# What values of l, m_l, and m_s are possible for n = 3?

Jan 18, 2015

Well, your set of quantum numbers is not "allowed" for a particular electron because of the value you have for $\text{l}$, the angular momentum quantum number.

The values the angular momentum quantum number is allowed to take go from zero to $\text{n-1}$, $\text{n}$ being the principal quantum number.

So, in your case, if $\text{n}$ is equal to 3, the values $\text{l}$ must take are 0, 1, and 2. Since $\text{l}$ is listed as having the value 3, this puts it outside the allowed range.

The value for ${m}_{l}$ can exist, since ${m}_{l}$, the **magnetic quantum number, ranges from $- \text{l}$, to $\text{+l}$.

Likewise, ${m}_{s}$, the spin quantum number, has an acceptable value, since it can only be $- \text{1/2}$ or $+ \text{1/2}$.

Therefore, the only value in your set that is not allowed for a quantum number is $\text{l} = 3$.

Jan 18, 2015

There are 4 quantum numbers which describe an electron in an atom.
These are:

$n$ the principal quantum number. This tells you which energy level the electron is in. $n$ can take integral values 1, 2, 3, 4, etc

$l$ the angular momentum quantum number. This tells you the type of sub - shell or orbital the electron is in. It takes integral values ranging from 0, 1, 2, up to $\left(n - 1\right)$.

If $l$ = 0 you have an s orbital.
$l = 1$ gives the p orbitals
$l = 2$ gives the d orbitals

$m$ is the magnetic quantum number. For directional orbitals such as p and d it tells you how they are arranged in space. $m$ can take integral values of $- l \ldots \ldots \ldots \ldots . 0. \ldots \ldots \ldots \ldots + l$.

$s$ is the spin quantum number. Put simply the electron can be considered to be spinning on its axis. For clockwise spin $s$= +1/2. For anticlockwise $s$ = -1/2. This is often shown as $\uparrow$ and $\downarrow$.

In your question $n = 3$. Let's use those rules to see what values the other quantum numbers can take:

$l = 0 , 1 \mathmr{and} 2$, but not 3.This gives us s, p and d orbitals.

If $l$ = 0 $m$ = 0. This is an s orbital
If $l$ = 1, $m$ = -1, 0, +1. This gives the three p orbitals. So $m$ = 0 is ok.
If $l$ = 2 $m$ = -2, -1, 0, 1, 2. This gives the five d orbitals.

$s$ can be +1/2 or -1/2.

These are all the allowed values for $n = 3$

Note that in an atom, no electron can have all 4 quantum numbers the same. This is how atoms are built up and is known as The Pauli Exclusion Principle.