Question

If $n + l = 6$, then total possible number of subshells would be:
(A) $3$
(B) $4$
(C) $2$
(D) $5$

Answer 1

Atomic orbitals are distinguished from each other with the help of quantum numbers. There are a total of four quantum numbers. There are total four quantum numbers that are: Principal quantum number, Azimuthal quantum number, magnetic quantum number and spin quantum number, they are represented as $n$, $l$, $ml$, $ms$ respectively.

Complete step by step answer:
Quantum numbers are important as they help us to determine the possible location of electrons and their configuration within the atoms.
We know about these quantum numbers before calculating the answer.
I. Principal quantum number ($n$): It determines the size and energy of an atom and also identifies the shell. The value of $n =$ $1,2,3......$
II. Azimuthal quantum number ($l$): It determines the $3 - D$ shape of the orbital.
Its value ranges from $0$ to $n - l$. The subshells corresponding to the values of $l$ are:

Value of $l$	Sub-shell
$0$	$s$
$1$	$p$
$2$	$d$
$3$	$f$
$4$	$g$

III. Magnetic quantum number ($ml$): It determines the spatial orientation of orbitals with respect to the standard set of coordinate axes.

The value of ml$$$$ = 2l + 1

IV. Spin quantum number ($ms$): It determines the spin of electrons.
Its value can either be $+ \dfrac{1}{2}$ or $- \dfrac{1}{2}$
Now, as we know about the various quantum numbers it will be easier to calculate the total number of possible subshells when $n + l = 6$.
We know that the value of $l$ can range between $0$ to $n - l$.
Therefore, the maximum value of $n$ can be $6$.
Let’s look at the possible values of $n$ and $l$

Value of $n$	Sub-shell $l$
$6$	$0$
$5$	$1$
$4$	$2$
$3$	$3$
$2$	$4$
$1$	$5$

The values i), ii), and iii) sets appropriately as in them, the sum of $n + l = 6$ and also the value of $l$ is ranging between $0$ to $n - l$.
But, in the cases iv), v) and vi) the values of $l$ are not ranging between $0$ and $n - l$.
Thus, three cases, that is
(i) $n = 6$; $l = 0$
(ii) $n = 5$; $l = 1$
(iii) $n = 4$; $l = 2$, are possible.
Hence the correct option is A, $3$.

Note:
The value of $l$ can never be equal to or greater than $n$. That means the Azimuthal quantum number is always smaller than the principal quantum number.
For the values of $n$ and $l$, values of $ml$ and $ms$ can be further calculated by using the formulae.