# 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 $(3/2) R$?

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.