Dulong Petit derivation.How do we get this formula?E=kT

[Outrageous]

Dulong Petit use energy,E=$k_B$T and the probability distribution as f(E)=1.
Internal energy,u=3N$k_B$T
$$C_v=∂u/∂T=3NkT$$
Three there because there is 3 modes in each atom.
Then my question is why do we use E=kT?
I understand 1 atom has 3 degree of freedom,and 1 freedom has kT/2.
A molecule has 5 degree of freedom at room temperature. Then why E=kT? Comes from?
Thanks

 

[DrDu]

kT/2 holds for a translational degree of freedom. For a vibration, you have rather kT. As an atom in a solid will vibrate in the cage formed by its neighbours, we get 3kT.

 

[Outrageous]

kT/2 holds for a translational degree of freedom. For a vibration, you have rather kT. As an atom in a solid will vibrate in the cage formed by its neighbours, we get 3kT.

Is that because in an atom in a solid, there are 6 neighbors so, (6/2)${k_B}$ T

 

[nasu]

Is that because in an atom in a solid, there are 6 neighbors so, (6/2)${k_B}$ T

No, this is not true in general and this is not the reason for the 3 in the formula even when it is true (simple cubic crystals). In most metals there are 8 or 12 nearest neighbors, for example.

An atom in a crystal has both kinetic and potential energy. In a gas it has only kinetic.
This is the reason for the different formulas.