# Degrees of freedom in a diatomic gas in 2-dimensions

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.

• Hmm, that is the number of degrees of freedom in three dimensions for a linear diatomic molecule. I don't see how the degrees of freedom in two dimensions could be the same. – Ryan Unger Apr 2 '15 at 14:58
• Yes, it seems to be the 3D answer. And vibration isn't always a degree of freedom (in real, elemental, 3d gases only chlorine has this). The 2D answer should be IMHO $\frac{3}{2}Nk_b$. – peterh - Reinstate Monica Apr 2 '15 at 23:17
• Completely classically a 3D gas should have a heat capacity of 7/2 $K_bT$. Three translational, two spin degrees of freedom and a vibrational potential and kinetic contribution. The reason real gasses often are 5/2 $K_bT$ is due to the vibrational contribution being 'frozen out' at lower temperatures – Chris Cundy Dec 31 '15 at 18:42

For each vibrational degree of freedom, the energy contained is $k_bT$, not $k_bT/2$.
The equipartition theorem says that each quadratic degree of freedom that appears in the energy function contributes $$\frac{1}{2}k T$$ to the internal energy. So all we have to do is to count the degrees of freedom:
• The translational kinetic energy of the diatomic molecule has two degrees of freedom because the molecule can move independently in the $$x$$ and $$y$$ direction.
Adding all of these together gives us the $$\frac{5}{2}kT$$ as expected from your solution manual. You obtain the heat capacity by differentiating this w.r.t temperature.