Question

The total spin resulting from a ${d^7}$ configuration is:
A.1
B.2
C.$\dfrac{5}{2}$
D.$\dfrac{3}{2}$

Answer 1

Each orbital in an atom is characterized by a unique set of values of the three quantum numbers n, $\ell$ , and m which respectively correspond to the electron's energy, angular momentum, the magnetic quantum number. Each such orbital can be occupied by a maximum of two electrons, each with its own spin quantum number ‘s’

Complete Step-by-Step Answer:
Before we move forward with the solution of the given question, let us first understand some important basic concepts.
All atoms in their elemental state have electrons present in them. Now these electrons are revolving around the nucleus. The movement of these electrons around the nucleus was initially explained by a concept known as energy shells. This stated that only a certain number of electrons could revolve around the nucleus at fixed distances from the nucleus. These paths were defined as circular orbits and were referred to as energy shells. They were called energy shells because the potential energy possessed by the electrons varied from one shell to another. The smallest shell had the highest energy level while the largest shell had the lowest energy level.
Even after this model was explained, there were certain qualms about the determination of the orientation of these electrons. This issue was addressed by introducing the concept of atomic orbitals. Atomic orbitals can be understood as a mathematical function which describes the location and wave like behaviour of an electron in an atom. There are 4 main types of atomic orbitals: s, p, d, f. These orbitals have the maximum capacity of 2, 6, 10, 14 electrons, respectively. Electrons are filled in these orbitals based on Hund’s rule of maximum multiplicity.
Hence, for the given question, the ${d^7}$ orbital can be represented as:

$\uparrow \downarrow$	$\uparrow \downarrow$	$\uparrow$	$\uparrow$	$\uparrow$

Where each arrow represents an arrow represents an electron and the direction of the arrow represents the spin of the electron. The first four electrons cancel out the spin of each other. Hence, the total spin of ${d^7}$ comes from the lone electrons in the last 3 subshells. Since each electron corresponds to a spin of $\dfrac{1}{2}$ , we can say that
The total spin of ${d^7} = \dfrac{1}{2} + \dfrac{1}{2} + \dfrac{1}{2} = \dfrac{3}{2}$
Hence, Option D is the correct option

Note: Hund’s multiplicity rule states that a given subshell is singly occupied by only one electron before any one orbital is doubly occupied, and all electrons in the singly occupied orbitals have the same spin.