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Thermal Properties I - Coggle Diagram

- - - - dU = dQ - dW
  - - - The experimental measurements of specific heat for solids are normally done at constant (atmospheric) pressure, yielding CP.
        
        However, theory calculates CV since constant internuclear distances are used. From the equation, we can relate CV to CP by easily measurable quantities
        
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- - - - At room temperature, in most cases, the value of the heat capacity is 3Nk = 3R per mole = 25 J mol−1 deg−1. This result, derived from data is called the Dulong and Petit law
        
        At lower temperatures the heat capacity drops markedly and approaches zero as T3
- - - - This is just the classical result that each particle has an energy of kBT/2 for each degree of freedom
        
        If we have Ν free particles in three dimensions, at thermal equilibrium the thermal average energy can be written out just as the preceding
- - - - where the first term on the right-hand side is the kinetic energy, involving the momentum p and mass m, and the second term is the potential energy, involving the displacement u and the force constant α.
    - - Each of the first and second terms yields kBT/2 proves the contribution from each degree of freedom, momentum or position, to the average thermal energy is kBT/2
        
        For Ν harmonic oscillators in three dimensions the average internal thermal energy is 3NkBT so CV = 3NkB, which for a mole of oscillators is just 3R, the Dulong and Petit value!