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In Ashcroft and Mermin Chapter 1, just above equation (1.50) and in the context of a classical ideal electron gas, it is said that the electronic specific heat at constant volume $c_v$ is defined by

$$c_v=\frac{\frac{dE}{dT}}{V}$$

which seemed highly irregular to me (I would have expected division by the mass $M$ of the system of electrons). Is there any reason for this that I am perhaps missing?

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Start with the definition of (extensive) specific heat:

$$C_v=\frac{dE}{dT}$$

from there, you can define three different intenstive quantities:

  • massic : $\displaystyle c_v=\frac{C_v}{m}$ in J/K/kg
  • molar : $\displaystyle c_v=\frac{C_v}{n}$ in J/K/mol
  • volumetric : $\displaystyle c_v=\frac{C_v}{V}$ in J/K/m$^3$

Depending on the book, you'll find different notations for those. All three quantities are in use, so if the name or notation isn't clear, have a look at their unit.

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  • $\begingroup$ Gotcha I think, so this is the last of the 3 right? “Specific” usually suggests division by mass no? $\endgroup$
    – EE18
    Commented Jun 19, 2022 at 16:04
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    $\begingroup$ @EE18 "Specific" actually means "independent of the size. In other words, it is an "intensive" counterpart of an extensive quantity. How this is obtained is a matter of convention. Depending on the context, it may refer to the "massic," "molar" or "volumetric" quantity in Miyase's answer. $\endgroup$
    – GiorgioP-DoomsdayClockIsAt-90
    Commented Jun 19, 2022 at 16:19

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