Question: What is the specific heat capacity at constant volume of a two-dimensional diatomic ideal gas of N particles at room temperature?
My answer: A diatomic gas can move in both directions, can vibrate, and can spin. This is 4 degrees of freedom and by the Equipartition theorem I know that each of these degrees of freedom have energy $k_bT/2$. Heat capacity at constant volume is defined as the change in energy per unit temperature, so my total comes to be:
$$C = \left (\frac{\partial U}{\partial T} \right )_V = \frac{\partial}{\partial T} \frac{4Nk_bT}{2} = 2Nk_b.$$
The actual answer is $(5/2)Nk_b$. I'm not sure where I'm missing the extra degree of freedom.
Best Answer
For each vibrational degree of freedom, the energy contained is $k_bT$, not $k_bT/2$.
See also: http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/eqpar.html