27. Quantum Numbers.

In the last post, we saw how quantum numbers were introduced to understand the structure of the atom. These quantum numbers described all the necessary parameters about an electron in a given orbit.Let us begin by understanding what these quantum numbers exactly tell us.

A] The Principal Quantum number (n)–

This parameter divulges information about the size of the orbit, its distance from the nucleus, and thus orbital energy.

e.g.- If n=1, then it indicates that we are looking at the orbit which is closest to the nucleus i.e the ‘K’ shell.

n=2 indicates the second orbit from the nucleus i.e ‘L’ shell. This orbit is farther from the nucleus when compared with the ‘K’Shell.

The maximum number of electrons allowed in any principal shell is given by 2n².

For the first K shell, n= 1, the allowed no of electrons is 2(1)² = 2.
For the second L shell, n= 2, the allowed no of electrons is 2(2)² = 8.
For the third M shell,  n= 3, the allowed no of electrons is 2(3)² = 18.
For the second N shell, n= 4, the allowed no of electrons is 2(4)²= 32.

B] Azimuthal or Orbital quantum number(ℓ)–

According to the Bohr-Sommerfield model, not only do electrons travel in certain orbits(quantization) but the orbits have different shapes. This quantum number denotes the shape or angular distribution of the orbital. It can take values from 0 to n-1, in steps of unity, where n is the principal quantum number.

When ℓ = 0, we have an s-orbital, which is spherical. This gives sharp lines in the spectra.
ℓ = 1 , indicates a p – orbital ,which is elliptical/elongated.
ℓ =2, indicates a d- orbital, which has a complex shape.
ℓ = 3, indicates an f – orbital, which again has a complex shape.

The spectroscopic notations for these are
s⇒ sharp lines
p⇒principle lines
d⇒diffuse lines
f⇒fine/fundamental lines.

These are just characterizations of the kind of lines that were observed by the spectroscopists for each orbital and thus the names s,p,d,f.

C] Magnetic Quantum number (m) –

Another concept improving the Bohr model was the discovery that the orbitals don’t have to lie in the same plane. They can be oriented in different directions in space. This orientation of the orbitals in space is described by the magnetic quantum number. So what if the p orbital is elongated and has a dumbbell shape! How does it look in 3- dimensions? How is that shape-oriented in space? The magnetic quantum number provides an answer to this question.m = –ℓ..0..+ℓ.

As the s-orbital is spherical, ℓ = 0,m=0. This means that the s-orbital is evenly distributed in space i.e it’s spherical.

But for p-orbital, ℓ=1. So, m= -1,0,+1. Thus, there are three types of orientations in space for the p-orbital(as shown in the figure below). It could either be on the x-axis or y-axis or z-axis. Thus, we have p_x, p_y, p_z orbitals.

m=-1 ↔ p_y.
m=0   ↔ pz.
m=+1  ↔px.

When  ℓ=1, m=-2,-1,0,1,2, which represent the five d-orbitals d_xy,d_yz,d_xz,d_x²_-y²,d_z².

The structure of the seven f-orbitals,namely , 4f_{y³ – 3x²y,} 4f_{xyz ,} 4f_{5yz² – yr²}  ,4f_5yz² _{– yr}²,4f_5xz² _{– xr}²,4f_zx² _{– zy}²,4f_x³ _{– 3xy}²,is more complex and can be represented diagrammatically  as follows –

The electrons occupy the f-orbitals only after the element Cerium(Atomic weight = 58).This f- orbitals are under the valence shell and rarely play an important role during reactions. Thus, they are not studied in detail at a preliminary level.

D]Spin Quantum number (s)–

This quantum number is required to explain the magnetic properties of substances. Just like the earth, the electron not only revolves around the nucleus but also spins around itself. It can spin in the clockwise or anti-clockwise direction. A moving charge always generates a magnetic field(the direction of the field can be found out by the right-hand rule where 4 fingers of the hand represent the direction of motion and the stretched thumb represents the direction of the magnetic field thus produced). Two orientations are possible for electrons depending on whether they are moving clockwise or counterclockwise as follows –

In the above figure, one electron is spinning clockwise and the other is moving anti-clockwise, thus the two electrons have opposite magnetic orientations (see North pole = N and South pole = S of the induced magnetic field). These two spins would cancel each other out. However, in the case of an unpaired electron, the spin is what generates a magnetic field, which gives rise to a number of magnetic properties.

Thus, for each value of magnetic quantum number(m), spin quantum number(s) has two values ,s= +1/2 and s=-1/2.

So, now we have 4 sets of quantum numbers that help us visualize the structure of the atom in a new light.

	Principal Quantum Number	Azimuthal/Orbital Quantum number	Magnetic Quantum Number	Spin Quantum Number
Symbols	n	ℓ	m	s
Spectroscopic notations	K,L,M,N..	s,p,d,f	–	–
What does it tell?	Size of the orbit, Distance from nucleus & Orbital energy	The shape of the orbital, no. of sub-shells present in the principal orbit/shell.	Orientations of the sub-shells(X, Y, Z-axis).	The direction of electron spin i.e Clockwise or Anti-clockwise.
Why is it required?	To explain the main spectral lines.	To explain the splitting and the hyperfine structure of spectral lines	To explain the splitting of lines in a magnetic field.	To explain the magnetic properties of substances.

How do we use these to write the orbital names? Check the table below –

n	ℓ	m	Orbital Name	Allowed no of electrons in the shell(no.of orbitals ✖️2)
n=1	ℓ=0  (s-orbital)	0	1s (one orbital)	2
n=2	ℓ = 0  (s-orbital). ℓ = 1  (p-orbital).	m= -1,0,+1 (py,pz,px)	2s 2py,2pz,2px (Four orbitals)	8
n=3	ℓ = 0 (s-orbital). ℓ = 1 (p-orbital). ℓ = 2 (d-orbital).	m=0 m= -1,0,+1. m= -2,-1,0,+1,+2.	3s 3py,3pz,3px 3dxy,3dyz,3dxz, 3dx2-y2,3dz2 (nine orbitals)	18

We shall begin discussing in the next post how these quantum numbers were useful in the further study of the elements. Till then,

Be a perpetual student of life and keep learning…

Good day!

Note – Please note that the tables above may not be completely visible on your mobile screen. Kindly open the post on an iPad/notebook or laptop screen to see the complete tables. I am trying to solve this issue but haven’t found a solution to this WordPress problem yet.

References and further reading –

1. Lecture 5, MIT  open courseware 3.091 by professor Donald Sadoway.
2. http://antoine.frostburg.edu/chem/senese/101/electrons/faq/f-orbital-shapes.shtml
3. http://www.tulane.edu/~sanelson/eens211/crystal_chemistry.htm
4. Precise Chemistry textbook by Sheth Prakashan Kendra.

Image source –

1) http://www.tulane.edu/~sanelson/eens211/crystal_chemistry.htm