# 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)

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.

No. of microstates:

No. of microstates:

No. of microstates:

Examples-

Number of Microstates of p

p

So, X = 2.

Since the number of orbitals in p sub-shell = 3

So, N = 2 x 3 = 6

Now from formula-

Number of Microstates of d

15 microstates indicates that there are 15 ways in which electrons can be arranged.

Similarly,

for d

for d

Example-

Number of microstates of p

For p

For d

### Q. One of the excited state of Ti has the electronic configuration [Ar]4s

Answer: B. 15

Hence, it can be calculated by the equation (2S+1) (2L+1)

Example-

Number of microstates of

For

(2S + 1) (2L + 1) = 2 (2 x 2 + 1) = 10

Number of microstates of

For

(2S + 1) (2L + 1) = 3 (2 x 3 + 1) = 21

### determine the number of microstates in term symbol of p

Number of microstates in

(2S + 1) = 3 and L = 1

or, (2S + 1) (2L + 1) = (3) (2 x 1 + 1) = 9

Number of microstates in

(2S + 1) = 1 and L = 2

(2S + 1) (2L + 1) = (1) (2 x 2 + 1) = 5

Number of microstates in

(2S+1) = 1 and L = 0

(2S + 1) (2L + 1) = (1) (2x0 + 1) = 1

Hence, total number of microstates in p

Example-

To calculate microstates of

For

J is 4

So, Microstates of

(2J + 1) = (2 x 4 + 1) = 9

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.

### Number of Microstates of p-System

No. of microstates:

### Number of Microstates of d-System

No. of microstates:

### Number of Microstates of f-System

No. of microstates:

Examples-

Number of Microstates of p

^{2}configuration-p

^{2}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 2012**Number of Microstates of d^{5}system-Number of Microstates of d

^{5}system = 25215 microstates indicates that there are 15 ways in which electrons can be arranged.

Similarly,

for d

^{1}configuration, the number of microstates is 10.for d

^{2}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-shellExample-

Number of microstates of p

^{1}d^{1}ConfigurationFor p

^{1}term, N = 2 x 3 = 6 and X = 1For d

^{1}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]4s^{2}3d^{1}4p^{1}. 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

^{2}DFor

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

Number of microstates of

^{3}FFor

^{3}F term, (2S + 1) is 3 and L is 3 So, number of Microstates of^{3}F term symbol-(2S + 1) (2L + 1) = 3 (2 x 3 + 1) = 21

### determine the number of microstates in term symbol of p^{2} configuration

^{3}P,^{1}D and^{1}S are the term symbol of p^{2}configuration.Number of microstates in

^{3}P term-(2S + 1) = 3 and L = 1

or, (2S + 1) (2L + 1) = (3) (2 x 1 + 1) = 9

Number of microstates in

^{1}D term-(2S + 1) = 1 and L = 2

(2S + 1) (2L + 1) = (1) (2 x 2 + 1) = 5

Number of microstates in

^{1}S term-(2S+1) = 1 and L = 0

(2S + 1) (2L + 1) = (1) (2x0 + 1) = 1

Hence, total number of microstates in p

^{2}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

^{2}P_{4}For

^{2}P_{4}term-J is 4

So, Microstates of

^{2}P_{4}term symbol-(2J + 1) = (2 x 4 + 1) = 9