Chapter 16, Problem 16.43E

Interpretation Introduction

(a)

Interpretation:

The diagram that represents the $z$ components of nuclear spin for ${}^{\text{1}}\text{H}\left(I=\frac{1}{2}\right)$ is to be drawn.

Concept introduction:

Nuclear spin shows the complete angular momentum corresponding to the nucleus. It is represented by $I$. The nucleus consists of protons and neutrons. It depends on the mass number of a nucleus. The odd mass number shows that the nucleus has half-integer spin like an electron. The even mass number shows that the nucleus has integer spin.

Answer to Problem 16.43E

The diagram that represents the $z$ components of nuclear spin for ${}^{\text{1}}\text{H}\left(I=\frac{1}{2}\right)$ is shown below.

Explanation of Solution

The $z$ component of nuclear spin is given by the formula shown below.

$L_z=m_I\hslash$…(1)

Where,

• $m_I$ is the magnetic quantum number.

The value of $m_I$ is given by the formula shown below.

$-I\le m_I\le +I$

The value of $I$ for ${}^{\text{1}}\text{H}$ is given as $\frac{1}{2}$. Therefore, the possible values of $m_I$ for ${}^{\text{1}}\text{H}$ are

$-\frac{1}{2}$, $0$ and $+\frac{1}{2}$.

Substitute value of $m_I=-\frac{1}{2}$ for ${}^{\text{1}}\text{H}$ in equation (1).

$L_z= -\frac{1}{2}\hslash$

Substitute value of $m_I=0$ for ${}^{\text{1}}\text{H}$ in equation (1).

$L_z= 0\hslash$

Substitute value of $m_I=\frac{1}{2}$ for ${}^{\text{1}}\text{H}$ in equation (1).

$L_z= +\frac{1}{2}\hslash$

Therefore, the $L_z$ values for ${}^{\text{1}}\text{H}$ are $-\frac{1}{2}\hslash$, $0\hslash$ and $+\frac{1}{2}\hslash$.

The diagram that represents the $z$ components of nuclear spin for ${}^{\text{1}}\text{H}\left(I=\frac{1}{2}\right)$ is shown below.

Figure 1

Conclusion

The diagram that represents the $z$ components of nuclear spin for ${}^{\text{1}}\text{H}\left(I=\frac{1}{2}\right)$ is shown in Figure 1.

Interpretation Introduction

(b)

Interpretation:

The diagram that represents the $z$ components of nuclear spin for ${}^{\text{35}}\text{Cl}\left(I=\frac{3}{2}\right)$ is to be drawn.

Concept introduction:

Nuclear spin shows the complete angular momentum corresponding to the nucleus. It is represented by $I$. The nucleus consists of protons and neutrons. It depends on the mass number of a nucleus. The odd mass number shows that the nucleus has half-integer spin like an electron. The even mass number shows that the nucleus has integer spin.

Answer to Problem 16.43E

The diagram that represents the $z$ components of nuclear spin for ${}^{\text{35}}\text{Cl}\left(I=\frac{3}{2}\right)$ is shown below.

Explanation of Solution

The $z$ component of nuclear spin is given by the formula shown below.

$L_z=m_I\hslash$…(1)

Where,

• $m_I$ is the magnetic quantum number.

The value of $m_I$ is given by the formula shown below.

$-I\le m_I\le +I$

The value of $I$ for ${}^{\text{35}}\text{Cl}$ is given as $\frac{3}{2}$. Therefore, the possible values of $m_I$ for ${}^{\text{35}}\text{Cl}$ are $-\frac{3}{2}$, $- \frac{1}{2}$, $0$, $+\frac{1}{2}$ and $+\frac{3}{2}$.

Substitute value of $m_I=-\frac{3}{2}$ for ${}^{\text{35}}\text{Cl}$ in equation (1).

$L_z= -\frac{3}{2}\hslash$

Substitute value of $m_I=-\frac{1}{2}$ for ${}^{\text{35}}\text{Cl}$ in equation (1).

$L_z= -\frac{1}{2}\hslash$

Substitute value of $m_I=0$ for ${}^{\text{35}}\text{Cl}$ in equation (1).

$L_z= 0\hslash$

Substitute value of $m_I=+\frac{1}{2}$ for ${}^{\text{35}}\text{Cl}$ in equation (1).

$L_z= +\frac{1}{2}\hslash$

Substitute value of $m_I=+\frac{3}{2}$ for ${}^{\text{35}}\text{Cl}$ in equation (1).

