Energy equipartition principle

The equipartition principle is a classic result which implies continuous energy states. Internal vibrations and to a lesser extent molecular rotations can only be understood in terms of quantized energy states. For the present discussion, this complication can be overlooked, since the sort of vibration a molecule experiences in a cage of other molecules is a sufficiently loose one (compared to internal vibrations) to be adequately approximated by the classic result. 

The mechanical modes whereby molecules may absorb and store energy are described by quadratic terms. For translational kinetic energy it involves the square of the linear momentum (E = p2/2m), for rotational motion it is the square of angular momentum (E = L2121) and for vibrating bodies there are both kinetic and potential energy (kx2/2) terms. The equipartition principle states that the total energy of a molecule is evenly distributed over all available quadratic modes. 

By the equipartition principle it now follows that each rotational degree of freedom can absorb energy of kT while each vibrational mode can absorb kT. By the same principle the heat capacity of an ideal gas... 

This interval is considerably more than the spacing between rotational levels, and since AEy kT at room temperature it is safe to conclude that most molecules in such a sample exist in the lowest vibrational state with only their zero-point energies. Here is the real reason for the breakdown of the classical equipartition principle. 

In /z-space the most probable kinetic energy ep — mCp = kT, which differs from the average energy according to the equipartition principle,... 

Hendrik Antoon Lorentz, from Leyden (Holland), presided the conference, whose general theme was the Theory of Radiation and the Quanta. The conference5 was opened with speeches by Lorentz and Jeans, one on Applications of the Energy Equipartition Theorem to Radiation, the other on the Kinetic Theory of Specific Heat according to Maxwell and Boltzmann. In their talks, the authors explored the possibility of reconciling radiation theory with the principles of statistical mechanics within the classical frame. Lord Rayleigh, in a letter read to the... 

These results are identical with those obtained from the equipartition principle ( 15d), so that, as a good approximation, classical methods can be used for the evaluation of the rotational energy and heat capacity of diatomic gases, except hydrogen and deuterium, at all temperatures above the very lowest. 

The energy and heat capacity at high temperatures should consequently be identical, as a first approximation, with the values given by the classical equipartition principle. 

By differentiation with respect to temperature of Qr, as given by equation (16.34), remembering that all the quantities except T are constant, it is readily found that Er is f/ZT and Cr is fiZ. These are the results to be expected from the equipartition principle for energy expressible in three square terms, as would be the case for a nonlinear molecule containing more than... 

Consider a polymer liquid subjected to a unit step strain at time t = 0. The equipartition principle states that kTjl of free energy is associated with each degree of freedom at equilibrium. Immediately following the unit... 

The equipartition principle was initially proposed by Maxwell [66] in 1867 who stated that the energy of a gas is equally divided between linear and rotational energy. The original theorem was later generalized by Boltzmann [6] in 1872 by showing that the internal energy is actually equally divided among all the independent components of motion in the system. 

The most obvious test of the validity of the equipartition principle is the attempt to calculate the energy content and thus the specific heat of substances composed of molecules simple enough for a fair guess at their structure and mechanics to be made. 

Furthermore, detailed experiment shows that all specific heats are functions of temperature, at least over certain ranges, and this is completely inexplicable in terms of the equipartition principle, which makes the energy proportional to T, so that the specific heat, the differential coefficient of the energy with respect to T, should be constant. 

The actual mode of variation of the specific heat is such as to suggest that in certain ranges of temperature some of the degrees of freedom pass entirely out of action. The principles so far introduced, then, need amplification by some quite fundamental new rules which provide reasons why sometimes degrees of freedom should be operative and sometimes not. These rules cannot be derived in any way except by the introduction of the quantum theory. In the meantime it appears that the equipartition principle is an incomplete statement. If the results it predicted were merely inaccurate in a numerical sense, the discrepancies could be attributed to causes within the framework of ordinary mechanics, for example, to the non-independence of rotations and vibrations, which would spoil the formulation of the energy as a sum of square terms, but the difficulty lies deeper. 

The hypothesis of the electron gas breaks down, however, in its application to specific heats. It implies that the thermal energies of electrons are equal to those of atoms, by the equipartition principle. But the specific heats of metals can be almost entirely accounted for by the atomic motions alone. Dulong and Petit s law, and indeed the Einstein and Debye relations, ignore any contribution firom the electrons to the energy, and yet in their respective spheres give a good enough account of the facts. 

Statistical methods are extensively employed at present. They make use of the postulates of statistical and quantum mechanics permitting, on the basis of the equipartition principle and of data concerning molecular structure, the calculation of the main energy contributions corresponding to individual types of motion. Structural data are obtained from spectra, recently predominantly from microwave spectra. Vibration frequency levels, which must be known if the contribution of vibrational motion is to be calculated, are likewise obtained from spectroscopic measurements. 

Space has three dimensions, and each dimension contributes ViRT to the translational energy of a gas. This concept is called the equipartition principle, the idea that each dimension (or degree of freedom ) contributes equally to the overall energy of the particle. Thus, the total translational energy of a gas is... 

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