Question

The total value of m for the electrons (n=4) is:
(A) 4
(B) 8
(C) 16
(D) 32

Answer 1

To solve this question, we first need to understand what quantum numbers are. The energy and position of an electron in an atom that has various shells, subshells, and orbitals are governed by quantum numbers.

Complete answer:
There are four quantum numbers.

1. Principal quantum number (n): This corresponds to the shells of an atom
   It has integer values n=1,2,3,4…
   The shell closest to the nucleus, K, has principal quantum number 1. The shell next to K is L and has principal quantum number 2, and so on.
   The maximum number of electrons in a shell is given by $2{{n}^{2}}$.
2. Azimuthal quantum number ($\ell$): This gives the number of subshells in a shell.
   The number of subshells in a shell range from 0 to (n-1).
   It has values $\ell$=0,1,2,3 corresponding to s, p, d, and f.
3. Magnetic quantum number (m): This gives the number of orbitals in a subshell.
   The number of orbitals in a subshell range from -$\ell$ to +$\ell$, including 0.
4. Spin quantum number (s): This states that an electron can have either a positive spin-up $(+\dfrac{1}{2})$ or a negative spin-down $(-\dfrac{1}{2})$.
   Now, it is given to us that the value of n = 4.
   - So, the value of $\ell$ will range from 0 to 3.
   $\ell$ = 0, 1, 2, 3
   - Now, the value of m will be
   For $\ell$ = 0 (s), m = 0
   For $\ell$ = 1 (p), m = -1, 0, 1
   For $\ell$ = 2 (d), m = -2, -1, 0, 1, -1
   For $\ell$ = 3 (f), m = -3, -2, -1, 0, 1, 2, 3
   So, the total number of orbitals is 16.

Hence, the total value of m for the electrons (n=4) is option (C) 16.

Note:
It should be noted that the filling of electrons in the shells of an atom is governed by
- The Aufbau Principle states that the electrons are filled in the increasing order of the energy of the orbitals.
- The Hund's Rule states that if two or more orbitals of the same energy are present, all of them must be half-filled before pairing up.
- The Pauli's exclusion principle states that two electrons in an atom cannot have the same values of all four quantum numbers. Hence there can only be two electrons in an orbital.