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:

\[E=N k_{B} T \quad \text { (internal energy of } N \text { oscillators). }\label{23.22}\]

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

\[C \equiv \frac{1}{M} \frac{d Q}{d T} \approx \frac{1}{M} \frac{d E}{d T}=\frac{k_{B}}{m} \quad \text { (specific heat of harmonic oscillators). }\label{23.23}\]

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.)