Chapter 11, Problem 11.73E

What are the values of $E$, $L$, and $L_z$ for an ${\text{F}}^{8+}$ atom whose electron has the following wavefunctions, listed as ${\Psi }_{n,l,m_l}$? (a) ${\Psi }_{1,0,0}$ (b) ${\Psi }_{3,2,2}$(c) ${\Psi }_{2,1,-1}$(d) ${\Psi }_{9,6,-3}$.

Interpretation Introduction

(a)

Interpretation:

The values of $E$, $L$, and $L_z$ for an ${\text{F}}^{8+}$ atom for the given wavefunctions are to be calculated.

Concept introduction:

In quantum mechanics, the wavefunction is given by $\Psi$. The wavefunction contains all the information about the state of the system. The wavefunction is the function of the coordinates of the particles and time.

Answer to Problem 11.73E

The value of $E$, $L$, and $L_z$ for an ${\text{F}}^{8+}$ atom for the given wavefunctions are $-\text{1}\text{.76}\times {10}^{-16}\text{J}$, $0$ and $0$ respectively.

Explanation of Solution

The energy for the given wave function is given by the formula,

$E=-\frac{Z^2{e}^4\mu }{8{\epsilon }_0^2{h}^2{n}^2}$ …$\left(1\right)$

Where,

• $Z$ is the atomic number.

• $E$ is the charge on electron.

• $\mu$ is the reduced mass.

• $n$ is the number of shell.

The total angular momentum is calculated by the formula,

$L=\sqrt{l\left(l+1\right)}\hslash$ …$\left(2\right)$

Where,

• $l$ is the angular momentum.

The $L_z$ is calculated by the formula,

$L_z=m_l\hslash$ …$\left(3\right)$

Where,

• $m_l$ is the magnetic quantum number.

The given wavefunctions for electron of ${\text{F}}^{8+}$ atom is ${\Psi }_{1,0,0}$. Therefore, the value of $n$, $l$ and $m_l$ is $1$, $0$ and $0$ respectively.

Substitute the value of $n$, $Z=9$ for ${\text{F}}^{8+}$ in equation $\left(1\right)$.

$\begin{array}{rll}E& =& -\frac{{\left(9\right)}^2{\left(1.602\times {10}^{-19}\right)}^4\left(9.1\times {10}^{-31}\right)}{8{\left(8.85\times {10}^{-12}\right)}^2{\left(6.626\times {10}^{-34}\right)}^2{\left(1\right)}^2}\\ & =& -\frac{4854.857\times {10}^{-107}}{27509.290\times {10}^{-92}}\text{J}\\ & =& -\text{1}\text{.76}\times {10}^{-16}\text{J}\end{array}$

Thus, the value of $E$ is $-\text{1}\text{.76}\times {10}^{-16}\text{J}$.

Substitute the value of $l$ in equation $\left(2\right)$.

$\begin{array}{l}L=\sqrt{0\left(0+1\right)}\hslash \\ L=0\end{array}$

Thus, the value of $L$ is $0$.

Substitute the value of $m_l$ in equation $\left(3\right)$.

$\begin{array}{rll}{L}_z& =& 0\times \hslash \\ & =& 0\end{array}$

Thus, the value of $L_z$ is $0$.

Hence, the value of $E$, $L$, and $L_z$ for an ${\text{F}}^{8+}$ atom for the given wavefunctions are $-\text{1}\text{.76}\times {10}^{-16}\text{J}$, $0$ and $0$ respectively.

Conclusion

The value of $E$, $L$, and $L_z$ for an ${\text{F}}^{8+}$ atom for the given wavefunctions are $-\text{1}\text{.76}\times {10}^{-16}\text{J}$, $0$ and $0$ respectively.

Interpretation Introduction

(b)

Interpretation:

The values of $E$, $L$, and $L_z$ for an ${\text{F}}^{8+}$ atom for the given wavefunctions are to be calculated.

Concept introduction:

In quantum mechanics, the wavefunction is given by $\Psi$. The wavefunction contains all the information about the state of the system. The wavefunction is the function of the coordinates of the particles and time.

Answer to Problem 11.73E

The value of $E$, $L$, and $L_z$ for an ${\text{F}}^{8+}$ atom for the given wavefunctions are $-\text{1}\text{.96}\times {10}^{-17}\text{J}$, $\sqrt{6}\hslash$ and $2\hslash$ respectively.

Explanation of Solution

The energy for the given wave function is given by the formula,

$E=-\frac{Z^2{e}^4\mu }{8{\epsilon }_0^2{h}^2{n}^2}$ …$\left(1\right)$

Where,

• $Z$ is the atomic number.

