1. Compare the collision systems H2(v = 1) + H2(v = 0) and D2(v = 1) + D2(v = 0) in the case of a pure gas at
identical thermal and density conditions.
(a) Compare the coupling, as predicted by Eq. (5.2), between the vibrational and translational motions of both
systems. Use Ehrenfest's adiabatic principle to predict the ratio o the relaxation rates for v = 1 vibrational-
to-translational energy transfer for a pure gas of H2 molecules and that in a pure gas of D2 molecules.
(b) Employ the Landau-Teller result of Eq. (5.4) to calculate this ratio directly.
(c) Calculate this ratio using Eq. (5.4) in the case that the collider gases, H2(v = 0) and D2(v = 0), are
replaced by Ar.
2. (a) Derive Eq. (5.14).
(b) Derive Eq. (5.16).
(c) Consider a collision between an incoming hard-sphere-like molecule with an initially stationary atom. Show
that to first order the amount of translational energy Et that is transferred upon collision is at least
Et = 1/2 m1v1(1 - F)(1 - D2/4) (see Eq. (1.24) from ref. [427]). Here m1 and m2 denote, respectively, the
mass of the molecule and the initially stationary atom, and v1 is the laboratory velocity of the incoming
molecule. D corresponds to the mass defect, defined as (m1 - m2)/m1, and F is the fraction of all the hard
sphere collisions with total collision cross section piRS2 that contribute to Et (see Eq. (5.23)). (Note that
this simple equation implies that if one has D = 0 and b = 0 all of the initial kinetic energy, 1/2m1v12, is
transferred to the atom, and, as follows also directly from Eq. (5.16), this leads to F = 0.)
(d) In the case of a mass mismatch, D = 10%, calculate the maximal fraction of the incoming kinetic energy
(i.e. at F = 0 which corresponds to b = 0 collision) that can be transferred to the atom. Next, show that
the fraction of collisions 1 - F that transfer more that 99% of the incoming translational energy of the
molecule with D = 10% is about 1%. (Note that if the incoming kinetic energy of the molecules is just
low enough that these 1% of molecules become trapped, the other 99% will be ejected from the trap. A
value of 1% for the fraction of the incoming molecules to remain in the trap strongly suggests that such
trapping experiments are feasible for kinematically slowed HBr that is subsequently trapped in a Rb
magneto-optical trap425 (see Study Box 12.1).)
3. Show that Eqs. (5.26) and (5.27) imply that j · â = j' · â' or (j' - j) · â = deltaj · â = 0, i.e. the projection of
the rotational angular momentum onto kinematic apse remains conserved for a hard shell type of collision.
(Note that rotationally inelastic state-to-state differential scattering experiments probing the alignment vector,
j', of the outgoing molecule for the closed shell Ne-Na2 system,344 and for the open shell rare gas-NO system,
have been shown to conform to this propensity rule in the quantum mechanical limit.428-430)
4. Derive Eqs. (5.28) to (5.35), starting from the conservation of energy and angular momentum, Eqs. (5.26)
and (5.27).
5. Explain why there is a summation over two terms in Eq. (5.64).
Above Problems are available as a PDF to print
Solutions to Chapter 5 Problems