Chapter 18, Problem 18.4E

Interpretation Introduction

(a)

Interpretation:

The numbers of terms for the summation of the electronic partition function for ${\text{N}}_{\text{2}}$ gas at standard temperature is to be stated.

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 electronic partition function is represented as,

$q_{\text{elect}}={\displaystyle \sum _i{g}_i{e}^{-{\in }_i/\text{kT}}}$

Answer to Problem 18.4E

The numbers of terms for the summation of the electronic partition function for ${\text{N}}_{\text{2}}$ gas at standard temperature are $46$.

Explanation of Solution

The term symbol of a diatomic compound is represented as,

${}^{2S+1}{\Lambda }_{\Omega }$

Where,

• $S$ represents the total spin of the molecule.

• $\Lambda$ represents the axial component of $L$.

• $\Omega$ represents the axial component of total angular quantum number $J$.

The term $\left(\Lambda \right)$ of diatomic molecule is represented by Greek symbols such as $\Sigma$, $\Pi$, $\Delta$ and $\Phi$. The term $\Omega$ is represented by either alphabet $\text{u}$ or $\text{g}$.

From the referred Table, the state of ${}^{\text{14}}{\text{N}}_{\text{2}}$ are represented as,

$\left[\begin{array}{l}{\text{x" (}}^{\text{1}}{\text{Σ}}_{\text{u}}{}^{\text{+}}\text{),}{\text{x' (}}^{\text{1}}{\text{Π}}_{\text{u}}\text{),}{\text{v }}^{\text{1}}{\text{Π}}_{\text{g}}\text{,}{\text{u}}_{\text{5}}\text{,}{\text{u}}_{\text{4}}\text{,}\text{s,}\text{r,}\text{q,}{\text{p,o}}_{\text{5}}{}^{\text{1}}{\text{Π}}_{\text{u}}\text{,}{\text{O}}_{\text{5}}{\text{ (}}^{\text{3}}{\text{Π}}_{\text{u}}\text{),}{\text{c'}}_{\text{n}}{}^{\text{1}}{\text{Σ}}_{\text{u}}{}^{\text{+}}\text{,}{\text{c}}_{\text{n}}{}^{\text{1}}{\text{Π}}_{\text{u}}\text{,}{\text{c}}_{\text{n}}{}^{\text{1}}{\text{Π}}_{\text{u}}\text{,}\\ {\text{o}}_{\text{4}}{}^{\text{1}}{\text{Π}}_{\text{u}}\text{,}{\text{O}}_{\text{4}}{\text{ (}}^{\text{3}}{\text{Π}}_{\text{u}}\text{),}{\text{c'}}_{\text{5}}{}^{\text{1}}{\text{Σ}}_{\text{u}}{}^{\text{+}}\text{,}{\text{c}}_{\text{4}}{}^{\text{1}}{\text{Π}}_{\text{u}}\text{,}{\text{c}}_{\text{4}}{}^{\text{1}}{\text{Π}}_{\text{u}}\text{,}{\text{z }}^{\text{1}}{\text{Δ}}_{\text{g}}\text{,}{\text{y }}^{\text{1}}{\text{Π}}_{\text{g}}{\text{,k }}^{\text{1}}{\text{Π}}_{\text{g}}\text{,}{\text{x }}^{\text{1}}{\text{Σ}}_{\text{g}}{}^{\text{-}}\text{,}{\text{d' }}^{\text{1}}{\text{Σ}}_{\text{u}}{}^{\text{-}}{\text{ or }}^{\text{1}}{\text{Δ}}_{\text{u}}\text{,}\\ {\text{o}}_3{}^{\text{1}}{\text{Π}}_{\text{u}}\text{,}{\text{H }}^{\text{3}}{\text{Φ}}_{\text{u}}\text{,}{\text{c'}}_{\text{4}}{}^{\text{1}}{\text{Σ}}_{\text{u}}{}^{\text{+}}\text{,}{\text{c}}_{\text{3}}{}^{\text{1}}{\text{Π}}_{\text{u}}\text{,}{\text{b' }}^{\text{1}}{\text{Σ}}_{\text{u}}{}^{\text{+}}\text{,}{\text{D }}^{\text{3}}{\text{Σ}}_{\text{u}}{}^{\text{+}}{\text{,b }}^{\text{1}}{\text{Π}}_{\text{u}}\text{,}{\text{a" }}^{\text{1}}{\text{Σ}}_{\text{g}}{}^{\text{+}}\text{,}{\text{C' }}^{\text{3}}{\text{Π}}_{\text{u}}\text{,}{\text{E }}^{\text{3}}{\text{Σ}}_{\text{g}}{}^{\text{+}}\text{,}{\text{C" }}^{\text{5}}{\text{Π}}_{\text{u}}\text{,}\\ {\text{C }}^{\text{3}}{\text{Π}}_{\text{u}}\text{,}{\text{G }}^{\text{3}}{\text{Δ}}_{\text{g}}{\text{,A' }}^{\text{5}}{\text{Σ}}_{\text{g}}{}^{\text{+}}\text{,}{\text{w }}^{\text{1}}{\text{Δ}}_{\text{u}}\text{,}{\text{a }}^{\text{1}}{\text{Π}}_{\text{g}}\text{,}{\text{a' }}^{\text{1}}{\text{Σ}}_{\text{u}}{}^{\text{-}}\text{,}{\text{B' }}^{\text{3}}{\text{Σ}}_{\text{u}}{}^{\text{-}}\text{,}{\text{W }}^{\text{3}}{\text{Δ}}_{\text{u}}\text{,}{\text{B }}^{\text{3}}{\text{Π}}_{\text{g}}\text{,}{\text{A }}^{\text{3}}{\text{Σ}}_{\text{u}}{}^{\text{+}}\text{,}{\text{X }}^{\text{1}}{\text{Σ}}_{\text{g}}{}^{\text{+}}\end{array}\right]$

