Quantum Numbers (Principal, Azimuthal, Magnetic, Spin)

In this chapter, we will learn about quantum numbers. As you know that there are a large number of persons in the world. If you have to search a particular person in this world, at least four things are needed:

• The country to which the person belongs,
• The city in that country to which the person resides,
• The street in that city where the person is residing, and
• The house or flat number.

Similarly, an atom contains a large number of atomic orbitals that vary in shape, size, and spatial orientation around the nucleus. To locate and identify a particular electron within an atomic orbital, four identification numbers are required. These identification numbers are called quantum numbers.

A set of four quantum numbers that provide complete information about an electron in an atom is called the quantum numbers. These quantum numbers provide information about the energy, orbital occupancy, size, shape, orientation of the orbital, and the direction of electron spin.

Bohr’s atomic model introduced a single quantum number, ‘n’ to describe an orbit. The quantum mechanical model of the atom uses three quantum numbers n, l, and m to describe an atomic orbital. These three quantum numbers are derived from Schrödinger’s wave equation. The quantum numbers which are required to completely signify the character of an electron in an atom are as follows:

• Principal quantum number (n)
• Azimuthal quantum number (l)
• Magnetic quantum number (m)
• Spin quantum number (s)

First three quantum numbers specify the location of an electron in an atom and the fourth quantum number gives the direction of the spin or self-rotation of the electron. Now let us understand these quantum numbers one by one.

Principal Quantum Number (n)

• The principal quantum number was proposed by Niels Bohr. It is denoted by n.
• This quantum number gives the information of circular orbits or shells which are designated by capital letters K, L, M . . . . For example, the shell for which n = 1 is called K shell (or 1st shell), the shell for which n = 2 is called L shell (or 2nd shell), and so on. The K shell is the closest to the nucleus, and as the value of n increases, the shells are farther from the nucleus. Thus, the value of n tells about the main energy level or shell in which an electron is residing.
• The value of principal quantum number n is a positive integer and can have infinite values i.e. n = 1, 2, 3, . . . . infinite.
• The principal quantum number n determines the average distance of the electron from the nucleus (radius of the orbit), which means it defines the size of the shell or orbit. As the value of n increases, the distance of electron from the nucleus also increases.
• It determines the location of the electron in a shell.
• It also determines the energy of the electron in a given orbit or shell. With the increase of value of n, the magnitude of the energy associated with an orbit or shell also increases.
• The principal quantum number n gives the information about the maximum number of electrons in a given shell or orbit. The maximum number of electrons in a shell is given by the formula 2n². Thus,
  • The maximum number of electrons in K-shell (1st shell) with (n = 1) = 2 * 1² = 2.
  • The maximum number of electrons in L-shell (2nd shell) with (n = 2) = 2 * 2² = 8.
  • The maximum number of electrons in M-shell (3rd shell) with (n = 3) = 2 * 3² = 18.
  • The maximum number of electrons in N-shell (4th shell) with (n = 4) = 2 * 4² = 32.

Azimuthal or Angular Momentum Quantum Number (l)

• Azimuthal quantum number was proposed by Arnold Sommerfeld. It is denoted by the letter “l”.
• This quantum number gives the information about the number of subshells or sublevels present within a given shell or main energy level.
• The value of l depends on the value of n. Therefore, for a given value of n, l can have the values from 0 to (n – 1). The value of l can have 0, 1, 2, 3, . . . . (n – 2), (n – 1). Thus, the value of l = (n – 1) always where ‘n’ represents the principal shell. It always lies between 0 to (n – 1). If
  • n = 1, l = 0 (one value).
  • n = 2, l = 0, l = 1 (two values)
  • n = 3, l = 0, l = 1, l = 2 (three values)
  • n = 4, l = 0, l =1, l = 2, l = 3 (four values)
  • Thus, for each value of n, there are n possible values of l.
• Different values of l represents different sub-shells which are designated by small letters s, p, d, f, . . . . These letters stand for sharp, principal, diffuse, and fundamental, which are spectral terms originally used to describe the series of lines observed in the atomic emission spectrum. For instance, the subshell for which l = 0 is called the s subshell, the one for which l = 1 is called the p subshell, l = 2 corresponds to the d subshell, and l = 3 corresponds to the f subshell, and so on.

