Question

For $_{24}Cr$ which of the following statements is/are correct?
(A) Number of electrons with principal quantum number $3$ is $13$
(B) Number of electrons with azimuthal quantum number $1$ is $12$
(C) Number of electrons with magnetic quantum number $\;0$ is $12$
(D) Number of minimum or maximum electrons with spin quantum number $+ \dfrac{1}{2}$ is $9$ or $15$

Answer 1

As we know that quantum number is a set of numbers with the help of which we can locate the position of an electron in an atom. There are four quantum numbers in total and using them we can calculate the number of electrons in a shell and subshell of an element.

Complete step by step solution:
We all know that there are four quantum numbers including:
Principal quantum number tells us about the size of the electron cloud and energy level of the electron and is represented by ‘n’. The shells K, L, M, N, O, P have the $n = 1,\;2,\;3,\;4,\;5\;,6$ respectively. The azimuthal quantum number tells us about the shape of electron could and subshell to which the electron belongs, represented as ‘l’ and subshells s, p, d, f, g have $l = 0,\;1,\;2,\;3,\;4$ respectively. The magnetic quantum number tells us about the orientation of the electron cloud, represented by ‘m’ and for a given value of l, m can be –l to 0 to +l and finally the spin quantum number which tells about the spin or rotation of the electron in its own axis.
So, for $_{24}Cr$whose electronic configuration in ground state is $1{s^2}2{s^2}2{p^6}3{s^2}3{p^6}4{s^2}3{d^4}$ and in excited state the is $1{s^2}2{s^2}2{p^6}3{s^2}3{p^6}4{s^1}3{d^5}$ hence we can say that the last electron is present in 3rd shell so $n$ for this element is $3$ and number of electrons it can accommodate is $2{n^2} = 2 \times {3^2} = 18$ and the 3rd shell accommodated $13$ electrons. So the first statement is correct.
The azimuthal quantum number can be calculated as $l = 0\;to\;(n - 1)$ so it can have the values as $0$,$1$ and $2$ and for $l = 1$, $n = 2$ so the number of electrons will be $8$. Hence the second statement is incorrect.
Now, for $m = 0$ the value of $l$ will be $l = 1$ so the number of electrons will be $8$. Hence the third statement is incorrect as well.
Lastly the spin of electrons can be positive and negative and one orbital can possess only two electrons therefore the number of electrons with positive spin $+ \dfrac{1}{2}$ can be $9$ or $15$. Hence this is the correct statement.

Therefore from the above explanation we can say that statements A and D are correct.

Note: In nutshell, the number of subshells in a principal shell is ‘n’, the number of orbitals in a shell is ${n^2}$ and maximum number of electrons in a shell is $2{n^2}$. The number of orbitals in a subshell is $2l + 1$ and maximum number of electrons in a subshell is $2(2l + 1)$.