How to Calculate Number of Microstates

How to Calculate the Number of Microstates

The configuration of atoms or ions or molecules is not a complete description of arrangement of electrons in a sub shell of an atom. In a given configuration, the electrons are arranged in many different way which have slightly different energy. These different arrangements are called microstates.

Microstates are determined by following different ways.
1. From electronic configuration of single sub-shell
2. From electronic configuration of two different sub-shell
3. From the term symbol without L-S coupling (J)
4. From the term symbol with L-S coupling (J)

Determine the microstates from electronic configuration of single sub-shell

The total number of possible arrangements of given number of electrons are the microstates of that electronic configuration.

Where N = Twice the number of orbitals in the sub shell = 2(2l + 1)
X = number of electrons in the given configuration
N! and X! are the factorials of N and X respectively.

Examples-
Number of Microstates of p² configuration-

p² configuration has two electrons.
So, X = 2.
Since the number of orbitals in p sub-shell = 3
So, N = 2 x 3 = 6
Now from formula-


CSIR NET/JRF June 2012Number of Microstates of d⁵ system-
Number of Microstates of d⁵ system = 252

15 microstates indicates that there are 15 ways in which electrons can be arranged.
Similarly,
for d¹ configuration, the number of microstates is 10.
for d² configuration,the number of microstates is 45.

Determine the microstates from electronic configuration of two different sub-shell

It can be calculated by multiplying microstate of one sub-shell to other sub-shell


Example-
Number of microstates of p¹d¹ Configuration
For p¹ term, N = 2 x 3 = 6 and X = 1
For d¹ term, N = 2 x 5 = 10 and X = 1


CSIR NET/JRF December 2012

Q. One of the excited state of Ti has the electronic configuration [Ar]4s²3d¹4p¹. The number of microstate with zero total spin (s) for this configuration is-
A. 9
B. 15
C. 27
D. 60

Answer: B. 15

Determine the microstates from the term symbol without L-S coupling (J)

Number of microstates is equal to the product of spin energy level (2S+1) and orbital energy level (2L+1) of the term.
Hence, it can be calculated by the equation (2S+1) (2L+1)
Example-
Number of microstates of ²D

For ²D term, (2S + 1) is 2 and L is 2 So, number of Microstates of ²D term symbol-
(2S + 1) (2L + 1) = 2 (2 x 2 + 1) = 10

Number of microstates of ³F
For ³F term, (2S + 1) is 3 and L is 3 So, number of Microstates of ³F term symbol-
(2S + 1) (2L + 1) = 3 (2 x 3 + 1) = 21

determine the number of microstates in term symbol of p² configuration

³P, ¹D and ¹S are the term symbol of p² configuration.

Number of microstates in ³P term-
(2S + 1) = 3 and L = 1
or, (2S + 1) (2L + 1) = (3) (2 x 1 + 1) = 9

Number of microstates in ¹D term-
(2S + 1) = 1 and L = 2
(2S + 1) (2L + 1) = (1) (2 x 2 + 1) = 5

Number of microstates in ¹S term-
(2S+1) = 1 and L = 0
(2S + 1) (2L + 1) = (1) (2x0 + 1) = 1

Hence, total number of microstates in p² configuration = 9 + 5 + 1 = 15

Determine the microstates from the term symbol with L-S coupling (J)

It can be calculated by the equation (2J + 1)
Example-
To calculate microstates of ²P₄
For ²P₄ term-
J is 4

So, Microstates of ²P₄ term symbol-
(2J + 1) = (2 x 4 + 1) = 9