For l = 4 and higher values, the subshells are designated by letters in alphabetical order after f. For example:

• l = 4 → g subshell
• l = 5 → h subshell
• l = 6 → i subshell, and so on.

These higher subshells are rarely encountered in most atomic structures but are relevant in advanced theoretical studies.

• The values of l allows us to calculate the total number of sub-shells present in a main shell (K, L, M, N). For example:
  • When n = 1 for K-shell (1st shell), the value of l = 0. Since l has only one value, K-shell has only one sub-shell which is designated as nl or 1s.
  • When n = 2 for L-shell (2nd shell), the value of l = 0, 1. Since l has two values, L-shell consists of two sub-shells which are designated as 2s and 2p.
  • When n = 3 for M-shell (3rd shell), the value of l = 0, 1, 2. Since l has three values, M-shell consists of three sub-shells which are designated as 3s, 3p, and 3d.
  • When n = 4 for N-shell (4th shell), the value of l = 0, 1, 2, 3. Since l has four values, N-shell consists of four sub-shells which are designated as 4s, 4p, 4d, and 4f.

Table: Designation of Sub-shells for n = 1 to n = 4

• The azimuthal quantum number gives the information about the shapes of the sub-shell, i.e. whether the electron cloud is spherical or dumb-bell shape or other shape.
• It also gives the information about the energies of different sub-shells present within the same main shell in the increasing order: s < p < d < f (Increasing order). When more than one electron is present in the same shell, the energy of a particular electron is determined by the value of both the principal quantum number (n) and the azimuthal quantum number (l). In other words, the energy of any electron depends on the value of n and l because the total energy = (n + l). The electron enters in that subshell whose (n + l) value or the value of energy is less. Thus, for a given main shell, the s subshell has the lowest energy, followed by p, d, and f in increasing order.
• The total number of electrons in a given subshell (s, p, d, and f sub-shells) is determined by the formula = 2(2l + 1) where l is the azimuthal quantum number. For each subshell:
  • s subshell (l = 0): 2(2(0)+1) = 2 electrons
  • p subshell (l = 1): 2(2(1)+1) = 6 electrons
  • d subshell (l = 2): 2(2(2)+1) = 10 electrons
  • f subshell (l = 3): 2(2(3)+1) = 14 electrons
• The azimuthal quantum number tells about the orbital angular momentum which is equal to √l(l + 1)h/2π. This formula measures the orbital angular momentum of an electron during its orbital motion about the nucleus. Sometimes, azimuthal quantum number is also called subsidiary quantum number.
• This quantum number also helps to explain the fine lines of spectrum.

Magnetic Quantum Number (m)

• The magnetic quantum number was introduced by Arnold Sommerfeld to explain the splitting of spectral lines in the presence of a magnetic field (Zeeman Effect). It is represented by the letter ‘m’.
• The magnetic quantum number provides information about the number of orientations or the number of orbitals in space.
• The value of m can have all integral values from – l to + l through 0.
• For s subshell (l = 0), m can have only one value, m = 0. This means that s-subshell has only one orbital. In other words, only one orientation is possible.
• For p-subshell (l = 1), m can have three possible values, +1, 0, and -1. This means that p-subshell has three orbitals. In other words, three orientations p_x, p_y, and p_z are possible.
• For d-subshell (l = 2), m can have five possible values, +2, +1, 0, -1, and -2. This means that d-subshell has five orbitals. In other words, five orientations are possible. These are d_xy, d_yz, d_zx, d_x²-y², and d_z².
• For f-subshell (l = 3), m can have seven possible values, +3, +2, +1, 0, -1, -2, and -3. This implies that f-subshell has seven orbitals.
• For a given value of ‘n’, the total value of m = n².
• For a given value of ‘l’, the total value of m = 2l + 1.
• Orbitals having the same energy are called degenerate orbitals. P_x, P_y, and P_z orbitals of p subshell are examples of degenerate orbitals.
• The number of degenerate orbitals of s subshell is zero.

