Question

Which of the following is not a valid set of four quantum numbers? How can you determine this?
A) $2,0,0, + \dfrac{1}{2}$
B) $2,1,0, - \dfrac{1}{2}$
C) $3,1, - 1, - \dfrac{1}{2}$
D) $1,0,0, + \dfrac{1}{2}$
E) $1,1,0, + \dfrac{1}{2}$

Answer 1

As we know that there are four quantum numbers which tell us about the shells, subshells, orbitals and spin of the electrons present in the valence shell. With the help of which we can locate the position of an electron in an atom. There are four quantum numbers in total and using them we can calculate the number of electrons in a shell and subshell of an element.

Complete step-by-step answer:
As we know that the principal quantum number tells us about the shell to which the electron belongs. It also tells about the size, energy level and maximum number of electrons that a shell can accommodate. The shells are represented by K, L, M, N and so on. The principal quantum number is represented by $n$.

Then we have azimuthal quantum number which tells us about the subshell to which the electron belongs and it is represented by $l$. The subshells are represented by s, p, d, f and so on.
Thirdly, we have a magnetic quantum number which is represented as ${m_l}$ and it tells us about the total number of orbitals present in a given subshell. For a given subshell there are $2l + 1$ values of magnetic quantum number.

Lastly, the spin quantum number which tells the spin of electrons can be positive and negative and one orbital can possess only two electrons therefore the number of electron with positive spin $+ \dfrac{1}{2}$ and with negative spin $- \dfrac{1}{2}$.

Now the first option is a valid set because $l$ can have a $0$ for $n = 2$ and similarly, ${m_l}$ can have $0$ when $l = 0$. Then the second option is also valid because the value of $l = 1$ is valid when $n = 2$ and ${m_l}$ can have a $0$ as well. Thirdly we again have a valid set for $n = 3$, the value of $l = 1$ and we can have ${m_l}$ as $- 1,0, + 1$ and the spin is also valid. Similar is the case with the fourth option.

But when we talk about the last option we can see that for a value of $n = 1$, the value of azimuthal quantum number cannot be $l = 1$ which is not possible, therefore the set is not valid and there is no description about the electron.

Therefore, the correct answer is option ‘E’.

Note: In nutshell, the number of subshells in a principal shell is $'n'$, the number of orbitals in a shell is ${n^2}$ and maximum number of electrons in a shell is $2{n^2}$. The number of orbitals in a subshell is $2l + 1$ and maximum number of electrons in a subshell is $2(2l + 1)$.