## Using the Equipartition Theorem to Estimate Heat Capacity (10 minutes)

• Recall that the equipartition theorem is the formula relating the temperature of a sample to it's average total energy.
• Example: solids at room temperature and above.
1. $k_{B}T$ is the energy stored in each normal mode of a crystal.
2. For each normal mode, $\frac{k_{B}T}{2}$ stored as kinetic energy, $\frac{k_{B}T}{2}$ stored as potential energy.
3. $\text{# of normal modes}= \left(\text{# atoms in crystal}\right) * \left(\text{# dimensions}\right)$
4. From the above information, $$U_{Tot}=\left(\text{# modes}\right) * k_{B}T \; \; .$$
• Now, using the equation for the total energy of any system, we can approximate the heat capacity of the system using

$$C_{\alpha} \, = \, \left(\frac{dU_{tot}}{dT}\right)_{\alpha} \; \; ,$$

Where $\alpha$ is the variable of the system being held constant (volume, pressure, etc.).

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