I think Vogelfrei80 is interested in the effect the lighter gas will have on reducing the overall moment of inertia of the wheel. Reducing that means you can accelerate/decelerate the wheel with less torque, and hence force on the pedals.

If you assume that the gas is a torus and you calculate the change in mass as in the post above, you can work out the change in moment-of-inertia using the lighter gas. Here is a handy calculator (it is amazing what folk put on the interweb

) :

https://www.vcalc.com/wiki/EmilyB/Mo...8about+axis%29
So for a particular linear acceleration of the unicycle you can work out the corresponding angular acceleration of the wheel and hence the extra torque you need to accelerate that difference in moment of inertia. Once you know the torque you can get the force on the pedals for a particular crank length.

This will only be the change in force due to the different gas in the tube, you'd have to judge if this is significant, or model yourself and the unicycle in more detail to figure out the absolute force and then determine if the change is significant.

Or at least that is my current thoughts on how to work this out

I think one of the main reasons they put nitrogen in tyres is that it keeps pressure better. As far as I know this is because the N2 molecule is very slightly larger than the O2 in the air it replaces. Putting helium in has the opposite effect, hence the question on using a FOSS tube instead of a rubber one since the polyurethane material is apparently less permeable.