Question

How many electrons can fill the energy shell described by principal quantum number $\mathbf{n}=\mathbf{4?}$

Answer 1

Principal quantum number $(n)$ is an electron shell. It describes the most probable distance of an arbitrary electron from the nucleus. So, larger the number of $n$, the number of allowed electrons increases. There are other types of quantum numbers that are also described by mechanical models and the number of electrons depends upon all quantum numbers.

Complete step by step answer:
The Schrödinger wave equation—It is also called a wave mechanical model or the quantum mechanical model of the atom.
This quantum mechanical model assigns four quantum number:
Principal quantum number $(n)$
Orbital Angular momentum quantum number $(l)\to$ Determine the shape of orbital.
$l=(n=1),$ if.
$n=1,\ l=0,$ S (Spherical shape artis)
$n=2,\ l=0,\,\ 1,$ P (dumb bell shape), Two possible subshell
$n=3,\ l=0,\ \,1,\ 2$ d, three possible subshell
The magnetic quantum number $(\text{ml})\to$ Determine the number of orbitals in subshell
Ml $=2l+l$
The electron spin quantum number $(\text{ms})=$ It determines the direction of electron spin, May have a spin of $\dfrac{+1}{2}$ represent by $\uparrow$ or $\dfrac{-1}{2}$ represented by $\downarrow$
The number of electrons in the subshell depends on the value of $l$. In each orbital. There is two electron of opposite spin so, if number of orbital i.e. $ml=2l+1,$then the number of electron will be
$2(2l+1)$
For ex: 25 subshell, 1 has a maximum of 2 electrons in it, since $2(2l+1)=2(0+1)=2$ for this subshell.
Some algebra shows that the maximum number of electrons that can be in a shell is $2{{n}^{2}}$.
So, according to above give question, $n=4$
The no. of electron will be
$2{{n}^{2}}=Q\times {{4}^{2}}=32,$ Mean that 4th energy shell can hold 32 electrons

Note:
Pauli’s exclusion principle state that each orbital can hold a maximum of 2 electrons and no. two electrons can have the same set of quantum number
$\psi ={{\psi }_{1}}(a)\ {{\psi }_{2}}(b)$
Where, $\psi =$ Probability amplitude o that electron 1 is in state $a$ and electron $2$ is in state $b$.
${{\psi }_{1}}(a)=$ probability amplitude that electron $1$ is in state $a$.
${{\psi }_{1}}(b)=$ probability amplitudes that electron $2$ is in state $b$.