1st EditionSarquis, J., Sarquis, M.2,183 explanations

Tom C. Hsu1,635 explanations

CHEMISTRYIf the vibration of a diatomic A - B is modelled using a harmonic oscillator, the vibrational frequency is given by $\omega=\left(k_{f} / \mu\right)^{1 / 2},$ where $\mu$ is the effective mass, $\mu=m_{A} m_{B} / (m_{A}+m_{B}) .$ If atom A is substituted by an isotope (for example $^{2} \mathrm{H}$ substituted for $^{1} \mathrm{H}).$ then to a good approximation the force constant remains the same. Why? (a) Show that when an isotopic substitution is made for atom A, such that its mass changes from $m_{A}$ to $m_{A},$ the vibrational frequency of $A^{2}-B, \omega_{A B},$ can be expressed in terms of the vibrational frequency of $A-B, \omega_{AB}$ as $\omega _{A B}=\omega _{AB} (\mu _{AB} / \mu _{AB} )^{1/2},$ where $\mu_{AB}$ and $\mu_{AB}$ are the effective masses of A - B and $A^{2}-B,$ respectively. (b) The vibrational frequency of $^{1}\mathrm{H}^{33} \mathrm{Cl}$ is $5.63 \times 10^{14}\ s^{-1}.$ Calculate the vibrational frequency of (i) $^{2} \mathrm{H}^{23} \mathrm{Cl}$ and (ii) $^{1}\mathrm{H}^{37} \mathrm{Cl}.$ Use integer relative atomic masses.