• $E$ is the charge on electron.

• $\mu$ is the reduced mass.

• $n$ is the number of shell.

The total angular momentum is calculated by the formula,

$L=\sqrt{l\left(l+1\right)}\hslash$ …$\left(2\right)$

Where,

• $l$ is the angular momentum.

The $L_z$ is calculated by the formula,

$L_z=m_l\hslash$ …$\left(3\right)$

Where,

• $m_l$ is the magnetic quantum number.

The given wavefunctions for electron of ${\text{F}}^{8+}$ atom is ${\Psi }_{3,2,2}$. Therefore, the value of $n$, $l$ and $m_l$ is $3$, $2$ and $2$ respectively.

Substitute the value of $n$, $Z=9$ for ${\text{F}}^{8+}$ in equation $\left(1\right)$.

$\begin{array}{rll}E& =& -\frac{{\left(9\right)}^2{\left(1.602\times {10}^{-19}\right)}^4\left(9.1\times {10}^{-31}\right)}{8{\left(8.85\times {10}^{-12}\right)}^2{\left(6.626\times {10}^{-34}\right)}^2{\left(3\right)}^2}\\ & =& -\text{1}\text{.96}\times {10}^{-17}\text{J}\end{array}$

Thus, the value of $E$ is $-\text{1}\text{.96}\times {10}^{-17}\text{J}$.

Substitute the value of $l$ in equation $\left(2\right)$.

$\begin{array}{l}L=\sqrt{2\left(2+1\right)}\hslash \\ L=\sqrt{6}\hslash \end{array}$

Thus, the value of $L$ is $\sqrt{6}\hslash$.

Substitute the value of $m_l$ in equation $\left(3\right)$.

$\begin{array}{rll}{L}_z& =& 2\times \hslash \\ & =& 2\hslash \end{array}$

Thus, the value of $L_z$ is $2\hslash$.

Hence, the value of $E$, $L$, and $L_z$ for an ${\text{F}}^{8+}$ atom for the given wavefunctions are $-\text{1}\text{.96}\times {10}^{-17}\text{J}$, $\sqrt{6}\hslash$ and $2\hslash$ respectively.

Conclusion

The value of $E$, $L$, and $L_z$ for an ${\text{F}}^{8+}$ atom for the given wavefunctions are $-\text{1}\text{.96}\times {10}^{-17}\text{J}$, $\sqrt{6}\hslash$ and $2\hslash$ respectively.

Interpretation Introduction

(c)

Interpretation:

The values of $E$, $L$, and $L_z$ for an ${\text{F}}^{8+}$ atom for the given wavefunctions are to be calculated.

Concept introduction:

In quantum mechanics, the wavefunction is given by $\Psi$. The wavefunction contains all the information about the state of the system. The wavefunction is the function of the coordinates of the particles and time.

Answer to Problem 11.73E

The value of $E$, $L$, and $L_z$ for an ${\text{F}}^{8+}$ atom for the given wavefunctions are $-4.41\times {10}^{-17}\text{J}$, $\sqrt{2}\hslash$ and $-\hslash$ respectively.

Explanation of Solution

The energy for the given wave function is given by the formula,

$E=-\frac{Z^2{e}^4\mu }{8{\epsilon }_0^2{h}^2{n}^2}$ …$\left(1\right)$

Where,

• $Z$ is the atomic number.

• $E$ is the charge on electron.

• $\mu$ is the reduced mass.

• $n$ is the number of shell.

The total angular momentum is calculated by the formula,

$L=\sqrt{l\left(l+1\right)}\hslash$ …$\left(2\right)$

Where,

• $l$ is the angular momentum.

The $L_z$ is calculated by the formula,

$L_z=m_l\hslash$ …$\left(3\right)$

Where,

• $m_l$ is the magnetic quantum number.

The given wavefunctions for electron of ${\text{F}}^{8+}$ atom is ${\Psi }_{2,1,-1}$. Therefore, the value of $n$, $l$ and $m_l$ is $2$, $1$ and $-1$ respectively.

Substitute the value of $n$, $Z=9$ for ${\text{F}}^{8+}$ in equation $\left(1\right)$.

$\begin{array}{rll}E& =& -\frac{{\left(9\right)}^2{\left(1.602\times {10}^{-19}\right)}^4\left(9.1\times {10}^{-31}\right)}{8{\left(8.85\times {10}^{-12}\right)}^2{\left(6.626\times {10}^{-34}\right)}^2{\left(2\right)}^2}\\ & =& -4.41\times {10}^{-17}\text{J}\end{array}$

Thus, the value of $E$ is $-4.41\times {10}^{-17}\text{J}$.

