Class 11 | Structure Of Atom | Quantum Numbers

Class 11 | Structure Of Atom | Quantum Numbers 

 The numbers which identify the state of an electron, specify the energy associated with it and its location around the nucleus are called quantum numbers.

These numbers are used to designate and distinguish electrons in atomic orbitals.

There are four quantum numbers.

1. Principal quantum number (n)

2. Azimuthal quantum number (l)

3. Magnetic quantum number (m)

4. Spin quantum number (s).

1. Principal quantum number (n)

i) This number represents the main shell to which electron belongs.

ii) It is designated by n.

iii) It has values n = 1, 2, 3, 4, 5, 6, 7.

iv) These levels are also represented by the letters K,  L,  M,  N, etc.

Each main shell contains a maximum of 2n² electrons. Thus,

K shell contains maximum 2 electrons

L  shell contains maximum 8 electrons

M shell contains maximum 18 electrons

N shell contains maximum 32 electrons.

Principal Quantum Number provide the following information :

1. The principal shell to which electron belongs.

2. The energy of the principal shell.

2. Azimuthal quantum number (l)

 i) Azimuthal quantum number is also called secondary, subsidiary, orbital or angular momentum quantum number.

ii) It is represented by letter l.

iii) This represents the subshell to which the electron belongs.

iv) Its values depend upon the value of principal quantum number.

v) It can have only positive values between 0 and n – 1. Thus the permissible values of l are 0,1,2…(n-1).

This quantum number provide the following information :

1. The subshell in which electron is present.

A principal shell in an atom is supposed to be composed of several sub levels known as subshells.

These subshellls are represented by letters s,  p,  d,  f.

The designation of subshells in terms of the values of l is given in the following table.

2. The number of subshells present in a principal shell

The values of l depend upon the value of n.

The azimuthal quantum number signifies the number of subshells presents in a principal shell.

The subshells of a particular shell are specified by putting the value of n before the symbol of subshell.

for example, s subshell of K shell  is represented as 1S while s subshell of L is represented as 2s. Thus,

n = 3 and l = 0 , we have 3s subshell

n = 4 and l = 2 , we have 4d subshell

3. the shape of subshell :

 The azimuthal quantum number is responsible for particular shape of particular shell because it defines the spatial distribution of the electron cloud around the nucleus.

4.The angular momentum of the electron :

the quantum number l is used to calculate the angular momentum of an electron in a particular subshell.

The orbital angular momentum L of an electron in a subshell with  azimuthal quantum number l is given by ,

The maximum number of electrons in a given subshell is given by 2 ( 2l + 1 ) Thus,

 s – subshell (l = 0) contains 2 electrons

p – subshell (l = 1) contains 6 electrons

d – subshell (l = 2) contains 10 electrons

f – subshell (l = 3) contains 14 electrons

3. Magnetic quantum number ( m ) :

• This quantum number is used to explain zeeman effect.

• It is represented by letter m. 

• The values of m are governed by values of l.

• It can have any integral value between - l and + l through zero.

• Thus for a given value of l, m will have (2l + 1) values.

• This number is also referred to as orientation quantum number because it gives the distribution of electron cloud in space around the nucleus.

• A subshell in an atom is supposed to be composed of several orbitals.

• The magnetic quantum number m tells to which orbital, an electron belongs.

• It also tells the number of orbitals which corresponds to a particular shell.

For example,

a) S - subshell :

I = 0 therefore m = 0.

• This implies that s subshell contains only one orbital because it has only one value of m orbital contained in s-subshell is called s-orbital.

b) p - subshell :

I = 1 therefore m = +1, 0, -1.

• Thus, a p-subshell contains three orbitals.

• They are designated as px, py, pz where subscripts x, y, z, refer to co-ordinate axes.

• Thus p-orbitals posses different orientations in space.

• px - orbitals is oriented along X axis, py along Y axis and pz along Z axis.

• In the absence of magnetic field, all the three p-orbitals are equivalent in energy and are said to be three fold degenerate or triply degenerate. 

c) d - subshell :

I = 2 therefore m = +2, +1, 0 -1, -2.

• Thus, d subshell possesses five orientations in space i.e. d subshell contains five d orbitals.

• These are designated as dxy, dyz, dxz, dx2-y2 and dz2.

• The three d orbitals dxy, dyz, dxz are planar orbitals.

• dx2-y2 and dz2 are axial orbitals. 

In the absence of magnetic field these orbitals are equivalent in energy and are said to be five fold degenerate.

 d) f - subshell :

I = 2 therefore m = +3, +2, +1, 0 -1, -2, -3.

• Thus f subshell possesses seven orientations in space.

• therefore f - subshell contains seven orbitals.

• f – orbitals are seven fold degenerate and are quite complicated in shape and designation.

4. Spin quantum number 

• This quantum number is represented by s.

• It accounts for the spin of electron since an electron can spin clockwise or anticlockwise in two opposite directions.

• It has only two values + ½ and - ½.

• For clockwise spin value is + ½ and for anti clockwise spin value is - ½ .

• Vertical Arrow pointing upwards represents clockwise spin while a Arrow pointing downwards represents anticlockwise spin.

• Two electrons with same sign of the spin quantum number are said to have parallel spin while those having opposite signs of the spin quantum number are said to have opposite spins or antiparallel spin.