I agree to a point with the other posts. I would suggest using helium in the tires, not the tubes. Make the tires impregnated with a polymer that would resist the (eventual) release of the helium and bond the tire to a single part extrusion rim. The advantages here - no weight of tubes, spokes or hub. Two valve stems should be used, placed 180 degrees from each other and set at 12 and 6 oclock while filling - one filling, other venting until pure helium exits, then pressurizing.
I would suggest doing the same to the frame. Completely sealed and a valvestem placed at each of the lower rearmost part of the frame and the other placed at the upper forward most part of the frame. One filling, one venting, then pressurizing
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The amount of weight lost by replacing pressurized air with pressurized helium is negligible. You just have to see the size of a helium sized balloon needed to lift a human cargo to get some concept of the volume of air needed to be replaced by helium to understand how small the helium in tires wouls affect the entire weight of a bicycle. Ideally the bike frame could be pumped out to leave a sealed vacuum inside but this still would make little practical difference.
Interesting, but pennywise and dollar foolish. Too complex for casual riders, too impractical (harder flats since no tubes) for hard-core tourers/commuters, too dependant on UCI ruling for racers. If you really want weight savings, get a materials science degree and make carbon fiber.
Dude do what every else dose to save weight and take all the grease out of the bearings, and don't forget the dust gaurds thats like a couple thousands of a gram. ;)
Assuming there is 1 Litter of gas in each tire and the tires are inflated to 65 psi.
PV = nRT
(4.42 atm)(1.0 L) = (n mol)(8.314 J/mol*K)(297 K)
n = 0.00179 moles
Helium = 4 N
Oxygen = 16 N
0.00179 mol * 4.0g = 0.007 g
0.00179 * 16g 0.028 g
0.028 - 0.007 = 0.021 g
Thus, you would decrease the weight around 0.021 grams per tire.