An electron is in a state with n = 3. What are (a) the number of possible values of I, (b) the number of possible values of ${\mathbf{m}}_{\mathbf{1}}$, (c) the number of possible values of ${\mathbf{m}}_{\mathit{s}}$, (d) the number of states in the n = 3 shell, and (e) the number of sub-shells in the n = 3shell?

An electron is in a state with n = 3.

Using the concept of possible values of orbital quantum number for every value of principal quantum number and the possible values of the magnetic quantum number for every value of l, it helps to get the required values. Again, for any possible values of m, every electron has two spins. Now, the number of sub-shells is decided by the principal quantum number. Now, define the total number of states of the shell using all these values in the given formula.

Formulas:

The number of possible ways of${\mathrm{m}}_1$ for every I value is,

${\mathrm{N}}_{\mathrm{I}}=\left(2\mathrm{I}+1\right)$ ….. (1)

The number of possible states of the shell for a given n value is,

${\mathrm{N}}_{\mathrm{n}}=2{\mathrm{n}}^2$ ….. (2)

For given value n, the total possible ways of I is n.

Hence, the total number of possible ways of I for n = 3 is 3 consisting of 0, 1, 2.

Using the above greatest value of I = 2 in equation (1), the total possible ways of${\mathrm{m}}_1$ is as follow.

$$\begin{gathered} {\mathrm{N}}_{\mathrm{I}}=\left(2\times 2\right)+1 \\ =5 \end{gathered}$$

The possible values of${\mathrm{m}}_1$ are:

${\mathrm{m}}_1=-\mathrm{I},-\left(\mathrm{I}-1\right),-\left(\mathrm{I}-2\right),..-2,-1,0,1,2$

For I = 2:

$$\begin{gathered} {\mathrm{m}}_2=-2,-1,0,1,2 \\ {\mathrm{m}}_1=-1,0,1 \\ {\mathrm{m}}_0=0 \end{gathered}$$

Hence, the total possible ways ${\mathrm{m}}_1$of is 5.

As ${\mathrm{m}}_s$refers to spin angular momentum, it only has values $\pm \frac{1}{2}$. Using the concept, the possible ways of${\mathrm{m}}_s$ irrespective of n, I, ${\mathrm{m}}_1$are $\pm \frac{1}{2}$.

Hence, there are two possible ways of ${\mathrm{m}}_{\mathrm{s}}$.

Using the given value n = 3 in equation (2), the total electron states for the shell is as given by,

$$\begin{gathered} {\mathrm{N}}_{\mathrm{n}}=2{\left(3\right)}^2 \\ =18 \end{gathered}$$

Hence, the value of the possible states is 18.

The subshells for a given n level are given by the different values of I. There are always the same number of I values as n.

Using the concept, the number of sub-shells for the given value n = 3 is 3 .

Hence, the number of sub-shells is 3.