Chapter 12, Problem 12.8P

(a)

Interpretation Introduction

Interpretation:

The electronic partition function of carbon at $10K$ and $298K$ has to be evaluated.

Concept Introduction:

The electronic partition function can be calculated using following equation.

$q^E=\Sigma g_j{e}^{\frac{-{\epsilon }_j}{kT}}$

Where,

    $\begin{array}{c}{q}^E=electronicpartitionfunction\\ g_j=\mathrm{deg}eneracy\\ {\epsilon }_j=energy\\ k=boltzmanncons\tan t\\ T=temperature\end{array}$

Degeneracy can be calculated using following formula.

$g_j=2J+1$

Where,

    $J=totalangularmomentum$

(b)

Interpretation Introduction

Interpretation:

An expression for electronic contribution to molar internal energy has to be derived and also it has to be plotted as a function of temperature.

Concept Introduction:

The total energy of a system can be represented as follows,

${\rm E}={\displaystyle \sum _i{N}_i{\epsilon }_i}$

Where,

    $\begin{array}{l}{N}_i=numberofmoleculeinthe{i}^{th}state\\ {\epsilon }_i=energyofthe{i}^{th}state\end{array}$

$Numberofmoleculeinthe{i}^{th}state$, $N_i={\displaystyle \sum _{}\frac{N{e}^{\frac{-{\epsilon }_i}{kT}}}{q}}.$

Where,

    $N{e}^{\frac{-{\epsilon }_i}{kT}}=populationof{i}^{th}state$

The expression for partition function q is ${\displaystyle \sum _i^{}{e}^{\frac{-{\epsilon }_i}{kT}}}$.

(c)

Interpretation Introduction

Interpretation:

The expression $E=\frac{N}{q}\left(k{T}^2\right).\frac{dq}{dT}$ has to be evaluated at ${25}^{o}C.$

Concept Introduction:

The internal energy expression in terms of partition function can be written as follows,

$E=\frac{N}{q}\left(k{T}^2\right).\frac{dq}{dT}$