Chapter 16, Problem 16.8E

Interpretation Introduction

(a)

Interpretation:

The values of $\Delta E$ in the transition energies for each individual transition for the ${}^{\text{1}}\text{S}{\to }^{\text{1}}\text{P}$ transition when a sample is exposed to a magnetic field of $2.\text{35 T}$ are to be calculated.

Concept introduction:

The phenomenon of splitting of a spectral line when a magnetic field is applied to it is known as Zeeman Effect. Magnetic field strength can be measured by using the Zeeman Effect. Applications of Zeeman Effect include NMR spectroscopy, MRI and electron spin resonance spectroscopy.

During an electronic transition, an electron from ground state moves straight to the excited state keeping the internuclear distance constant.

The change in the energy of the state, $\Delta E$ is calculated by the formula shown below.

$\Delta E={\mu }_{\text{B}}\cdot M_L\cdot B$

Answer to Problem 16.8E

The values of $\Delta E$ in the transition energies for ${}^{\text{1}}\text{P}$ in the ${}^{\text{1}}\text{S}{\to }^{\text{1}}\text{P}$ transition are $-2.18\times {10}^{-23}\text{ J}$, $0\text{ J}$ and $+2.18\times {10}^{-23}\text{ J}$.

Explanation of Solution

In case of ${}^{\text{1}}\text{S}{\to }^{\text{1}}\text{P}$ transition, the splitting of ${}^{\text{1}}\text{S}$ state does not take place because for this state, the value of $L=0$ and so $M_L=0$. Therefore, splitting of only ${}^{\text{1}}\text{P}$ state occurs due to the presence of degenerate $M_L=-1\text{, }0$ and $+1$ states. The value of $\Delta E$ is calculated by the formula shown below.

$\Delta E={\mu }_{\text{B}}\cdot M_L\cdot B$…(1)

Where,

• ${\mu }_{\text{B}}$ is the Bohr magneton $\left(9.274\times {10}^{-24}\text{J}/\text{T}\right)$.

• $M_L$ is the magnetic quantum number.

• $B$ is the magnetic field.

The value of magnetic field is $2.\text{35 T}$.

For, $M_L=-1$.

Substitute the values of ${\mu }_{\text{B}}$, $B$ and $M_L=-1$ in equation (1) to calculate the change in energy.

$\begin{array}{rll}\Delta E& =& 9.274\times {10}^{-24}\text{J}/\text{T}\times -1\times 2.3\text{5 T}\\ & =& -2.179\times {10}^{-23}\text{ J}\\ & \approx & -2.18\times {10}^{-23}\text{ J}\end{array}$

For, $M_L=0$

Substitute the values of ${\mu }_{\text{B}}$, $B$ and $M_L=0$ in equation (1) to calculate the change in energy.

$\begin{array}{rll}\Delta E& =& 9.274\times {10}^{-24}\text{J}/\text{T}\times 0\times 2.3\text{5 T}\\ & =& 0\text{ J}\end{array}$

For, $M_L=+1$.

Substitute the values of ${\mu }_{\text{B}}$, $B$ and $M_L=+1$ in equation (1) to calculate the change in energy.

$\begin{array}{rll}\Delta E& =& 9.274\times {10}^{-24}\text{J}/\text{T}\times +1\times 2.3\text{5 T}\\ & =& +2.179\times {10}^{-23}\text{ J}\\ & \approx & +2.18\times {10}^{-23}\text{ J}\end{array}$

Therefore, the values of $\Delta E$ in the transition energies for ${}^{\text{1}}\text{P}$ in the ${}^{\text{1}}\text{S}{\to }^{\text{1}}\text{P}$ transition are $-2.18\times {10}^{-23}\text{ J}$, $0\text{ J}$ and $+2.18\times {10}^{-23}\text{ J}$.

Conclusion

The values of $\Delta E$ in the transition energies for ${}^{\text{1}}\text{P}$ in the ${}^{\text{1}}\text{S}{\to }^{\text{1}}\text{P}$ transition are $-2.18\times {10}^{-23}\text{ J}$, $0\text{ J}$ and $+2.18\times {10}^{-23}\text{ J}$.

