# Helium is found to diffuse four times more rapidly than an unknown gas. What is the approximate molar mass of the unknown gas?

##### 1 Answer

Around

You can start from any expression for the speed. The RMS speed is a fine choice:

#v_(RMS) = sqrt((3RT)/M)# ,where

#R# and#T# are from the ideal gas law, and#M# is the molar mass of the gas in#"kg/mol"# .

The ratio of two speeds *ratio of the gas diffusion rates*

#(v_(RMS,B))/(v_(RMS,A)) = color(green)(z_B/z_A = sqrt(M_A/M_B))# which is

Graham's law of diffusion; the gas with more mass per particle diffuses more slowly. (In this case, the molar mass can now be in#"g/mol"# and it won't matter.)

Since helium is known to diffuse **as fast** as unknown gas

Let's check mathematically.

#z_B/z_A = sqrt(M_A/M_B)#

If we assign helium as

#=> 1/4 = sqrt(4.0026/M_B)#

#=> 1/16 = 4.0026/M_B#

#=> color(blue)(M_B) = 4.0026 xx 16 ~~ color(blue)("64 g/mol")#

So the unknown gas has about