Ludwig Boltzmann: The Man Who Trusted Atoms by Cercignani Carlo

Author:Cercignani, Carlo [Cercignani, Carlo]
Language: eng
Format: mobi
Publisher: Oxford University Press
Published: 2006-01-11T16:00:00+00:00

8.3 Specific heats again

Let us return now to the problem of specific heats, which should be a problem concerning equilibrium states and hence easily solvable. The difficulties typically arise from the impossibility of applying the theorem of equipartition of energy. If we accept the idea that the gas is in an equilibrium state with a Maxwell-Boltzmann distribution, then every quadratic term—in the velocities or coordinates—appearing in the expression of the total (kinetic + potential) energy of a polyatomic molecule not under the action of external forces contributes with a term kT (where, as usual, k is the Boltzmann constant and T the absolute temperature) to the thermal energy per molecule, which would consequently equal kT, if ν is the number of terms of the kind indicated above. In this calculation one must consider only the terms coupled with each other in the collision, i.e. actually able to exchange energy; otherwise even in the case of smooth hard spheres (a typical model of a monatomic gas) one should consider the terms of rotational kinetic energy, which however is not convertible into any other form of energy. Thus for diatomic molecules shaped like a hard rod, previously discussed, one neglects the rotational kinetic energy about the axis joining the two atoms and one must take ν = 5, rather than ν = 6. Obviously one would have a discontinuity of behaviour between the case of a perfectly hard rod and a slightly elastic one, because the distance between the two atoms might change in the second case and the molecule would acquire at least one extra vibrational degree of freedom, with the consequence that the thermal energy of the gas would suddenly vary from kT to 3kT. As in many other problems of classical statistical mechanics, it is commonly thought that this question too is solved by quantum mechanics. In fact, the Planck distribution, which allows the vibration frequency a basic role in the energy distribution, permits gradual unfreezing of the vibrational degree of freedom, thus restoring, in agreement with experience, the continuity of the behaviour of the specific heat when the temperature varies.

However, the problem shows up in another form [14]: why do we not take into account, in the quantum treatment, other degrees of freedom, such as those of the electrons or of the various constituents of the nucleus? The answer is that these fast degrees of freedom are decoupled from the slow ones, and hence their energy cannot be exchanged during a time interval relevant for a macroscopic observation. Now this was exactly the answer given by Boltzmann [15] for the corresponding classical problem of the freezing of the degrees of freedom:

But how can the molecules of a gas behave as rigid bodies? Are they not composed of smaller atoms? Probably they are; but the vis viva of their internal vibrations is transformed into progressive and rotational motion so slowly, that, when the gas is brought to a lower temperature the molecules may retain for days, or even for years, the higher vis viva of their internal vibrations corresponding to the original temperature.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.