Chapter 18, Problem 18.1E

Interpretation Introduction

(a)

Interpretation:

The value of $q_{\text{nuc}}$ for ${}^{12}\text{C}$ atoms is to be determined.

Concept introduction:

Different linearly independent wavefunctions that have same energy are called degenerate. This is expressed in terms of degeneracy. If two functions are having same energy then they are called doubly degenerate and so on. The degeneracy for nuclear state is given as,

$g_1=2I+1$

Where,

• $I$ represents the nuclear spin.

Answer to Problem 18.1E

The value of $q_{\text{nuc}}$ for ${}^{12}\text{C}$ atoms is $1$.

Explanation of Solution

From Appendix $5$,

The nuclear spin of ${}^{12}\text{C}$ atoms is $0$.

The degeneracy for nuclear state is given as,

$g_1=2I+1$ …(1)

Where,

• $I$ represents the nuclear spin.

Substitute the value of $I$ in the equation (1).

$\begin{array}{rll}{g}_1& =& 2\left(0\right)+1\\ & =& 1\end{array}$

The partition function for nucleus is given as,

$q_{\text{nuc}}=g_1$ …(2)

Substitute the value of $g_1$ in the equation (2).

$q_{\text{nuc}}=1$

Therefore, the value of $q_{\text{nuc}}$ for ${}^{12}\text{C}$ atoms is $1$.

Conclusion

The value of $q_{\text{nuc}}$ for ${}^{12}\text{C}$ atoms is $1$.

Interpretation Introduction

(b)

Interpretation:

The value of $q_{\text{nuc}}$ for ${}^{56}\text{Fe}$ atoms is to be determined.

Concept introduction:

Different linearly independent wavefunctions that have same energy are called degenerate. This is expressed in terms of degeneracy. If two functions are having same energy then they are called doubly degenerate and so on. The degeneracy for nuclear state is given as,

$g_1=2I+1$

Where,

• $I$ represents the nuclear spin.

Answer to Problem 18.1E

The value of $q_{\text{nuc}}$ for ${}^{56}\text{Fe}$ atoms is $1$.

Explanation of Solution

From Appendix $5$,

The nuclear spin of ${}^{56}\text{Fe}$ atoms is $0$.

The degeneracy for nuclear state is given as,

$g_1=2I+1$ …(1)

Where,

• $I$ represents the nuclear spin.

Substitute the value of $I$ in the equation (1).

$\begin{array}{rll}{g}_1& =& 2\left(0\right)+1\\ & =& 1\end{array}$

The partition function for nucleus is given as,

$q_{\text{nuc}}=g_1$ …(2)

Substitute the value of $g_1$ in the equation (2).

$q_{\text{nuc}}=1$

Therefore, the value of $q_{\text{nuc}}$ for ${}^{56}\text{Fe}$ atoms is $1$.

Conclusion

The value of $q_{\text{nuc}}$ for ${}^{56}\text{Fe}$ atoms is $1$.

Interpretation Introduction

(c)

Interpretation:

The value of $q_{\text{nuc}}$ for ${}^1\text{H}$ atoms is to be determined. The corresponding answer is to be explained.

Concept introduction:

Different linearly independent wavefunctions that have same energy are called degenerate. This is expressed in terms of degeneracy. If two functions are having same energy then they are called doubly degenerate and so on. The degeneracy for nuclear state is given as,

$g_1=2I+1$

Where,

• $I$ represents the nuclear spin.

Answer to Problem 18.1E

The value of $q_{\text{nuc}}$ for ${}^1\text{H}$ atoms is $2$.The nuclear degeneracy of ${}^1\text{H}$ is due to the presence of only one proton.

Explanation of Solution

From Appendix $5$,

The nuclear spin of ${}^1\text{H}$ atoms is $\frac{1}{2}$.

The degeneracy for nuclear state is given as,

$g_1=2I+1$ …(1)

Where,

• $I$ represents the nuclear spin.

Substitute the value of $I$ in the equation (1).

$\begin{array}{rll}{g}_1& =& 2\left(\frac{1}{2}\right)+1\\ & =& 2\end{array}$

The partition function for nucleus is given as,

$q_{\text{nuc}}=g_1$ …(2)

Substitute the value of $g_1$ in the equation (2).

$q_{\text{nuc}}=2$

Therefore, the value of $q_{\text{nuc}}$ for ${}^1\text{H}$ atoms is $2$.

Hydrogen is the only atom that has no neutrons in its nucleus. Therefore, the nuclear degeneracy of ${}^1\text{H}$ is due to the presence of only one proton.

Conclusion

The value of $q_{\text{nuc}}$ for ${}^1\text{H}$ atoms is $2$. The nuclear degeneracy of ${}^1\text{H}$ is due to the presence of only one proton.

Interpretation Introduction

(d)

Interpretation:

The value of $q_{\text{nuc}}$ for $\text{D}$ atoms is to be determined. The corresponding answer is to be explained.

Concept introduction:

Different linearly independent wavefunctions that have same energy are called degenerate. This is expressed in terms of degeneracy. If two functions are having same energy then they are called doubly degenerate and so on. The degeneracy for nuclear state is given as,

$g_1=2I+1$

Where,

• $I$ represents the nuclear spin.

Answer to Problem 18.1E

The value of $q_{\text{nuc}}$ for $\text{D}$ atoms is $3$. The nuclear degeneracy of deuterium atom is due to the presence of a proton and a neutron.

Explanation of Solution

From Appendix $5$,

The nuclear spin of $\text{D}$ atoms is $1$.

The degeneracy for nuclear state is given as,

$g_1=2I+1$ …(1)

Where,

• $I$ represents the nuclear spin.

Substitute the value of $I$ in the equation (1).

$\begin{array}{rll}{g}_1& =& 2\left(1\right)+1\\ & =& 3\end{array}$

The partition function for nucleus is given as,

$q_{\text{nuc}}=g_1$ …(2)

Substitute the value of $g_1$ in the equation (2).

$q_{\text{nuc}}=3$

Therefore, the value of $q_{\text{nuc}}$ for $\text{D}$ atoms is $3$.

A deuterium atom contains one proton and one neutron. The spin of proton and neutron does not cancel each other but combines with each other. Therefore, the nuclear degeneracy of deuterium atom is due to the presence of a proton and a neutron.

Conclusion

The value of $q_{\text{nuc}}$ for $\text{D}$ atoms is $3$. The nuclear degeneracy of deuterium atom is due to the presence of a proton and a neutron.