2) The fundamental vibrational frequency of the \( \mathrm{H}_{2} \) molecule is \( 4200 \mathrm{~cm}^{-1} \). Calculate the energy level spacing and the ZPE, in \( \mathrm{J} \) and in \( \mathrm{kJ} / \mathrm{mol} \). \[ \begin{array}{l} \mathrm{h}=1.05 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s} \\ \mathrm{c}=3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}=3.00 \times 1010 \mathrm{~cm} / \mathrm{s} \\ \mathrm{N}_{\mathrm{A}}=6.02 \times 1023 \mathrm{~mol}^{-1} \end{array} \]
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Later, in Table $13.4,$ we will find that the following molecules have the indicated vibrational frequencies: \[ \begin{array}{c} ^{35} \mathrm{Cl}_{2}\left(560 \mathrm{cm}^{-1}\right) \\ ^{1} \mathrm{H}_{2}\left(4401 \mathrm{cm}^{-1}\right) \end{array}^{39} \mathrm{K}^{35} \mathrm{Cl}\left(281 \mathrm{cm}^{-1}\right) \] (a) What are the force constants for these molecules if we treat them as harmonic oscillators? $(b)$ Assuming that the force constant for $^{37} \mathrm{Cl}_{2}$ is the same as for $^{35} \mathrm{Cl}_{2}$, predict the fundamental vibrational frequency of $^{37} \mathrm{Cl}_{2}$.
The force constant of the ${ }^{1} \mathrm{H}^{19} \mathrm{~F}$ molecule is approximately $966 \mathrm{~N} \mathrm{~m} .$ (a) Find the frequency of vibration of the molecule. (b) The bond length in ${ }^{1} \mathrm{H}^{19} \mathrm{~F}$ is approximately $0.92 \mathrm{~nm}$. Plot the potential energy of this molecule versus intemuclear distance in the vicinity of $0.92 \mathrm{~nm}$ and show the vibrational energy levels as in Fig. $8.20 .$
Vibrations of the hydrogen molecule $\mathrm{H}_{2}$ can be modeled as a simple harmonic oscillator with the spring constant $\quad k=1.13 \times 10^{3} \mathrm{N} / \mathrm{m} \quad$ and $\quad$ mass $m=1.67 \times 10^{-27} \mathrm{kg} .$ (a) What is the vibrational frequency of this molecule? (b) What are the energy and the wavelength of the emitted photon when the molecule makes transition between its third and second excited states?
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