The relative motion of two atoms in a molecule can be described as the motion of
ID: 1492244 • Letter: T
Question
The relative motion of two atoms in a molecule can be described as the motion of a single body of mass m moving in one dimension, with potential energy U(r) = A/r^12 - B/r^6 where r is the separation between the atoms and A and B are positive constants. This is called a Lennard Jones potential. Find the equilibrium separation rO of the atoms, in terms of the constants A and B. A positive total energy trajectory of this system describes a scattering event, in which the two atoms approach each other from large separation, reach a minimum separation, then move again. Find the radius of minimum separation (turning point) on such a trajectory, in terms of the constants A and B and the total energy E greater than 0.Explanation / Answer
1) For equilibrium separation ,ro , -dU(r)/dr = 0
=> 12A/r13 - 6B/r7 = 0
=> For r = ro
=> 12A/ro13 - 6B/ro7 = 0
=> 12A/ro6 - 6B = 0
=> ro = (2A/B)1/6 ----------------------> equilibrium separation of atoms
2) For radius of minimum separation
=> A/r12 - B/r6 = 0
=> A/r12 = B/r6
=> r = (A/B)1/6