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3-3: The Periodic Law of Elements

In the preceding section, we have learned that the structure of hydrogen atom can completely be explained by quantum mechanics. Then, how about the other kinds of elements? To see this, the periodic law of elements seems one of the most suitable subjects. In the present section, let us see that the periodic law is beautifully derived from the shell structure of atoms.

[The Periodicity]
A Russian chemist, D. I. Mendeleev (Russia, 1834 - 1907) discovered the periodic law of elements that the properties of the elements are periodic functions of their atomic weights (1869), i.e., by arranging the elements in order of increasing atomic weights, elements having similar properties occur at fixed intervals. On the basis of this periodicity of elements, the periodic table arranged the elements in order of atomic numbers to emphasize the chemical relationships between the elements with similar configurations.
Since the periodic table must be very familiar to you in textbooks of chemistry in your highschool time, its details will be omitted here except for a few examples of the periodicity of the elements as shown below.

Some examples of the periodicity of the elements are shown in Fig. (A) and Fig. (B). We can find similar features in other various experimental data.
What Fig. (A). shows is the ionization energy which is necessary to ionize an atom by removing an electron out of an atom.
The elements He, Ne, Ar, Kr, Xe and Rn, are especially difficult to be ionized and they are called the noble gases or rare gases.
On the other hand, the neighboring elements whose atomic numbers are larger by one than those of the noble gases are the alkali metals, Li, Na, K, Rb, Cs and Fr, which have high chemical activity and are quite easy to combine with other elements. Fig. (B) shows the atomic radii estimated from the interatomic distances in crystals. This data shows that the atomic radii of the alkali metals are remarkably large.
We can see from these data that the atomic numbers Z = 2, 10, 18, 36, 54 and 86 may have a special meaning which can be explained by quantum mechanics as discussed below.

[Shell Structure of Atoms]
As mentioned before, the eigenstates of a hydrogen atom are specified by a set of quantum numbers, n , l , and m and the energy eigenvalues, , are determined only by an integer n. (See Eq. (5) on the preceding page.)
For example, the ground state is of n = 1 and l = m = 0. The first excited states are of n = 2, but there are four possible states of l = m = 0, and l = 1 and m = -1, 0, 1.
These four states have the same energy eigenvalue. Namely, the first excited states are "fourfold". Generally, such "multifold" states are called degenerate states, and its multiplicity is called degeneracy. So that the first excited states of a hydrogen atom are quadruply degenerate.
In the hydrogen atom, all the eigenstates with the same quantum number n are degenerate, and their degeneracy is given by .
The reason why such degenerate excited states occur in the hydrogen atom is because the force exerted on the electron is purely the Coulomb force from the nucleus (proton). In the general kinds of atoms other than hydrogen, the situation is somewhat different. There the force exerted on electrons are not pure Coulomb atractive force between the electron and the nucleus (proton), but there are many interactions being worked from other electrons.
Even in general atoms other than hydrogen, although the eigenstates of electrons are still specified by the quantum numbers , the energy levels would be splitted. Although the energy eigenvalues in the hydrogen atom are determined only by n , those in general atoms other than hydrogen are determined by the quantum numbers n and l , (but do not depend on m ). The energy levels of the general atoms are listed in Table 1 below in ascending order from the lowest one. And their schematic drawing is shown in Fig. (C).

Fig. (C): The energy levels of electrons in hydrogen atom and other atoms in general
The energy levels in the hydrogen atom and the general atom are compared in this figure. Here, the energy levels on the right-hand side are of the hydrogen atom and those on the left-hand side the general atoms other than hydrogen.
The detailed explanation is given below.

