Electron Configuration, Atomic Structure and the Pascal Triangle core points for IITJEE and AIEEE

Electron Configuration and the Pascal Triangle for IITJEE

The common character of theories of atomic constitution has been the endeavour to find configurations and motions of the electrons which would seem to offer an interpretation of the variations of the chemical properties of the elements with the atomic number as they are so clearly exhibited in the well-known periodic law. A consideration of this law leads directly to the view that the electrons atom are arranged in distinctly separate groups, each containing a number of electrons equal to one of the periods in the sequence of the elements, arranged according to increasing atomic number. In the first attempts to obtain a definite picture of the configuration and motion of the electrons in these groups it was assumed that the electrons within each group at any moment were placed at equal angular intervals on a circular orbit with the nucleus at the centre, while in later theories this simple assumption has been replaced by the assumptions that the configurations of electrons with in the various groups do not possess such a simple axial symmetry, but exhibit a higher degree of symmetry in space, it being assumed, for instance, that the configuration of the electrons at any moment during their motions possesses polyhedral symmetry.
The elecrtron can be analyzed in several ways. Presently, the Bohr model of the atom will be used which uses the shell model for evaluating electrons. The shell model has electrons residing in various energy levels about the nucleus. The first shell (K Shell) can have up to 2 electrons. This shell is filled first with electrons, thus any atom with greater than 2 protons or atomic number will have this shell filled. The next shell is the (L Shell) and can have up to 8 electrons in it. So on with the (M Shell). When the number of electrons starts filling the (N Shell) they then start packing in the inner shells. Electron shells have limited capacity for electrons. The farther an electron shell is from nucleus, the larger it is.

An easy way to calculate the total number of electrons that can be held by a given energy level is to use the formula 2*n² , where n equals the number of the electron shell. For example, for the 1st electron shell n=1 and 2*1² = 2, telling us that the capacity of the 1st shell is 2 electrons as we have already seen. For the 2nd shell ( n=2 ) and 2*2² = 8. For an atom to fill its 2nd electron shell, 10 electrons would be needed : 2 to fill the 1st shell and 8 to fill the 2nd.

The maximum number of electrons is double square number.The square numbers can be found in the next triangle's form of the natural numbers :

We may show a connection between square numbers and the tetrahedral numbers :

As you might expect, the 3rd shell has a total capacity of 2*3² = 18 electrons. But things get a bit tricky here. Electron shells actually have sublevels. The first sublevel ( the s sublevel ) holds 2 electrons. The second, p , sublevel holds 6.The third, d , sublevel holds 10. When levels 3s and 3p are filled, electron shell#3 acts as if it has reached capacity with only 8 total electrons. In other words, in an atom with 20 electrons ( which is the element calcium) the first 2 electrons are located in the 1st shell, the next 8 in shell#2, the following 8 in shell#3 and remaining 2 electrons are located in shell#4.

The number of sublevels that an energy level can contain is equal to the principle quantum number of that level. The first sublevel is called s sublevel. s sublevels have one orbital, which can hold up to two electrons. The second sublevel is called a p sublevel. p sublevels have three orbitals, each of which can hold 2 electrons, for a total of 6. The third sublevel is called a d sublevel and d sublevels have 5 orbitals, for a possible total of 10 electrons. The fourth sublevel is called an f sublevel. f sublevels, with 7 orbitals, can hold up to 14 electrons. Although energy levels that are higher than 4 would contain additional sublevels, these sublevels have not been named because no known atom in its ground state would have electrons that occupy them. The information about the sublevels is summarized in the table below :

The maximum numbers of electrons in sublevels may be represented as the following triangle's form of numbers :

The law of the electron configurations is the law of the geometrical area. There is a connection between odd numbers and the square numbers :

There is also a connection between odd numbers and the triangular numbers :

The odd numbers , square numbers and pyramidal numbers can be found as columns in Pascal triangle of the second kind :

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An electron configuration is a method of indicating the arrangement of electrons about a nucleus. A typical electron configuration consists of numbers, letters and superscripts with the following format:
1. A number indicates the energy level.( The number is called the principal quantum number.)
2.A letter indicates the type of orbital: s,p,d,f...
3.A superscript indicates the number of electrons in the orbital.
To write an electron configuration:
1.Determine the total number of electrons to be represented.
2.Use the Aufbau process to fill the orbitals with electrons. The Aufbau process requires that electrons fill the lowest energy orbitals first. In another words, atoms are built from the ground upwards.
3.The sum of the superscripts should equal the total number of electrons.