Chapter 16, Problem 16.22E

Interpretation Introduction

(a)

Interpretation:

The spin and number of equivalent nuclei required to produce the given ESR spectrum are to be stated.

Concept introduction:

Electron spin resonance is the method to study the materials with an unpaired electron. An unpaired electron absorbs the microwave radiation under the strong magnetic field. The difference between two energy levels is calculated by the formula shown below.

$\Delta E_{\text{mag}}=g_{\text{e}}\cdot {\mu }_{\text{B}}\cdot B$

Answer to Problem 16.22E

The spin and number of equivalent nuclei required to produce the given ESR spectrum are $\frac{1}{2}$ and $1$ respectively.

Explanation of Solution

The give ESR spectrum is shown below.

Figure 1

The number of spins in the ESR spectrum is shown below.

$\text{number of signals}=\underset{\begin{array}{l}i=\text{number}\\ \text{of atoms}\end{array}}{\Pi }2N_i{I}_i+1$…(1)

Where,

• $N_i$ is the number of equivalent nuclei $i^{\text{th}}$ atom.

• $I$ is the nuclear spin of equivalent nuclei $i^{\text{th}}$ atom.

The total number of signal in the given ESR spectrum is $3$.

Three different peaks with same intensity indicate that $2$ nuclei are present of the same atom and have of nuclear spin are present.

Substitute the values of $N$ and number of signal in the equation (1).

$\begin{array}{rll}\text{3}& =& 2\left(1\right)I+1+2\left(1\right)I\\ \text{3}& =& 4I+1\\ \text{2}& =& 4I\end{array}$

The above equation is further simplified for the value of $I$ as shown below.

$I=\frac{1}{2}$

Therefore, the spin and number of equivalent nuclei required to produce the given ESR spectrum are $\frac{1}{2}$ and $1$ respectively.

Conclusion

The spin and number of equivalent nuclei required to produce the given ESR spectrum are $\frac{1}{2}$ and $1$ respectively.

Interpretation Introduction

(b)

Interpretation:

The spin and number of equivalent nuclei required to produce the given ESR spectrum are to be stated.

Concept introduction:

Electron spin resonance is the method to study the materials with an unpaired electron. An unpaired electron absorbs the microwave radiation under the strong magnetic field. The difference between two energy levels is calculated by the formula shown below.

$\Delta E_{\text{mag}}=g_{\text{e}}\cdot {\mu }_{\text{B}}\cdot B$

Answer to Problem 16.22E

The spin and number of equivalent nuclei required to produce the given ESR spectrum are $\frac{1}{2}$ and $2$ respectively.

Explanation of Solution

The give ESR spectrum is shown below.

Figure 2

The number of spins in the ESR spectrum is shown below.

$\text{number of signals}=\underset{\begin{array}{l}i=\text{number}\\ \text{of atoms}\end{array}}{\Pi }2N_i{I}_i+1$…(1)

Where,

• $N_i$ is the number of equivalent nuclei $i^{\text{th}}$ atom.

• $I$ is the nuclear spin of equivalent nuclei $i^{\text{th}}$ atom.

The total number of signal in the given ESR spectrum is $3$.

Three peaks with different intensities indicate that two equivalent nuclei are present.

Substitute the values of $N$ and number of signal in the equation (1).

$\begin{array}{rll}\text{3}& =& 2\left(2\right)I+1\\ \text{3}& =& 4I+1\\ \text{2}& =& 4I\end{array}$

The above equation is further simplified for the value of $I$ as shown below.

$I=\frac{1}{2}$

Therefore, spin and number of equivalent nuclei required to produce the given ESR spectrum are $\frac{1}{2}$ and $2$ respectively.

Conclusion

The spin and number of equivalent nuclei required to produce the given ESR spectrum are $\frac{1}{2}$ and $2$ respectively.

Interpretation Introduction

(c)

Interpretation:

The spin and number of equivalent nuclei required to produce the given ESR spectrum are to be stated.

Concept introduction:

Electron spin resonance is the method to study the materials with an unpaired electron. An unpaired electron absorbs the microwave radiation under the strong magnetic field. The difference between two energy levels is calculated by the formula shown below.

$\Delta E_{\text{mag}}=g_{\text{e}}\cdot {\mu }_{\text{B}}\cdot B$

Answer to Problem 16.22E

The spin and number of equivalent nuclei required to produce the given ESR spectrum are $\frac{1}{2}$ and $1$ respectively.

Explanation of Solution

The give ESR spectrum is shown below.

Figure 3

The number of spins in the ESR spectrum is shown below.

$\text{number of signals}=\underset{\begin{array}{l}i=\text{number}\\ \text{of atoms}\end{array}}{\Pi }2N_i{I}_i+1$…(1)

Where,

• $N_i$ is the number of equivalent nuclei $i^{\text{th}}$ atom.

• $I$ is the nuclear spin of equivalent nuclei $i^{\text{th}}$ atom.

The total number of signal in the given ESR spectrum is $5$.

Five different peaks with same intensity represents that $4$ nuclei are present of the same atom and have of nuclear spin are present.

Substitute the values of $N$ and number of signal in the equation (1).

$\begin{array}{rll}\text{5}& =& 2\left(1\right)I+1+2\left(1\right)I+1+2\left(1\right)I+1+2\left(1\right)I+1\\ \text{5}& =& 8I+1\\ \text{4}& =& 8I\end{array}$

The above equation is further simplified for the value of $I$ as shown below.

$I=\frac{1}{2}$

The spin and number of equivalent nuclei required to produce the given ESR spectrum are $\frac{1}{2}$ and $1$ respectively.

Conclusion

The spin and number of equivalent nuclei required to produce the given ESR spectrum are $\frac{1}{2}$ and $1$ respectively.

Interpretation Introduction

(d)

Interpretation:

The spin and number of equivalent nuclei required to produce the given ESR spectrum are to be stated.

Concept introduction:

Electron spin resonance is the method to study the materials with an unpaired electron. An unpaired electron absorbs the microwave radiation under the strong magnetic field. The difference between two energy levels is calculated by the formula shown below.

$\Delta E_{\text{mag}}=g_{\text{e}}\cdot {\mu }_{\text{B}}\cdot B$

Answer to Problem 16.22E

The spin and number of equivalent nuclei required to produce the given ESR spectrum are $\frac{1}{2}$ and $4$ respectively.

Explanation of Solution

The give ESR spectrum is shown below.

Figure 4

The number of spins in the ESR spectrum is shown below.

$\text{number of signals}=\underset{\begin{array}{l}i=\text{number}\\ \text{of atoms}\end{array}}{\Pi }2N_i{I}_i+1$…(1)

Where,

• $N_i$ is the number of equivalent nuclei $i^{\text{th}}$ atom.

• $I$ is the nuclear spin of equivalent nuclei $i^{\text{th}}$ atom.

The total number of signal in the given ESR spectrum is $5$.

Five different peaks with different intensities indicate that four equivalent nuclei are present.

Substitute the values of $N$ and number of signal in the equation (1).

$\begin{array}{rll}\text{5}& =& 2\left(4\right)I+1\\ \text{5}& =& 8I+1\\ \text{4}& =& 8I\end{array}$

The above equation is further simplified for the value of $I$ as shown below.

$I=\frac{1}{2}$

Therefore, the spin and number of equivalent nuclei required to produce the given ESR spectrum are $\frac{1}{2}$ and $4$ respectively.

Conclusion

The spin and number of equivalent nuclei required to produce the given ESR spectrum are $\frac{1}{2}$ and $4$ respectively.