QUANTUM NUMBERS: WHAT AND WHAT ARE, SOLVED EXERCISES - CHEMISTRY

The quantum numbers are those that describe the allowed energy states for particles. In chemistry they are used especially for the electron within atoms, assuming that their behavior is that of a standing wave rather than a spherical body orbiting the nucleus.

Considering the electron as a standing wave, it can only have concrete and non-arbitrary vibrations; which in other words means that their energy levels are quantized. Therefore, the electron can only occupy the places characterized by an equation called the three-dimensional wave function ѱ.

Source: Pixabay

The solutions obtained from the Schrödinger wave equation correspond to specific places in space where electrons travel within the nucleus: the orbitals. Hence, considering also the wave component of the electron, it is understood that only in orbitals is there the probability of finding it.

But where do quantum numbers for the electron come into play? Quantum numbers define the energetic characteristics of each orbital and, therefore, the state of the electrons. Its values adhere to quantum mechanics, complex mathematical calculations and approximations made from the hydrogen atom.

Consequently, quantum numbers take on a range of predetermined values. The set of them helps to identify the orbitals through which a specific electron transits, which in turn represents the energy levels of the atom; and also the electronic configuration that distinguishes all the elements.

An artistic illustration of atoms is shown in the image above. Although a bit over exaggerated, the center of the atoms have a higher electron density than their edges. This means that as the distance from the nucleus increases, the lower the probability of finding an electron.

Likewise, there are regions within that cloud where the probability of finding the electron is zero, that is, there are nodes in the orbitals. Quantum numbers represent a simple way to understand orbitals and where electronic configurations arose from.

What and what are quantum numbers in chemistry?

Quantum numbers define the position of any particle. In the case of the electron, they describe its energetic state, and therefore, in which orbital it is located. Not all orbitals are available for all atoms, and they are subject to the principal quantum number n.

Principal quantum number

It defines the main energy level of the orbital, so all lower orbitals must adjust to it, as well as their electrons. This number is directly proportional to the size of the atom, because the greater the distances from the nucleus (larger atomic radii), the greater the energy required by the electrons to move through these spaces.

What values can n take? Whole numbers (1, 2, 3, 4,…), which are their allowed values. However, by itself it does not provide enough information to define an orbital, only its size. To describe orbitals in detail, you need at least two additional quantum numbers.

Azimuth, angular, or secondary quantum number

It is denoted by the letter l, and thanks to it, the orbital acquires a definite shape. Starting from the principal quantum number n, what values does this second number take? Since it is the second, it is defined by (n-1) up to zero. For example, if n is equal to 7, then l is (7-1 = 6). And its range of values is: 6, 5, 4, 3, 2, 1, 0.

Even more important than the values of l are the letters (s, p, d, f, g, h, i…) associated with them. These letters indicate the shapes of the orbitals: s, spherical; p, weights or ties; d, clover leaves; and so on with the other orbitals, whose designs are too complicated to be associated with any figure.

What is the usefulness of it so far? These orbitals with their proper forms and in accordance with the approximations of the wave function, correspond to the subshells of the main energy level.

Hence, a 7s orbital indicates that it is a spherical subshell at level 7, while a 7p orbital indicates another with the shape of a weight but at the same energy level. However, neither of the two quantum numbers yet accurately describe the "probabilistic whereabouts" of the electron.

Magnetic quantum number

The spheres are uniform in space, no matter how much they are rotated, but the same is not the case with "weights" or "clover leaves." This is where the magnetic quantum number ml comes into play, which describes the spatial orientation of the orbital on a three-dimensional Cartesian axis.

As just explained, ml depends on the secondary quantum number. Therefore, to determine its allowed values, the interval (- l, 0, + l) must be written and completed one by one, from one extreme to the other.

For example, for 7p, p corresponds to = 1, so its ml are (-1, o, +1). It is for this reason that there are three p orbitals (p _x, p, _and p _z).

A direct way to calculate the total number of ml is by applying the formula 2 l + 1. Thus, if l = 2, 2 (2) + 1 = 5, and since l is equal to 2 it corresponds to the d orbital, there is therefore both five d orbitals.

Additionally, there is another formula to calculate the total number of ml for a principal quantum level n (that is, ignoring l): n ². If n is equal to 7, then the number of total orbitals (no matter what their shapes are) is 49.

Spin quantum number

Thanks to the contributions of Paul AM Dirac, the last of the four quantum numbers was obtained, which now refers specifically to an electron and not to its orbital. According to the Pauli exclusion principle, two electrons cannot have the same quantum numbers, and the difference between them lies in the moment of spin, ms.

What values can ms take? The two electrons share the same orbital, one must travel in one direction of space (+1/2) and the other in the opposite direction (-1/2). So ms has values of (± 1/2).

The predictions made for the number of atomic orbitals and defining the spatial position of the electron as a standing wave, have been confirmed experimentally with spectroscopic evidence.

Solved exercises

Exercise 1

What is the shape of the 1s orbital of a hydrogen atom, and what are the quantum numbers that describe its lone electron?

First, s denotes the secondary quantum number l, whose shape is spherical. Since s corresponds to a value of l equal to zero (s-0, p-1, d-2, etc.), the number of states ml is: 2 l + 1, 2 (0) + 1 = 1 That is, there is 1 orbital that corresponds to the subshell l, and whose value is 0 (- l, 0, + l, but l is worth 0 because it is subshell s).

Therefore, it has a single 1s orbital with unique orientation in space. Why? Because it is a sphere.

