DIFFERENTIAL ELECTRON: QUANTUM NUMBERS AND EXAMPLES - CHEMISTRY

The differential or differentiating electron is the last electron placed in the sequence of the electron configuration of an atom. What is its name? To answer this question, the basic structure of an atom is necessary: its nucleus, the vacuum and the electrons.

The nucleus is a dense and compact aggregate of positive particles called protons, and of neutral particles called neutrons. Protons define the atomic number Z and, together with neutrons, make up the atomic mass. However, an atom cannot carry only positive charges; therefore the electrons orbit around the nucleus to neutralize it.

Thus, for every proton that joins the nucleus, a new electron joins its orbitals to counteract the increasing positive charge. In this way, the newly added electron, the differential electron, is closely related to the atomic number Z.

The differential electron is in the outermost electronic shell: the valence shell. Therefore, the further away you are from the nucleus, the greater the energy associated with it. It is this energy that is responsible for their participation, as well as that of the rest of the valence electrons, in the characteristic chemical reactions of the elements.

Quantum numbers

Like the rest of the electrons, the differential electron can be identified by its four quantum numbers. But what are quantum numbers? They are "n", "l", "m" and "s".

The quantum number "n" denotes the size of the atom and the energy levels (K, L, M, N, O, P, Q). «L» is the secondary or azimuthal quantum number, which indicates the shape of the atomic orbitals, and takes values of 0, 1, 2 and 3 for the orbitals «s», «p», «d» and «f», respectively.

"M" is the magnetic quantum number and indicates the spatial orientation of the orbitals under a magnetic field. Thus, 0 for the orbital «s»; -1, 0, +1, for the "p" orbital; -2, -1, 0, +1, +2, for the "d" orbital; and -3, -2, -1, 0, +1, +2, +3, for the "f" orbital. Finally, the spin quantum number «s» (+1/2 for ↑, and -1/2 for ↓).

Therefore, a differential electron has associated the previous quantum numbers ("n", "l", "m", "s"). Because it counteracts the new positive charge generated by the extra proton, it also provides the element's atomic number Z.

How to know the differential electron?

The image above represents the electron configurations for elements from hydrogen to neon gas (H → Ne).

In this, the electrons of the open shells are indicated by the color red, while those of the closed shells are indicated by the color blue. The layers refer to the quantum number "n", the first of the four.

Thus, the valence configuration of H (↑ in red) adds another electron with opposite orientation to become that of He (↓ ↑, both blue because now level 1 is closed). This added electron is then the differential electron.

Thus, graphically it can be seen how the differential electron adds to the valence shell (red arrows) of the elements, differentiating them from each other. The electrons fill the orbitals respecting Hund's rule and Pauling's exclusion principle (perfectly observed from B to Ne).

And what about quantum numbers? These define each arrow - that is, each electron - and their values can be corroborated with the electron configuration to know whether or not they are those of the differential electron.

Examples in multiple elements

Chlorine

In the case of chlorine (Cl), its atomic number Z is equal to 17. The electron configuration is then 1s ² 2s ² sp ⁶ 3s ² 3p ⁵. The orbitals marked in red correspond to those of the valence shell, which has an open level 3.

The differential electron is the last electron to be placed in the electron configuration, and the chlorine atom is that of the 3p orbital, whose arrangement is as follows:

↑ ↓

3px 3py 3pz

(-1) (0) (+1)

Respecting Hund's rule, the 3p orbitals of equal energy are filled first (an up arrow in each orbital). Second, the other electrons pair with the lone electrons from left to right. The differential electron is represented in a green frame.

Thus, the differential electron for chlorine has the following quantum numbers: (3, 1, 0, -1/2). That is, "n" is 3; "L" is 1, orbital "p"; "M" is 0, because it is the middle "p" orbital; and "s" is -1/2, since the arrow points down.

Magnesium

The electron configuration for the magnesium atom is 1s ² 2s ² sp ⁶ 3s ², representing the orbital and its valence electron in the same way:

↑ ↓

This time, the differential electron has the quantum numbers 3, 0, 0, -1/2. The only difference in this case with respect to chlorine is that the quantum number «l» is 0 because the electron occupies an orbital «s» (the 3s).

Zirconium

The electron configuration for the zirconium (transition metal) atom is 1s ² 2s ² sp ⁶ 3s ² 3p ⁶ 4s ² 3d ¹⁰ 4p ⁶ 5s ² 4d ². In the same way as the previous cases, the representation of the orbitals and valence electrons is as follows:

Thus, the quantum numbers for the differential electron marked in green are: 4, 2, -1, +1/2. Here, since the electron occupies the second "d" orbital, it has a quantum number "m" equal to -1. Also, because the arrow points up, its spin number "s" equals +1/2.

Unknown element

The differential electron quantum numbers for an unknown element are 3, 2, +2, -1/2. What is the atomic number Z of the element? Knowing Z you can figure out what the element is.

This time, since "n" is equal to 3, it means that the element is in the third period of the periodic table, with "d" orbitals as the valence shell ("l" equal to 2). Therefore, the orbitals are represented as in the previous example:

↑ ↓

The quantum numbers "m" equal to +2, and "s" equal to -1/2, are key to correctly locating the differential electron in the last 3d orbital.

Thus, the element being sought has full 3d ¹⁰ orbitals, as well as its internal electronic shells. In conclusion, the element is the metal zinc (Zn).

However, the quantum numbers of the differential electron cannot distinguish between zinc and copper, because the latter element also has full 3d orbitals. Why? Because copper is a metal that does not comply with the rules for filling electrons for quantum reasons.

References

Jim Branson. (2013). Hund's Rules. Retrieved on April 21, 2018, from: quantummechanics.ucsd.edu
Lecture 27: Hund's rules. Retrieved on April 21, 2018, from: ph.qmul.ac.uk
Purdue University. Quantum Numbers and Electron Configurations. Retrieved on April 21, 2018, from: chemed.chem.purdue.edu
Salvat Encyclopedia of Sciences. (1968). Física Salvat, SA de Ediciones Pamplona, volume 12, Spain, pages 314-322.
Walter J. Moore. (1963). Physical Chemistry. In particles and waves. Fourth edition, Longmans.

DIFFERENTIAL ELECTRON: QUANTUM NUMBERS AND EXAMPLES - CHEMISTRY - 2025