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Quantum Numbers

what is Quantum Numbers

An atom contains large number of shells and subshells. These are distinguished from one another on the basis of their size, shape and orientation (direction) in space. The parameters are   expressed in terms of different numbers calledquantum numbers.

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

i)   Principal quantum number(n):

It tells the main shell in which the electron resides and the approximate distance of the electron from the nucleus. This value determines to a large extent energy of the orbital. It also tells the maximum number of electrons a shell can accommodate is  2n2, where n is the principal quantum number.          

                                           Shell  K      L      M      N

 Principal quantum number (n)   1      2      3        4

 Maximum number of electrons   2      8      18      32

Permissible values of n: all positive integers.

ii)   Azimuthal or angular momentum quantum number( l ):

This represents the number of subshells present in the main shell. These subsidiary orbits within a shell will be denoted as s,p,d,f… This tells the shape of the sub shells. The orbital angular momentum of the electron is given as:
(or) for a particular value of  .

∙   For a given value of n, possible values of vary from 0 to n – 1. This means that there arepossible shapes in the shell.


iii)   Themagnetic quantum number(m):

An electron due to its angular motion around the nucleus generates an electric field. This electric field is expected to produce a magnetic field. Under the influence of external magnetic field, the electrons of a subshell can orient themselves in certain preferred regions of space around the nucleus called orbitals. The magneticquantum number determines the number of preferred orientations of the electron present in a subshell. The values  allowed depends on the value of, the angular  momentum quantum number, m can, assume all integral values between –  to + including zero. Thus m can be –1, 0, +1 for = 1. Total values of m associated with a particular value of  are given by 2 + 1.

iv)   The spin quantum number(s):Just like earth not only revolves around the sun but also spins about its own axis, an electron in an atom not only revolves around the nucleus but also spins about its own axis. Since an electron can spin either in clockwise direction or in anticlockwise direction, therefore, for any particular value of magnetic quantum number, spinquantum number can have two values, i.e., +1/2 and –1/2 or these are represented by two arrows pointing in the opposite directions, i.e., ↑ and ↓. When an electron goes to a vacant orbital, it can have a clockwise or anticlockwise spin. This quantum number helps to explain the magnetic properties of the substances.

Spin angular momentum where s = ½.

   Another term, defined as multiplicity is given as 2|S|+1 where |S| is total spin

   = no. of unpaired electrons × 1/2.

   Can you derive the following expressions?

   1.   no. of orbital = for ‘n’ respectively

   2.      no. of e- = 2n2 or   for ‘n’  respectively  

The atom is built up by filling electrons in various orbitals according to the following rules.

Aufbau Principle:This principle states that the electrons are added one by one to the various orbitals in order of their increasing energy starting with the orbital of lowest energy. The increasing order of energy of various orbital is

1s,2s, 2p,3s, 3p,4s, 3d,4p, 5s,4d, 5p,6s, 4f,5d ,6p,5f, 6d,7p……… ……………

How to remember such a big sequence?  To make it simple we are giving you the method to write the increasing order of the orbitals. Starting from the top, the direction of the arrows gives the order of filling of orbitals.

         

Alternatively, the order of increasing energies of the various orbitals can be calculated on the basis of (n+l) rule.

 The energy of an orbital depends upon the sum of values of the principal quantum number (n) and the azimuthal quantum number (l). This is called (n+l) rule. According to this rule,

“In neutral isolated atom, the lower the value of (n+l) for an orbital, lower is its energy. However, if the two different types of orbitals have the same value of (n+l), the orbitals with lower value of n has lower energy’’.

Illustration of (n +l) rule

 

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