**More on Degrees of Freedom**

There are actually 6 degrees of freedom for a diatomic molecule: three for translation in 3 dimensions, two for rotation around two axes, and one for vibration along the axis between the two atoms. So why isn't the energy 3 kT instead of 5/2 kT? This was one of the great mysteries in physics in the late 1800's, and one of the early successes of quantum mechanics (see Feynman for example).

The two atoms in a diatomic molecule act as if they
are connected by a spring, and form a harmonic oscillator with a resonant frequency
f. According to quantum mechanics, energy comes in lumps of magnitude hf, where
h is Planck's constant. The energy
cannot be zero; the minimum energy, for the *ground state*, is hf/2. So
the two atoms can be in the ground state, or vibrating with energy 3hf/2, 5hf/2,
etc. The significant point is that it cannot vibrate with an energy between
hf/2 and 3hf/2. The probability of finding a molecule in an excited state with
energy E is proportional to exp(-E/kT), where k is Boltzmann's constant, and
T is the temperature in degrees Kelvin. For nitrogen and oxygen the resonant
frequency is quite high, the energy increment between the ground state and 1^{st}
excited state is large, and at room temperature the probability of a molecule
being in the first excited state is very close to zero. Therefore for all practical
purposes there is no coupling of energy into this degree of freedom.