Question

(a) How many $l$ values are associated with $n=3$ ?
(b) How many $m_l$​ values are associated with $l=1$?

Answer 1

This question can only be answered after having a profound knowledge of principle quantum numbers and orbital number along with the magnetic quantum number and the relation between each.

Complete answer:
(a) Starting from the beginning, the principal quantum number $n$ is the relative overall energy of each orbital. As the distance from the nucleus increases, the energy level of each orbital increases. And in the question, it is given that $n$ is $3$. The quantum number or $l$ determines the shape of the electron cloud and there are various such forms starting from the $s$ orbital. The value of $l$ starts from zero. It is also termed as angular momentum. The relation between $l$ and $n$ is that $l$ ranges from $0$ to $n-1$.

So the answer to the first part of the question is that $l$ has three values and they are $0$, $1$ and $2$.

(b) The magnetic quantum number or ${{m}_{l}}$ determines the projection of the orbital angular momentum along a specified axis. The relation of magnetic quantum number and the quantum number is that magnetic quantum number ranges from the positive value of the quantum number to the negative value of the quantum number.

Therefore the answer to the second part of the question is that the magnetic quantum number will have three values and the values are $-1$, $0$ and $1$.

Note: Apart from these, there is another term known as spin quantum numbers which describe the projection of the spin angular momentum along the specified axis and also the intrinsic angular momentum of the electrons with the orbital.