# Derivation of Specific Heat of an Ideal Monatomic Gas at a Fixed Volume from Maxwell-Boltzmann Distribution

How do I derive the specific heat of an ideal monatomic gas from the Maxwell-Boltzmann distribution?

I think I understand the derivation based on degrees of freedom, but I'm supposed to be able to derive it from the Maxwell-Boltzmann distribution.

My understanding on the M-B distribution equation is that it simply finds the probability that a single molecule has a particular speed, but I'm not sure how to connect that to specific heat.

The Maxwell-Boltzmann distribution is also a tool to compute average values out of physical quantities written in terms of the velocity of the particles. For instance, the average kinetic energy is given by $$\langle K\rangle=\int\frac{1}{2}mv^2F(v)dv=\frac{3}{2}kT,$$ where $F(v)$ is the Maxwell-Boltzmann distribution.
The last step is to compare the thermodynamic internal inergy $dU=nc_vdT$ with the internal energy obtained by the kinetic theory. This would give you the specific heat $$c_v=\frac{3}{2}R.$$