Question

How many electron states are in these sub-shells: (a) n = 4,l = 3 ; (b) n = 3, l = 1 ; (c) n = 4, l = 1 ; (d) n = 2, l = 0 ?

Short answer

1. There are 14 electron states in the subshell n = 4, l = 3 .
2. There are 6 electron states in the subshell n = 3, l = 1 .
3. There are 6 electron states in the subshell n = 4, l = 1 .
4. There are 2 electron states in the subshell n = 2, l = 0 .

Step-by-step solution

The given data:

The given quantum numbers of the subshell are:

1. n = 4, l = 3
2. n = 3, l = 1
3. n = 4, l = 1
4. n = 2, l = 0

Understanding the concept of electron states

Using the concept of electron states of every value of orbital quantum number l , we get (2l + l) values of magnetic quantum number m, ranging from -l to +l. Now, for each given value of m, there are 2 different spin orientations of the electron. Hence, there are 2(2l+l) possible values for each value of l.

Formula:

The total electron states for a given valu ….. (1)

(a) Calculation of electron states for n = 4, l = 3 :

Using the given data l = 3 in equation (1), the possible values of electron states is as follow.

$$\begin{gathered} {\mathrm{N}}_{\mathrm{l}}=2\left(2\times 3+1\right) \\ =14 \end{gathered}$$

Hence, there are 14 electron states in this subshell.

(b) Calculation of electron states for n = 3, l =1

Using the given data l = 1 in equation (1), the possible values of electron states is as follow.

$$\begin{gathered} {\mathrm{N}}_{\mathrm{l}}=2\left(2\times 1+1\right) \\ =6 \end{gathered}$$

Hence, there are 6 electron states in this subshell.

(c) Calculation of electron states for n = 4, l = 1 :

Using the given data l = 1 in equation (1), the possible values of electron states is as follow:

$$\begin{gathered} {\mathrm{N}}_{\mathrm{l}}=2\left(2\times 1+1\right) \\ =6 \end{gathered}$$

Hence, there are 6 electron states in this subshell.

(d) Calculation of electron states for  :

Using the given data l = 0 in equation (1), the possible values of electron states is as follow.

$$\begin{gathered} {\mathrm{N}}_{\mathrm{l}}=2\left(2\times 0+1\right) \\ =2 \end{gathered}$$

Hence, there are 2 electron states in this subshell.