Chapter 15, Problem 15.22E

Interpretation Introduction

Interpretation:

The total number of states in an atom which has $d^6$ electron configuration for its valence subshell is to be predicted.

Concept introduction:

A term symbol describes the orbital, spin, and total angular momenta of an electronic state. The symbol, $L$, represents the vector sum of orbital angular momentum. The symbol, $S$, represents the vector sum of spin angular momentum. The symbol, $J$, represents the total angular momentum. Therefore, the term symbol is represented as given below.

${}^{2S+1}{L}_J$

Answer to Problem 15.22E

The total number of states in an atom which has $d^6$ electron configuration for its valence subshell is $208$.

Explanation of Solution

The value of an azimuthal quantum number, $l$, for each $d$ electron is $2$. The possible term symbols for $d^6$ electron configuration is ${}^1\text{S}$, ${}^1\text{S}$, ${}^1\text{D}$, ${}^1\text{D}$, ${}^1\text{F}$, ${}^1\text{F}$, ${}^1\text{G}$, ${}^1\text{G}$, ${}^1\text{I}$, ${}^3\text{P}$, ${}^3\text{P}$, ${}^3\text{D}$, ${}^3\text{F}$, ${}^3\text{F}$, ${}^3\text{G}$, ${}^3\text{H}$, and ${}^5\text{D}$ from Table 15.1. For the complete term symbols, the value of $J$ is calculated first. The value of $J$ can be calculated by the formula shown below.

$J=L+S\to \left|L-S\right|$

Where $J$, $L$, and $S$ are quantum numbers.

Each spin can contribute $s=1/2$. The value of $S$ can be calculated from the superscript of the term symbols, which is $2S+1$. The values of $2S+1$ in the given term symbols are $1$, $3$, and $5$. The values of $S$ for the given term symbols are calculated as shown below.

$\begin{array}{rll}2S+1& =& 1\\ S& =& 0\\ 2S+1& =& 3\\ S& =& 1\\ 2S+1& =& 5\\ S& =& 2\end{array}$

The value of $J$ for the ${}^1\text{S}$ term symbol is shown below.

$\begin{array}{l}J=0+0\to \left|0-0\right|\\ J=0\end{array}$

The value of $J$ for the ${}^1\text{S}$ term symbol is $0$. The term symbol is ${}^1{\text{S}}_0$.

The value of $J$ for the ${}^1\text{D}$ term symbol is shown below.

$\begin{array}{l}J=2+0\to \left|2-0\right|\\ J=2\end{array}$

The value of $J$ for the ${}^1\text{D}$ term symbol is $2$. The term symbol is ${}^1{\text{D}}_2$.

The value of $J$ for the ${}^1\text{F}$ term symbol is shown below.

$\begin{array}{l}J=3+0\to \left|3-0\right|\\ J=3\end{array}$

The value of $J$ for the ${}^1\text{F}$ term symbol is $3$. The term symbol is ${}^1{\text{F}}_3$.

The value of $J$ for the ${}^1\text{G}$ term symbol is shown below.

$\begin{array}{l}J=4+0\to \left|4-0\right|\\ J=4\end{array}$

The value of $J$ for the ${}^1\text{G}$ term symbol is $4$. The term symbol is ${}^1{\text{G}}_4$.

The value of $J$ for the ${}^1\text{I}$ term symbol is shown below.

$\begin{array}{l}J=6+0\to \left|6-0\right|\\ J=6\end{array}$

The value of $J$ for the ${}^1\text{I}$ term symbol is $6$. The term symbol is ${}^1{\text{I}}_6$.

The value of $J$ for the ${}^3\text{P}$ term symbol is shown below.

$\begin{array}{l}J=1+1\to \left|1-1\right|\\ J=2\to 0\end{array}$

The value of $J$ for the ${}^3\text{P}$ term symbol is $2,1$, and $0$. The term symbols are ${}^3{\text{P}}_2$, ${}^3{\text{P}}_1$ and ${}^3{\text{P}}_0$.

The value of $J$ for the ${}^3\text{D}$ term symbol is shown below.

