# What is the number of the lowest energy level that has a p sublevel?

Jun 23, 2017

$n = 2$.

If we seek the lowest energy level (given by the principal quantum number $n$) for which the $p$ orbital sublevel exists, we look for the minimum $n$ for which $l$, the angular momentum quantum number, is valid.

The principal quantum number is defined as:

$n = 1 , 2 , . . .$, in integer steps

and the angular momentum quantum number is defined as:

$l = 0 , 1 , 2 , . . . , n - 1$, in integer steps, where ${l}_{\max} = n - 1$.

Since $p$ orbitals have $l = 1$, we must have that if ${l}_{\max} = n - 1 = 1$ is the maximum possible $l$ we can achieve for a given $n$, then $\boldsymbol{n = 2}$ is the minimum required quantum level for $p$ orbitals to exist.

Likewise...

• $n = 1$ is the minimum $n$ for which $s$ orbitals exist.
(${l}_{\max} = n - 1 = 0$.)
• $n = 3$ is the minimum $n$ for which $d$ orbitals exist.
(${l}_{\max} = n - 1 = 2$.)
• $n = 4$ is the minimum $n$ for which $f$ orbitals exist.
(${l}_{\max} = n - 1 = 3$.)

etc.