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So why is the gas constant in $C=3R/M$? How does this relate to the specific heat capacity of a solid element?

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The equipartition theorem in thermodynamics states that (in D.V. Schroeder's notation), $$U_{thermal} = \frac{1}{2}Nfk_bT$$ where $U$ is the total thermal energy, $f$ is the number of degrees of freedom, $k_b$ is Boltzmann's constant, and $T$ is temperature.

For the heat capacity of a substance at constant volume, we assumer there is no work being done on the system as work, in general, acts to change volume (think compression in a bicycle pump, for example). We remind ourselves that the definition of heat capacity is $$ C \equiv \frac{Q}{\Delta T}$$

The first law of thermodynamics states that $$\Delta U = Q -W$$. With no work, we may simply re-arrange for $U$: $$C_V = \left(\frac{\Delta U}{\Delta T}\right)_V = \left(\frac{\partial U}{\partial T} \right)_V $$

Now,

$$C_V = \left(\frac{\partial U}{\partial T} \right) = \frac{\partial}{\partial T}\left(\frac{Nfk_bT}{2}\right) = \frac{Nfk_b}{2}$$

We have the relationship that $$ nR = Nk_b$$

where R is the gas constant and n is the number of moles. The law of Dulong and Petit says that there are 6 degrees of freedom for a solid, so the heat capacity per mole of a substance should be $3R$.

The heat capacity of a substance per unit mass, or $c$, is defined as

$$c \equiv \frac{C}{m} $$

where $m$ is the mass of whatever substance you're measuring. Thus, the heat capacity per unit mass per mol of a solid would give you $$c = \frac{3R}{m}$$.

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  • $\begingroup$ what is the upside down 9 symbol mean next to the U and T and what is the subscript V means? $\endgroup$ – Fred Weasley Apr 20 at 6:06
  • $\begingroup$ Those are partial derivatives! They're a lot like total derivatives, which you might have seen, but not exactly. You can check wiki for more about properties of the partial derivative. $\endgroup$ – swickrotation Apr 20 at 6:12

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