Potential energy in the equipartition theorem.

[center o bass]

The equipartition theorem states that the average energy of any quadratic degree of freedom is $$1/2 k T$$, such that the total thermal energy is $$U_{thermal} = Nf\frac{1}{2}kT$$ where f is the number of degrees of freedom.

When the temperature is high enough the vibration mode is excited and then we are to include a potential energy term for the energy coming from the bond between molecules, $$\frac{1}{2}kx^2$$ (At least according to my book).

I thought that the _thermal_ energy of a system was only associated with the kinetic energy of the molecules, but if this is right, what then is that potential energy term doing in my calculations?

 

[Jano L.]

Hello center o bass,
it is hard to define "thermal energy" precisely. It is better to use "total energy", which is total kinetic and potential energy of the molecules.

The reason temperature is said to be connected to translational energy of the molecules (which is only a part of total energy) is the circumstance that this kinetic energy is always (for non-relativistic gas) quadratic function of momenta; this is true for any gas, even if its molecules have non-quadratic interaction energy.

Other energies (internal, potential...) are quadratic only in special cases, or due to simplified model, so their energy is connected to temperature in a more complicated way.


Jano