Chapter 5.3, Problem 31SSC

Interpretation Introduction

Interpretation:

The sequence in which ten electrons occupy the five orbitals related to an atom’s $d$ subshell needs to be described.

Concept introduction: In an atom,  subshell is defined as the set of states in a given shell that have the same azimuthal quantum number $\left(\mathcal{l}\right)$.

Answer to Problem 31SSC

The $d$ -subshell can contain $10$ electrons.

Therefore, firstly the $5$ atomic orbitals are occupied with one electron each $d_z{}^2\uparrow , d_{xy}\uparrow , d_{xz}\uparrow , d_{yz}\uparrow ,and d_x{}^2-y^2\uparrow$

Then the remaining five electrons, results in the pairing of electrons in each d-atomic orbital.

$d_z{}^2\uparrow \downarrow , d_{xy}\uparrow \downarrow , d_{xz}\uparrow \downarrow , d_{yz}\uparrow \downarrow ,and d_x{}^2-y^2\uparrow \downarrow$

Explanation of Solution

In an atom,  subshell is defined as the set of states in a given shell that have the same azimuthal quantum number $\left(\mathcal{l}\right)$.  

For a  $d$ -subshell, the value of azimuthal quantum number $\left(\mathcal{l}\right)$ is $3$. Also, a d-subshell contains $5$ atomic orbitals,  namely,   $d_z{}^2, d_{xy}, d_{xz}, d_{yz} ,$ and  $d_x{}^2-y^2$ that can be occupied by $2$ electrons each.

The filling of electrons in a subshell, such as the $d$ -subshell, of a given shell is governed by the Hund's Rule of maximum multiplicity.

According to this rule, firstly all the $5$ atomic orbitals of the d-subshell are singly filled and then the electrons are paired.

Since, a $d$ -subshell can contain $10$ electrons.

Therefore, firstly the $5$ atomic orbitals are occupied with one electron each $d_z{}^2\uparrow , d_{xy}\uparrow , d_{xz}\uparrow , d_{yz}\uparrow ,and d_x{}^2-y^2\uparrow$

Then the remaining five electrons, results in the pairing of electrons in each d-atomic orbital.

$d_z{}^2\uparrow \downarrow , d_{xy}\uparrow \downarrow , d_{xz}\uparrow \downarrow , d_{yz}\uparrow \downarrow ,and d_x{}^2-y^2\uparrow \downarrow$

Conclusion

The $d$ -subshell can contain $10$ electrons.

Therefore, firstly the $5$ atomic orbitals are occupied with one electron each $d_z{}^2\uparrow , d_{xy}\uparrow , d_{xz}\uparrow , d_{yz}\uparrow ,and d_x{}^2-y^2\uparrow$

Then the remaining five electrons, results in the pairing of electrons in each d-atomic orbital.

$d_z{}^2\uparrow \downarrow , d_{xy}\uparrow \downarrow , d_{xz}\uparrow \downarrow , d_{yz}\uparrow \downarrow ,and d_x{}^2-y^2\uparrow \downarrow$