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The Six Degrees of Freedom of a Diatomic Molecule

For a diatomic molecule at sufficiently high temperatures, the equipartition of energy theorem distributes an equal portion of the energy, equal to , among each quadratic term in the Hamiltonian. Here is the Boltzmann constant and is the absolute temperature. Three of the degrees of freedom are translations, two are rotations, and one is vibration. The vibrational degree of freedom contributes a total energy since the associated kinetic energy and potential energy are both quadratic forms. The temperature should not be too high (say, K), otherwise electronic degrees of freedom might be excited.

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For the translations, the kinetic energy is equal to in each direction , , and ; for the rotations, for angles θ and ϕ, where is the moment of inertia of the molecule and is the angular velocity. Finally, the energy associated with the vibrational degree of freedom, in the harmonic oscillator approximation, can be written as , where is the velocity along the direction of the chemical bond, is a spring constant associated with the chemical bond, is the elongation or compression, and is the equilibrium position. Adding the energies of all six degrees gives a theoretical value for the molar heat capacity at constant volume , where is the universal gas constant.
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