Clausius molecular speed

Rudolf Clausius made fundamental contributions to thermodynamics and to the kinetic theory of gases. In a key paper that described heat as the random motion of molecules, he included an estimate of the speeds of several kinds of gas molecules at "the temperature of melting ice," namely

oxygen 461 m/s ,
nitrogen 492 m/s ,
hydrogen 1844 m/s .

1) Use an appropriate formula from the kinetic model of gases to compute the root mean square speed of O₂, N₂, and H₂ at the melting temperature of ice.

2) Clausius derived the following formula for the root mean square speed of the molecules of a gas:

u = (485 m/s) [T/(ρ^.273 K)]^1/2 ,

where ρ is the density of the gas relative to air and T the absolute temperature. Show that Clausius' formula is equivalent to the formula in your text by deriving Clausius' version from the textbook version. Take the "molar mass" of air to be 28.964 g/mol.

Reference

Rudolf Clausius, "On the nature of the motion which we call heat," Annalen der Physik 100, 353-380 (1857); translation published in Philosophical Magazine 14, 108-127 (1857)

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