Equipartition of energy theorem

The equipartition of energy theorem states that the energy in a molecule is, on average, distributed evenly among all molecular degrees of freedom. This theorem can be used to predict the heat capacities of gases. 

Each hamionic temi in the Hamiltonian contributes k T to the average energy of the system, which is the theorem of the equipartition of energy. Since this is also tire internal energy U of the system, one can compute the heat capacity... 

The theorem of the equipartition of energy when extended to the thermal equilibrium between matter and ether was very well confirmed as far as the infrared part of black-body radiation was concerned. Its extension to the ultraviolet domain, however, leads to absurd results, so that, at least for the time being, one is unable to derive the Boltzmann-Stefan law and Wien s displacement law without reference to thermodynamical results. At the present time one cannot see how these difficulties can be solved.217... 

The theorem of the equipartition of energy can now be applied to the one-dimensional motion referred to by < >... 

In this equation, p is the total momentum of particle i and m is its mass. According to the theorem of the equipartition of energy each degree of freedom contributes k T/2. If there are N particles, each with three degrees of freedom, then the kinetic energy should equal 3Nk T/2. Nc in Equation (6.14) is the number of constraints on the system. In a molecular dynamics simulation the total linear momentum of the system is often constrained to a value of zero, which has the effect of removing three degrees of freedom from the system and so would be equal to 3. Other types of constraint are also possible as we shall discuss... 

The law of equipartition of energy reveals, once again, that each degree of rotational freedom contributes kT per molecule or RT per mole to the internal energy of a diatomic ideal gas, relative to its ground-state energy. This theorem also applies to polyatomic nonlinear ideal gases, because rotation and also... 

Table 7.2. Heat capacities at volume constant in the classical theorem of equipartition of energy...

The classical theorem of equipartition of energy states that if a molecular variable occurs in the classical energy in a quadratic form (to the second power) the contribution to the ensemble average system energy corresponding to that variable is equal to Nk- T/l. Equation (27.5-9) conforms to this theorem. We can verify... 

It can be seen from Fig. 5 that the specific heat approaches the classical (Dulong and Petit) value oi3R when r as predicted by the theorem of the Equipartition of Energy. At lower temperatures, however, the specific heat decreases rapidly towards zero in accordance with the third law. For 7 it is found, both experimentally and theoretically, that CyOcT. ... 

Using the theorem of equipartition of energy, mv /2 = k T/2, we can rewrite the third term on the right-hand side of Equation 13.5 as... 

Besides, the equipartition energy theorem allows us to write for the average of the potential energy of the harmonic oscillator ... 

Cf. V 23, Section 6. In the case of sinusoidal oscillations the time average of the potential energy is equal to that of the kinetic energy. The theorem of equipartition of kinetic energy therefore determines also the total energy content for harmonic oscillators. 

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