Caloric capacity is the heat or energy \ delta that is required to raise the temperature by dT , which is expressed as\\n\\n

$\delta Q = C_V dT$

\\n\\nIf the energy of n moles, with N_A the Avogadro number, k_B is the Boltzmann constant and T the temperature is\\n\\n

$U=\displaystyle\frac{f}{2}nN_Ak_BT$

\\n\\nso if the volume is kept constant\\n\\n

$dU=\delta Q=\displaystyle\frac{f}{2}nN_Ak_BdT$

\\n\\nso with\\n\\n

$R=k_BN_A$

you have

equation

Specific heat corresponds to caloric capacity per mass

$c_V=\displaystyle\frac{C_V}{M}$

If m is the mass of an atom, the mass M will be

$M=nN_Am$

with n the number of moles and N_A the number of Avogadro. As the caloric capacity is

$C_V=\displaystyle\frac{f}{2}nkN_A$

so the specific heat is

equation