Chapter 15, Problem 15.26NP

Interpretation Introduction

Interpretation: The bond length of $O_2$ molecule should be determined when the standard molar enthalpy is $S^{\circ }=205.14\text{ kJ/mol}$ .

Concept Introduction: Entropy is the measure of probability and the molecular disorder in a macroscopic system.

The formula to calculate the molecular entropy is:

  $S^{\circ }=S_{trans}+S_{rot}+S_{vib}+S_{elec}$

The formula given by Sackur−Tetrode equation to calculate translational entropy is:

  $S_{trans}=R\ln \left(\frac{kT{e}^{5/2}}{\Lambda P^{\circ }}\right)$

Formula to calculate entropy due to ground electronic state $S_{elec}$ ,

  $S_{elec}=R\ln \left(\text{ground electronic state degeneracy}\right)$

Formula to calculate rotational partition function $q_{rot}$ ,

  $R\ln \left(q_{rot}\right)=S_{rot}-R$

Formula to calculate characteristic temperature ${\theta }_{rot}$ ,

  ${\theta }_{rot}=\frac{T}{2q_{rot}}$

Formula to calculate the moment of inertia $I$ ,

  $I=\frac{h^2}{8{\pi }^{2}k{\theta }_{rot}}$

Formula to calculate bond length,

  $\begin{array}{l}{R}_e=\sqrt{\frac{I}{\mu }}\\ R_e=\sqrt{\frac{I}{\left(\frac{1}{2}\times \text{Molar mass}\times \text{1 amu}\right)}}\end{array}$