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Degrees of freedom of diatomic molecule and Fermi energy

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We have at our disposal an amount of energy equaling 3.5kJ and we wish to allocate 1 mole of gas which occupies volume V.

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We explain why the internal energy of a gas of diatomic moleculs is 5/2 N k T near or below room temperature and we explain the derivation of the formula for the Fermi energy of an electron gas.

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In A) you made mistake in the first part. The internal energy for a diatomic molecule near or below room temperature is
5/2 N k T = 5/2 n R T.

This is because the molecule has besides kinetic energy also internal energy in the form of rotational energy. In a purely classical approach, it should also have vibrational energy, but that contribution is frozen at and below room temperatures. You can see this as follows. You can count the degrees of freedom for the molecule in two ways. Any possible motion of the molecule, can always be described as the motion of the two atoms separately. Then because each atom has 3 degrees of freedom (each atom can move in 3 independent directions), the total number of degrees of freedom is 6. Now these 6 degrees of freedom can also be accounted of by consideing that the two atoms form a molecule and then considering the possible degrees of freedom for the molecule.

The center of mass of the molecule has 3 degrees of freedom.

There are 2 independent ways to choose a rotation axis for the molecule ...

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