundamental building blocks in nature. The great importance of the elements in the periodic table to the fields of physics, chemistry, and biology prompts one to ask the following rather philosophical question. Why is it that our universe contains approximately one hundred elements rather than, for example, ten, an order of magnitude less, or one thousand, an order of magnitude more? The ouroose of this article is to answer this auestion and to show &at the size of our periodic table is not a result of chance but deoends on two of the fundamental forces in nature: the electromagnetic force the nuclear or strong force A Balance between Forces The rationale for the approximate calculations to be presented is that the size of the periodic table depends on a delicate balance between the long-range repulsive and short-range attractive forces in the nucleus. For an idealized spherical nucleus, the repulsive Coulomb (E:) energy is given by
A Uniformly Sized Liquid Drop Assuming a uniformly charged liquid drop model (1)of the nucleus, the attractive nuclear energy is equivalent to the surface energy, which is the product of the spherical area of the nucleus and the nuclear surface tension per unit area y. Equations 1 and 2 can be rewritten in terms of the atomic number Z and mass number A using the empirical relationship between the nuclear radius (R,) and the mass number (A) for spherical nuclei (21, namely R, = r d L a
(3)
where r. is a radius parameter (the value of this constant is givenlater, indepindenr of mass number A. Substituting eq 3 into eqs 1 and 2 gives
and where the Coulombic (CJ and nuclear coupling (C,) constants are
and Distortions of the Nuckar Drop The potential energy of the nuclear drop is altered by a change in its nuclear shape. The upper limit of the periodic table is reached by an element with certain properties: The nucleus of the most stable isotope is spontaneously unstable to small deformations and disintegrates in nuclear times. Let us consider small distortions of a sphere. For small, axially symmetric distortions the nuclear radius can be written in terms of the spherical radius R,.
where Z is the number of protons; e is the protonic charge (maenitude eaual to the charee of the electron but o~oosite signf; and R. is the radius of particular spherical &leus (with Z protons) idealized as an incompressible fluid.
where 0 is the angle of the radius vector from the symmetry axis; and a2is a parameter describing the amount of quadrupole distortion. This expression is only valid for small distortions because higher order terms are neglected. Bohr and Wheeler (1)showed in their seminal paper that the Coulomb and surface energies for small distortions of the nuclear drop are given by
Ths paper is part ofa senes to appear m the Journald Chemml Educatron on nuclear and rad~ocnemlstry It was prepared for the Commlnee on Nbc ear and Radlochem~stly01 the Nat~onalAcademy of Sciences--National Research Council.
and
a
730
Journal of Chemical Education
where E: and E: are the Coulomb and surface energies of a spherical nucleus. (See eqs 1and 2, respectively.) The nuclear drop becomes unstable to small deformations when its total energy ET at a small but finite deformation becomes equal to the total energy of the spherical nucleus.
where r, is the nuclear radius wnstant (see eq 3); y is the nuclear surface tension per unit area; and e is the protonic charge. The wnstants r. and y can be estimated from experimental data, and for the present purpose need be known only ualitatively. Here, I assume r, = 1.15 fm and 7 = 1MeVIfmq. In cgs units,
1 MeV fm2
y =--
The total energy of the deformed drop is the sum of the Coulomb and surface energies given by eqs 9 and 10. Y=
1.602 x
1.602 x 10" erg M~V
loz6fm2 em2
loz0ergs
mz
Rearranging eq 11, we get Substituting these values into eq 16, we get
As can be seen from eq 12, the instability limit, defined by the condition is reached when
It is the value of this ratio of the nuclear to Coulomb coupling constant that determines the size of the periodic table. Combining eqs 15 and 17, we get ZL= 110. Discussion of Results
Hence, when the Coulomb energy of the spherical nuclear drop is equal to twice its surface energy, the nucleus is no longer stable with respect to deformations of the simplest type. eqs 4 and 5 into eq 13, we get the limiting Substitu% value (1)of ( /A)
The Limiting Value of the Atomic Number
To estimate the limiting value of the atomic number ZL in the periodic table, assume a relationship betweenA and Z for the heaviest nuclei as the nuclear stability limit is approached. Based on the heaviest known nuclei, an approximate value for this A/Z ratio is 512. Inserting this value into eq 14, we get the following qualitative estimate of the limiting value of the atomic number ZL.
Although the above approximate value of ZL, obtained by the calculations described here, is subject to considerable uncertainty, its order of magnitude of 10' is to be taken seriously. It is known from many studies that reliable results are obtained by treating the nucleus as a charged liquid drop of incompressible fluid (3).However, in reality the nucleus is known to contain internal structure. Particular numbers of protons and neutrons lead to nuclear shells (2) that give the nucleus extra stability. For example, proton and neutron numbers of 50,82, and 126 are known shells. Several decades ago calculations that included the effects of such nuclear shell structure had already been used to predict that superheavy nuclei may exist. This postulated, new island of nuclei that is stabilized by shell structure has not been found to date. However, one must rewgnize that the above estimate of the size of the periodic table was made using the charged liquid drop model of the nucleus. Summary
where ZLdepends on the ratio of the nuclear to Coulomb coupling constant. This ratio can be calculated from eqs 6 and 7 and is given by
c, zo&
-=-
CC
3e2
(16)
Elementam calculations have been resented to show why the periLdic table has approximaGly one hundred elements. This size is a direct result of the relative maenitudes of the nuclear and electromagnetic forces.
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Literature Cited 1. Bob, N.;Wheeler, J. A. Phys Re". 1939,56,426. 2. See,foreram.le,fiedlander; Kennedy;Madas;Miuer.Nucl~rondRad~hrhrrstri, 3rded.; John W h y and Srms (1981);p 29. 3. See, for example, Vmdenborh:Hviaenga Nudmr h i o n : AeademicRess, 1973.
Volume 70 Number 9 September 1993
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