The total number of states of ${}^{\text{14}}{\text{N}}_{\text{2}}$ is $46$.

The electronic partition function is represented as,

$q_{\text{elect}}={\displaystyle \sum _i{g}_i{e}^{-{\in }_i/\text{kT}}}$

Where,

• $g_i$ represents the degeneracy.

• ${\in }_i$ represents the energy of $i^{th}$ microstate.

• $k$ represents the Boltzmann constant with value $1.381\times {10}^{-23}\text{J}/\text{K}$.

• $T$ represents the temperature $\left(\text{K}\right)$.

The number of terms in the electronic partition function will be equal to the number of states. Therefore, the numbers of terms for the summation of the electronic partition function for ${\text{N}}_{\text{2}}$ gas at standard temperature are $46$.

Conclusion

The numbers of terms for the summation of the electronic partition function for ${\text{N}}_{\text{2}}$ gas at standard temperature are $46$.

Interpretation Introduction

(b)

Interpretation:

The numbers of terms for the summation of the electronic partition function for ${\text{O}}_{\text{2}}$ gas at standard temperature is to be stated.

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 electronic partition function is represented as,

$q_{\text{elect}}={\displaystyle \sum _i{g}_i{e}^{-{\in }_i/\text{kT}}}$

Answer to Problem 18.4E

The numbers of terms for the summation of the electronic partition function for ${\text{O}}_{\text{2}}$ gas at standard temperature are $37$.

Explanation of Solution

The term symbol of a diatomic compound is represented as,

${}^{2S+1}{\Lambda }_{\Omega }$

Where,

• $S$ represents the total spin of the molecule.

• $\Lambda$ represents the axial component of $L$.

• $\Omega$ represents the axial component of total angular quantum number $J$.

The term $\left(\Lambda \right)$ of diatomic molecule is represented by Greek symbols such as $\Sigma$, $\Pi$, $\Delta$ and $\Phi$. The term $\Omega$ is represented by either alphabet $\text{u}$ or $\text{g}$.

