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

  • Context: Graduate 
  • Thread starter Thread starter Outrageous
  • Start date Start date
Click For Summary
SUMMARY

The Dulong-Petit law is derived from the relationship between energy and temperature, expressed as E=kT, where k represents Boltzmann's constant. The internal energy of a solid is given by u=3NkBT, indicating that each atom contributes three degrees of freedom due to vibrational motion. The specific heat capacity at constant volume is defined as Cv=∂u/∂T=3Nk, highlighting the distinction between translational and vibrational degrees of freedom. The formula E=kT is applicable in the context of vibrational energy in solids, where atoms oscillate within a lattice structure formed by neighboring atoms.

PREREQUISITES
  • Understanding of Boltzmann's constant (kB) and its role in statistical mechanics
  • Familiarity with the concepts of internal energy and heat capacity
  • Knowledge of atomic degrees of freedom in solids and gases
  • Basic principles of crystal structures and atomic interactions
NEXT STEPS
  • Study the derivation of the Dulong-Petit law in solid-state physics
  • Explore the concept of degrees of freedom in different states of matter
  • Learn about the relationship between kinetic and potential energy in crystalline structures
  • Investigate the differences in heat capacity between solids and gases
USEFUL FOR

Physicists, materials scientists, and students studying thermodynamics and solid-state physics will benefit from this discussion, particularly those interested in the behavior of atoms in crystalline materials.

Outrageous
Messages
373
Reaction score
0
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
 
Science news on Phys.org
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.
 
  • Like
Likes   Reactions: 1 person
DrDu said:
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
 
Outrageous said:
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.
 

Similar threads

Graduate What's the deal with plasma physicists setting heat cap. ratio to 3?
  • · Replies 2 ·
Replies
2
Views
1K
Graduate The association of degrees of freedom with temperature
  • · Replies 17 ·
Replies
17
Views
3K
Graduate Einstein's theory for specific heat
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Understanding the Energy Modes of Springs in a Solid
  • · Replies 1 ·
Replies
1
Views
1K
Derive Dulong-Petit Law: Classical Kinetic Theory & Equipartition of Energy
  • · Replies 2 ·
Replies
2
Views
5K
Graduate Average Energy and the Equipartition Theorem
  • · Replies 5 ·
Replies
5
Views
4K
Graduate Temperature, Kinetic Energy, Boltzmann Factors, and SHO
  • · Replies 6 ·
Replies
6
Views
3K
Specific heat for a triatomic gas
  • · Replies 2 ·
Replies
2
Views
7K
Graduate Kirkendall effect: vacancy concentration and pair interaction energy
  • · Replies 1 ·
Replies
1
Views
2K
A special case of the grand canonical ensemble
  • · Replies 1 ·
Replies
1
Views
2K