In Problem 4.P.44 you can find the effective spring stiffness corresponding to the interatomic force for aluminum and lead.
Let's assume for the moment that, very roughly, other atoms have similar values.
(a) What is the (very) approximate frequency f for the vibration of H2, a hydrogen molecule?
(b) What is the (very) approximate frequency f for the vibration of O2, an oxygen molecule?
(c) What is the approximate vibration frequency f of the molecule D2, both of whose atoms are deuterium atoms (that is,
each nucleus has one proton and one neutron)?
(d) Explain why the ratio of the deuterium frequency to the hydrogen frequency is quite accurate, even though you have
estimated each of these quantities very approximately, and the effective spring stiffness is normally expected to be
significantly different for different atoms. (Hint: What interaction is modeled by the effective “spring”?)