For a mono-atomic ideal gas, there is only three translational modes with no rotational and vibrational modes.
– There is no rotational modes because the moment of inertia about any axis is zero.
– There is no vibrational modes because an atom cannot vibrate by itself as its centre of mass will be moving.(which is translational) Hence, you need at least 2 atoms.
Number of atoms = 2
Total number of degrees-of-freedom = 2*3 = 6.
Note: If its has 3 atoms, the total number of degrees-of-freedom will be 3*3 = 9.
Number of translation degrees-of-freedom = 3
Number of rotational degrees-of-freedom = 2
(This is because the moment of inertia about the crossed out axis in the above picture is 0)
Number of vibrational modes = 6 – 3 – 2 = 1