Question

How many total orbitals in shell $n = 4?$ What is the relationship between the total number of shells and the quantum number $n$ for that shell?

Answer 1

The principal quantum number $n$ describes the energy of an electron and is always a positive integer. Each atom has in general many orbitals associated with each value of $n,$ these orbitals together are sometimes called electron shells. The azimuthal quantum number $l$ describes the orbital angular momentum of each electron and is non-negative integer. Within a shell where $n$ in some integer ${n_0},l$ ranges across all values satisfying the relation $0 \leqslant l \leqslant {n_0} - 1$.

Complete step by step answer:
We have ${n^2}$ orbitals in one energy level and $n$ subshell in one energy level
Number of orbitals in a shell is given by ${n^2}$
Given number of orbitals in the given shell $= {n^2} = {4^2} = 16$
Number of subshells in the shell are equal to the principal quantum number of the shell.
Y Pauli's exclusion principle: Maximum number of electrons $32$ and an orbital can contain two electrons only.
$l = n - 1 = 4 - 1 = 3.$
Number of orbitals is $= 2l + 1 = 2 \times 3 + 1 = 7$
Therefore factor it can accommodate a total of electron i.e. $2 \times 7 = 14.$
For $n = 4$ there are $16$ orbitals, $4$ subshell, $32$ electrons and $14$ electrons with $l = 3,$
Subshells are usually identified by their $n -$ and $l -$values. $n$ is represented by its numerical value, but $l$ is represented as follows:
$0$ is represented by $'s'$, $'p'$, $2$ by $'d'$, $3$ by $'f'$ and $4$ by $'9'$

Additional Information:
The magnetic quantum number, ${m_l}$ describes the magnetic moment of an electron in arbitrary direction and is also always an integer. Within a subshell where $l$ is some integer ${l_0},{m_l}$ ranges thus.
$- {l_0} \leqslant {m_1} \leqslant {l_0}.$ Servable rules govern the placement of electrons in orbitals. The first dictates that no two electrons in an atom may have the set of values of quantum numbers. These quantum numbers include the three that defend orbitals as well $s,$ or spin quantum number. Thus, two electrons may occupy a single orbital, so long as they have different values of $s.$ However only two electrons, because of their spin can be associated with each orbital. Additionally an electron always tends to fall to the lowest possible energy state. It is possible for it to occupy any orbital so long as it does not violate the Pauli Exclusion Principle, but if lower energy orbitals are available, this condition is unstable. The electron will eventually lose energy and drop into the lower orbital. Thus electrons fill orbitals in the order given above.

Note: The Pauli exclusion principle states that no two electrons in an atom can have the same values of all four quantum numbers. If there are two electrons in an orbital with given values for three quantum numbers, $\left( {n,l,m} \right)$ these two electrons must differ in their spin. The electrons do not orbit the nucleus in the manner of a planet orbiting the sun, but instead exist as standing waves. Thus the lowest possible energy an electron can take is similar to the fundamental frequency of a wave on a string. The electrons are never in a single point loration, although the probability of interacting with an electron at a single point can be found can be found from the wave function of the electron.