Chapter 16, Problem 16.35E

Interpretation Introduction

Interpretation:

The ESR spectrum for the ${\text{ND}}_{\text{2}}$ radical at a magnetic field of $\text{3482}\text{gauss}$ is to be stated.

Concept introduction:

The full form of ESR spectroscopy is electron-spin resonance spectroscopy. The ESR spectroscopy is used to observe those materials which contain unpaired electrons. ESR spectroscopy also works like NMR spectroscopy, but, in ESR, electron spin in spite of atomic nuclei spin gets excited.

Answer to Problem 16.35E

The ESR spectrum for the ${\text{ND}}_{\text{2}}$ radical at a magnetic field of $\text{3482}\text{gauss}$ has 7 hyperfine absorptions at the values $3716\text{gauss}$, $3638\text{gauss}$, $3560\text{gauss}$, $3482\text{gauss}$, $3404\text{gauss}$, $3326\text{gauss}$ and $3248\text{gauss}$.

Explanation of Solution

The spin value of $I(\text{D})$ or $I{(}^2\text{H})$ is $1$.

The number of $\text{H}$ atom is $2$.

The spin value of $I(\text{N})$ is $1$.

The number of $\text{N}$ atom is $1$.

The given hyperfine coupling constant $a$ for deuterium, $\text{D}$ is $7\text{8}\text{gauss}$.

The given magnetic field is $\text{3482}\text{gauss}$.

The value of total $m_I$ for ${\text{ND}}_{\text{2}}$ is $\left(1\right)\left(2\right)+\left(1\right)\left(1\right)=3$.

The expression that is use to calculate the ESR signals of the ${\text{ND}}_{\text{2}}$ radical is given below.

$\text{Number}\text{of}\text{signal}=2m_I+1$…(1)

Where,

• $m_I$ is the total spin value.

Substitute the value of total spin values and number of atoms of $\text{N}$ and $\text{D}$ in equation (1).

$\begin{array}{rll}\text{Number}\text{of}\text{signal}& =& 2\times \left(1\right)\left(2\right)+1+2\times \left(1\right)\left(1\right)+1\\ & =& 5+3\\ & =& 7\end{array}$

So, the total signal of ${\text{ND}}_{\text{2}}$ radical is $7$.

Thus, the value of $m_I$ for ${\text{ND}}_{\text{2}}$ varies from $+3$ to $-3$.

The value of $B_{\text{Local}\text{environemnt}}$ for different values of $m_I=+3$ is calculated by formula given below.

$B_{\text{Local}\text{environemnt}}=B_{\text{Mag}\text{field}}+a\cdot m_I$…(1)

Where,

• $B_{\text{Mag}\text{field}}$ is the magnetic field.

• $a$ is the hyperfine coupling constant.

Substitute the values $B_{\text{Mag}\text{field}}$ and $a$ in equation (1).

$\begin{array}{rll}{B}_{\text{Local}\text{environemnt}}& =& \text{3482}\text{gauss}+7\text{8}\text{gauss}\times 3\\ & =& 3716\text{gauss}\end{array}$

The value of $B_{\text{Local}\text{environemnt}}$ for different values of $m_I=+2$ is calculated by formula given below.

Substitute the values $B_{\text{Mag}\text{field}}$ and $a$ in equation (1).

$\begin{array}{rll}{B}_{\text{Local}\text{environemnt}}& =& \text{3482}\text{gauss}+7\text{8}\text{gauss}\times 2\\ & =& 3638\text{gauss}\end{array}$

The value of $B_{\text{Local}\text{environemnt}}$ for different values of $m_I=+1$ is calculated by formula given below.

Substitute the values $B_{\text{Mag}\text{field}}$ and $a$ in equation (1).

$\begin{array}{rll}{B}_{\text{Local}\text{environemnt}}& =& \text{3482}\text{gauss}+7\text{8}\text{gauss}\times 1\\ & =& 3560\text{gauss}\end{array}$

The value of $B_{\text{Local}\text{environemnt}}$ for different values of $m_I=0$ is calculated by formula given below.

Substitute the values $B_{\text{Mag}\text{field}}$ and $a$ in equation (1).

$\begin{array}{rll}{B}_{\text{Local}\text{environemnt}}& =& \text{3482}\text{gauss}+7\text{8}\text{gauss}\times 0\\ & =& 3482\text{gauss}\end{array}$

The value of $B_{\text{Local}\text{environemnt}}$ for different values of $m_I=-1$ is calculated by formula given below.

Substitute the values $B_{\text{Mag}\text{field}}$ and $a$ in equation (1).

$\begin{array}{rll}{B}_{\text{Local}\text{environemnt}}& =& \text{3482}\text{gauss}+7\text{8}\text{gauss}\times -1\\ & =& 3404\text{gauss}\end{array}$

The value of $B_{\text{Local}\text{environemnt}}$ for different values of $m_I=-2$ is calculated by formula given below.

Substitute the values $B_{\text{Mag}\text{field}}$ and $a$ in equation (1).

$\begin{array}{rll}{B}_{\text{Local}\text{environemnt}}& =& \text{3482}\text{gauss}+7\text{8}\text{gauss}\times -2\\ & =& 3326\text{gauss}\end{array}$

The value of $B_{\text{Local}\text{environemnt}}$ for different values of $m_I=-3$ is calculated by formula given below.

Substitute the values $B_{\text{Mag}\text{field}}$ and $a$ in equation (1).

$\begin{array}{rll}{B}_{\text{Local}\text{environemnt}}& =& \text{3482}\text{gauss}+7\text{8}\text{gauss}\times -3\\ & =& 3248\text{gauss}\end{array}$

Therefore, the ESR spectrum for the ${\text{ND}}_{\text{2}}$ radical at a magnetic field of $\text{3482}\text{gauss}$ contains the 7 hyperfine absorptions at the values $3716\text{gauss}$, $3638\text{gauss}$, $3560\text{gauss}$, $3482\text{gauss}$, $3404\text{gauss}$, $3326\text{gauss}$ and $3248\text{gauss}$.

Conclusion

The ESR spectrum for the ${\text{ND}}_{\text{2}}$ radical at a magnetic field of $\text{3482}\text{gauss}$ has 7 hyperfine absorptions at the values $3716\text{gauss}$, $3638\text{gauss}$, $3560\text{gauss}$, $3482\text{gauss}$, $3404\text{gauss}$, $3326\text{gauss}$ and $3248\text{gauss}$.