Translational motion, average energy

FIGURE 6.17 The translational and rotational modes of atoms and molecules and the corresponding average energies of each mode at a temperature T. (a) An atom or molecule can undergo translational motion in three dimensions, (b) A linear molecule can also rotate about two axes perpendicular to the line of atoms, (c) A nonlinear molecule can rotate about three perpendicular axes. 

In the above expressions for C(t), the averaging over initial rotational, vibrational, and electronic states is explicitly shown. There is also an average over the translational motion implicit in all of these expressions. Its role has not (yet) been emphasized because the molecular energy levels, whose spacings yield the characteristic frequencies at which light can be absorbed or emitted, do not depend on translational motion. However, the frequency of the electromagnetic field experienced by moving molecules does depend on the velocities of the molecules, so this issue must now be addressed. 

We can use Eq. 8.78 and the formulas that we just derived for q to find the average energy of the different types of motion. First, from Eq. 8.59 the thermal energy of translational motion of a gas (in three dimensions) is... 

A fundamental theorem of classical mechanics called the equipartition theorem (which we shall not derive here) states that the average energy of each degree of freedom of a molecule in a sample at a temperature T is equal to kT. In this simple expression, k is the Boltzmann constant, a fundamental constant with the value 1.380 66 X 10-21 J-K l. The Boltzmann constant is related to the gas constant by R = NAk, where NA is the Avogadro constant. The equipartition theorem is a result from classical mechanics, so we can use it for translational and rotational motion of molecules at room temperature and above, where quantization is unimportant, but we cannot use it safely for vibrational motion, except at high temperatures. The following remarks therefore apply only to translational and rotational motion. 

The well-known Maxwell-Boltzmann distribution for the velocity or momentum associated with the translational motion of a molecule is valid not only for free molecules but also for interacting molecules in a liquid phase (see Appendix A.2.1). The average kinetic energy of a molecule at temperature T is, accordingly, (3/2)ksT. For the molecules to react in a bimolecular reaction they should be brought into contact with each other. This happens by diffusion when the reactants are dispersed in a solution, which is a quite different process from the one in the gas phase. For fast reactions, the diffusion rate of reactant molecules may even be the limiting factor in the rate of reaction. 

The average energy of translational motion of gas molecules is proportional only... 

The Kelvin scale is so designed that the absolute temperature, measured from the absolute zero, is propoitional to the increase of the average kinetic energy of translational motion of the molecules of a... 

It does not matter which substance is selected. The laws of molecular motion are such that the average kinetic energy of translational motion of a gas of any substance is exactly the same as that for any other substance at the same temperature, that is, in tliermal equilibrium with it. 

Figure 14. Difference of averaged energies E(i) between total translational and total rotational motions plotted as a function of the system size. The difference was measured at l — 320 ps. Filled diamonds, squares, and crosses represent water, oxygen and alcohol molecules, respectively. The number of molecules is 216 and the temperature is 305 K.

In gaseous hydrogen, for example, H2 the molecules will be moving freely from one location to another this is called translational motion, and the molecules therefore possess translational kinetic energy KEtrans = in which v stands for the average velocity of the molecules you may recall from your... 

One of the differences of fundamental importance between x-ray photons and neutrons is in the energies of the particles. Whereas the energy of an x-ray photon is, as mentioned before, of the order of 10 keV, the kinetic energy of a thermal neutron is of the order of 10 meV. The average energy associated with the motion of atoms, arising from vibrational, rotational, and translational motions of molecules, is of the order of kT. At ambient temperatures kT is about 20 meV. 

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