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Diatomic Vibrations revision 2013

Page history last edited by Ben Dreyfus 6 years, 5 months ago

 

Physicists and physical chemists claim that we can model the vibration of diatomic molecules as that of a simple harmonic oscillator (SHO). 

Doing so can help us begin to think about what it means to say that a molecule has "vibrational energy". 

 

Let's consider the diatomic molecule HCl (with 35Cl, the most common isotope of chlorine).  The potential energy of this diatomic molecule is modeled by the familiar curve below, where r represents the distance between the two atoms.  Let r1 = 0.12 nm, r2 = 0.13 nm, and r3 = 0.14 nm. 

 

 

 

Using the SHO model, we can imagine a diatomic molecule as represented by two masses connected by a spring. In the case that one of the atoms is significantly larger than the other, as is the case for HCl, we can treat the vibration as if only the smaller atom (M1 in the pictures below, representing H) were moving relative to the larger one (M2 in the picture below, representing Cl). It is as if the Cl atom is fixed in place.  Only the H atom oscillates back and forth under this (quite reasonable) assumption.  (Remember that even when the force between two objects is the same, the more massive one has a much smaller acceleration.) 

 

1) First consider the movement of the hydrogen atom over time. Simulate this movement with your two fists, using one to represent the Cl atom and the other to represent the H atom.

 

2) Below is a series of frames showing the position of the hydrogen atom as a function of time for critical points in the motion (the times are given in the table).  At t1, the two atoms are separated by their equilibrium distance, r2.  At time t2, the two atoms are maximally separated, by the distance r3.  Fill in the remaining rows of the table with the appropriate values for r.  Note that all the times are given in the left-hand column.

 

t1 = 0
 
t2 = 2.9 x 10-15 sec

t3 = 5.8 x 10-15 sec

 

 

 

 

 

 

 

t4 = 8.7 x 10-15 sec

 

 

 

 

 

 

t5 = 1.16 x 10-14 sec

 

 

 

 

 

 

t6 = 1.45 x 10-14 sec

 

 

 

 

 

 

 

 

 

 

3) Use the time and inter-atomic distance information given in the above table to draw the graph of r as a function of time for the hydrogen atom below.

 

 

 

4)  Determine the frequency, amplitude, and period for this oscillation.

 

 

 

 

 

5) Consider how the graph you just drew is related to the potential energy curve. 

 

     a) How do the critical points on the position vs. time graph map onto the L-J potential energy curve?

 

 

 

 

     b) Are there places on the potential energy curve where the SHO model would not apply?  What would such places on the curve represent physically?  Are there other ways in which you believe the SHO model is not an appropriate one to apply to diatomic vibration?

 

 

 

 

 

6) Draw energy bar charts (total energy, potential energy, and kinetic energy) for the following three points:

 

  • r = r1

 

 

  • r = r2

 

 

  • r = r3

 

 

7)  Suppose that, instead of defining U = 0 as in the graph above (where U = 0 when the atoms are far apart), you define the bottom of the well as U = 0.   For this choice, draw energy bar charts for the same points as in question 6.

 

 

 

 

 

8)  Under what circumstances is it useful to make each choice about where U = 0 (at the bottom of the well vs. when the atoms are separated by a large distance)?

 

 

 

9)  What is the kinetic energy of the molecule when r = r3?  What is the kinetic energy of the molecule when r = r2?

 

 

 

Bonus: How much less likely is it that the diatomic molecule HCl would have an energy of 2E1 than an energy of E1?  Assume room temperature and consider the Boltzmann distribution!  Assume U = 0 at the bottom of the well.

 

 

 

**[Extra Note:  If you calculate the frequency for the HCl vibration, it should lie in the infrared (IR) region of the spectrum.  You may have encountered IR spectroscopy in chemistry courses or elsewhere as a way of measuring the difference between vibrational energy levels in diatomic molecules.  This information in turn can be used to identify the identity of the particular diatomic under investigation.  The full description of such spectroscopic techniques requires a discussion of quantum mechanics, but suffice to say that the vibrational energy levels of a diatomic molecule provide a sort of "molecular fingerprint," the spacing between levels being unique to that particular molecule.] 

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