Differential Electron: Quantum Numbers, How to Know It and Examples

He differential electron or differentiator is the last electron placed in the sequence of the electronic configuration of an atom. Why is his name? To answer this question, the basic structure of an atom is necessary: ​​its nucleus, vacuum and electrons.

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

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

The differential electron is in the most external electronic layer: the valence layer. Therefore, the farther away you are from the nucleus, the greater the energy associated with it. It is this energy that is responsible for its participation, as well as that of the rest of the valence electrons, in the chemical reactions characteristic 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 the 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"s"orbital; -1, 0, +1, for the"p"orbital; -2, -1, 0, +1, +2, for the orbital"d"; and -3, -2, -1, 0, +1, +2, +3, for the"f"orbital. Finally, the quantum number of spin"s"(+1/2 for ↑, and -1/2 for ↓).

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

How to know the differential electron?

In the upper image, the electronic configurations for the elements from hydrogen to neon gas (H → Ne) are represented.

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

In this way, the valence configuration of H (↑ of red color) 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 observed how the differential electron is added to the valence layer (red arrows) of the elements, differentiating them from each other. The electrons fill the orbitals respecting the rule of Hund and the principle of exclusion of Pauling (perfectly observed from the B to the Ne).

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

Examples in several elements

Chlorine

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

The differential electron is the last electron that is placed in the electronic configuration, and the chlorine atom is that of the 3p orbital, whose disposition is the following:

↑ ↓ ↑ ↓ ↑

3px 3py 3pz

(-1) (0) (+1)

Respecting Hund's rule, first fill the 3p orbitals of equal energy (one arrow up in each orbital). Second, the other electrons pair with the solitary 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"p"orbital of the medium; and"s"is -1/2, since the arrow points downwards.

Magnesium

The electronic 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:

↑ ↓

3s

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"s"orbital (the 3s).

Zirconium

The electronic configuration for the zirconium atom (transition metal) 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 valence orbitals and electrons is as follows:

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

Unknown element

The quantum numbers of the differential electron for an unknown element are 3, 2, +2, -1/2. What is the atomic number Z of the element? Knowing Z, you can decipher 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 layer ("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 keys to correctly locate the differential electron in the last 3d orbital.

Thus, the element that is sought has the 3d orbitals ¹⁰ full, just like its internal electronic layers. In conclusion, the element is zinc metal (Zn).

However, the quantum numbers of the differential electron can not discern between zinc and copper, because the latter 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

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