23.5: Specific Heat
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How can we compute the specific heat of a collection of harmonic oscillators? Starting from the temperature of a brick, as given by equation (23.21), we solve for the brick’s internal energy:
E=NkBT (internal energy of N oscillators).
Recall that the specific heat is the heat required per unit mass to increase the temperature of the brick by one degree. For a solid body, essentially all the heat added to the body goes into increasing its internal energy. Thus, if the mass of the brick is M=Nm where m is the mass per oscillator, then the predicted specific heat of the brick is
C≡1MdQdT≈1MdEdT=kBm (specific heat of harmonic oscillators).
This formula is in reasonable agreement with measurements when the temperature is high enough so that all the harmonic oscillators are in excited states, i. e., with r>1. (We equate dQ=dE using the first law of thermodynamics, since no work is being done by the brick.)