Question

Answer the following questions for Zn atom:
How many electrons have $n + l + m = 4?$

Answer 1

Atomic number of zinc is $30$ . Using the atomic number, we would write the electronic configuration of Zinc. An understanding of the quantum numbers is vital here to get the answer.

Complete step by step solution:
Firstly let us write the electronic configuration of Zinc (atomic number $= 30$ ).
$1{s^2}2{s^2}2{p^6}3{s^2}3{p^6}3{d^{10}}4{s^2}$
Or $\left[ {Ar} \right]3{d^{10}}4{s^2}$
Now, let us look at the quantum numbers:
There are four quantum numbers which are used to define the location and spin of any electron in an atom. Quantum numbers give us the address of an electron, they can give us information which are general or even very specific.
So, let me first tell you about the quantum numbers in brief to refresh your memory, I will explain the three quantum numbers that are required to solve this problem and also the values of each for this question:
$n$ is the principal quantum number which gives us the information that which energy level does the electron lie in. Its value ranges from $1$ to a shell containing the outermost electrons. For Zinc, the values of n can be $1,{{ }}2,{{ }}3{{ }} and {{ }}4.$
$l$ is the angular momentum quantum number that gives us the information about the sublevel to which a particular electron belongs to.
The value of $l$ is from $0{{ }} to {{ }}n - 1$ .
$s$ subshell has $l = 0{{ }}\left( {n - 1 = 1 - 1 = 0} \right)$ , $p$ has $l{{ }} = {{ }}1{{ }}\left( {n - 1 = 2 - 1 = 1} \right)$ , $d$ has $l = 2{{ }}\left( {n - 1 = 3 - 1 = 2} \right).$
$m$ is the magnetic quantum number tells which orbital the electron is in. Its value ranges from $+ l{{ }}to {{ }}-l$ . Thus for Zinc the value of m could be from $+ 3{{ }} to {{ }} - 3.$
Now, let us see how many electrons can have $n + l + m = {{ }}4$
Case one : $n = 4,l = 0,m = 0$ means $\left[ {4s} \right]$ which have two electrons.
Case two : $n = 3,l = 1,m = 0$ means $\left[ {3p} \right]\;$ which have six electrons.
Case three : $n = 2,l = 1,m = 1$ means $\left[ {3p} \right]\;$ which have six electrons.
So total $14$ electrons can have $n + l + m = 4$
So the answer is $14$ electrons.

Additional Information: Each electron has its own set of four quantum numbers. Something similar like people have unique fingerprints. Quantum number "is telling" us everything about electrons.

Note:
While solving such type of questions we should take the help of the table below:

Number	Symbol	Possible values
Principal quantum number	$n$	$1,2,3,4, \ldots$
angular momentum quantum number	$l$	$0,1,2,3, \ldots {{ }}\left( {{{ }}0{{ }}to{{ }}n - 1} \right)$
magnetic quantum number	$m$	$- l{{ }}to{{ }} + l$
spin quantum number	${{s }} or {{ }}{m_s}$	$+ 1/2,{{ }} - 1/2$