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

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SUMMARY

The average kinetic energy of a solid is expressed as 3/2*RT due to the contributions from vibrational motion in three dimensions. Each vibrational mode accounts for two degrees of freedom: one for potential energy and one for kinetic energy. Therefore, in a solid, the total degrees of freedom from vibrations is 3, leading to the formula 3RT for vibrational energy. This understanding is crucial for analyzing the thermodynamic properties of solids at appropriate temperature ranges.

PREREQUISITES
  • Understanding of kinetic energy and thermodynamic principles
  • Familiarity with the concept of degrees of freedom in statistical mechanics
  • Knowledge of vibrational modes in solid-state physics
  • Basic grasp of the ideal gas law and its relation to temperature (R and T)
NEXT STEPS
  • Study the role of vibrational modes in solid-state physics
  • Learn about the equipartition theorem and its applications
  • Explore the implications of degrees of freedom on heat capacity in solids
  • Investigate the behavior of solids at different temperature ranges and their thermal properties
USEFUL FOR

Students of physics, particularly those studying thermodynamics and solid-state physics, as well as educators and researchers interested in the kinetic energy of solids and its implications in material science.

Cisneros778
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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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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.
 
Thanks a lot!
 

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