Relationship among Values of n, l, and m

The relationship between principal, azimuthal, and magnetic quantum numbers is summed up in the below table:

Spin Quantum Number (s)

• The spin quantum number was proposed by George Uhlenbeck and Samuel Goudsmit. It is denoted by the symbol ‘s’.
• This quantum number gives the information about the spin of the electrons around its own axis. The electron in an atom revolves around the nucleus and also spins about its own axis. Spin quantum number describes this rotation of an electron on its own axis.
• The value of s can have only two values for an electron, i.e. +1/2 and -1/2, which specifies the spin or rotation or direction of an electron on its axis during the movement.
• For clockwise / up spin (↑) of the electron: +1/2
• For anticlockwise / down spin (↓) of the electron: -1/2
• Each orbital which contains two electrons always has opposite sign.
• The spin quantum number gives the value of spin angular momentum, which is equal to √[s(s + 1)](h/2π), where s = +1/2 or -1/2 for an electron.
• Maximum spin of an atom = 1/2 * number of unpaired electron

Relation between Various Quantum Numbers

Quantum numbers define the unique state of an electron in an atom. The relationships between these quantum numbers help to determine electron configuration and orbital characteristics. Here is the relationship between various quantum numbers in table form:

Solved Examples on Quantum Numbers

Example 1: Write down all the other quantum numbers of an electron for the principal quantum number n = 3.

Solution: Principal quantum number n =3 (given)

The possible values of azimuthal quantum number l = 0 to (n – 1) = 0, 1, 2

The possible values of magnetic quantum number m = -l to +l (including 0)

• For l = 0, m = 0
• For l = 1, m = -1, 0, +1
• For l = 2, m = -2, -1, 0, +1, +2

Example 2: An electron is in 3d-orbital. What are the possible values of quantum numbers n, l, m, and s?

Solution: For 3d-orbital, Principal quantum number n = 3

For d-orbital or d-subshell, Azimuthal quantum number l = 2

Magnetic quantum number m = -2, -1, 0, +1, +2

Spin quantum number s = +1/2 and -1/2 for each value of m.

Example 3: Can we have 5g-subshell? How many orbitals are possible for this subshell?

Solution: For 5g-subshell, n = 5

l = 0, 1, 2, 3, and 4

Thus, the corresponding subshells are: 5s, 5p, 5d, 5f, and 5g. Hence, we can have 5g subshell theoretically.

For 5g-subshell, l = n -1 = 5 – 1 = 4

The values of m = -4, -3, -2, -1, 0, +1, +2, +3, +4. Thus, there are 9 orbitals in 5g-subshell.

You can also calculate the number of orbitals by using formula:

Number of orbitals = 2l + 1

Each orbital can hold 2 electrons, so the 5g subshell can hold up to 18 electrons.

Example 4: Which of the following sets of quantum numbers are not allowed?

(1) n = 2, l = 2, m = -1, s = +1/2

(2) n = 2, l = 1, m = -1, s = -1/2

(3) n = 2, l = 0, m = 0, s = 0

(4) n = 2, l = 1, m = 2, s = +1/2

Solution:

(1) Since the value of l cannot be equal to l, therefore, the first set of quantum numbers is not permitted.

(2) The second set of quantum numbers is permitted.

(3) The third set of quantum numbers is not permitted because the value of spin quantum number cannot be zero.

(4) The fourth set of quantum numbers is also not allowed because the value of m cannot be greater than 1.