$\begin{array}{l}J=2+1\to \left|2-1\right|\\ J=3\to 1\end{array}$

The value of $J$ for the ${}^3\text{D}$ term symbol is $3,2$, and $1$. The term symbols are ${}^3{\text{D}}_3$, ${}^3{\text{D}}_2$, and ${}^3{\text{D}}_1$.

The value of $J$ for the ${}^3\text{F}$ term symbol is shown below.

$\begin{array}{l}J=3+1\to \left|3-1\right|\\ J=4\to 2\end{array}$

The value of $J$ for the ${}^3\text{F}$ term symbol is $4,3$ and $2$. The term symbols are ${}^3{\text{F}}_4$, ${}^3{\text{F}}_3$, and ${}^3{\text{F}}_2$.

The value of $J$ for the ${}^3\text{G}$ term symbol is shown below.

$\begin{array}{l}J=4+1\to \left|4-1\right|\\ J=5\to 3\end{array}$

The value of $J$ for the ${}^3\text{G}$ term symbol is $5,4$, and $3$. The term symbols are ${}^3{\text{G}}_5$, ${}^3{\text{G}}_4$, and ${}^3{\text{G}}_3$.

The value of $J$ for the ${}^3\text{H}$ term symbol is shown below.

$\begin{array}{l}J=5+1\to \left|5-1\right|\\ J=6\to 4\end{array}$

The value of $J$ for the ${}^3\text{H}$ term symbol is $6,5$, and $4$. The term symbols are ${}^3{\text{H}}_6$, ${}^3{\text{H}}_5$, and ${}^3{\text{H}}_4$.

The value of $J$ for the ${}^5\text{D}$ term symbol is shown below.

$\begin{array}{l}J=2+2\to \left|2-2\right|\\ J=4\to 0\end{array}$

The value of $J$ for the ${}^5\text{D}$ term symbol is $4,3,2,1$, and $0$. The term symbols are ${}^5{\text{D}}_4$, ${}^5{\text{D}}_3$, ${}^5{\text{D}}_2$, ${}^5{\text{D}}_1$, and ${}^5{\text{D}}_0$.

The degeneracy is calculated by the formula $2J+1$. The total term symbols are ${}^1{\text{S}}_0$, ${}^1{\text{S}}_0$, ${}^1{\text{D}}_2$, ${}^1{\text{D}}_2$, ${}^1{\text{F}}_3$, ${}^1{\text{F}}_3$, ${}^1{\text{G}}_4$, ${}^1{\text{G}}_4$, ${}^1{\text{I}}_6$, ${}^3{\text{P}}_2$, ${}^3{\text{P}}_1$, ${}^3{\text{P}}_0$, ${}^3{\text{P}}_2$, ${}^3{\text{P}}_1$, ${}^3{\text{P}}_0$ ${}^3{\text{D}}_3$, ${}^3{\text{D}}_2$, ${}^3{\text{D}}_1$, ${}^3{\text{F}}_4$, ${}^3{\text{F}}_3$, ${}^3{\text{F}}_2$, ${}^3{\text{F}}_4$, ${}^3{\text{F}}_3$, ${}^3{\text{F}}_2$, ${}^3{\text{G}}_5$, ${}^3{\text{G}}_4$, ${}^3{\text{G}}_3$, ${}^3{\text{H}}_6$, ${}^3{\text{H}}_5$, ${}^3{\text{H}}_4$ ${}^5{\text{D}}_4$, ${}^5{\text{D}}_3$, ${}^5{\text{D}}_2$, ${}^5{\text{D}}_1$, and ${}^5{\text{D}}_0$. The degeneracy of the term symbols are $1$, $1$, $5$, $5$, $7$, $7$, $9$, $9$, $13$, $5$, $3$, $1$, $7$, $5$, $3$, $9$, $7$, $5$, $9$, $7$, $5$, $11$, $9$, $7$, $13$, $11$, $9$, $9$, $7$, $5$, $3$, and $1$ The total number of states is calculated by taking the sum of the degeneracy of each state. The total number of states are $208$.

Conclusion

The total number of states in an atom which has $d^6$ electron configuration for its valence subshell is $208$.