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Quantum Number

Principal Quantum Number, n
The quantum number n (known as the principal quantum number) along with quantum number l describes the radial distribution of an electron. By the term radial distribution, we mean the probability distribution of the electron as a function of its distance r from the nucleus. The principal quantum number n can have only integral values of 1, 2, 3, ... In a one-electron system (such as H, He+ and Li2+), the energy of the electron is determined by the value of the principal quantum number only through the expression
 

The negative sign in the above expression means that electrostatic forces hold the electrons and the nucleus together; hence, it is more stable than the nucleus and electron taken separately. The energy of the electron increases with an increase in the value of the quantum number n, which implies that the electron is bound by the electrostatic forces of attraction to the nucleus. It has zero energy when n is infinity - a situation in which electron does not experience the electrostatic forces of attraction by the nucleus.

 

Since the energy of the electron depends on its distance from the nucleus, it is obvious that the principal quantum number also tells us something about the size of the orbital. By the term size of an orbital we mean the region around the nucleus in which there exists 90 to 95% probability of its existence. Obviously, the size of the orbital increases as the value of the principal quantum number is increased.
 
The principal quantum number represents the different energy shells in which the electron can exist. These shells are also designated by the symbols K, L, M, ..., depending upon the value of n equal to 1, 2, 3, ..., respectively. In a multi-electron atom, the energy of the shell, besides depending upon the quantum number n, is also determined by the quantum number l.
 
Azimuthal Quantum Number, l
The quantum number l (known as azimuthal or subsidiary quantum number) represents the quantized values of angular momentum of the electron in an orbital and is given by the expression
 (4.13)
where h is Planck's constant (= 6.626 × 10-34 J s).
The permitted values of l are 0, 1, 2, ..., (n - 1) , i.e., it can have more than one value depending upon the value of n Table.

Table Permitted values of l for a given value of n
 

Value of n

permitted values of l

1

0

2

0 and 1

3

0, 1 and 2

4

0, 1, 2 and 3

.... and so on

 


In general, this quantum number represents the shape of an orbital (whether it is spherical or dumb-bell shaped and so on). The value of l is also designated by the symbols. These, along with the approximate shape of the orbital are described in the following Table:

 

Table Designation of quantum number l by the symbols

Value of l

Designation

Approximate shape

0

s

spherical

1

p

dumb-bell

2

d

double dumb-bell

3

f

complicated shapes

4

g

...

...and so on


Magnetic Quantum Number, m
The angular momentum, being a vector quantity has magnitude as well as direction. The magnitude of angular momentum is determined by the azimuthal quantum number l. The direction of angular momentum is determined from the magnetic quantum number m. It is for this reason; the quantum number m describes what is known as space quantization of angular momentum. The z-component is conventionally fixed by the direction of the external magnetic field. When an atom is put in a magnetic field, the angular momentum of the electron takes only some specific orientations with respect to the z-axis such that the component of the angular momentum also has quantized values and is given by the expression
 

where m is the magnetic quantum number, the permitted values of which are 0,  1,2,..., l. The number of allowed values of m for a given value of l is 2l + 1.

Permitted values of m for a given value of l

 

Azimuthal quantum number, l

Permitted values of m

Their number
(2l + 1)

0

0

1

1

+ 1, 0, - 1

3

2

+ 2, + 1, 0, - 1, - 2

5

3

+ 3, + 2, + 1,0, - 1, - 2, - 3

7

.. and so on,


In other words, the total permitted values of m for a given value of l gives the number of orbitals of one type within a sub shell. 
 
For example, there is one orbital of s-type if the value of l is zero, three p-orbitals if the value of l is one, five d-orbitals if the value of l is two and so on.
 
Spin Quantum Number, s
An electron present in an orbital also spins around its own axis. This spinning produces angular momentum whose value is given by the expression
 (4.15)

where s is known as spin quantum number and has a value of 1/2. The z-component of this angular momentum is also quantized and is given by the expression

Lz = ms (h/2p ) (4.16)

where ms is either + 1/2 or - 1/2. The spinning of the electron with ms = + 1/2 is conventionally known as a-spin and is represented by a vertical arrow (). The spinning of the electron with ms = - 1/2 is conventionally known as b-spin and is represented by a downward arrow ().




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