# Question #a505f

May 15, 2017

${n}^{2}$ orbitals.

#### Explanation:

The principal quantum number, $n$, tells you the energy level on which an electron resides inside an atom.

The number of orbital present on a given energy level can be determined by looking at the value of the angular momentum quantum number, $l$, which tells you the subshell in which the electron resides.

As you can see, for any value of $n$, you have

$l = 0 , 1 , 2 , \ldots \left(n - 1\right)$

The number of orbitals present in a given subshell is given by the magnetic quantum number, ${m}_{l}$, which can take the values

${m}_{l} = \left\{- l , - \left(l - 1\right) , \ldots , - 1 , 0 , + 1 , \ldots , + \left(l - 1\right) , + l\right\}$

Now, for the first energy level, you have

$n = 1$

This implies that

$l = 0 \to$ one subshell

and

${m}_{l} = 0 \to$ one orbital

You can thus say that the first energy level holds $1$ subshell and $1$ orbital.

For the second energy level, you have

$n = \textcolor{red}{2}$

This implies

$l = \left\{0 , 1\right\} \to$ two subshells

and

• $l = 0 \implies {m}_{l} = 0 \to$ one orbital
• $l = 1 \implies {m}_{l} = \left\{- 1 , 0 , + 1\right\} \to$ three orbitals

Therefore, you can say that the second energy level holds $2$ subshells and a total of $4$ orbitals, so

$\text{no. of orbitals} = 4 = {\textcolor{red}{2}}^{2}$

For the third energy level, you have

$n = \textcolor{red}{3}$

This implies

$l = \left\{0 , 1 , 2\right\} \to$ three subshells

and

• $l = 0 \implies {m}_{l} = 0 \to$ one orbital
• $l = 0 \implies {m}_{l} = \left\{- 1 , 0 , + 1\right\} \to$ three orbitals
• $l = 2 \implies {m}_{l} = \left\{- 2 , - 1 , 0 , + 1 , + 2\right\} \to$ five orbitals

You can thus say that the third energy level holds a total of $9$ orbitals, so

$\text{mo. of orbitals} = 9 = {\textcolor{red}{3}}^{2}$

You can go on if you want, but at this point, it should be clear that the number of orbitals present for a given energy level, i.e. for a given value of the principal quantum number, is given by

$\textcolor{\mathrm{da} r k g r e e n}{\underline{\textcolor{b l a c k}{\text{no. of orbitals} = {n}^{2}}}}$