Physicists and physical chemists claim that we can model the vibration of diatomic molecules as that of a simple harmoni

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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" (which you may have heard about in chemistry). Let's consider the diatomic molecule HCI (with 3C1, the most common isotope of chlorine). The potential energy of this diatomic molecule is modeled by the curve at right (which you may have seen in the "Energy Skate Park" tutorial), where U represents potential energy and r represents the distance between the two atoms. Let - 0.12 nm, r=0.13 nm, and r = 0.14 nm. E 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 (M, in the pictures below, representing H) were moving relative to the larger one (M, in the picture below, representing CI). It is as if the Cl atom is fixed in place. Only the Hatom 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 latom and the other to represent the H atom. 2) On the next page 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 ts, the two atoms are separated by their equilibrium distance, 13. At time to the two atoms are maximally separated, by the distance ri. Fill in the remaining rows of the table with the appropriate values forr. Note that all the times are given in the left-hand column.
All M t = 0 12 ali to = 2.9 x 10 sec га t3 = 5.8 x 10-15 sec 14 = 8.7 x 10° sec ts = 1.16 x 10 sec to-1.45 x 10 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 (maximum, minimum, etc.) on the position vs. time graph map onto the potential energy graph?
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: • mm • 12 • rar 7) Suppose that, instead of defining PE-0 as in the graph above (where PE - 0 when the atoms are far apart). you define the bottom of the well as PE=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 PE = 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-r? What is the kinetic energy of the molecule when rr2?