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

Question

Ellie Andrews · Accepted Answer

Around #"64 g/mol"#.

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 #v# is directly proportional to the ratio of the gas diffusion rates #z# (the #3RT# cancels out):

#(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 #4# times as fast as unknown gas #B#, then off the top of my head, helium is probably #4^2 = 16# times as light (one-sixteenth the molar mass of #B#).

Let's check mathematically.

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

If we assign helium as #A#, then the unknown gas is #B# and #z_B/z_A = 1/4#.

#=> 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 #bb16# times higher of a molar mass.

Avery Adams · Answer

undefined The approximate molar mass of the unknown gas is about 16 g/mol.