# Statistical mechanics - average particle energy, average kinetic energy

I'm looking at derivations for average particle energy giving $E=kT$:

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/bolapp.html

And average particle kinetic energy giving $K_E=\dfrac{3}{2}kT$:

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/molke.html#c1

The concept of potential energy doesn't enter the above derivations.

I'm confused by the difference in the two situations. Specifically in the first derivation, the average value is obtained by integrating over $dE$, whereas the second is obtained by integrated over $dv$. Is there a reason for the difference?

Basically I don't understand why the first derivation doesn't equally apply to the second situation, leading to an average kinetic energy of $kT$.

• have you tried looking at other derivations to help clarify? The average particle kinetic energy for example can be derived another way by using particle momenta. – Jaywalker Apr 19 '16 at 16:25
• I haven't. I think I might have to get a Statistical Mechanics book so I can learn this properly. – Ameet Sharma Apr 19 '16 at 16:27
• It looks like the first form comes from the canonical ensemble and the second is associated with the velocity moments for an ideal gas, which I discuss at length here http://physics.stackexchange.com/a/218643/59023. – honeste_vivere May 22 '16 at 19:23