## Physics Friday 72

When we previously derived the ideal gas law PV=NkT=nRT from a particle bouncing in a box, we used the Maxwell speed distribution, which was in turn derived using the Boltzmann factor with regards to the kinetic energy of the translational motion of the molecules.
Now, if we want to consider the internal energy U of the gas, and the properties deriving from it, such as specific heat capacity, we must consider not only the translatation of the molecules, but also other possible motions of the molecule; in particular, the number of degrees of freedom.
For a molecule of n atoms, there are 3n degrees of freedom; 3 of these are translation of the center of mass of the molecule (as space is three-dimensional). The remaining 3n-3 are internal degrees of freedom, such as rotations, bond length vibrations, and bond angle changes. (See here.)
We note that the internal degrees of freedom are in fact quantized, and thus, if the energy levels for a mode are spaced wide enough, that mode may not be accessisible at energies, and thus temperatures, below certain levels. So, we must consider the available degrees of freedom; the equipartition theorem tells us that in thermal equilibrium, all the available degrees of freedom should have the same average kinetic energy.
To give examples, we first consider a monatomic gas, such as the inert gases: there are only 3 degrees of freedom, and the internal energy is simply the total kinetic energy . For nitrogen gas, a diatomic molecule, there are six degrees of freedom: three translational modes, two rotational modes, and vibration of the bond length. At room temperature, all of these except for the vibration are accessable, and each of the available modes has an average energy per molecule of , for internal energy , five-thirds that of a monatomic gas at the same temperature.