Interpretation Introduction

(b)

Interpretation:

The values of $\Delta E$ in the transition energies for each individual transition for the ${}^{\text{1}}\text{P}{\to }^{\text{1}}\text{D}$ transition when a sample is exposed to a magnetic field of $2.\text{35 T}$ are to be calculated.

Concept introduction:

The phenomenon of splitting of a spectral line when a magnetic field is applied to it is known as Zeeman Effect. Magnetic field strength can be measured by using the Zeeman Effect. Applications of Zeeman Effect include NMR spectroscopy, MRI and electron spin resonance spectroscopy.

During an electronic transition, an electron from ground state moves straight to the excited state keeping the internuclear distance constant.

The change in the energy of the state, $\Delta E$ is calculated by the formula shown below.

$\Delta E={\mu }_{\text{B}}\cdot M_L\cdot B$

Answer to Problem 16.8E

The values of $\Delta E$ in the transition energies for ${}^{\text{1}}\text{P}$ in ${}^{\text{1}}\text{P}{\to }^{\text{1}}\text{D}$ transition are $-2.18\times {10}^{-23}\text{ J}$, $0\text{ J}$ and $+2.18\times {10}^{-23}\text{ J}$.

The values of $\Delta E$ in the transition energies for ${}^{\text{1}}\text{D}$ in ${}^{\text{1}}\text{P}{\to }^{\text{1}}\text{D}$ transition are $-4.358\times {10}^{-23}\text{ J}$, $-2.18\times {10}^{-23}\text{ J}$, $0\text{ J}$, $+2.18\times {10}^{-23}\text{ J}$ and $+4.35\times {10}^{-23}\text{ J}$.

Explanation of Solution

In case of ${}^{\text{1}}\text{P}{\to }^{\text{1}}\text{D}$ transition, the splitting of ${}^{\text{1}}\text{P}$ state occurs due to the presence of degenerate $M_L=-1\text{, }0$ and $+1$ states. The value of $\Delta E$ is calculated by the formula shown below.

$\Delta E={\mu }_{\text{B}}\cdot M_L\cdot B$…(1)

Where,

• ${\mu }_{\text{B}}$ is the Bohr magneton $\left(9.274\times {10}^{-24}\text{J}/\text{T}\right)$.

• $M_L$ is the magnetic quantum number.

• $B$ is the magnetic field.

The value of magnetic field is $2.\text{35 T}$.

For, $M_L=-1$.

Substitute the values of ${\mu }_{\text{B}}$, $B$ and $M_L=-1$ in equation (1) to calculate the change in energy.

$\begin{array}{rll}\Delta E& =& 9.274\times {10}^{-24}\text{J}/\text{T}\times -1\times 2.3\text{5 T}\\ & =& -2.179\times {10}^{-23}\text{ J}\\ & \approx & -2.18\times {10}^{-23}\text{ J}\end{array}$

For, $M_L=0$

Substitute the values of ${\mu }_{\text{B}}$, $B$ and $M_L=0$ in equation (1) to calculate the change in energy.

$\begin{array}{rll}\Delta E& =& 9.274\times {10}^{-24}\text{J}/\text{T}\times 0\times 2.3\text{5 T}\\ & =& 0\text{ J}\end{array}$

For, $M_L=+1$.

Substitute the values of ${\mu }_{\text{B}}$, $B$ and $M_L=+1$ in equation (1) to calculate the change in energy.

$\begin{array}{rll}\Delta E& =& 9.274\times {10}^{-24}\text{J}/\text{T}\times +1\times 2.3\text{5 T}\\ & =& +2.179\times {10}^{-23}\text{ J}\\ & \approx & +2.18\times {10}^{-23}\text{ J}\end{array}$

Therefore, the values of $\Delta E$ in the transition energies for ${}^{\text{1}}\text{P}$ are $-2.18\times {10}^{-23}\text{ J}$, $0\text{ J}$ and $+2.18\times {10}^{-23}\text{ J}$.