The energy levels in the hydrogen atom and the general atom are compared in Fig. (C), in which the energy levels on the right-hand side are of the hydrogen atom and those on the left-hand side the general atoms other than hydrogen. Every level is specified by the quantum numbers n and l . The states with l = 0, 1, 2, 3, ... are denoted by symbols, s, p, d, f, ... , respectively.
In the hydrogen atom, the energy eigenvalues of the states with the same n but different l are degenerate (of a equal energy). However, in general atoms other than hydrogen, this degeneracy is broken and these energy levels are splitted.
The energy splitting are seen in Fig. (C), in which the relative position or order of the levels is roughly drawn and you should ignore the absolute values of the energy eigenvalues.
The circled numbers in Fig. (C), show the total electron numbers below which all levels are occupied or filled up by electrons. These figures exactly reproduce the atomic numbers of the noble gases.
The same number of electrons as the atomic number Z are filled in the levels in ascending (or increasing) order of energies of the levels and thus an atom might be made up. Roughly speaking, the "spatial" positions of these levels would be queued up (or layered) from inner to outer layers. This might look something like the layer structure in an onion. Each layer is called a shell and this type of structure the shell structure.

[The Pauli Principle, Spin]
How many electrons can be filled in each shell ?
In quantum mechanics, different electrons are not distinguished. Not only electrons but all kinds of identical particles are indistinguishable in quantum mechanics when being interchanged. Considering this indistinguishability of identical particles, all kinds of particles are classified into two groups; one is the group of fermions and the other the group of bosons, and all particles in quantum mechanics are either fermions or bosons. Which group, the fermion group or the boson group, a particle belongs to is uniquely dependent on the kind of the particle. Such a property of particles is called the statistics. For example, electrons and protons are fermions, and photons are bosons.
Now, let us go back to electrons that are fermions. In the case of fermions, no two identical fermions can be in the same quantum state. In other words, only 0 or 1 fermion can enter into one quantum state. This rule is called the Pauli principle, which was first proposed in 1925 by W. Pauli (Austria, Switzerland, 1900 - 58). It is also called the Pauli exclusion principle. Readers who want to learn more details of such statistics are recommended to read some other textbooks on quantum mechanics.
The lowest-energy level is 1s as seen in Fig. (C). This 1s level is not degenerate. There is only one state with the same energy eigenvalue. According to the Pauli principle, only one electron can therefore stay in this 1s level. However, two electrons can occupy this level in practice. The reason is because there is another degree of freedom called spin which was proposed by G. Uhlenbeck (The Netherlands) and S. A. Goudsmit (The Netherlands) in 1925. More details concerning the spin would be referred to other textbooks of quantum mechanics.
Thus, if both the Pauli principle and the spin are taken into account, then one state characterized by the quantum numbers n , l and m could contain a maximum of two electrons. The maximum numbers of electrons which can be contained in a level characterized by the quantum numbers n and l are listed in the fourth column, "Max of electron number", in the above Table 1.

[Closed Shells, the Noble Gases, the Periodic Law]
Filling Z electrons into the electron levels in Table 1 or in Fig. (C) in ascending order from the lowest-energy level, we can make up the electron configuration of an atom with the atomic number Z.
For example, filling one electron in the lowest-energy 1s shell (level), we have a hydrogen atom, H. If we fill two electrons in the same 1s shell, then this shell would be filled up to be a closed shell according to the Pauli principle. This is a helium atom, He. Because the energy of the next level is rather high, the helium atom, He, with Z = 2 is quite stable. This is the most typical noble gas.
If we put an electron in the next 2s shell (level), we would have the electron configuration of the lithium atom, Li. Lithium is one of the alkali metals which is of very high chemical activity. If the 2p shell is filled up to be the closed shell after putting electrons successively in the levels, the total electron number would be 10 and we have the next noble gas of Z = 10, which is neon, Ne. The next is a sodium atom, Na, whose configuration is that one electron is contained in the next 3s shell. Sodium is also one of the alkali metals and has quite high chemical activity.
Thus, as the atomic number increases, we would successively have the closed shells which correspond to the noble gases, and one electron being added, we would have very active elements, i.e. the alkali metals, and another electron being increased in addition, we have the alkaline earth metals that are of rather high activity.
The elements just before the closed shells are also chemically very active and called the halogens which have one less electrons than the noble gases. In this way, as the electron number increases, the property of the element varies periodically. This is the periodic law of elements.
Thus we conclude that the periodic law is completely explained by quantum mechanics.
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