Quantum Numbers

Quantum Numbers

An atom contains a large number of orbitals. These are distinguished from each other on the basis of their shape, size and orientation in space. These characteristics of an orbital are expressed in terms of three numbers, called principal, azimuthal and magnetic quantum number.

Quantum numbers may be defined as a set of 4 numbers with the help of which we can get complete information about all the electrons in an atom, i.e. location, energy, the type of Orbital occupied, space and orientation of that orbital.

Principal Quantum Number

It tells the principal energy level or shell to which the electron belongs.

It is donated by the  letter n and can have any integral value except 0 i.e. n=1,2,3,4…. etc.

The various principal energy shells are also designated by the letters K,L,M,N,O….starting from the nucleus.

This number helps to explain the main lines of the spectrum on the basis of the electronic jump between these shell.

(a) It gives the average distance of the electron from the nucleus, i.e. it largely determined the size of the electron cloud.

(b) It completely determine the energy of the electron in hydrogen atom and hydrogen like particles.

 

For the first principal shell( K),n=1 which means that this energy shell is lowest energy and lies closest to the nucleus.

For the second principal shells( L),n=2 and for the third principal shell (M), n=3 and so on.

The energies of the various principal shells follow the sequence:

K<L<M<N<O…..

1<2<3<4<5……

The maximum number of electrons present in any principle shell is given by 2n² Where n is the number of principal shell.

Azimuthal or Angular momentum Quantum Number

Within the same principal Shell, there are present a number of sub shells or sub levels of energy. As a result, the number of electronic jumps increases and so is the number of lines.

Thus, this number helps to explain the fine lines of the spectrum.

Azimuthal quantum number tells about the :

(1) Number of sub shells present in the main shells.

(2) The angular momentum of the electron present in any sub shell.

(3) The relative energies of the various sub shells.

(4) The shapes of the various sub shells present within the same principal Shell.

 

For a given value of n, it can have any integral value ranging from 0 to n-1.

For 1st Shell (K), n=1,  l can have only one value i.e., l=0

For the 2nd Shell (L), n=2, l can have two values  i.e. l=0 and 1

For the 3rd Shell ( M), n= 3, l can have three values i.e. l=0, 1, 2

For the 4th shells ( N ), n=4, l can have 4 values i.e. l=0, 1, 2, 3

Depending upon the values of l, i.e. l = 0, 1, 2 and 3,the different sub shells  are designated as s,p,d and f .These notations are the initial letters of the words, sharp, principal, diffused and fundamental formerly used to describe different spectral lines.

l = 4 is called g sub shell, l = 5 is called is h sub shell.

(1) First principal shell (K shell or n=1 ) has only one sub shell called the s sub shell.

(2) Second principal shell (L shell or n=2)  has only 2 sub shells i.e. s sub shell ( l = 0 ) and p sub shell (l=1)

(3) Third principal shell (M shell or n= 3) has three sub shells i.e. s sub shell ( l=0 ), p  sub shell (l= 1) and d sub shell (l=2 )

(4) Fourth principal shell (N shell or n=4 ) has four sub shell i.e. s sub shell ( l=0), p sub shell ( l= 1), d sub shell ( l=2) and f sub shell ( l=3).

The number of sub shells present in any principal shell is equal to the number of principal Shell or the principal quantum number.

The energies of different sub shells present within the same principal are found to be in order

s < p< d < f

i.e. an electron in the s – sub shell has lower energy than that in the p sub shell of the same principal Shell.

The maximum number of electrons in the s, p, d and f sub shell are 2, 6,10 and 14.

 

Magnetic Quantum Number

This quantum number is required to explain the fact that when the source producing the line spectrum is placed in a magnetic field, each spectral line splits up into a number of lines.

An electron due to its orbital motion around the nucleus generates an electric field. This electric field in turn produces a magnetic field which can interact with the external magnetic field. Thus under the influence of external magnetic field, the electrons of a sub shell can orient themselves in certain preferred regions of space around the nucleus called orbitals.

The magnetic quantum number determines the number of orbitals present in any sub shell.

The magnetic quantum number determines the number of preferred orientation of the electron present in a sub shell.

The magnetic quantum number is denoted by the letter m or m_l and for a given value of l, it can have all the values ranging from -l to + l including zero.

For every value of l, m  has 2l + 1 values.

(1) For l=0, m can have only one value. This means that s sub shell has only one orientation in space. s sub shell has only one orbital called s – orbital.

(2) For l=1( p sub shell), m can have three values i.e. m = -1,0, +1.p sub shell has 3 orbitals. Since these 3 orbitals are oriented along x axis, y axis and z axis, therefore they are commonly referred to as p_x, p_y and p_z.

(3) For l=2( d sub shell), m can have five values i.e. m =-2,  -1,0, +1, +2 .d sub shell has 5 orbitals.

(4) For l=3 (f sub shell ), m can have 7 values i.e. m = -3, – 2,  -1,0, +1, +2, +3  there are 7 different orientation of f sub shells. f sub shell has 7 orbitals.

 

All the three p orbitals of a particular principal shell have the same energy in the absence of a magnetic field. All the five d orbitals of a particular shell have the same energy and all the 7 f orbitals have same energy.

These orbitals of the same sub shell having equal energy are called degenerate orbitals.

In the presence of an external magnetic field, this degeneracy is broken and orbitals of the same sub shell acquire slightly different energy. This cause the splitting of a given spectral line into many.

Spin Quantum Number

The electron in an atom not only moves around the nucleus but also spin about its own axis. This number give the information about the direction of spinning of the electron present in any orbital.

It is represented by s or m_s.

Since the electron in an orbital can spin either in clockwise direction or in the anticlockwise direction, hence for a given value of m, s can have only two values i.e. + ½ and – ½  or these are very often represented by two arrows pointing in the opposite direction i.e. ↑ or ↓.

This quantum number helps to explain the magnetic properties of the substance. A spinning electron behaves like a micro magnet with a definite magnetic moment. If an orbital contains 2 electrons, the two magnetic moment opposes and cancel each other.

In an atom, if all the orbitals are fully filled, net magnetic moment is zero and the substance is diamagnetic.

If some half filled orbitals are present, the substance has a net magnetic moment and is paramagnetic.