Question

Question #a505f

Answer 1

`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, ... (n-1)`

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 = {-l, - (l-1), ..., -1, 0, +1, ..., +(l-1), + l}`

Now, for the first energy level, you have

`n=1`

This implies that

`l = 0 ->` one subshell

and

`m_l = 0 ->` 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 = color(red)(2)`

This implies

`l = {0, 1} ->` two subshells

and

• `l = 0 implies m_l = 0 ->` one orbital
• `l=1 implies m_l = {-1, 0 ,+1} ->` three orbitals

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

`"no. of orbitals" = 4 = color(red)(2)^2`

For the third energy level, you have

`n = color(red)(3)`

This implies

`l = {0, 1, 2} ->` three subshells

and

• `l=0 implies m_l = 0 ->` one orbital
• `l=0 implies m_l = {-1, 0 ,+1} ->` three orbitals
• `l=2 implies m_l = {-2, -1, 0, +1, +2} ->` five orbitals

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

`"mo. of orbitals" = 9 = color(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

`color(darkgreen)(ul(color(black)("no. of orbitals" = n^2)))`