Question

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?

Answer 1

The answer is `(e)`.

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, `+1/2` and `-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.

`(a)" "n=2, l=0, m_l = 0, m_s = +1/2" "color(green)(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 `2s` orbital, that has spin-up.

`(b)" " n=2, l=1, m_l = 0, m_s = -1/2" "color(green)(sqrt())`

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

`l = 0, 1, ..., (n-1)`

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

`(c)" "n=3, l=1, m_l=-1, m_s = -1/2" "color(green)(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 = {-1, color(white)(-)0, +1}`

This set represents an electron located on the third energy level, in the p-subshell, in the `3p_x` orbital, that has spin-down.

`(d)" "n=1, l=0, m_l = 0, m_s = +1/2" "color(green)(sqrt())`

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

`(e)" "n=1, l=1, m_l = 0, m_s = +1/2 " "color(red)(xx)`

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

`n=1" "` and `" "l=1`

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