In case of ${}^{\text{1}}\text{P}{\to }^{\text{1}}\text{D}$ transition, the splitting of ${}^{\text{1}}\text{D}$ state occurs due to the presence of degenerate $M_L=-2\text{, }-1\text{, }0\text{, }+1$ and $+2$ states. The value of $\Delta E$ is calculated by the formula shown below.

$\Delta E={\mu }_{\text{B}}\cdot M_L\cdot B$…(1)

Where,

• ${\mu }_{\text{B}}$ is the Bohr magneton $\left(9.274\times {10}^{-24}\text{J}/\text{T}\right)$.

• $M_L$ is the magnetic quantum number.

• $B$ is the magnetic field.

The value of magnetic field is $2.\text{35 T}$.

For, $M_L=-2$.

Substitute the values of ${\mu }_{\text{B}}$, $B$ and $M_L=-2$ in equation (1) to calculate the change in energy.

$\begin{array}{rll}\Delta E& =& 9.274\times {10}^{-24}\text{J}/\text{T}\times -2\times 2.3\text{5 T}\\ & =& -4.358\times {10}^{-23}\text{ J}\end{array}$

For, $M_L=-1$.

Substitute the values of ${\mu }_{\text{B}}$, $B$ and $M_L=-1$ in equation (1) to calculate the change in energy.

$\begin{array}{rll}\Delta E& =& 9.274\times {10}^{-24}\text{J}/\text{T}\times -1\times 2.3\text{5 T}\\ & =& -2.179\times {10}^{-23}\text{ J}\\ & \approx & -2.18\times {10}^{-23}\text{ J}\end{array}$

For, $M_L=0$

Substitute the values of ${\mu }_{\text{B}}$, $B$ and $M_L=0$ in equation (1) to calculate the change in energy.

$\begin{array}{rll}\Delta E& =& 9.274\times {10}^{-24}\text{J}/\text{T}\times 0\times 2.3\text{5 T}\\ & =& 0\text{ J}\end{array}$

For, $M_L=+1$.

Substitute the values of ${\mu }_{\text{B}}$, $B$ and $M_L=+1$ in equation (1) to calculate the change in energy.

$\begin{array}{rll}\Delta E& =& 9.274\times {10}^{-24}\text{J}/\text{T}\times +1\times 2.3\text{5 T}\\ & =& +2.179\times {10}^{-23}\text{ J}\\ & \approx & +2.18\times {10}^{-23}\text{ J}\end{array}$

For, $M_L=+2$.

Substitute the values of ${\mu }_{\text{B}}$, $B$ and $M_L=+2$ in equation (1) to calculate the change in energy.

$\begin{array}{rll}\Delta E& =& 9.274\times {10}^{-24}\text{J}/\text{T}\times +2\times 2.3\text{5 T}\\ & =& +4.35\times {10}^{-23}\text{ J}\end{array}$

Therefore, the values of $\Delta E$ in the transition energies for ${}^{\text{1}}\text{D}$ are $-4.358\times {10}^{-23}\text{ J}$, $-2.18\times {10}^{-23}\text{ J}$, $0\text{ J}$, $+2.18\times {10}^{-23}\text{ J}$ and $+4.35\times {10}^{-23}\text{ J}$.

Conclusion

The values of $\Delta E$ in the transition energies for ${}^{\text{1}}\text{P}$ in ${}^{\text{1}}\text{P}{\to }^{\text{1}}\text{D}$ transition are $-2.18\times {10}^{-23}\text{ J}$, $0\text{ J}$ and $+2.18\times {10}^{-23}\text{ J}$.

The values of $\Delta E$ in the transition energies for ${}^{\text{1}}\text{D}$ in ${}^{\text{1}}\text{P}{\to }^{\text{1}}\text{D}$ transition are $-4.358\times {10}^{-23}\text{ J}$, $-2.18\times {10}^{-23}\text{ J}$, $0\text{ J}$, $+2.18\times {10}^{-23}\text{ J}$ and $+4.35\times {10}^{-23}\text{ J}$.