Degrees of Freedom: Why 3/2*RT for Kinetic Energy in a Solid?

In summary, the discussion is about the average kinetic energy for a solid, which is determined by the number of degrees of freedom. The possible degrees of freedom are translation, rotation, and vibration. While there is only 1 degree of freedom for vibration in a solid, it is possible to have vibrations in all three dimensions, resulting in a total of 3 degrees of freedom and an average kinetic energy of 3/2*RT.
  • #1
Cisneros778
48
0

Homework Statement



I want to know why the average kinetic energy FOR A SOLID is 3/2*RT.

Homework Equations



For every degree of freedom = 1/2*RT
Possible Degrees of freedom are:
Translation, rotation, vibration

The Attempt at a Solution



In a solid, I am certain there is vibrational energy because of the spring like forces from interacting particles in a lattice. This would only account for 1 degree of freedom so far.
I am against saying there is any translation or rotation because the particles are fixed in their respective positions since it is a solid.

So I am confused why the average kinetic energy is 3/2*RT when I see it as 1/2*RT.

(1 degree of freedom) x 1/2*RT = 1/2*RT
 
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  • #2
You can have vibrations in 3 dimensions.
Every vibration dimension corresponds to 2 degrees of freedom (potential and kinetic energy), so the result is 3RT, for an appropriate temperature range and in a crystal.
 
  • #3
Thanks a lot!
 

FAQ: Degrees of Freedom: Why 3/2*RT for Kinetic Energy in a Solid?

How is the 3/2*RT equation derived for kinetic energy in a solid?

The 3/2*RT equation for kinetic energy in a solid is derived from the equipartition theorem, which states that at thermal equilibrium, each degree of freedom in a molecule will have an average energy of 1/2*k*T. In a solid, there are 3 translational degrees of freedom and 2 rotational degrees of freedom, hence the 3/2 factor in the equation.

Why is the 3/2*RT equation specifically used for solids?

The 3/2*RT equation is used for solids because in a solid, the molecules are tightly packed and restricted in their movements, resulting in only 3 translational and 2 rotational degrees of freedom. This is in contrast to gases, which have 3 translational and 3 rotational degrees of freedom, and liquids, which have 3 translational, 2 rotational, and 1 vibrational degree of freedom.

What is the significance of the 3/2 factor in the equation?

The 3/2 factor in the equation represents the average energy per degree of freedom in a solid. This means that for every degree of freedom, there is an average energy of 3/2*k*T, where k is the Boltzmann constant and T is the temperature. This factor is derived from the equipartition theorem and is specific to solids due to their unique molecular structure.

Does the 3/2*RT equation apply to all solids?

Yes, the 3/2*RT equation applies to all solids, regardless of their chemical composition or molecular structure. This is because the equation is based on the fundamental principles of thermodynamics and the equipartition theorem, which apply to all systems in thermal equilibrium.

Can the 3/2*RT equation be used to calculate kinetic energy in other states of matter?

No, the 3/2*RT equation is specific to solids and cannot be used to calculate kinetic energy in other states of matter. As mentioned earlier, gases and liquids have additional degrees of freedom, and therefore, their kinetic energies are calculated using different equations. The 3/2*RT equation is only applicable to solids due to their unique molecular structure and restricted degrees of freedom.

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