Question

In other problems and examples in the textbook we found 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 oscillation ("vibration") of H₂, a hydrogen molecule containing two hydrogen atoms? Remember that frequency is defined as the number of complete cycles per second or "hertz": f = 1/T. There is no one correct answer, since we're just trying to calculate the frequency approximately. However, just because we're looking for an approximate result doesn't mean that all answers are correct! Calculations that are wildly in disagreement with what physics would predict for this situation will be counted wrong.
f =    ??? cycles/s (hertz)

(b) What is the (very) approximate frequency f for the vibration of O₂, an oxygen molecule containing two oxygen atoms?
f = ??????    cycles/s (hertz)

(c) What is the approximate vibration frequency f of D₂, a molecule both of whose atoms are deuterium atoms (that is, each nucleus has one proton and one neutron)?
f =    ????? cycles/s (hertz)

(d) Which of the following statements are true? (Select all that apply.)

The estimated frequency for O₂ is quite accurate because the mass of an oxygen atom is similar to the mass of an aluminum atom. The true vibration frequency for D₂ is lower than the true vibration frequency for H₂, because the mass is larger but the effective "spring" stiffness is nearly the same, since it is related to the electronic structure, which is nearly the same for D₂ and H₂. The estimated frequencies for D₂ and H₂ are both quite accurate, because these are simple molecules, and the effective spring stiffness is expected to be the same as we found inside a block of metal. Neither of the estimated frequencies for D₂ and H₂ is accurate, but the ratio of the D₂ frequency to the H₂ frequency is quite accurate, because the "spring" represents the interatomic force, which is nearly the same for atoms with similar chemical structure (number of electrons).

Answer 1

This question requires a lot of data from your text book like young's modulus for alumnimium and k value.

(a) we can use the formula

w = sqrt (K / m )

where m is mass of hydrogen atom, K is spring siffness for aluminium

then use, f = w/2*pi to get the frequency

k for alumnimum is around 15.618  

Using this and using hydrogen mass (taken from internet)

f = 1.3e13 Hz

(b) similarly, by same methid,

f (O₂) = 3.5e12 Hz

(c) For deuterium

f = 1e13 Hz

(d) The true vibration frequency for D2 is lower than the true vibration frequency for H2, because the mass is larger but the effective "spring" stiffness is nearly the same, since it is related to the electronic structure, which is nearly the same for D2 and H2  

  Neither of the estimated frequencies for D2 and H2 is accurate, but the ratio of the D2 frequency to the H2frequency is quite accurate, because the "spring" represents the interatomic force, which is nearly the same for atoms with similar chemical structure