Question

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

Answer 1

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.