# Which of the following is not a valid set of four quantum numbers? How can you determine this?

## a. 2,0,0, + 1/2 b. 2,1,0, -1/2 c. 3,1, -1, -1/2 d. 1,0,0, +1/2 e. 1,1,0, +1/2?

Jul 1, 2016

The answer is $\left(e\right)$.

#### Explanation:

Start by making sure that you're familiar with the valid values each quantum number can take.

As you can see, the principal quantum number, $n$, determines the possible values of the angular momentum quantum number, $l$, which in turn determines the possible values of the magnetic quantum number, ${m}_{l}$.

The spin quantum number is independent of the of the values taken by the other three quantum numbers and can only have two possible values, $+ \frac{1}{2}$ and $- \frac{1}{2}$.

Now, take a look at the relationship between the value of $n$ and the value of $l$ for each of those five sets of quantum numbers.

$\left(a\right) \text{ "n=2, l=0, m_l = 0, m_s = +1/2" } \textcolor{g r e e n}{\sqrt{}}$

This set is valid because $l$ can take the value $0$ when $n = 2$. The same can be said about ${m}_{l}$, which can take the value $0$ when $l = 0$.

This quantum number set represents an electron located on the second energy level, in the s-subshell, in the $2 s$ orbital, that has spin-up.

$\left(b\right) \text{ " n=2, l=1, m_l = 0, m_s = -1/2" } \textcolor{g r e e n}{\sqrt{}}$

This set is valid because $l = 1$ is still within the accepted value of

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

when $n = 2$. This time, the set represents an electron that is located on the second energy level, in the p-subshell, in the $2 {p}_{z}$ orbital, that has spin-down.

$\left(c\right) \text{ "n=3, l=1, m_l=-1, m_s = -1/2" } \textcolor{g r e e n}{\sqrt{}}$

Once again, you're dealing with a valid set. All the quantum numbers are well within their accepted values. Notice that when $l = 1$, you can have

${m}_{l} = \left\{- 1 , \textcolor{w h i t e}{-} 0 , + 1\right\}$

This set represents an electron located on the third energy level, in the p-subshell, in the $3 {p}_{x}$ orbital, that has spin-down.

$\left(d\right) \text{ "n=1, l=0, m_l = 0, m_s = +1/2" } \textcolor{g r e e n}{\sqrt{}}$

This set is valid and it represents an electron located on the first energy level, in the s-subshell, in the $1 s$ orbital, that has spin-up.

$\left(e\right) \text{ "n=1, l=1, m_l = 0, m_s = +1/2 " } \textcolor{red}{\times}$

This is not a valid set of quantum numbers. Notice that $l$ cannot take the value of $n$, but here

$n = 1 \text{ }$ and $\text{ } l = 1$

This means that the set cannot describe an electron located in an atom, i.e. it's not a valid set.