Equipartition law

Consider a gas of N non-interacting diatomic molecules moving in a tln-ee-dimensional system of volume V. Classically, the motion of a diatomic molecule has six degrees of freedom—tln-ee translational degrees corresponding to the centre of mass motion, two more for the rotational motion about the centre of mass and one additional degree for the vibrational motion about the centre of mass. The equipartition law gives (... 

Application of the equipartition law shows that for a molecule in thennal equilibrium. 

KiQ) X Q, we see that T represents an effective temperature measured by the average amplitude of fluctuations of the reaction coordinate Q. As noted in the last equation of Eq. 34, it approaches real temperature T at high temperatures of k iT boj, where Eq. 34 reduces to the equipartition law of energy in classical statistical mechanics. At low temperatures of k iT liw, on the other hand, T approaches a nonvanishing value reflecting the zero-point vibration of Q, given by the last equation of Eq. 34. 

We see that the resonance vibrations of the wall cause an effective cooling of the lowest electromagnetic modes (provided y < 1). The total number of quanta and the total energy in this example are formally infinite, due to the equipartition law of the classical statistical mechanics. In reality both these quantities are finite, since v (0) [Pg.331]

By differentiating the molar energies with respect to temperature, we obtain the molar heat capacities, Q, predicted by the equipartition law. 

If we examine the heat capacities of polyatomic gases. Table 4.3, we find two points of disagreement between the data and the equipartition law prediction. The observed heat capacities (1) are always substantially lower than the predicted values, and (2) depend noticeably on temperature. The equipartition principle is a law of classical physics, and these discrepancies were one of the first indications that classical mechanics was not... 

The MaxweU-Boltzmann law is of fundamental importance. We shall begin by applying it to the derivation of the equipartition law, which plays so prominent a part in determining the equilibrium of material systems, e, the energy of a molecule, is the sum of terms 7j, 7, corresponding to the different coordinates p, p, p",.,. which describe the motions. Let attention be fixed upon one particular type of coordinate, p, and for this purpose let (7) be rewritten... 

As will have been seen, the derivation of the equipartition law depends upon certain rather general, and perhaps somewhat abstract, assumptions about molecular dynamics. It is desirable, therefore, to bring the result as soon as possible into relation with experimentally measurable matters. The most striking method of doing this is afforded by the study of specific heats. 

There can be little doubt, therefore, that in many respects the equipartition law gives a rather accurate account of what happens. Its successes leave little doubt that when translations, rotations, and vibrations do exist, they are reasonably well describable as sums of independent square terms. 

To obtain the equipartition law in the form which gave, apart... 

From the formula for the pressure of a perfect gas, p = and the relation pV — NkT, it follows that the kinetic energy in the three translational degrees of freedom is pT. The allocation for each is thus kT. The equipartition law provides that where the energy is shared between s square terms, the average amount in a molecule is skT. The molecular heat, C , is therefore... 

The magnitude of the random force is greater for a particle with a larger friction coefficient, typically a larger particle. This random force satisfies the requirement of the equipartition law thermal energy per degree of freedom is k T/2 (Problem 3.13). 

By application of the Boltzmann equipartition law, this leads to a depletion of chains in the porous volume by a factor. 

Planck realized that the equipartition law, which assigns equal energy to each standing wave, could not be valid he also realized that the roll-off in the energy distribution at high frequencies could be obtained with the assumption that the energy of a harmonic oscillator cannot take on any value, as is assumed in the classical equipartition law, but that it is quantized a harmonic oscillator can absorb and emit energy only in finite steps. 

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