From the referred Table, the state of ${}^{\text{16}}{\text{O}}_{\text{2}}$ are represented as,

$\left[\begin{array}{l}{\text{Z (}}^{\text{2}}{\text{Π}}_{\text{u}}\text{), Y,}\text{W}\left({}^{\text{3}}{\text{Σ}}_{\text{u}}{}^{\text{+}}\right)\text{ ,}\text{V, U,}\text{R,}\text{Q,}\text{P,}{}^{\text{1}}{\text{Φ}}_{\text{u}}\text{,}\text{I", I', I,}{\text{H (}}^{\text{3}}{\text{Π}}_{\text{u}}\text{),}\text{4f,}\text{L}\left({}^{\text{3}}{\text{Π}}_{\text{u}}\right)\text{,}\text{K}\left({}^{\text{1}}{\text{Δ}}_{\text{u}}\right)\text{,}\\ \text{J}\left({}^{\text{1}}{\text{Σ}}_{\text{u}}{}^{\text{+}}\right)\text{,}\text{G}\left({}^{\text{3}}{\text{Σ}}_{\text{u}}{}^{\text{+}}\right)\text{,}\text{i}\left({}^{\text{1}}{\text{Δ}}_{\text{2u}}\right),\text{i}\left({}^{\text{3}}{\text{Δ}}_{\text{2u}}\right)\text{,}\text{h}\left({}^{\text{1}}{\text{Π}}_{\text{u}}\right)\text{,}{\text{g(}}^{\text{1}}{\text{Π}}_{\text{u}}\text{),}{\text{F}}^{\text{*}}\text{,}{\text{F}}^{\text{3}}{\text{Π}}_{\text{u}}\text{,}\text{E}{}^{\text{3}}{\text{Σ}}_{\text{u}}{}^{\text{-}}\text{,}\text{F}{}^{\text{1}}{\text{Σ}}_{\text{u}}{}^{\text{+}}\text{,}\\ \text{D}\left({}^{\text{3}}{\text{Σ}}_{\text{u}}{}^{\text{+}}\right)\text{, e}\left({}^{\text{1}}{\text{Δ}}_{\text{2u}}\right)\text{,}{\text{e}}^{\text{*}}\left({}^{\text{3}}{\text{Δ}}_{\text{2u}}\right)\text{,}\text{d}\left({}^{\text{1}}{\text{Π}}_{\text{g}}\right)\text{,}\text{c}\left({}^{\text{3}}{\text{Π}}_{\text{g}}\right)\text{,}\text{B}{}^{\text{3}}{\text{Σ}}_{\text{u}}{}^{\text{-}}\text{,}\text{A}{}^{\text{3}}{\text{Σ}}_{\text{u}}{}^{\text{+}}\text{,}\text{A}{}^{\text{3}}{\text{Δ}}_{\text{u}}\text{,}\text{c}{}^{\text{1}}{\text{Σ}}_{\text{u}}{}^{\text{-}}\text{,}\\ \text{b}{}^{\text{1}}{\text{Σ}}_{\text{g}}{}^{\text{+}}\text{,}\text{a}{}^{\text{1}}{\text{Δ}}_{\text{g}}\text{,}\text{x}{}^{\text{3}}{\text{Σ}}_{\text{g}}\end{array}\right]$

The total number of states of ${\text{O}}_{\text{2}}$ is $37$.

The electronic partition function is represented as,

$q_{\text{elect}}={\displaystyle \sum _i{g}_i{e}^{-{\in }_i/\text{kT}}}$

Where,

• $g_i$ represents the degeneracy.

• ${\in }_i$ represents the energy of $i^{th}$ microstate.

• $k$ represents the Boltzmann constant with value $1.381\times {10}^{-23}\text{J}/\text{K}$.

• $T$ represents the temperature $\left(\text{K}\right)$.

The number of terms in the electronic partition function will be equal to the number of states. Therefore, the numbers of terms for the summation of the electronic partition function for ${\text{O}}_{\text{2}}$ gas at standard temperature are $37$.

Conclusion

The numbers of terms for the summation of the electronic partition function for ${\text{O}}_{\text{2}}$ gas at standard temperature are $37$.