What is the spin of that electron? According to Hund's rule, it must be oriented as +1/2, as it is the first to occupy the orbital. Thus, the four quantum numbers for the 1s ¹ electron (hydrogen electron configuration) are: (1, 0, 0, +1/2).

Exercise 2

What are the subshells that would be expected for level 5, as well as the number of orbitals?

Solving for the slow way, when n = 5, l = (n -1) = 4. Therefore, there are 4 sublayers (0, 1, 2, 3, 4). Each subshell corresponds to a different value of l and has its own values of ml. If the number of orbitals were determined first, then it would be enough to double it to obtain that of the electrons.

The available sublayers are s, p, d, f, and g; hence, 5s, 5p, 5d, 5d, and 5g. And their respective orbitals is given by the interval (- l, 0, + l):

(0)

(-1, 0, +1)

(-2, -1, 0, +1, +2)

(-3, -2, -1, 0, +1, +2, +3)

(-4, -3, -2, -1, 0, +1, +2, +3, +4)

The first three quantum numbers are enough to finish defining the orbitals; and for that reason the ml states are named as such.

To calculate the number of orbitals for level 5 (not the atom totals), it would be enough to apply the formula 2 l + 1 for each row of the pyramid:

2 (0) + 1 = 1

2 (1) + 1 = 3

2 (2) + 1 = 5

2 (3) + 1 = 7

2 (4) + 1 = 9

Note that the results can also be obtained simply by counting the integers in the pyramid. The number of orbitals is then the sum of them (1 + 3 + 5 + 7 + 9 = 25 orbitals).

Fast way

The above calculation can be done in a much more direct way. The total number of electrons in a shell refers to its electronic capacity, and can be calculated with the formula 2n ².

Thus, for exercise 2 we have: 2 (5) ² = 50. Therefore, shell 5 has 50 electrons, and since there can be only two electrons per orbital, there are (50/2) 25 orbitals.

Exercise 3

Is the existence of a 2d or 3f orbital likely? Explain.

The subshells d and f have 2 and 3 for the main quantum number. To know if they are available, it must be verified if these values fall within the interval (0,…, n-1) for the secondary quantum number. Since n is 2 for 2d, and 3 for 3f, its intervals for l are: (0,1) and (0, 1, 2).

From them it can be observed that 2 does not enter (0, 1) or 3 does not enter (0, 1, 2). Therefore, the 2d and 3f orbitals are not energetically allowed and no electrons can transit through the region of space defined by them.

This means that the elements in the second period of the periodic table cannot form more than four bonds, while those belonging to period 3 onwards can do so in what is known as expansion of the valence shell.

Exercise 4

Which orbital corresponds to the following two quantum numbers: n = 3 and l = 1?

Since n = 3, we are in layer 3, and l = 1 denotes the p orbital. Therefore, the orbital simply corresponds to 3p. But there are three p orbitals, so it would take the magnetic quantum number ml to discern a specific orbital among them.

Exercise 5

What is the relationship between quantum numbers, electron configuration, and the periodic table? Explain.

Because quantum numbers describe the energy levels of electrons, they also reveal the electronic nature of atoms. The atoms, then, are arranged in the periodic table according to their number of protons (Z) and electrons.

The groups of the periodic table share the characteristics of having the same number of valence electrons, while the periods reflect the energy level in which these electrons are found. And what quantum number defines the energy level? The main one, n. As a result, n is equal to the period that an atom of the chemical element occupies.

Likewise, from the quantum numbers the orbitals are obtained which, after being ordered with the Aufbau construction rule, gives rise to the electronic configuration. Therefore, quantum numbers are in the electron configuration and vice versa.

For example, the electron configuration 1s ² indicates that there are two electrons in an s subshell, of a single orbital, and in shell 1. This configuration corresponds to that of the helium atom, and its two electrons can be differentiated using the quantum number of the spin; one will have the value of +1/2 and the other of -1/2.

Exercise 6

What are the quantum numbers for the 2p ⁴ subshell of the oxygen atom?

There are four electrons (the 4 over the p). All of them are at level n equal to 2, occupying the subshell l equal to 1 (the orbitals with weight shapes). Until then, the electrons share the first two quantum numbers, but differ in the remaining two.

Since l is equal to 1, ml takes the values (-1, 0, +1). Therefore, there are three orbitals. Taking into account Hund's rule of filling the orbitals, there will be a paired pair of electrons and two of them unpaired (↑ ↓ ↑ ↑).

The first electron (from left to right of the arrows) will have the following quantum numbers:

(2, 1, -1, +1/2)

The other two remaining

(2, 1, -1, -1/2)

(2, 1, 0, +1/2)

And for the electron in the last 2p orbital, the arrow to the far right

(2, 1, +1, +1/2)

Note that the four electrons share the first two quantum numbers. Only the first and second electrons share the quantum number ml (-1), since they are paired in the same orbital.

References

Whitten, Davis, Peck & Stanley. Chemistry. (8th ed.). CENGAGE Learning, p 194-198.
Quantum Numbers and Electron Configurations. (sf) Taken from: chemed.chem.purdue.edu
Chemistry LibreTexts. (March 25, 2017). Quantum Numbers. Recovered from: chem.libretexts.org
Helmenstine MA Ph.D. (April 26, 2018). Quantum Number: Definition. Recovered from: thoughtco.com
Orbitals and Quantum Numbers Practice Questions.. Taken from: utdallas.edu
ChemTeam. (sf). Quantum Number Problems. Recovered from: chemteam.info

QUANTUM NUMBERS: WHAT AND WHAT ARE, SOLVED EXERCISES - CHEMISTRY - 2025