# Why use dimensionless heat capacity?

Perhaps this is blindly obvious, but in typical discussions of statistical mechanics (with, say, constant volume) one often finds that, rather than using the heat capacity

$$C_V = \frac{\partial E}{\partial T}\bigg|_V = T\frac{\partial S}{\partial T}\bigg|_V,$$

(where the 2nd equality comes from the 1st Law of Thermodynamics), the dimensionless heat capacity

$$\hat{C}_V = \frac{C_V}{Nk}$$

(where $N$ is the # of particles in the system, and $k$ is the Boltzmann constant) is introduced and used instead. So.... why? How is the dimensionless version more useful than the original?

• in physics, it's cool to get rid of the pesky units! It helps in making nice graphical plots, among other things...
– Cham
Aug 30, 2019 at 16:15

• This explains the choice to use the specific heat $c_v = C_V/N$, but doesn't really explain why Boltzmann's constant is thrown in there. Feb 22, 2017 at 17:26
I think this also comes from the fact that using equipartition theorem, each quadratic degree of freedom in global hamiltonian contributes for $$\frac{1}{2}k_BT$$ in energy and so for $$\frac{1}{2}k_B$$ in $$C_V$$, so basically $$\hat{C_V}=\frac{3}{2}$$ means 3 quadratic degree of freedom (monoatomic ideal gas).
(Note that equipartition theorem is valid only when quantization of energy level can be neglicted in front of $$k_BT$$)