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.

 

[PJC01]

for your question about the Dulong Petit derivation. The formula E=kT is derived from statistical mechanics, specifically the Boltzmann distribution. This distribution describes the probability of a particle having a certain amount of energy at a certain temperature. In the case of Dulong Petit, we are looking at a system of particles (atoms or molecules) at a certain temperature. The probability distribution for this system is f(E)=1, which means that all energy levels are equally likely. This leads to the formula for internal energy, u=3N$k_B$T, where N is the number of particles and $k_B$ is the Boltzmann constant. This formula takes into account the 3 degrees of freedom for each atom in the system, which is why we use E=kT.

As for your question about why a molecule has 5 degrees of freedom at room temperature, this is because molecules can also rotate and vibrate in addition to their translational motion. However, at room temperature, these additional degrees of freedom are not fully excited and contribute less to the internal energy compared to the translational motion. This is why we still use E=kT, as it takes into account the average energy of all degrees of freedom, including the additional ones for molecules. I hope this helps clarify the derivation of the Dulong Petit formula for you.