Question

How many electron states are there in a shell defined by the quantum number n = 5 ?

Short answer

There are 50 electron states in the shell defined by the quantum number n = 5 .

Step-by-step solution

The given data

The given quantum number, n = 5

Understanding the concept of electron states in a shell

Using the concept of possible electron states for a given quantum number, we can get the required value of the number of the electron states in a shell.

Formulae:

The number of possible electron states for each value of $\mathcal{l}$is, $N_{\mathcal{l}}=2(2\mathcal{l}+1)$ …...(i)

The total number of possible electron states for a given n, $N_n=\sum _1^{n-1}N\mathcal{l}$ ……(ii)

Calculation of the number of electron states in a shell

We know that for a given quantum number n, there are n possible values of $\mathcal{l}$, ranging from 0 to (n-1) .

Thus, the total number of electron states ( $m_s=\pm \frac{1}{2}$) can be given using equation (i) in equation (ii) with the given value n = 5 as follows:

$$\begin{gathered} N_n=\sum _1^{n-1}\left[2\left(2\mathcal{l}+1\right)\right] \\ =2\sum _1^{n-1}\left(2\mathcal{l}+1\right) \\ =2n^2 \\ =2{\left(5\right)}^2=50 \end{gathered}$$

Hence, the total number of electron states is 50 .