Energy Distribution of 3D Rotating and Vibrating Triatomic Molecular Gas Using A Lennard-Jones Type Potential for Molecular Binding

Here we use attractive forces that are mediated by a Lennard-Jones (LJ) type potential as the binding force between the atoms of the diatomic molecule. The form of the LJ potential is where epsilon is the minimum (negative) potential, r is the distance between atom centers and sigma is a scaling length which will usually be approximately equal to r. The exponent b is chosen to be 6 for the usual LJ potential but can be modified for either tighter binding or more vibrational motion. The LJ potential is strongly repelling at close range and weakly attracting at longer ranges. For this simulation the power used was b=3. From the LJ potential a force, FLJ, can by computed by taking the derivative with respect to r: and this force is used iteratively in Newton's laws of motion to update the momentum and speed of the mobile atoms. When the atoms have initial rotation speed, v, about their common center point, the equation for a stable rotation radius becomes: where r is the separation and r/2 is the rotation radius of the atoms. If we multiply equation 3 by r, we can solve it for both the stable r as well as the maximum rotation speed. If we set the initial separation to be less than or greater than this stable radius, then the motion of the atoms will have sinusoids (i.e. the separation of the atoms will vibrate) superimposed on the stable radius. For this animation we set sigma equal to twice the hard disc radius of the atoms. For r>sigma we use the forces from the above equation to compute the speed changes. The learner has access to sliders that allow adjustments, within limits, of the value of epsilon, the initial tangential speed, separation divided by the stable separation and the atom radius. The form of the kinetic energy distribution for simple atoms is, with 2 dimensions, a simple exponential: where a is a normalization constant and Eavg is the average kinetic energy of the atoms. The form of the kinetic energy distribution for the diatomic molecules with 2 dimensional motion, turns out to be the following exponential form: where Eavg is the average kinetic energy. Conventional thermal physics would say that the difference in these two distributions is that we have add two additional degrees of freedom. Actually we have added a single variable constraint, the binding energy between the two atoms of the diatomic molecule. By adding this constraint we have increased the number of modes of energy exchange by two, vibration and rotation, for a total of 4. That gives rise to the 2 in the exponent exp(-2E/Eavg) and the linear term in the (2E/Eavg) expression above. For the CO2 molecules, internally, we now have 4 more modes of energy exchange and that gives rise to the term (E/Eavg)^3 multiplying their exponential energy distribution, exp(-4E/Eavg)