What is the ratio predicted from Graham's law for rates of diffusion for #NH_3#/#HCl# ?

Question

Naomi Aronson · Accepted Answer

Graham's Law of Diffusion just bases the ratio of diffusion rates #z# on the reciprocal ratio of the square root of the molar masses #M#. If we normalize one molar mass to #1# and the diffusion rate of that gas to #1#, then

#z^"*" prop 1/sqrt(M^"*")#.

Or more explicitly, with either gas having #z# and #M# not #1#,

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

You can see this answer for a more explicit derivation.

(The molar masses here can be used as #"g/mol"#, despite the molar masses in, say, the RMS speed expression, being in #"kg/mol"#, since the factors of #1000# cancel out.)

#=> color(blue)(z_(NH_3)/(z_(HCl))) = sqrt(M_(HCl)/M_(NH_3))#

#= sqrt("36.4609 g/mol"/"17.0307 g/mol")#

#= color(blue)(1.463)#

So, ammonia gas diffuses a bit less than #1.5# times as fast as hydrogen chloride gas.

Another way to do this is to get the ratio of their molar masses right away:

#"17.0307 g/mol"/"36.4609 g/mol" = 0.467#

and as such, we normalize #M_(NH_3)# to #0.467# and #M_(HCl)# to #1#, as well as #z_(HCl) = 1#.

Ammonia then has a rate of diffusion that is...

#z_(NH_3) prop 1/sqrt(0.467) => 1.463# times as fast.

Whichever way works for you. I would suggest the first way, which is perhaps a bit less confusing.

Ellie Andrews · Answer

undefined The ratio predicted from Graham's law for the rates of diffusion of NH₃ to HCl is approximately 1.54:1.