What is the point of including the gas constant in the dulong and petit law? So why is the gas constant in $C=3R/M$? How does this relate to the specific heat capacity of a solid element? 
 A: 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 $$c = \frac{3R}{m}$$.
