(a) Given that the spacing between vibrational energy levels of the HCl molecule is 0.36 eV, calculate the effective force constant. (b) Find the classical temperature associated with this difference between vibrational energy levels in HCl.
Strategy (a) Because \kappa=\mu \omega^{2} we first need to find μ and ω. The reduced mass μ is given by \mu=m_{1} m_{2} /\left(m_{1}+m_{2}\right). We also know that \Delta E=h f=\hbar \omega, \text { so } \omega=\Delta E / \hbar. At the lowest level n=0 \text { and } E_{\text {vibr }}=\hbar \omega / 2.
(b) Two degrees of freedom are associated with a onedimensional oscillator, one from the kinetic energy and one from the potential (see Section 9.3). Therefore we can say that
\Delta E=\hbar \omega=2\left(\frac{k T}{2}\right)=k T(where k is Boltzmann’s constant), and so T=\Delta E / k.
