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

Jul 28, 2017

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 $\text{g/mol}$, despite the molar masses in, say, the RMS speed expression, being in $\text{kg/mol}$, since the factors of $1000$ cancel out.)

$\implies \textcolor{b l u e}{{z}_{N {H}_{3}} / \left({z}_{H C l}\right)} = \sqrt{{M}_{H C l} / {M}_{N {H}_{3}}}$

$= \sqrt{\text{36.4609 g/mol"/"17.0307 g/mol}}$

$= \textcolor{b l u e}{1.463}$

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:

$\text{17.0307 g/mol"/"36.4609 g/mol} = 0.467$

and as such, we normalize ${M}_{N {H}_{3}}$ to $0.467$ and ${M}_{H C l}$ to $1$, as well as ${z}_{H C l} = 1$.

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

${z}_{N {H}_{3}} \propto \frac{1}{\sqrt{0.467}} \implies 1.463$ times as fast.

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