Substitute the value of $l$ in equation $\left(2\right)$.

$\begin{array}{l}L=\sqrt{1\left(1+1\right)}\hslash \\ L=\sqrt{2}\hslash \end{array}$

Thus, the value of $L$ is $\sqrt{2}\hslash$.

Substitute the value of $m_l$ in equation $\left(3\right)$.

$\begin{array}{rll}{L}_z& =& -1\times \hslash \\ & =& -\hslash \end{array}$

Thus, the value of $L_z$ is $-\hslash$.

Hence, the value of $E$, $L$, and $L_z$ for an ${\text{F}}^{8+}$ atom for the given wavefunctions are $-4.41\times {10}^{-17}\text{J}$, $\sqrt{2}\hslash$ and $-\hslash$ respectively.

Conclusion

The value of $E$, $L$, and $L_z$ for an ${\text{F}}^{8+}$ atom for the given wavefunctions are $-4.42\times {10}^{-17}\text{J}$, $\sqrt{2}\hslash$ and $-\hslash$ respectively.

Interpretation Introduction

(d)

Interpretation:

The values of $E$, $L$, and $L_z$ for an ${\text{F}}^{8+}$ atom for the given wavefunctions are to be calculated.

Concept introduction:

In quantum mechanics, the wavefunction is given by $\Psi$. The wavefunction contains all the information about the state of the system. The wavefunction is the function of the coordinates of the particles and time.

Answer to Problem 11.73E

The value of $E$, $L$, and $L_z$ for an ${\text{F}}^{8+}$ atom for the given wavefunctions are $-2.18\times {10}^{-18}\text{J}$, $\sqrt{42}\hslash$ and $-3\hslash$ respectively.

Explanation of Solution

The energy for the given wave function is given by the formula,

$E=-\frac{Z^2{e}^4\mu }{8{\epsilon }_0^2{h}^2{n}^2}$ …$\left(1\right)$

Where,

• $Z$ is the atomic number.

• $E$ is the charge on electron.

• $\mu$ is the reduced mass.

• $n$ is the number of shell.

The total angular momentum is calculated by the formula,

$L=\sqrt{l\left(l+1\right)}\hslash$ …$\left(2\right)$

Where,

• $l$ is the angular momentum.

The $L_z$ is calculated by the formula,

$L_z=m_l\hslash$ …$\left(3\right)$

Where,

• $m_l$ is the magnetic quantum number.

The given wavefunctions for electron of ${\text{F}}^{8+}$ atom is ${\Psi }_{9,6,-3}$. Therefore, the value of $n$, $l$ and $m_l$ is $9$, $6$ and $-3$ respectively.

Substitute the value of $n$, $Z=9$ for ${\text{F}}^{8+}$ in equation $\left(1\right)$.

$\begin{array}{rll}E& =& -\frac{{\left(9\right)}^2{\left(1.602\times {10}^{-19}\right)}^4\left(9.1\times {10}^{-31}\right)}{8{\left(8.85\times {10}^{-12}\right)}^2{\left(6.626\times {10}^{-34}\right)}^2{\left(9\right)}^2}\\ & =& -2.18\times {10}^{-18}\text{J}\end{array}$

Thus, the value of $E$ is $-2.18\times {10}^{-18}\text{J}$.

Substitute the value of $l$ in equation $\left(2\right)$.

$\begin{array}{l}L=\sqrt{6\left(6+1\right)}\hslash \\ L=\sqrt{42}\hslash \end{array}$

Thus, the value of $L$ is $\sqrt{42}\hslash$.

Substitute the value of $m_l$ in equation $\left(3\right)$.

$\begin{array}{rll}{L}_z& =& -3\times \hslash \\ & =& -3\hslash \end{array}$

Thus, the value of $L_z$ is $-3\hslash$.

Hence, the value of $E$, $L$, and $L_z$ for an ${\text{F}}^{8+}$ atom for the given wavefunctions are $-2.18\times {10}^{-18}\text{J}$, $\sqrt{42}\hslash$ and $-3\hslash$ respectively.

Conclusion

The value of $E$, $L$, and $L_z$ for an ${\text{F}}^{8+}$ atom for the given wavefunctions are $-2.18\times {10}^{-18}\text{J}$, $\sqrt{42}\hslash$ and $-3\hslash$ respectively.