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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?

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One benefit of scaling the heat capacity with another extensive variable is that you end up with an intensive property -- heat capacity per # of particles. Similarly specific heat refers to the heat capacity per unit mass so that the value of the intensive property can be compared between samples of the same material but with different sizes or geometries for example.

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  • $\begingroup$ 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. $\endgroup$ – dmckee Feb 22 '17 at 17:26

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