Question

The molar specific heat at constant volume for monoatomic gas is

Answer 1

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 1/2R1/2R worth of heat capacity. Therefore, you have 3/2R3/2R.

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 cv=5/2Rcv=5/2R and therefore cp=7/2Rcp=7/2R giving the typical specific heat ratio of air as γ=7/5=1.4γ=7/5=1.4. At high temperature, if you assume the vibrational mode is fully excited, you get γ=8/6=1.3333γ=8/6=1.3333 which can be used for calorically perfect, high temperature gases.