$L_z= +\frac{3}{2}\hslash$

Therefore, the $L_z$ values for ${}^{\text{35}}\text{Cl}$ are $-\frac{3}{2}\hslash$, $- \frac{1}{2}\hslash$, $0\hslash$, $+\frac{1}{2}\hslash$ and $+\frac{3}{2}\hslash$.

The diagram that represents the $z$ components of nuclear spin for ${}^{\text{35}}\text{Cl}\left(I=\frac{3}{2}\right)$ is shown below.

Figure 2

Conclusion

The diagram that represents the $z$ components of nuclear spin for ${}^{\text{35}}\text{Cl}\left(I=\frac{3}{2}\right)$ is shown in Figure 2.

Interpretation Introduction

(c)

Interpretation:

The diagram that represents the $z$ components of nuclear spin for ${}^{\text{118}}\text{Sn}\left(I=0\right)$ is to be drawn.

Concept introduction:

Nuclear spin shows the complete angular momentum corresponding to the nucleus. It is represented by $I$. The nucleus consists of protons and neutrons. It depends on the mass number of a nucleus. The odd mass number shows that the nucleus has half-integer spin like an electron. The even mass number shows that the nucleus has integer spin.

Answer to Problem 16.43E

The diagram that represents the $z$ components of nuclear spin for ${}^{\text{118}}\text{Sn}\left(I=0\right)$ is shown below.

Explanation of Solution

The $z$ component of nuclear spin is given by the formula shown below.

$L_z=m_I\hslash$…(1)

Where,

• $m_I$ is the magnetic quantum number.

The value of $m_I$ is given by the formula shown below.

$-I\le m_I\le +I$

The value of $I$ for ${}^{\text{118}}\text{Sn}$ is given as $0$. Therefore, the possible value of $m_I$ for ${}^{\text{118}}\text{Sn}$ is $0$.

Substitute value of $m_I=0$ for ${}^{\text{118}}\text{Sn}$ in equation (1).

$L_z= 0\hslash$

Therefore, the $L_z$ value for ${}^{\text{118}}\text{Sn}$ is $0\hslash$.

The diagram that represents the $z$ components of nuclear spin for ${}^{\text{118}}\text{Sn}\left(I=0\right)$ is shown below.

Figure 3

Conclusion

The diagram that represents the $z$ components of nuclear spin for ${}^{\text{118}}\text{Sn}\left(I=0\right)$ is shown in Figure 3.

Interpretation Introduction

(d)

Interpretation:

The diagram that represents the $z$ components of nuclear spin for ${}^{\text{14}}\text{N}\left(I=1\right)$ is to be drawn.

Concept introduction:

Nuclear spin is the total angular momentum of the nucleus. It is represented by $I$. The nucleus consists of protons and neutrons. It depends on the mass number of a nucleus. The odd mass number shows that the nucleus has half-integer spin like an electron whereas the even mass number shows that the nucleus has integer spin.

Answer to Problem 16.43E

The diagram that represents the $z$ components of nuclear spin for ${}^{\text{14}}\text{N}\left(I=1\right)$ is shown below.

Explanation of Solution

The $z$ component of nuclear spin is given by the formula shown below.

$L_z=m_I\hslash$…(1)

Where,

• $m_I$ is the magnetic quantum number.

The value of $m_I$ is given by the formula shown below.

$-I\le m_I\le +I$

The value of $I$ for ${}^{\text{14}}\text{N}\left(I=1\right)$ is given as $0$. Therefore, the possible values of $m_I$ for ${}^{\text{14}}\text{N}$ are $-1$, $0$ and $+1$.

Substitute value of $m_I=-1$ for ${}^{\text{14}}\text{N}$ in equation (1).

$L_z=-1\hslash$

Substitute value of $m_I=0$ for ${}^{\text{14}}\text{N}$ in equation (1).

$L_z= 0\hslash$

Substitute value of $m_I=1$ for ${}^{\text{14}}\text{N}$ in equation (1).

$L_z=+1\hslash$

Therefore, the $L_z$ values for ${}^{\text{14}}\text{N}$ are $-1\hslash$, $0\hslash$ and $+1\hslash$.

The diagram that represents the $z$ components of nuclear spin for ${}^{\text{14}}\text{N}\left(I=1\right)$ is shown below.

Figure 4

Conclusion

The diagram that represents the $z$ components of nuclear spin for ${}^{\text{14}}\text{N}\left(I=1\right)$ is shown in Figure 4.