Why is molar specific heat at constant volume of a monatomic ideal gas a constant? I thought specific heat varies depending on the substance. Why is it always $(3/2) R$?
 A: The specific heat of a molecule depends on the number of degrees of freedom the molecule has. There are several degrees of freedom available: translation (3), rotation (3), vibration (depends on the number of bonds in a molecule) and electronic modes. 
Now, for something that is monatomic, you have 3 translational modes (x,y,z directions), zero rotation modes (because energy contained in the rotation about each axis for a single atom is negligible), 0 vibration modes (because there are no bonds) and because it is an ideal gas, there are no electronic modes. 
So you have 3 degrees of freedom. The translational degrees of freedom are "fully excited" at very low temperatures, in the ten kelvin range. So at reasonable temperatures (ie. greater than a few kelvin) each of these degrees of freedom provides constant heat capacity. 
Each degree of freedom contributes $1/2 R$ worth of heat capacity. Therefore, you have $3/2 R$. 
Continuing this logic, a diatomic molecule will add 2 rotational modes at normal temperatures. Technically there is also a vibrational mode that is added, but this takes high temperatures to be activated. So at room temperature, you will get $c_v = 5/2 R$ and therefore $c_p = 7/2 R$ giving the typical specific heat ratio of air as $\gamma = 7/5 = 1.4$. At high temperature, if you assume the vibrational mode is fully excited, you get $\gamma = 8/6 = 1.3333$ which can be used for calorically perfect, high temperature gases. 
