If Graham's law doesn't apply to the ammonia and hydrogen chloride diffusion in a glass tube demonstration, is there a law that does?

A comment below [What exactly are "white fumes" and why does holding a bottle of ammonia (conc) next to a bottle of HCl (conc) make them?](https://chemistry.stackexchange.com/q/163982/16035) links to The
Royal Society Of Chemistry video [Diffusion in action | The reaction of ammonia and hydrogen chloride demonstration](https://youtu.be/SWByFMo32Qg)



> 
> Declan Fleming revisits the classic diffusion demo of the reaction of ammonia with hydrogen chloride – providing some great tips for doing it even better.
> 
> 
> Download the instructions: <https://rsc.li/3xsjpxS>
> 
> 
> 


Fleming says:



> 
> As Graham's law doesn't apply to this situation I tend to leave things here...
> 
> 
> 


and



> 
> Although the presence of air molecules makes Graham's law inapplicable...
> 
> 
> 


and the instructions linked in the video's description includes the following:



> 
> However, this method neglects the fact that the assumptions behind Graham’s law cannot apply. The different sizes of the molecules and their collision cross sections with other molecules come into play. In other words, the [rate of diffusion depends](https://pubs.acs.org/doi/abs/10.1021/ed044p745) not only on the molecules of HCl and NH3, but also on the properties of the (mainly) nitrogen and oxygen molecules into which they are diffusing. As such, the observed ratio [may be slightly less than that predicted](https://pubs.acs.org/doi/abs/10.1021/ed074p1069) in the simplistic model above – the demo is perhaps best kept as a qualitative one at this level.
> 
> 
> 


The ratio of the distances from the boundary where smoke particles are produced and the two sources is 15.5 cm / 8 cm ~ 1.94 and the square root of the mass ratio (if it did apply, which it does not) is (36.46 / 17.03)1/2 ~ 1.46.


Both gasses are diffusing in air, which is not at all the same as *[effusing](https://en.wikipedia.org/wiki/Effusion)* from an aperture smaller than the mean free path (of order 10 microns). So I understand why [Graham's law](https://en.wikipedia.org/wiki/Graham%27s_law) does not apply here. So I'd like to ask:


**Question:** If Graham's law doesn't apply to the ammonia and hydrogen chloride diffusion in a glass tube demonstration, is there a law that does, or at least gets close?




---


Annotated screenshot from The Royal Society Of Chemistry video "Diffusion in action | The reaction of ammonia and hydrogen chloride demonstration" <https://youtu.be/SWByFMo32Qg>


[![Annotated screenshot from The Royal Society Of Chemistry video "Diffusion in action | The reaction of ammonia and hydrogen chloride demonstration" https://youtu.be/SWByFMo32Qg](https://i.stack.imgur.com/viT5P.jpg)](https://i.stack.imgur.com/viT5P.jpg)




The experiment and modeling of the position using differential equations is described in a [paper behind a paywall](https://aip.scitation.org/doi/10.1063/1.5083927?af=R) (doi:10.1063/1.5083927).


Here is the abstract:



> 
> Vapor-phase ammonia, NH3(g), and hydrochloric acid, HCl(g), undergo a series of complex reactions, including nucleation and growth, to form solid ammonium chloride, NH4Cl(s). The counterdiffusional experiment, whereby HCl(g) and NH3(g) diffuse from opposite ends of a tube and react to form spatiotemporally complex patterns, has a rich history of study. In this paper, we combine experimental data, molecular simulations, and analysis and simulations of a partial differential equation model to address the questions of where the first unobserved vapor product NH4Cl(g) and visually observable precipitate NH4Cl(s) form and how these positions depend on experimental parameters. These analyses yield a consistent picture which involves a moving reaction front as well as previously unobserved heterogeneous nucleation, wall nucleation, and homogeneous nucleation. The experiments combined with modeling allow for an estimate of the heterogeneous and homogeneous nucleation thresholds for the vapor-to-solid phase transition. The results, synthesized with the literature on this vapor-to-particle reaction, inform a discussion of the details of the reaction mechanism, including the role of water, which concludes the paper.
> 
> 
> 


Another [paper](https://acp.copernicus.org/articles/14/9233/2014/acp-14-9233-2014.pdf) has diffusion coefficients for hydrogen chloride and for ammonia in nitrogen (or air). The diffusion coefficient for ammonia is about 1.5 larger than that for hydrogen chloride, explaining while ammonia travels further before the reactants meet.


In a [random walk](https://www.mit.edu/%7Ekardar/teaching/projects/chemotaxis(AndreaSchmidt)/random.htm), the average distance from the source increases with the square of the time. It is not clear to me what happens in the cylindrical geometry, but the rate of transport probably is linear after a steady state has been reached.


There are several complications that would explain why the distance traveled might not be inversely proportional to the gas diffusion coefficients. First, the substances have to undergo a phase transition from dissolved to gas. Second, there might be reactions in the gas phase with water in air. Third, you don't see the ammonium chloride in the gas phase; rather, it is the crystalline ammonium chloride you see, first as smoke and then as solid on the wall of the glass tube.


The simplest empirical law would be "8 cm away from the source of HCl". Once you start varying parameters (temperature, pressure, distance, geometry, humidity, isotopes, different reactants, gas different than air) you would have to include these parameters in the law.


Or you go to the local university library and check out the theoretical model described in the literature cited above and how it compares to experimental results.


 1